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+FireFly: A High-Throughput and Reconfigurable
+Hardware Accelerator for Spiking Neural Networks
+Jindong Li
+, Guobin Shen
+, Dongcheng Zhao
+, Qian Zhang
+, Zeng Yi
+Abstract—Spiking neural networks (SNNs) have been widely
+used due to their strong biological interpretability and high
+energy efficiency. With the introduction of the backpropagation
+algorithm and surrogate gradient, the structure of spiking neural
+networks has become more complex, and the performance gap
+with artificial neural networks has gradually decreased. However,
+most SNN hardware implementations for field-programmable
+gate arrays (FPGAs) cannot meet arithmetic or memory effi-
+ciency requirements, which significantly restricts the development
+of SNNs. They do not delve into the arithmetic operations
+between the binary spikes and synaptic weights or assume un-
+limited on-chip RAM resources by using overly expensive devices
+on small tasks. To improve arithmetic efficiency, we analyze
+the neural dynamics of spiking neurons, generalize the SNN
+arithmetic operation to the multiplex-accumulate operation, and
+propose a high-performance implementation of such operation by
+utilizing the DSP48E2 hard block in Xilinx Ultrascale FPGAs.
+To improve memory efficiency, we design a memory system to
+enable efficient synaptic weights and membrane voltage memory
+access with reasonable on-chip RAM consumption. Combining
+the above two improvements, we propose an FPGA accelerator
+that can process spikes generated by the firing neuron on-the-fly
+(FireFly). FireFly is implemented on several FPGA edge devices
+with limited resources but still guarantees a peak performance
+of 5.53TSOP/s at 300MHz. As a lightweight accelerator, FireFly
+achieves the highest computational density efficiency compared
+with existing research using large FPGA devices.
+Index Terms—Spiking Neural Networks, Field-programmable
+gate array, Hardware Accelerator
+I. INTRODUCTION
+S
+PIKING neural networks (SNNs) are considered the third
+generation of artificial neural networks (ANNs) [1]. They
+were developed to mimic the operational mechanism in the
+Manuscript created January 1, 2023. This work was supported by the
+Strategic Priority Research Program of the Chinese Academy of Sciences
+(Grant No. XDB32070100). (Corresponding authors: Qian Zhang; Yi Zeng.)
+Jindong Li and Qian Zhang are with the Research Center for Brain-
+Inspired Intelligence, Institute of Automation, Chinese Academy of Sciences,
+Beijing 100190, China, and also with the School of Artificial Intelligence,
+University of Chinese Academy of Sciences, Beijing 100049, China (e-mail:
+[email protected], [email protected]).
+Guobin Shen is with the Research Center for Brain-Inspired Intelli-
+gence, Institute of Automation, Chinese Academy of Sciences, Beijing
+100190, China, and also with the School of Future Technology, University
+of Chinese Academy of Sciences, Beijing 100049, China (e-mail: shen-
+[email protected]).
+Dongcheng Zhao is with the Research Center for Brain-inspired Intel-
+ligence, Institute of Automation, Chinese Academy of Sciences, Beijing
+100190, China (e-mail: [email protected]).
+Yi Zeng is with the Research Center for Brain-inspired Intelligence, the
+National Laboratory of Pattern Recognition, at the Institute of Automation,
+Chinese Academy of Sciences, Beijing 100190, China, and University of
+Chinese Academy of Sciences, Beijing 100049, China, and Center for
+Excellence in Brain Science and Intelligence Technology, Chinese Academy
+of Sciences, Shanghai 200031, China (e-mail: [email protected]).
+human brain, where information is communicated via spikes
+among neurons. Surrogate gradient algorithms have been
+introduced for SNNs tackling nondifferentiable problems to
+enhance the learning capability of SNNs. [2], [3]. Recent
+advances in SNNs have demonstrated comparable performance
+to non-spiking ANNs [4]–[8]. However, compared to the
+extensive work on ANN accelerators [9]–[11], the existing
+SNN hardware accelerator still lags, limiting the practical
+applications of SNNs.
+Most research ignores the importance of efficiently imple-
+menting arithmetic operations in SNN accelerators. In Field-
+programmable gate array (FPGA) design, using the built-
+in dedicated hard block to implement arithmetic operations
+can achieve considerably higher performance than its general
+logic fabric counterparts. Fabric-only implementations in an
+arithmetic-extensive application can lead to a compromised
+clock frequency and even routing failures when the fabric
+consumption is high. However, in the SNN accelerator design,
+the register transfer level (RTL) description of the SNN
+arithmetic operation cannot be automatically synthesized into
+the dedicated arithmetic hard block. Therefore, most SNN
+accelerators adopt the fabric-only implementation without
+further optimizations. Although a single arithmetic operation
+unit in an SNN accelerator consumes considerably fewer
+resources than a multiply-accumulate (MAC) unit in an ANN
+accelerator design, hardware optimization of such operation
+can still significantly impact the system’s performance when
+the unit is instantiated hundreds or even thousands of times.
+In the Xilinx Ultrascale FPGA, the dedicated arithmetic hard
+block, or the DSP48E2, enhances the speed and efficiency of
+many operations, including multiplication, addition, wide bus
+multiplexing, pattern detection, and single instruction multiple
+data (SIMD) operations. It is possible to generalize the SNN
+computation to the arithmetic operations that the DSP48E2
+can provide.
+Another important aspect of the SNN accelerator design
+is the memory system. When scaling the parallelism, the
+memory bandwidth imbalance between the binary input-output
+spikes, the multi-bit synaptic weights, and the multi-bit mem-
+brane voltage becomes problematic. While the computational
+complexity and the memory footprint of the binary spikes
+decrease, the memory access requirements of synaptic weights
+and membrane voltage do not. The off-chip memory access
+bandwidth needed by the weights and membrane voltage
+cannot fully support the increased parallelism brought by
+the hardware-friendly synaptic operations and storage-friendly
+binary spikes without further exploration of the reuse mecha-
+nism. Most hardware accelerators assume large on-chip mem-
+arXiv:2301.01905v1  [cs.NE]  5 Jan 2023
+
+ID2
+ory, store all the synaptic weights, and accumulate membrane
+voltage on-chip to ease the harsh bandwidth requirement. This
+method is not scalable, especially when the model gets larger
+and targets edge FPGA devices. A scalable memory system
+for synaptic weights and membrane voltage balancing, as well
+as off-chip data access and on-chip data buffering, should be
+developed.
+At present, most existing neuromorphic hardware or accel-
+erators focus on brain simulation tasks. While these hard-
+ware designs claim to support event-driven processing, they
+are inefficient in terms of resource utilization, computational
+density, and scalability. In real-world SNN applications, it is
+not feasible to use overly expensive and large FPGA devices.
+A lightweight and high-performance SNN accelerator targeting
+resource-constrained edge scenarios should be developed.
+Focusing on these aspects, we propose FireFly, a high
+throughput and reconfigurable FPGA accelerator that can
+achieve both arithmetic and memory efficiency. Our contri-
+butions can be summarized as follows.
+1) We generalize the SNN arithmetic operation to the
+multiplex-accumulate operation and propose a high-
+performance implementation of such an operation by
+utilizing the DSP48E2 hard block in Xilinx Ultrascale
+FPGAs.
+2) We design a synaptic weight delivery hierarchy and
+a partial sum and membrane voltage (Psum-Vmem)
+unified buffer to balance the off-chip memory access
+bandwidth and on-chip RAM consumption.
+3) We evaluate multiple deep SNN models on various
+datasets and achieve faster inference speed and higher
+classification accuracy than the existing research. We
+implement FireFly on several commercial off-the-shelf
+FPGA edge devices with limited resources, bringing
+hope for real-world SNN applications in edge scenarios.
+II. RELATED WORK
+The existing dedicated neuromorphic hardware designed for
+SNN can be categorized into three types.
+The majority of neuromorphic hardware constructs its hard-
+ware substrates in a Network on Chip fashion. Loihi [12],
+Tianji Chip [13], Spinnaker [14] and TrueNorth [15] fall into
+this category. In these hardware designs, neurons are grouped
+into multiple neurocores, which communicate via spikes
+through the Network-on-Chip (NoC), and spike messages are
+scheduled by dedicated routers. These hardware architectures
+are compatible with the event-driven nature of SNNs, as spike
+events are generated, transferred, and processed only if the
+neuron fires. However, these neuromorphic hardware designs
+place rigid restrictions on the network. The SNN networks
+are distributed among the neurocores, and the total number of
+neurons in the model cannot exceed the maximum capacity
+of the hardware, not to mention the harsh fan-in and fan-out
+hardware limitations of the network.
+The second type of neuromorphic hardware explores emerg-
+ing devices. The BrainScale [16] developed by Heidelberg
+University emulated spiking neural networks on analog neu-
+romorphic hardware and achieved several advantages over
+conventional computers. Some research explores new materials
+like mem-resistors and optics [17]–[19]. However, the low
+precision and uncertain nature of the hardware prevent them
+from being used in practice.
+The third type of neuromorphic hardware follows the
+scheme of the ANN accelerator design except for construct-
+ing dedicated hardware for synaptic operations and explores
+optimal dataflow for SNNs specifically [20]–[26]. These types
+of work require less area cost and achieve higher computing
+resource utilization. Fine-grained parallelism of the accelerator
+can enable high-performance computing of the SNN compared
+with the sequential spike processing mechanism of the NoC
+counterparts. This type of hardware has the fewest restrictions
+on the network models and can quickly adapt to emerging
+neuromorphic research. FPGA platforms are the ideal choice
+for this type of hardware due to their flexibility and reconfig-
+urability.
+While FireFly belongs to this category, its contributions of
+FireFly are largely complementary to the existing work.
+SyncNN [21] proposed a novel synchronous event-driven
+SNN reconfigurable inference engine and evaluated multiple
+SNN models on multiple FPGA devices. Fang et al. [27]
+proposed a holistic optimization framework for the encoder,
+model, and architecture design of FPGA-based neuromorphic
+hardware. However, these designs are based on high-level
+synthesis, thus inducing large resource redundancy.
+Lee et al. [28], [29] and Chen et al. [30] explored spatial-
+temporal parallelism by unrolling the computations in both the
+spatial and time dimensions and achieved significant accel-
+eration. However, parallelization across multiple time points
+violates the time-related sequential nature of the membrane
+voltage update behavior.
+SpinalFlow [25] achieved significant sparsity acceleration
+by adopting a different input/output spike representation to
+skip the non-spike computations. SATO [31] achieved high-
+speed inference by incorporating a temporal-oriented dataflow
+and a bucket-sort-based dispatcher to balance the workload.
+However, these techniques only work for temporal coding
+SNNs, limiting the accuracy of the SNN models.
+DeepFire [23] was the first research migrating DSP48E2s
+into neuron core design. However, they did not delve into the
+function of DSP48E2 and still induce large fabric overhead.
+We argue that with careful register transfer level (RTL)
+design, focusing on optimizing spatial parallelism on FPGA,
+adopting regular and simple time-step CNN-like processing,
+and fully utilizing the multi-function DSP48E2, we can still
+achieve impressive inference throughput on small FPGA edge
+devices. FireFly is more applicable in real-world applications
+where design space exploration is constrained by limited
+resources.
+III. SNN BASICS
+A. Spiking Neuron Model
+Spiking neurons are the basic units of SNNs, which are con-
+nected through weighted synapses and transmit information
+through binary spikes. Although more complex and detailed
+neuron models such as Izhikevich [32] and Hodgkin–Huxley
+
+3
+[33] can accurately model a biological neuron’s behavior,
+simpler models such as Integrate and Fire (IF) [34] and Leaky
+Integrate and Fire (LIF) [35] are used more often in current
+SNN applications.
+An IF neuron integrates its inputs over multiple timesteps
+and generates a spike whenever the integrated membrane
+voltage surpasses a firing threshold. A LIF neuron acts the
+same except for the leaky behavior of the membrane voltage.
+The neural dynamics of a LIF neuron membrane potential u
+can be described as:
+τm
+du
+dt = −u + R · I(t),
+u < Vth
+(1)
+where Vth denotes the threshold, I denotes the input current,
+R denotes the resistance, and τm is the membrane time
+constant. A spike is generated when u reaches Vth and u is
+reset to resting potential urest, which is set to 0 in this work.
+The membrane potential’s neural dynamics can be divided into
+three phases, and each phase can be described in a discrete
+computational form::
+Input current integration phase. All the presynaptic currents
+generated by the presynaptic spikes are integrated at each
+discrete timestep.
+I[t] =
+�
+j
+wijsj[t] + bi
+(2)
+where the subscript i represents the ith neuron, wij is the
+synaptic weight from neuron j to neuron i, and bi is a bias.
+Membrane potential update phase. The membrane potential
+of each neuron is updated by the integrated presynaptic
+currents at each timestep.
+vi[t] = (1 − 1
+τm
+)ui[t] + I[t]
+(3)
+where (1 −
+1
+τm ) < 1 denotes the leaky term, which is ignored
+when using the IF model.
+Output spike generation phase. Whenever the membrane
+potential reaches the firing threshold, the neuron generates an
+output spike and resets its membrane potential.
+(ui[t + 1], si[t + 1]) =
+� (vi[t], 0), vi[t] < Vth
+(0, 1),
+vi[t] ≥ Vth
+(4)
+In these three phases, we have two key observations. The
+input current integration phase completely dominates the total
+computational cost due to the high degree of synaptic connec-
+tivity and a large number of neurons. The membrane potential
+update phase has the harshest storage requirement because the
+membrane potential is read and written back and forth in every
+timestep. We will focus on these two aspects in the following
+sections.
+B. Dataflow and Parallelism Scheme for SCNN
+Similar to convolutional neural networks (CNNs), convolu-
+tional layers dominate the total computational cost in spiking
+convolutional neural networks (SCNNs). We mainly focus on
+the dataflow optimizations of the convolutional layers and
+show that the dataflow can be migrated to fully connected
+layers.
+Algorithm 1: Pseudo Code of FireFly Architecture.
+Input: Given the binary spike map size (H, W),
+input-output channels (Cin, Cout), kernel size
+(Kh, Kw), total timestep T, leaky factor λ,
+threshold Vth and parallelism factor P. Divide
+the input output channels into
+(ci = ⌈ Cin
+P ⌉, co = ⌈ Cout
+P ⌉) groups.
+Input: T × ci fragments of I[P][H × W] streams,
+each stream passes the hardware for co times.
+Output: co × T fragments of O[P][H × W] streams.
+1 Create buffer for synaptic weights:
+W[P][Cin][Kh][Kw];
+2 Create buffer for Psum/Vmem: V [P][H × W];
+3 for po ← 0 to co do
+4
+Load Weights: W[P][Cin][Kh][Kw];
+5
+for t ← 0 to T do
+6
+for pi ← 0 to ci do
+7
+for s ← 0 to H × W do
+8
+Unroll and pipeline;
+9
+for o ← 0 to P do
+10
+for i ← 0 to P do
+11
+w = W[o][pi × P + i][0 →
+Kh][0 → Kw];
+12
+i = neighbour (I[i][s]);
+13
+V [o][s]+ = w · i;
+14
+end
+15
+end
+16
+if pi = ci − 1 then
+17
+V [o][s]× = (1 − λ);
+18
+if V [o][s] > Vth then
+19
+V [o][s] = 0, O[o][s] = 1;
+20
+else
+21
+O[o][s] = 0;
+22
+end
+23
+if t = T − 1 then
+24
+V [o][s] = 0;
+25
+end
+26
+end
+27
+end
+28 end
+Input/Output spike representation varies in different neu-
+romorphic hardware. Most SNN hardware implementations
+adopt the Address-Event-Representation (AER) data format to
+transmit spikes between neurons. The standard AER package
+for one spike includes the spiking neuron’s input location
+and the spike’s timestamp. Although the AER data format is
+compatible with the event-driven nature of SNNs, multiple bits
+are needed to express the original single-bit spike event. The
+logic and storage overhead may not be worth it.
+This paper adopts the original single-bit format to represent
+the binary spikes. At any discrete timestep t in the digitalized
+SCNN, the output spikes of all the neurons in one channel of
+the convolutional layer can be considered a timestep snapshot
+in the form of a binary map [36]. In this case, the input-
+current integration phase computation process of the SNNs is
+
+4
+AXI DataMover
+Read Addr
+InSpike Stream
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+W
+W
+W
+W
+W
+W
+W
+W
+FIFO
+FIFO
+FIFO
+FIFO
+CONV or MLP
+AXI DataMover
+Write Addr
+OutSpike Stream
+AXI DataMover
+Read Addr
+Weight Stream
+Maxpool En?
+FIFO
+MAX
+BRAM
+...
+...
+...
+LineBuffer for Conv
+Shift Reg for MLP
+CONV or MLP
+Maxpool En?
+Bypass
+Lv1
+Lv2
+Lv3
+Lv4
+Weight Delivery Hierchy
+...
+BRAM
+BRAM
+BRAM
+Update Engine
+Psum-Vmem 
+Unified Buffer
+8
+...
+DSP48E2 Chain
+PS
+PL
+ARM
+CPU
+DDR4
+AXI
+Interconnect
+DDR
+Controller
+Mem
+Fabric(LUT)
+Register(FF)
+DSP48E2
+Fig. 1. FireFly Architecture.
+almost the same as that of the traditional ANNs except for
+the additional time dimension and the changed operation. The
+set of computations for the complete SNN convolutional layer
+that receives a single batch of input can be formulated as a
+loop nest over these 7 variables. All permutations of these
+6 loop variables, except for the timestep variable, are legal.
+Permutations of the loop variables open up the possibility of
+different dataflow choices. The tiling of the loop variables
+opens up the possibility of different parallelism schemes.
+Different permutations of the loop variables adopt different
+kinds of dataflow. Different dataflow schemes for convolution
+have been extensively studied by Eyeriss [9]. The key con-
+sideration is how to minimize data movement and maximize
+data reuse. In SCNN, synaptic connection weights need to
+be fetched and membrane voltage needs to be updated at
+every time timestep, due to the unique time dimension in SNN
+computation. Therefore, output and weight stationary dataflow
+can minimize the data movement of the multi-bit membrane
+voltage and synaptic weight data between on-chip logic and
+off-chip memory.
+Different tiling strategies for the loop variables enable
+different parallelism schemes. The tiling of the loop variables
+can induce data reordering or data segmentation. We argue that
+it is important to keep the input and output spike arrangements
+the same to enable spikes to be processed in an on-the-fly
+fashion without complicated data reaarangement. We chose
+the spatial tiling of the input and output channel dimensions
+rather than tiling within the same spike feature map to avoid
+data rearranging or irregular off-chip data access.
+Adopting the dataflow and parallelism scheme above, the
+pseudo-code of the FireFly is described in Algorithm 1.
+IV. HARDWARE ARCHITECTURE
+A. Architecture Overview
+In this section, the digital design of SNNs is discussed in
+detail. Fig.1 shows the overall system design of FireFly.
+FireFly targets heterogeneous Zynq Ultrascale devices. The
+central processing unit (CPU) of the processing system (PS)
+acts as the controller for system state control and external
+memory access. The programmable logic (PL) accelerates the
+SNN inference.
+AXI DataMover IP, instead of AXI DMA IP, enables high-
+throughput and low-latency data transactions between the off-
+chip DRAM and on-chip memory storage. The unique store
+and forward feature of AXI DataMover is enabled to allow
+multiple outstanding requests.
+The weight-stationary systolic array is responsible for the
+acceleration of SNN arithmetic operations. The systolic array
+consists of several DSP48E2 chains and multiple adder trees.
+A weight matrix delivery hierarchy is proposed to enable
+efficient weight loading to the systolic array. Two separate
+datapaths for convolutional and fully connected layers are de-
+signed to generate binary spike vectors for the systolic array. A
+Psum-Vmem unified buffer and update engine is constructed to
+support back-and-forth membrane potential update and IF/LIF
+neuron dynamics. An optional MaxPooling unit is placed on
+the output spike datapath to support on-the-fly pooling.
+The designs of the systolic array, the spike vector generation
+unit, the synaptic weight delivery hierarchy, and the Psum-
+Vmem unified buffer are elaborated in detail below.
+B. Synaptic Operations Featured by DSP48E2
+As shown in Fig.2A, DSP48E2 is the dedicated digital
+signal processing logic block in the Xilinx Ultrascale series
+FPGA. Most FPGA neuromorphic hardware simply treats
+them as multipliers and leaves them underutilized. However,
+they enhance the speed and efficiency of many applications
+far beyond multiplication-based digital signal processing [37].
+Considering customizing arithmetic operations for the SNN
+model, the mathematical dot product operation between the
+binary spike and the synaptic weight can be modeled as a
+multiplex-accumulate operation which in this paper, we call
+the synaptic operation. The spike acts like the control signal of
+
+5
+=
+Y
+Z
+X
+W
+18
+18
+18
+30
+30
+27
+5
+4
+2
+30
+18
+18
+30
+48
+48
+9
+48
+48
+48
+4
+27
+3
+4
+8
+0
+0
+0
+1
+0
+B
+A
+D
+C
+INMODE
+CARRYIN
+OPMODE
+CARRYINSEL
+BCIN*
+ACIN*
+BCOUT*
+ACOUT*
+A:B
+RND
+U
+V
+17Bit Shift
+17Bit Shift
+ALUMODE
+CARRYOUT
+PATTERNDETECT
+PATTERNBDETECT
+PCIN*
+CARRYCASCIN*
+MULTSIGNIN*
+CREG/C Bypass/Mask
+MULTSIGNOUT*
+PCOUT*
+XOR OUT
+CARRYCASCOUT*
+P
+Dual B Register
+Dual A D
+and Pre-Adder
+MULT
+27x18
+=
+Y
+Z
+X
+W
+18
+18
+18
+30
+30
+27
+5
+4
+2
+30
+18
+18
+30
+48
+48
+9
+48
+48
+48
+4
+27
+3
+4
+8
+0
+0
+0
+1
+0
+B
+A
+D
+C
+INMODE
+CARRYIN
+OPMODE
+CARRYINSEL
+BCIN*
+ACIN*
+BCOUT*
+ACOUT*
+A:B
+RND
+U
+V
+17Bit Shift
+17Bit Shift
+ALUMODE
+CARRYOUT
+PATTERNDETECT
+PATTERNBDETECT
+PCIN*
+CARRYCASCIN*
+MULTSIGNIN*
+CREG/C Bypass/Mask
+MULTSIGNOUT*
+PCOUT*
+XOR OUT
+CARRYCASCOUT*
+P
+Dual B Register
+Dual A D
+and Pre-Adder
+MULT
+27x18
+A1
+A1
+A2
+A2
+B2
+B2
+B1
+B1
+C
+SIMD
+Add
+P
+OP
+W
+X
+Y
+Z
+PCIN
+PCOUT
+OPMODE
+9
+48
+30
+18
+48
+0
+0
+0
+0
+A1
+A2
+B2
+B1
+C
+SIMD
+Add
+P
+OP
+W
+X
+Y
+Z
+PCIN
+PCOUT
+OPMODE
+9
+48
+30
+18
+48
+0
+0
+0
+0
+Shared
+OPMODE
+SIMD=4
+4x4=16
+Spike 
+Operations 
+C)
+A)
+B)
+Fig. 2. Implementing Synaptic operations Using DSP48E2. A) The functional
+circuit diagram of a single DSP48E2 slice [37]. B) A simplified functional
+circuit diagram of the DSP48E2 performing spike-based computations. C) An
+equivalent circuits of the DSP48E2 when SIMD mode is enabled.
+the multiplexer, switching the synaptic weight on or off when
+the neuron is firing or resting. The following adder sums up
+all the synaptic weights coming from the firing neuron.
+In traditional ANNs, one operation usually refers to one
+two-operand multiplication or two-operand addition. In SNNs,
+we define one synaptic operation as one 2:1 multiplexing or
+two-operand addition. We show that the dedicated DSP48E2
+unit can provide up to 16 synaptic operations at high speed.
+This technique is described in detail below.
+When the first stage multiplier in DSP48E2 is disabled,
+ALUMODE control bits are all cleared and carry inputs are
+ignored, the simplified DSP slice operation shown in Fig. 2B
+in the ALU stage can be expressed as:
+Post Adder Out = W + X + Y + Z.
+where W, X, Y and Z are four built-in 48-bit wide bus
+multiplexers. Moreover, the post-adder can be statically con-
+figured into SIMD mode, supporting a single 48-bit adder, dual
+independent 24-bit adders, or quad independent 12-bit adders.
+The outputs of the four multiplexers are always added
+together by the post-adder. There are dozens of combinations
+of inputs to these multiplexers: one of them can be: either C
+or all 0s on the X multiplexer; either A:B or all 0s on the X
+multiplexer; all 0s on the Y multiplexer; either P, PCIN, or
+all 0s on the Z multiplexer. The 30-bit A and 18-bit B data
+inputs can optionally be registered once or twice to construct a
+pipeline stage, while the 48-bit C data inputs can be optionally
+staged once. The post-adder’s output can be staged into the P
+register, and the PCIN is the cascade input from a lower DSP
+slice. A nine-bit control input named OPMODE contains fields
+TABLE I
+RESOURCE UTILIZATION COMPARISON.
+DSP48E2
+LUT
+FF
+CARRY8
+DSP
+1
+0
+0
+0
+Fabric
+0
+86
+114
+8
+for W, X, Y, and Z multiplexer selects and can be dynamically
+changed.
+Utilizing the wide bus multiplexer, the cascade datapath,
+and the SIMD mode of the post-adder in DSP48E2, we can
+pack up to 16 sets of synaptic operations into a single DSP
+slice.
+In this work, the synaptic connection weights are quantized
+into INT8 by the well-established post-training quantization or
+quantization-aware training methods developed in traditional
+neural networks (NNs).
+Four sets of INT8 weights are resized to INT12 and concate-
+nated into 48-bit. The upper 30 bits are assigned to the input
+port A while the lower 18 bits are assigned to input port B. A
+and B get concatenated and multiplexed by the X multiplexer.
+In NNs, the input activations are shared by different sets of
+weights to generate different channels. In this case, one spike
+is fetched to dynamically switch the X multiplexer between
+the four sets of weights (A:B) and all 0s, performing four 2:1
+multiplex operations simultaneously.
+Similarly, another four sets of INT8 weights are resized,
+concatenated, and directly assigned to the C data input. another
+spike is fetched to dynamically switch the W multiplexer
+between C and all 0s, performing another four 2:1 multiplex
+operations.
+The Z multiplexer selects the PCIN inputs and the partial
+sum from the lower DSP slice. The Y multiplexer outputs
+are set to all 0s. The post-adder is set to SIMD mode and
+acts as four independent 12-bit adders, summing the four
+multiplexers, and performing an equivalent number of eight
+addition operations. Therefore, as shown in Fig.2C, a single
+slice of DSP48E2 can contribute 16 synaptic operations in
+total without general fabric logic overhead.
+Direct access to the specific features in DSP48 is achieved
+by directly instantiating the DSP48E2 primitive. The straight-
+forward implementation of the synaptic operations described
+above will consume 86 Look-up-tables, 114 Flip-flops and
+8 Carry chains. Though it might not seem expensive on a
+small scale, it is considerably less efficient than the proposed
+approach and will lead to a compromised clock frequency.
+C. Systolic Array for Synaptic Operations
+The systolic array is a specialized mesh of homogeneous
+PEs designed to process massive parallel computations. It has
+the potential to run at a high frequency due to its regular
+and adjacent interconnections. However, designing systolic
+arrays is not trivial. Previous neuromorphic hardware adopting
+a systolic array architecture failed to achieve satisfactory
+performance, either in resource efficiency or clock frequency.
+Most systolic arrays targeting FPGA devices are implemented
+in low-speed general fabrics. In this paper, we design a
+
+6
+high-performance systolic array featured by the DSP48E2 for
+SNNs.
+A more straightforward representation of the aforemen-
+tioned synaptic operations featured by a single DSP48E2 slice
+can be expressed as follow:
+pi = si · Wi + pi−1, p−1 = 0.
+where si is the 1 × 2 binary spike vector, and wi is the
+2 × 4 INT8 synaptic weights matrix, pi is the 1 × 4 partial
+sum vector, and the pi−1 is the partial sum vector contributed
+by the lower DSP slice with the same shape as pi. · represents
+the spikes-weights vector-matrix multiplication.
+The 12-bit representation of each channel in pi allows
+up to eight DSP48E2 slices to cascade in a row without
+possible numeric overflow. In this way, the extended synaptic
+operations featured by a cascaded DSP48E2 chain can be
+expressed as follows:
+p =
+7
+�
+i=0
+si · Wi = s · W .
+Where s is the 1 × 16 binary spike vector, and W is the
+16 × 4 8-bit-integer (INT8) synaptic weights matrix, p is the
+1 × 4 partial sum vector.
+The cascaded DSP48E2 chain is the basic processing ele-
+ment (PE) in our systolic array design. A PE consists of eight
+cascaded DSP48E2 slices. A M × N systolic array consists
+of M
+4 columns of PE, with each column consisting of N
+16 PEs
+and an adder tree. Each column in the systolic array computes
+N
+16 1 × 16 binary spike vector and 16 × 4 weight matrix
+multiplication, while the adder tree sums up the results from
+N
+16 PEs, generating four output channels. With M
+4 columns,
+the systolic array generates M outputs channels in total.
+Each PE in the systolic array contains different sets
+of synaptic weights. Adopting a weight-stationary scheme,
+synaptic weights remain cached in a PE until they are no
+longer needed. The same 1 × N binary spike vector is shared
+across columns horizontally, and M partial sums flow out of
+the systolic array vertically.
+D. Spike Vector Generation for Convolution by Line Buffer
+Similar to ANN, 2-D convolution is the basic operation in
+a digitalized SCNN. We incorporate the traditional line buffer
+design [38] to generate the spike window needed for the spike-
+map convolution. The line buffer is commonly seen in CNN
+accelerator design because it can efficiently achieve kernel-
+level parallelism and ensure good reuse of image data.
+When FireFly is configured to SCNN mode, Cin channels of
+binary spike map are bundled together and stream into the line
+buffer. The Kh×Kw spikes-bundle window is then flattened to
+a Kh×Kw ×Cin vector and sent to the systolic array. In most
+of the established CNN architectures, 3 × 3 convolution with
+stride 1 and the same padding is the most common configu-
+ration. The SCNN architecture follows this scheme. Ideally,
+general neuromorphic hardware for SNN should support all
+types of convolutional layers with different configurations.
+But the hardware would not work efficiently for all types
+of convolution configuration and such design would cause
+hardware overhead, thus might not be feasible. Therefore, we
+design specialized line buffer logic for 3 × 3 convolution.
+Nevertheless, the methods discussed here are compatible with
+other kernel sizes. Using the Dynamic Function Exchange
+features in FPGA, hardware supporting different types of
+convolutional layers can be dynamically deployed in FPGA
+during runtime.
+When FireFly is configured for multi-layer perception
+(MLP) topology mode, the line buffer datapath for SCNN is
+left idle and the shift register datapath for MLP is switched
+on. The shift register forms a serial-to-parallel stream width
+adapter by combining the Cin input spikes of Kh × Kw input
+transactions into one. The length of the binary spike vector in
+SCNN and MLP datapaths is the same, compatible with the
+height of the systolic array.
+E. Synaptic Weight Delivery in a Multi-level Hierachy
+An M × N systolic array configured in weight stationary
+mode needs M × N sets of weights. Switching the current
+set of stationary synaptic weights with the next set of weights
+can be problematic. The instantaneous switching bandwidth is
+extremely high but switching occurs when weights expire.
+The main idea of our solution is that the instantaneous
+bandwidth needed when switching to the next set of weights
+needs to be amortized over an idle period when the weights
+are kept stationary.
+As shown in Fig.3D, we propose a 4-level synaptic weight
+memory hierarchy to enable on-the-fly delivery of weights
+with minimum resource consumption. First, the synaptic
+weight stream coming from the AXI DataMover is adapted
+by the Lv1 stream width adapter. The adapted weight stream
+flows into the Lv2 Partial Reuse FIFO and is reused T times.
+The weight stream from the Partial Reuse FIFO stage its way
+through the Lv3 width adapter and then gets cached in the Lv4
+skid buffer. The systolic array holds the current set of weights
+stationary by applying back pressure to the skid buffer and
+releasing the pressure when the current set of weights is no
+longer needed.
+A stream width adapter converts the N-bit input stream to
+a N × M-bit output stream by allocating M elements of the
+input stream and firing them all at once. A skid buffer is the
+smallest Pipeline FIFO Buffer. It decouples two sides of a
+ready/valid handshake to allow back-to-back transfers without
+a combinational path between input and output, thus pipelining
+the path.
+The Partial Reuse FIFO is the key component in this 4-level
+synaptic weight delivery hierarchy.
+Most designs utilize the dual-port RAM to build a ping-
+pong buffer (shown in Fig. 3A) or a FIFO, to hide the latency
+of the data transfer process. However, the traditional ping-
+pong buffer mechanism can be problematic and the FIFO
+mechanism does not support data reuse.
+The switching of the ping-pong buffer may complicate the
+controller design. Ping-pong buffers are costly and inefficient.
+The depth of the buffer must be large enough to support
+the most storage-expensive cases, not to mention the buffer
+
+7
+In Stream
+Out Stream
+PushPtr
+PopPtr
+when(Overflow)
+   Start=End
+PushPtr
+PopPtr
+In Stream
+Out Reused Stream
+Available
+Reusing
+To be Reused
+Active
+Current
+Write
+Next
+Read
+Current
+Read
+Next
+Write
+A)
+B)
+Lv3 Serial to 
+Parallel
+Lv4 Skid
+Buffer
+To Systolic
+Array
+From DDR
+when(last step)
+   Start=End
+Batch 1
+Batch 2
+Batch 1
+Lv2 Partial 
+Reuse FIFO
+1 8 7 2 9 3 -7 -5 ...
+-1 -2 … 4 -7 9 7 1
+-7 -1 … 7 -5 3 2 8
+… 4 -7 9 7 1
+… 7 -5 3 2 8
+Batch 1
+1  8  7  2
+9  3 -5 -7
+1 7 9 -7 4 … -2 -1
+8 2 3 -5 7 … -1 -7
+2 4 6 1 ...
+9 3 1 7 ...
++
+=
+C)
+D)
+Lv1 Serial to 
+Parallel
+Fig. 3. Different Approaches for Hiding Data Transfer Latency to Improve Throughput. A) Ping-pong buffer. B) Synchronous FIFO. C) The Proposed Patrial
+Reuse FIFO. D) A four-level synaptic weights delivery hierarchy to enable synaptic weights reuse, reduce off-chip memory bandwidth and hide the weight
+loading latency to the systolic array.
+size has to be doubled for ping-pong operation. However, the
+worst-case scenario will not occur in most cases. Only a small
+portion of the ping-pong buffer is occupied most of the time.
+While the aforementioned problems are negligible in ANN
+accelerator design, we cannot afford to “double the size”
+in SNN neuromorphic hardware design because the memory
+bandwidth needed has already increased multiple times.
+Ideally, the on-chip buffer that stores the synaptic weights
+in SNN should have the following properties:
+1) We do not need to double the buffer size and split the
+buffer into two regions for ping-pong operation just
+to guarantee no read-write collision will happen. No
+manual switching of the split buffers is needed.
+2) In SNN, the same synaptic weights need to be accessed
+at every timestep. We expect the data in the buffer can
+be read several times before they expire and are replaced
+by new data.
+3) The depth of the buffer is set to support the most storage-
+expensive cases, but multiple batches of data can be
+preloaded into the available large RAM spaces when
+the storage requirements are less expensive.
+We propose Partial Reuse FIFO, to address the above
+requirements and enable data reuse and space exploration
+without complex control logic.
+As shown in Fig. 3B, traditional synchronous FIFO can
+be described using a ring. The circumference of the ring
+represents the depth of the FIFO. The width of the ring
+represents the data width of the FIFO. A push pointer is used
+to mark the write address of the incoming data. A pop pointer
+is used to mark the read address of the output data. When the
+push pointer and the pop pointer point to the same address,
+the FIFO is either full or empty, depending on whether the
+occupancy of the FIFO is rising or falling. When the FIFO is
+full, the ready signal to the inputs AXI-Stream is clear. When
+the FIFO is empty, the valid signal to the outputs AXI-Stream
+is clear.
+As shown in Fig.3C, the mechanism of the Partial Reuse
+FIFO is the same as the traditional synchronous FIFO, except
+that a partial region in the FIFO ring cannot be flushed by
+incoming data until it is reused T times, where T is a control
+register of the Partial Reuse FIFO.
+The reuse region of the FIFO is labeled by Start and End.
+The pop pointer jumps back to the Start position whenever it
+reaches the end. The reuse counter increases whenever the
+pop pointer jumps back to Start. The Start label stays the
+same when the region is still being reused. When the counter
+reaches T, the counter is reset, label End becomes the next
+label Start and the next label End is set by Start+L-1, where
+L is another control register of the partial reuse FIFO. Unlike
+the traditional synchronous FIFO, when the push pointer meets
+the label Start, the Partial Reuse FIFO is full and the ready
+signal to the inputs AXI-Stream is clear. When label End is
+ahead of the push pointer, the Partial Reuse FIFO is considered
+empty until the reuse sector of the FIFO is filled by the input
+stream.
+The partial reuse FIFO satisfies the aforementioned prop-
+erties. Using the valid-ready handshake protocol of the AXI-
+Stream, the function of the partial reuse FIFO is self-contained,
+with only two control registers exposed. The partial reuse
+FIFO contains only a monolithic RAM and does not need
+to be split. The push-pop pointer in the FIFO control logic
+ensures no read-write collision. The reuse sector protected by
+the Start-End label enables data reuse. New data from multiple
+batches can be pushed to the partial reuse FIFO sequentially
+as long as the FIFO is not full.
+F. Psum-Vmem Unified Buffer and Spike Generation Logic
+A classic systolic array consumes data from the inputs and
+weights data domain and feeds data to the outputs data domain.
+If one data domain stays stationary, the other two must flow
+through the computing logic. This metric holds for the three
+classic input, weight and output stationary dataflows.
+Our architecture adopts the weight stationary dataflow. In
+this case, synaptic weights remain stationary in the systolic
+array, and the input binary spikes and the output flow in and
+out of the systolic array. The flowing spike vector is generated
+by the line buffer mechanism, and the outputs are stored in
+the proposed Psum-Vmem Unified Buffer.
+In our architecture, the synaptic operations in SNN are
+spatially parallelized. However, it is unlikely to flatten a
+whole layer spatially onto the area-power-restricted hardware
+substrates. Therefore, certain tiling strategies need to be imple-
+mented. We adopt the channel tiling strategy to accommodate
+layers with a large number of channels to the same systolic
+array. Input spike map channels are split into multiple tiles
+to fit into the height of the systolic array. Output spike map
+
+8
+Acc
+Phase
+Thresh
+Phase
+Last
+Step
+Last
+Tile
+Finish
+Finish
+Clear
+Phase
+Psum-Vmem
+Unified
+Buffer
+Stored
+Psum/Vmem
+Read Addr
+Psum-Vmem
+Update
+Engine
+Updated
+Psum/Vmem
+Updated
+Psum/Vmem
+New Psum
+Acc
+Unit
+New Psum
+Stored 
+Threshold
+Leak Coeff
+Leak En
+Reset En
+Updated
+Spike
+Leak
+Unit
+Threshold
+Unit
+C)
+Psum/Vmem
+Psum/Vmem
+B)
+A)
+Write Addr
+Spikes
+Fig. 4.
+Psum-Vmem Update Mechanism. A) The finite-state-machine per-
+forming the Psum-Vmem update. B) The proposed Psum-Vmem unified buffer
+and Psum-Vmem update engine. C) The hardware implementation details of
+the Psum-Vmem update engine.
+channels are calculated N at a time according to the width of
+the systolic array.
+In each single timestep, the partial sums of the N output
+spike map channels are stored on-chip and are not fully
+accumulated until all tiles of the input spike map channels
+are calculated. In each layer, the membrane voltage of the N
+output spike map channels are also needed to be stored on-
+chip until all timesteps are iterated. Instead of instantiating
+a separate buffer for partial sum and membrane voltage, we
+propose the Psum-Vmem Unified Buffer to reduce RAM
+consumption.
+Since tiles of input spike map channels in a single timestep
+are sent to the computing array one by one and the temporal
+dimension of SNN is kept in its natural way of executing in a
+sequential manner, the partial sum accumulating process and
+the membrane voltage update process can be scheduled using
+a finite state machine. There are three states specified in the
+FSM: accumulating phase, thresholding Phase, and clearing
+phase.
+During the accumulating phase, Psum extracted from the
+Psum-Vmem unified buffer is accumulated by the computing
+results from the systolic array. When the last tile of the input
+spike map channel in the current timestep arrives and the
+current timestep is not the last, the FSM switches to the
+thresholding phase. The extracted Psum is first accumulated,
+then processed by the optional leaky unit and the thresholding
+unit, and eventually written back to the unified buffer. The
+accumulated Vmem will be subtracted from a fixed portion of
+its value by the optional leaky unit to support the LIF neuron
+dynamics. The thresholding unit will compare the Vmem with
+the threshold, generate a spike, and reset the Vmem if it
+exceeds the threshold. All of the computations are pipelined to
+improve timing. The FSM switches back to the accumulating
+phase when this phase finishes. When the last tile of the
+input spike map channel in the last timestep arrives, the FSM
+switches to the Clearing Phase. The computation process is the
+same as the thresholding phase, except that the Vmem value
+will be cleared to reset the unified buffer for the next SNN
+layer.
+V. IMPLEMENTATION AND EXPERIMENTS
+A. Experiments Setup
+Most neuromorphic hardware uses expensive large FPGA
+devices, ignoring the feasibility of deploying such hardware
+in the real world. FireFly is mapped onto several off-the-shelf
+commercially available Xilinx Zynq Ultrascale FPGAs, in-
+cluding the Ultra96v2, KV260 and ZCU104 FPGA evaluation
+boards, bringing hope of SNN real-world applications in an
+edge scenario. The FPGA chips of the three evaluation boards
+are xczu3eg, xczu5ev, and xczu7ev, respectively.
+Our proposed FireFly is designed using SpinalHDL, a
+hardware description language equipped with object-oriented
+programming and functional programming. Compared with
+an HLS-based code template, parameterized Verilog, or Sys-
+temVerilog, SpinalHDL can offer a higher level of abstraction
+and reconfigurability. The Verilog codes generated by the
+SpinalHDL compiler are synthesized and implemented in the
+Xilinx Vivado 2021.1 with ML-Based design optimization to
+achieve a higher clock rate and faster timing closure. Power
+consumption estimates and timing results are obtained after
+place-and-route using the power analysis and timing summary
+tools in the Vivado Design Suite, which provides detailed
+analysis and accurate estimation. Throughput performance is
+obtained by recording the timer value on the PS side of Zynq
+while the PL runs the benchmark tasks.
+B. Bridging the Gap between Peak and Avg. GSOP/s
+The theoretical peak GSOP/s of an SNN accelerator is given
+as:
+Peak GSOP/s = 2 × f × M × N.
+(5)
+where f is the system clock frequency, and M ×N denotes
+the size of the systolic array. The peak GSOP/s calculation is
+the same as [20] and [24]. In FireFly, M denotes the number
+of columns in the systolic array, while N denotes the rows.
+The peak performance should be proportional to the systolic
+array size. However, the actual throughput, or average GSOP/s,
+can be degraded due to insufficient bandwidth and inefficient
+controller design.
+In our design, the line buffer mechanism enables binary
+spike map reuse, the partial reuse FIFO enables synaptic
+weight reuse, and the Psum-Vmem buffer is used to avoid
+back-and-forth fetch and store. The memory bandwidth needed
+for off-chip data transfer is minimized, and thus not a bottle-
+neck of the system’s average performance.
+We argue that the communication between the controller
+and the accelerator significantly impact the system’s actual
+
+9
+TABLE II
+COMPARISON WITH OTHER WORKS IN RESOURCE UTILIZATION.
+Work
+Device
+Slice LUTs
+Slice Registers
+BRAM/URAM
+DSP48
+Frequency
+Peak GSOP/s
+Used
+Utilization
+Used
+Utilization
+Used
+Utilization
+Used
+Utilization
+[39]
+xc7vx690t
+53k
+12.20%
+100k
+11.50%
+65
+4.40%
+0
+0%
+100
+/
+[22]
+xc7k325t
+170k
+83.70%
+113k
+27.70%
+254
+57.10%
+0
+0%
+135
+3.2
+[24]
+xcvu440
+302k
+11.90%
+421k
+8.30%
+192
+7.60%
+0
+0%
+200
+1562.5
+[40]
+xcku115
+585k
+88.20%
+232k
+17.40%
+432
+20%
+0
+0%
+140
+253
+[25]
+28nm ASIC
+/
+/
+/
+/
+/
+/
+/
+/
+200
+684.5
+[31]
+28nm ASIC
+/
+/
+/
+/
+/
+/
+/
+/
+200
+3970.1
+ours1
+xczu3eg
+15k
+21.40%
+53k
+37.50%
+162
+75%
+288
+80%
+300
+1382.4
+ours2
+xczu7ev
+42k
+18.20%
+196k
+42.60%
+25/40
+11.5/41.6%
+1152
+66.60%
+300
+5529.6
+ours3
+xczu5ev
+32k
+27.35%
+112k
+47.86%
+16/24
+11.1%/37.5%
+576
+46.20%
+300
+1382.4×2
+1 FireFly with a 16 × 144 systolic array implemented on Ultra96v2.
+2 FireFly with a 32 × 228 systolic array implemented on ZCU104.
+3 FireFly with two 16 × 144 systolic arrays implemented on KV260.
+throughput. Note that we choose the Zynq devices as the
+system platforms. The built-in host CPU controller enables
+fast deployment of different SNN networks without the need
+to change the PL logic. In most Zynq-based SNN acceler-
+ators such as Cerebron [20], the host program in the Zynq
+processing system sends synaptic weights and binary input
+spike maps into the Zynq programmable logic and collects
+the output spike maps in different SNN layers. However,
+the control command sequence traveling between PS and PL
+through the low-performance AXI-Lite protocol induces non-
+negligible latency, leaving the systolic array idle and reducing
+the average throughput. In FireFly, the host program generates
+a command sequence in advance and sends the commands to
+PL through a high-performance AXI-Stream to the internal
+command queue of the AXI DataMover. In this way, the req-
+ack waiting clock cycles between commands are eliminated.
+The average throughput can go a step further.
+C. Performance Analysis
+The size of the systolic can be statically reconfigured
+in FireFly according to the on-chip resources on different
+evaluation boards. A M ×N systolic array in FireFly receives
+N presynaptic inputs and produces partial sum for M neurons,
+where M = P and N = Kh × Kw × P. The resource con-
+sumption, memory bandwidth and acceleration performance is
+linearly proportional to the parallelism factor P. P can be any
+value as long as the systolic array can fit in the target device.
+As P is also the tiling factor of the input and output channels
+in a convolutional layer, it is preferable to set P to a power of
+2 because the number of channels in most convolutional layers
+is a power of two. Therefore, we evaluate two representative
+configurations, 16 × 144 and 32 × 288 to demonstrate the
+reconfigurability of FireFly.
+The usage of DSP48 to implement synaptic operations sig-
+nificantly reduces the fabric overhead and achieves significant
+GSOP/s improvements compared with most existing hardware.
+The performance of FireFly is still impressive. FireFly with
+a 16 × 144 systolic array can achieve a peak performance of
+1382.4GSOP/s, and FireFly with a 32×288 systolic array can
+achieve a peak performance of 5529.6GSOP/s, as shown in
+Table II.
+To the best of our knowledge, SIES [24] achieves the
+highest GSOP/s among all the existing FPGA-based acceler-
+ators. Compared with SIES [24], FireFly mapped on xczu3eg
+consumes only
+1
+20 LUTs and 1
+8 FFs but still achieve similar
+GSOP/s, whereas FireFly mapped on xczu7ev consumes only
+1
+7 LUTs and
+1
+2 LUTs FFs and achieves a ×3.5 speed up.
+Additionally, we map two heterogeneous FireFly cores onto
+xczu5ev to support the concurrent inference of two indepen-
+dent SNNs.
+We can still achieve higher throughput when compared
+with SpinalFlow and SATO, which are state-of-the-art SNN
+hardware accelerators built in 28nm ASIC. We are well aware
+that it is difficult to make an apples-to-apples comparison
+with the hardware adopting different design methodologies,
+supporting different types of neurons, using different synaptic
+weight precisions or implementing on different platforms,
+FireFly can still be called a high-performance SNN accelerator
+due to its excellent GSOP/s performance.
+D. Benchmark Evaluations
+We deploy several state-of-the-art SNN networks trained by
+backpropagation algorithms [4] on FireFly to test the inference
+performance. We evaluate not only the static datasets such as
+MNIST, CIFAR10 and CIFAR100 but also the neuromorphic
+datasets such as DVS-CIFAR10 and DVS-Gesture.
+The models are trained using surrogate functions like
+quadratic gate and arctangent gradient. Direct coding and
+backpropagation through time algorithm significantly reduce
+the total timesteps of the SNNs. In our experiment, the
+timesteps are scaled down to four without a significant ac-
+curacy drop.
+We first apply batchnorm fusion to merge the batch normal-
+ization layer with the preceding convolutional layer to deploy
+the Pytorch-Trained SNN model to FireFly. Then we adopt
+post-training quantization techniques to convert the Float32
+synaptic weights to INT8 and the Float32 threshold to INT18.
+Note that the performance drop of post-training quantization
+without further retraining or fine-tuning is negligible in SNN
+because no scaling errors of multiplications are introduced.
+FireFly shows reconfigurability on different SNN models for
+different image classification tasks. We evaluate four different
+SNN model structures with 5, 7, 9, and 11 convolutional
+
+10
+TABLE III
+COMPARISON WITH RELATED WORK FOR MULTIPLE IMAGE CLASSIFICATION TASKS USING SNNS FOR MULTIPLE DATASET.
+Work
+Network
+Dataset
+Latency
+Accuracy
+GSOP/s
+Device
+Frequency
+power
+TVLSI’14
+[41]
+784-500-500-10
+MNIST
+9.25ms
+94.2
+/
+xc6slx150t
+75MHz
+1.5W
+ICCAD’20
+[27]
+28x28-32c3-p2-32c3-p2-256-10
+MNIST
+7.53ms
+99.42
+/
+xczu9eg
+125MHz
+4.5W
+TCAD’22
+[30]
+28x28-16c-32c-8c-10
+MNIST
+45us
+98.5
+22.6
+xc7z045
+200MHz
+0.96W
+TCAS-I’21
+[42]
+784-200-100-10
+MNIST
+3.15ms
+92.93
+xc7vx485t
+100MHz
+/
+JCST’20
+[24]
+28x28-12c5-p2-64c5-p2-10
+MNIST
+/
+99.16
+1562.5
+xcvu440
+200MHz
+/
+TCAD’21
+[22]
+32x32-32c3-p2-32c3-p2-256-10
+SVHN
+1.21 ms
+82.15
+3.2
+xc7k325t
+100MHz
+0.699W
+784-512-256-128-64-10
+FMNIST
+0.14 ms
+89.01
+200MHz
+0.982W
+TRETS’22
+[43]
+28x28-32c3-p2-32c3-p2-256-10
+MNIST
+77us
+99.17
+/
+xczu9eg
+200MHz
+24.5W
+32x32-(192c5-192c1-192c1-p3)*2-
+192c5-192c1-10c1-AP-10
+CIFAR10
+6.8ms
+88.19
+DATE’22
+[26]
+144x144-p4-32c-p2-
+32c-p2-512-512-11
+NMNIST
+3.83ms
+97.81
+51.2
+22nm
+ASIC
+400MHz
+0.11W
+DVS-Gesture
+7.1ms
+92.4
+ours
+SCNN-51
+MNIST
+0.491ms
+98.12%
+91%5
+xczu3eg
+300MHz
+2.55W
+SCNN-72
+CIFAR10
+1.035ms
+91.36%
+89%5
+SCNN-113
+CIFAR100
+2.125ms
+64.28%
+86%5
+SCNN-94
+DVS-CIFAR10
+3.541ms
+72.40%
+87%5
+SCNN-94
+DVS-Gesture
+3.541ms
+89.29%
+87%5
+1 SCNN-5: 28x28-16c3-64c3-p2-128c3-p2-256c3-256c3-10
+2 SCNN-7: 32x32-16c3-64c3-p2-128c3-128c3-p2-256c3-256c3-p2-512c3-10
+3 SCNN-9: 48x48-16c3-64c3-64c3-p2-128c3-128c3-p2-256c3-256c3-p2-512c3-512c3-10
+4 SCNN-11: 32x32-16c3-64c3-64c3-p2-128c3-128c3-128c3-p2-256c3-256c3-256c3-p2-512c3-512c3-100
+5 The GSOP/s utilization ratio: Actual measured GSOP/s divided by the peak GSOP/s. The peak GSOP/s is 1382.4 on xczu3eg.
+layers on five different datasets, shown in Table III. Note
+that our chosen device, xczu3eg, is an edge device having the
+fewest resources among all the listed hardware, but still, Fire-
+Fly shows significant improvement in all these benchmarks.
+Compared with [27], FireFly achieves a ×15 speed up and
+similar accuracy on the MNIST dataset. Compared with [21],
+FireFly achieves higher accuracy and a ×6 inference speed
+up on CIFAR10 dataset. Compared with ASIC design [26],
+FireFly achieves a ×2 speed up and similar accuracy on DVS-
+Gesture dataset. Note that our SNN models are considerably
+bigger and deeper than the listed benchmarks.
+When using a larger xczu7ev device, all the inference per-
+formances listed above are improved by ×4 because xczu7ev
+supports higher parallelism and has a peak performance of
+5.523TSOP/s. Our system also supports multiple heteroge-
+neous cores running different SNN models concurrently. When
+targeting xczu5ev, two FireFly cores can be deployed indepen-
+dently to support multiple real-world tasks.
+E. Discussion
+We argue that for FPGA-based SNN accelerator design,
+the benefits of designing complicated hardware supporting
+spike sparsity may not make up for the losses of irregular
+interconnect and underutilization of the dedicated hard block.
+The system clock frequency can have a significant impact
+on inference performance. Compared with ASICs, routing in
+FPGAs contributes more delay time since logic elements are
+connected through a series of switching matrices instead of
+direct physical wires. A complex digital design with irregular
+interconnect can easily violate the timing requirements even
+in the most state-of-the-art FPGA devices. Most existing
+FPGA-based SNN accelerators can only satisfy the timing
+requirement of at most 200MHz even on the expensive Virtex
+Ultrascale+ device.
+An important aspect of FPGA low-power system design is
+to utilize the existing dedicated hard block rather than build
+one from scratch. Implementing the same function using the
+dedicated hard block in FPGAs usually consumes less energy
+than using the general fabric counterparts. However, most
+existing FPGA-based SNN accelerators fail to delve into the
+features provided by the existing dedicated hard block and
+adopt a no-brainer implementation of spike computation using
+low-speed fabric.
+In this paper, FireFly provides a different perspective on
+designing dedicated neuromorphic hardware for spiking neural
+networks targeting FPGA devices. We are well aware that
+it is important to design hardware that supports sparsity
+acceleration. However, to our best knowledge, only few studies
+[25] [31] targeting ASICs can show significant speed-ups
+considering this inherent nature of SNNs, not to mention
+the large majority of FPGA-based designs. Instead of design-
+ing complicated circuits to support the sparsity acceleration,
+FireFly consists of a monolithic systolic array and adopts a
+straightforward weight stationary dataflow. The acceleration
+comes from the clock frequency improvement brought by
+the regular and simple interconnect of the systolic array, the
+pipelined arithmetic computations, and, most importantly, the
+flexible use of the multi-function DSP48E2s.
+In fact, the potential of the DSP48E2 is still far from being
+fully realized. Wu et al. [11] proposed a high-throughput pro-
+cessing array for matrix multiplication based on DSP supertile
+and achieved peak DSP clock rates on Xilinx UltraScale (741
+MHz) and UltraScale+ (891 MHz) devices. SNN accelerators
+can incorporate the DSP supertile design and achieve even
+higher performance.
+The potential of other dedicated hard blocks on FPGA
+is also yet to be exploited. Scaling the Cascades [10] fully
+utilized the dedicated cascade interconnect of the DSP48E2,
+BRAM36K, and URAM288K and achieved nearly 100 %
+usage of these hard blocks, delivering incredible inference
+speed on MLPerf benchmarks. It is necessary to migrate the
+existing hardware optimization techniques of ANN accelerator
+
+11
+design to SNN neuromorphic hardware research.
+Nevertheless, we agree that ideally, the main advantage of
+new SNN accelerators compared to ANNs on digital hard-
+ware comes primarily from exploiting the sparsity of spikes
+and not from the replacement of MAC operations with AC
+operations [44]. Future neuromorphic hardware design should
+exploit spike sparsity and migrate existing FPGA optimization
+techniques simultaneously.
+VI. CONCLUSIONS
+In this work, we introduced a high-throughput and recon-
+figurable hardware accelerator for spiking neural networks.
+To achieve high-performance inference of SNN, we fully
+exploited the features of the dedicated DSP48E2 embedded in
+the FPGA and achieved the highest GSOP/s compared with the
+existing accelerator designs. To improve memory efficiency,
+we designed a synaptic weight delivery hierarchy and a Psum-
+Vmem unified buffer to support the high parallelism. To
+demonstrate FireFly’s reconfigurability, we evaluated multiple
+deep SNN models on various datasets. To make SNN appli-
+cations more convenient, we used off-the-shelf commercially
+available FPGA edge devices, offering a more feasible solution
+than any other existing hardware. In the future, we will try to
+migrate more optimization techniques targeting FPGAs while
+exploring sparsity acceleration to enable more energy-efficient
+SNN software and hardware co-design.
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+

29A0T4oBgHgl3EQfNP8E/content/tmp_files/2301.02142v1.pdf.txt

@@ -0,0 +1,1364 @@
+TRANSPORTATION SCIENCE
+Vol. 00, No. 0, Xxxxx 0000, pp. 000–000
+issn 0041-1655|eissn 1526-5447|00|0000|0001
+INFORMS
+doi 10.1287/xxxx.0000.0000
+© 0000 INFORMS
+Authors are encouraged to submit new papers to INFORMS journals by means of
+a style file template, which includes the journal title. However, use of a template
+does not certify that the paper has been accepted for publication in the named jour-
+nal. INFORMS journal templates are for the exclusive purpose of submitting to an
+INFORMS journal and should not be used to distribute the papers in print or online
+or to submit the papers to another publication.
+Playing hide and seek: tackling in-store picking
+operations while improving customer experience
+F´abio Neves-Moreira, Pedro Amorim
+[email protected], [email protected]
+INESC TEC, Faculty of Engineering, University of Porto, Porto 4200–465, Portugal
+The evolution of the retail business presents new challenges and raises pivotal questions on how to reinvent
+stores and supply chains to meet the growing demand of the online channel. One of the recent measures
+adopted by omnichannel retailers is to address the growth of online sales using in-store picking, which allows
+serving online orders using existing assets. However, it comes with the downside of harming the offline
+customer experience. To achieve picking policies adapted to the dynamic customer flows of a retail store, we
+formalize a new problem called Dynamic In-store Picker Routing Problem (diPRP). In this relevant problem
+– diPRP – a picker tries to pick online orders while minimizing customer encounters. We model the problem
+as a Markov Decision Process (MDP) and solve it using a hybrid solution approach comprising mathematical
+programming and reinforcement learning components. Computational experiments on synthetic instances
+suggest that the algorithm converges to efficient policies. Furthermore, we apply our approach in the context
+of a large European retailer to assess the results of the proposed policies regarding the number of orders picked
+and customers encountered. Our work suggests that retailers should be able to scale the in-store picking
+of online orders without jeopardizing the experience of offline customers. The policies learned using the
+proposed solution approach reduced the number of customer encounters by more than 50% when compared
+to policies solely focused on picking orders. Thus, to pursue omnichannel strategies that adequately trade-off
+operational efficiency and customer experience, retailers cannot rely on actual simplistic picking strategies,
+such as choosing the shortest possible route.
+Key words :
+omnichannel retail; in-store picking; Markov decision process; reinforcement learning;
+real-world application
+1
+arXiv:2301.02142v1  [cs.LG]  5 Jan 2023
+
+Author: Article Short Title
+2
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+1.
+Introduction
+The future of retail is very uncertain. This uncertainty is strongly connected to the pace at which
+customer consumption patterns, channel shifts, and expectations around speed and convenience
+are changing. The Covid-19 pandemic fuelled this unpredictable behavior, but these changes are
+here to stay. At the moment, several pivotal questions posed by investors, entrepreneurs, business
+professionals, and academics connected to retail remain unanswered (Caro et al. 2020).
+One of the most critical challenges faced by large retailers is related to the future of physical
+retail stores (Adhi et al. 2022). While some believe that physical retail may be “dead”, others
+have many reasons to defend their existence as a way to differentiate from competition, offering
+remarkable experiential shopping based on good customer service, exuberant stores, and online-to-
+physical channels harmonization (Dennis 2018). Nowadays, many retailers recognize that focusing
+on omnichannel retailing is the way to go as a means for enriching customer value proposition
+and improving operational efficiency (Gao and Su 2017a). In particular, the physical channel still
+remains important for a substantial part of grocery retail customers, who find it convenient to
+place an order online and have their groceries picked up at a nearby retail location (Glaeser et al.
+2019, Vyt et al. 2022). However, many challenges emerge from this offline-online dichotomy. For
+example, it is not clear how to provide the online level of information to offline customers (Gao
+and Su 2017b), to which channel should the most stock be committed (Jia et al. 2021), and which
+assortment decisions should be made in the offline and online channels (Chen et al. 2021).
+What is clear is that physical retail stores offer tremendous value in terms of convenience and
+speed, particularly in food retail (Barbee et al. 2021). For example, these assets support the recent
+trend in omnichannel retailing to shift from e-commerce to quick commerce (Q-Commerce), where
+the focus is on providing small quantities of goods ranging from groceries, stationeries to over-
+the-counter medicines in short delivery times of 10 to 30 minutes. This type of service, sometimes
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+3
+called on-demand delivery, requires faster fulfillment strategies that require a shift from traditional
+warehouses located on the outskirts to micro-warehouses or physical retail stores located near the
+delivery locations. This means that the number of in-house and third-party pickers inside retail
+stores should increase considerably in the upcoming times, so that retailers can face the growth
+in the number of online and pick-in-store orders that need to be delivered in short delivery times.
+These omnichannel fulfillment strategies should be further propelled when fully digitized stores are
+deployed, allowing retailers to profit from recent technologies, such as Internet of Things (IoT),
+advanced robotics, and digital twins (Olsen and Tomlin 2020).
+Motivated by the real case of a European retailer, this paper will focus on the satisfaction of
+online orders that have to be picked by in-store pickers. These orders are posted on the website
+of the retailer and can be delivered at home or picked in-store by the customer. This context is a
+very challenging one, and aiding pickers during their picking routes is becoming more and more
+important due to four main reasons. First, the online business is drastically increasing in recent
+times, as customers have been almost forced to experiment online shopping (Richter 2021). Second,
+most physical stores do not have sufficient backroom space to settle a warehouse optimized for
+picking all products that can appear in online orders (Pires et al. 2021). Third, since more and
+more third-party pickers (i.e., Uber, Glovo, and Delivery Hero) are now shopping in the stores due
+to the recent interest in Q-Commerce (Nierynck 2020), it is important to provide a service that
+can aid them to navigate through retail stores when they are not properly aware of the locations of
+every product. Fourth, and more importantly, with the referred influx of internal and third-party
+pickers, the in-store customer experience is at risk, conflicting with online-offline strategies followed
+by grocery retailers.
+In this paper, to achieve picking policies that can be adapted to the dynamic customer flows of a
+retail store and help to mitigate the aforementioned challenges, we introduce a new problem called
+Dynamic In-store Picker Routing Problem (diPRP). This dynamic problem considers a dynamic
+retail store environment and consists in making decisions so that a picker maximizes the number of
+
+Author: Article Short Title
+4
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+orders picked while minimizing the number of customer encounters. Note that modeling uncertain
+customer flows inside a store is a core element of this challenge. Traditional methods for aiding
+pickers based on stochastic mathematical programming are likely to result in intractable Mixed
+Integer Programming (MIP) models and are not suited to capture the dynamics of a sequential
+decision problem. Therefore, the diPRP is modeled as a Markov Decision Process (MDP) and
+solved through a well-known reinforcement learning technique. The picking policies are obtained
+employing a hybrid Q-learning algorithm that maximizes immediate and future rewards (i.e., a
+positive reward is received when a product is picked and a negative reward one is received when
+a customer is encountered) by providing the best actions (i.e., picker movements) to be performed
+by the picker. The objective is to pick the largest number of orders (operational efficiency) while
+avoiding customer encounters (shopping experience).
+We provide evidence on how one can profit from real-world data and orchestrate mathematical
+programming, simulation, and machine learning techniques to solve a relevant practical problem
+in omnichannel retail - the diPRP. The scientific contributions of this work are threefold:
+1. We formalize a new and relevant problem in retail where a picker picks online and pick-in-store
+orders inside a retail store, while avoiding physical customer encounters - the diPRP.
+2. We solve the dynamic problem using a hybrid Q-learning algorithm, which includes MIP and
+MDP components to determine a picking policy that is adapted to the dynamic environment
+of an in-store picking operation.
+3. We provide computational experiments on the algorithm convergence using synthetic instances
+and derive managerial insights on four different picking policies that were validated on a real-
+world application partnered with a large European retailer. These insights aim at helping
+retailers to make better decisions related to their omnichannel strategies. Namely, proving a
+solid ground to scale new fulfillment options.
+The remainder of this paper is organized as follows. Section 2 reviews the relevant literature related
+to in-store picker routing problems. In Section 3 we describe the problem and present the inherent
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+5
+retail store simulation environment to be used in our study. Section 4 details the proposed solution
+approach based on a Q-learning algorithm. Section 5 presents the computational experiments
+performed on synthetic instances, which showcases the algorithm convergence. Section 6 exposes
+a real-world application in a European retail store. Finally, in Section 7, we present the main
+conclusions of this work and suggest future research directions.
+2.
+Literature Review
+In the following subsections, we review recent approaches dealing with picker routing problems
+(Section 2.1), dynamic routing problems Section 2.2, and dynamic picker routing problems Section
+2.3. We consider that these topics cover the modeling and solution approach techniques needed to
+tackle our problem - diPRP -, which consists of picking products in a dynamic store environment,
+a new type of picking operation.
+2.1.
+Picker routing problems
+The picker or order routing problem was introduced by Ratliff and Rosenthal (1983) and is typi-
+cally modeled as a Traveling Salesman Problem (TSP) on a Steiner graph (Letchford et al. 2013,
+Cambazard and Catusse 2018, Rodr´ıguez-Pereira et al. 2019). Several problem extensions have
+been proposed in the literature to adapt the standard model to real-world cases with specific busi-
+ness constraints (Chabot et al. 2017, Quader and Castillo-Villar 2018, Ardjmand et al. 2018) and
+to combine subsequent problems, such as storage and batching (van Gils et al. 2018). In terms of
+solution methods, this problem is usually tackled with heuristics (Theys et al. 2010), but important
+problem properties were discovered to increase the efficiency of exact methods based on mathemati-
+cal formulations (Cornu´ejols et al. 1985, Pansart et al. 2018). Recently, new formulations have been
+proposed to increase the size of the instances solved exactly (Scholz et al. 2016) and to integrate the
+order picking problem with other activities in the supply chain (Roodbergen et al. 2015), as this
+remains to be a relevant problem in warehouse operations management. Picker routing problems
+are typically solved within narrow aisle warehouses (Roodbergen and de Koster 2001, Roodbergen
+and Koster 2001, de Koster et al. 2007, Chabot et al. 2018, Masae et al. 2020) and not inside
+
+Author: Article Short Title
+6
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+stores. Considering layouts that are not optimized for picking operations and can be crowded with
+customers lead to a new context that has never been studied in the order picking literature. In a
+store, product layouts may be very irregular and most graph properties assumed in order picking
+literature are not applicable. Furthermore, customer shopping paths can be very chaotic and are
+not known apriori, meaning that picking problems are more stochastic and dynamic.
+2.2.
+Dynamic routing problems
+Although stochastic dynamic routing has a 30-year history, the majority of the papers published
+in this field are recent. More than half of the papers in the literature review of Ulmer et al. (2020)
+have been published after 2010. Other literature reviews on this topic have been published by
+Berbeglia et al. (2010), Pillac et al. (2013), Ritzinger et al. (2016), Psaraftis et al. (2016), and Ojeda
+Rios et al. (2021). It is worth mentioning that multiple trends (e.g., e-commerce growth, sharing
+economies, and sustainability) and technological advances (e.g., digital connectivity, big data, and
+automation) are fostering the development of new approaches to tackle dynamic routing problems
+(Savelsbergh and Van Woensel 2016). To solve these problems, the unified framework for stochastic
+programming proposed by Powell (2019) has been determinant, as researchers have been adapting
+it to tackle dynamic problems modeled as MDPs. To turn dynamic programming applicable to
+large real-world problems, the approximate dynamic programming approaches presented in Powell
+(2007) have also been a preponderant reference in this field.
+More recently, several attempts have also been tried to solve routing problems modeled as MDPs
+through Reinforcement Learning techniques (Nazari et al. (2018), Li et al. (2021), Kullman et al.
+(2022)). However, despite their potential to be used in new contexts, these methods are still seen
+as unexplored by the routing community, as suggested by Hildebrandt et al. (2021).
+2.3.
+Dynamic picker routing problem
+In some business contexts, it is interesting to consider a dynamic variant of the picker routing
+problem. These problems arise when new information allows for better planning or execution of the
+picking operations (i.e. a new order arrived). With respect to dynamic picker routing problems, the
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+7
+literature is very scarce. Gong and Koster (2008) consider picking operations where picking infor-
+mation can dynamically change in a picking cycle. Based on a stochastic polling theory approach,
+the authors suggest that shorter order throughput times and higher on-time service completion
+ratios can be achieved. Lu et al. (2016) consider dynamic order-picking strategies that allow for
+changes in the picking lists during a picking route. The authors allow for the re-optimization of
+the optimal route during the picking operation and achieve better solution quality when compared
+to the static version of the problem.
+Note that these authors consider that dynamic information arriving is only related to customer
+arrivals. However, in a store, there can be two types of entities performing picking, the offline
+customers and the online order pickers. Pickers can gather information on the location of the
+customers as they perform their picking paths. This maps into a new dynamic picking routing
+problem. However, the interactions between in-store pickers and in-store customers have not been
+considered yet, that is, no approach has been proposed for solving the dynamic in-store picker
+routing problem.
+Despite the considerable number of articles dealing with picker routing problems, no publication
+has considered the new environment that is faced by in-store picking teams. Indeed, the pandemic
+context of 2020 brought new operations to be managed and new challenges to be tackled while
+pursuing different objectives. For instance, constraints imposed on the number of customers inside
+of a store may turn retailers interested in guiding customers through their shopping paths to
+accelerate customers’ shopping process, as opposed to making them stay for longer inside the store
+(Hui et al. 2013, Boros et al. 2015, Hirpara and Parikh 2021). Each customer in-store is also solving
+its picker routing problem to define its shopping path. The fact that customers do not know the
+exact position of the products and that some customers may have picking sequence preferences
+(Larson et al. 2005) can induce chaotic flows of customers through the aisles of a store (Chen
+et al. 2015, Li et al. 2017). This means that the state of the store is not known apriori when the
+picking route of each picker is being defined, it is stochastic. Therefore, in case we are interested in
+
+Author: Article Short Title
+8
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+considering picking paths that do not disturb customers’ shopping experience, the picker routing
+problem proposed approaches fall short on tackling this issue. This is a gap that we address in this
+work.
+3.
+Problem description
+In this section, we formalize the diPRP by describing all the relevant elements included in the
+considered retail store environment, the picker agents that will make decisions inside the store, and
+the MDP that is proposed to model the associated dynamic problem. Hence, Section 3.1 details the
+retail store environment, including the store graph, the models for simulating the shopping paths
+of physical customers, and the online order arrivals composed of a set of picking positions. Section
+3.2 clarifies the behavior of a picker considering the information that is available in each step (e.g.,
+next product to be picked and physical customers nearby). Section 3.3 describes the MDP that
+models the dynamic problem we aim to solve, including decision epochs, states, actions, transition
+function, rewards, and objective function.
+3.1.
+Retail Store Environment
+To model a realistic retail store environment, we resort to a simulation environment considering
+physical customer arrivals and online order arrivals. The store is modeled as a sparse graph G =
+(V,E) where the set of vertices (locations in-store) is partitioned into vertex 0, which is the entrance
+of the store where the shopping paths start, vertices {1,...,v}, corresponding to v positions inside
+the store, and vertex v + 1, which is the position where the shopping paths end (i.e., a cashier or
+an order preparation zone). Physical customers arrive at the store following a Poisson distribution
+with a mean value of λStore customers per time period. Each physical customer c ∈ C moves through
+the aisles of the store navigating through the graph G at a speed speedCustomer and according to
+a given shopping path θc containing a sequence of locations in-store. A physical customer takes
+service timeCustomer time periods to pick a product located at a certain node. The maximum
+number of customers inside the store can never surpass the capacity of the store |C| and the
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+9
+maximum number of customers in the same store node is m. The arrival rate of online orders is
+also a random variable, which follows a Poisson distribution with a mean value of λOnline orders
+per time period. Each online order o ∈ O is composed of a set of picking positions Lo ⊂ V that need
+to be visited to pick the ordered products.
+3.2.
+Picker agents
+Whenever an online order o arrives it needs to be allocated to an idle picker. At that moment,
+a picking sequence θo is defined according to a business decision rule (e.g., shortest route, heavy
+items first, and frozen products at the end). In this work, we start by assuming that these pick-
+ing sequences are minimizing the traveled distance, i.e., shortest route, - maximizing operational
+efficiency, which is the common case in practice. The picker follows a given picking sequence, yet
+he is interested in avoiding customer encounters not to disturb their shopping experience. There-
+fore, between every pair of products to be picked, the picker solves a shortest path problem, while
+avoiding customers currently in the store. The dynamic component of this problem corresponds to
+solving a series of dynamic shortest path problems. The fundamental trade-off that is inherent to
+this problem comes from the fact that the picker follows a predefined sequence of picking positions
+that is provided by a decision rule adopted by the retail store (e.g., shortest routes), but he/she
+needs to avoid customer encounters in the path between two picking positions. The position of the
+customers in-store is not known in advance and it is only revealed when the picker looks through
+the aisle. Throughout the day, each picker receives several online orders, and thus, several picking
+sequences to execute.
+3.3.
+Dynamic picker routing problem as a Markov decision process
+To formulate the diPRP as finite horizon MDP with T time periods, consider a decision epoch
+k ∈ 0,1,..,T at which a decision for the next position node to visit inside the store should be made.
+A new decision epoch k is triggered whenever the picker arrives at a new position. Each decision
+epoch k is associated with a decision state sk. At the initial state s0 and initial decision epoch
+k = 0, the picker is located at the starting depot node 0. At the terminal decision epoch k = T,
+
+Author: Article Short Title
+10
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+the picker returns to the finishing depot node v + 1 after picking the order that is currently being
+picked. At a state sk, the system receives updated information on the number of customers at the
+same position of the picker and adjacent positions (connected by an arc). Based on the system
+information about current online orders, an action ak (i.e., the next position to visit in targeting a
+product or the depot) needs to be decided. Each position that is visited increases the total travel
+time and, potentially, the number of customer encounters. The goal of the picker is to find optimal
+paths, short and quick to traverse while minimizing the number of customer encounters. The picker
+is positively rewarded for each product picked. Overall, the problem faced by the in-store picker
+can be modeled as a MDP composed of four components, namely, (1) a set of states S; (2) a set
+of actions A; (3) a reward function R; and the transition probabilities P.
+3.3.1.
+Decision epochs A decision epoch k begins when a picker arrives at a new location
+in-store.
+3.3.2.
+States The state of the system at decision epoch k is defined by the tuple sk =
+(nk,zk,tk), where nk ∈ V is the picker’s current location, zk = (zk(1),zk(2),...,zk(|Lo|)) is a vector
+containing the status of each picking location of an online order o at decision epoch k, and tk ∈ [0,T]
+is the arrival time at location nk. For each picking location of an online order, zk(.) takes on a
+value in the set {0,1}:
+zk(.) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0,
+if picking location has not been visited by time tk
+1,
+if picking location has been visited by time tk
+(1)
+In the initial state s0 = (0,z0,0), the picker is located at the order preparation zone and z0(n) is
+0 or 1 for each picking location n ∈ V \ {0}. The final decision epoch is attained when the picker
+reaches the terminal state sK in the set {(0,zk,tk) : tk ∈ [0,T],zk ∈ {0,1}|Lo|}, where the picker has
+returned to the preparation zone by time T and the status of the picking locations of the online
+order being served is 0 or 1. The state space is the set S = V × [0,T] × {0,1}|Lo|.
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+11
+3.3.3.
+Actions An action at decision epoch k is an assignment of the picker to an in-store
+location belonging to V. When the picker is at state sk, the set of feasible actions, corresponding
+to the possible arcs leaving nk is given by
+A(sk) = {ak ∈ {n ∈ V : (sk,n) ∈ E}} :
+(2)
+3.3.4.
+Transition Following the selection of action ak from state sk, the process transitions
+to a new state sk+1 in decision epoch k +1. There is no uncertainty regarding the position in which
+the picker will be in the next decision epoch. However, the unknown information regarding the
+number of customers encounter is revealed. This information is a realization of Ωk+1 and is given
+by ωk+1. Let �Rk+1(sk,ak,ωk+1) be the random negative reward accrued at decision epoch k when
+selecting action ak from state sk and observing random information ωk+1.
+3.3.5.
+Rewards When the picker occupies state sk and performs an action ak ∈ A, a reward
+is accrued depending on the conditions of the in-store location to which the picker moves. The
+reward is composed of four components, (1) fixed negative reward for each picker step, (2) number
+of in-store customers in the same location of the picker, (3) number of in-store customers in the
+locations reachable from the picker location, and (4) a positive reward for a picked product. The
+number of steps given from the last decision epoch is given by φ1, the number of customers in
+the same location is given by φ2, the number of customers nearby is given by φ3, and the number
+products picked is given by φ4. These four components can be adjusted using weights w1 to w4,
+respectively. The reward Rk(sk,ak,ωk+1), which is a stochastic variable depending on the store
+environment, is defined as
+Rk(sk,ak,ωk+1) = −w1 φ1(ωk+1) − w2 φ2(ωk+1) − w3 φ3(ωk+1) + w4 φ4(ωk+1)
+(3)
+3.3.6.
+Objective The objective is to maximize the expected sum of rewards across decision
+epochs, that is, if weights are properly set, trying to pick as much orders as possible while mini-
+mizing in-store customer encounters. Let π be a sequence of actions a0,a1,...,aT for every decision
+
+Author: Article Short Title
+12
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+epoch k = 0,...,T. The aim is to find an optimal policy π to maximize the total expected reward.
+The objective function, conditional on initial state s0, is the following
+min
+π E
+�
+T
+�
+k=0
+Rk|s0
+�
+(4)
+4.
+Solution approach
+The diPRP defined in Section 3 demands solution techniques with different scopes and purposes:
+(1) whenever an online order arrives, a Shopping Routing Problem (SRP) model, an MIP, needs
+to be solved to obtain a picking sequence according to predefined decision rules; and (2) for each
+picking sequence and for every pair of products on that sequence, a sequential decision problem is
+solved and recursively updates the policy via a Q-learning algorithm. Figure 1 presents an overview
+of the devised approach, showing where each technique is needed and how the picker agent interacts
+with the store environment.
+In Section 4.1, we describe the problem, the mathematical formulation and the solution approach
+that are used to obtain the picking sequences to feed the picker agent related procedures. In Section
+4.2, we describe the proposed Q-learning approach to learn an optimal policy for efficiently picking
+orders while avoiding in-store customer encounters. Section 4.3 provides an example of picking
+paths obtained with different policies and an example of what a policy obtained with Q-Learning
+looks like.
+4.1.
+SRP definition
+To introduce the standard SRP, consider the retail store environment described in Section 3.1.
+Each edge (i,j) ∈ E is a pair of positions associated with a sequence of grid steps corresponding to
+the shortest path between picking position i and j. Edges are weighted by a factor wij, typically
+representing the distance or the duration of the arc. Again, let Lo ⊂ V be a subset of vertices
+corresponding to the picking positions that have to be visited to pick a given shopping list o. The
+main objective of the picker is to pick all the products in the shopping list while minimizing the
+distance (or duration) of its shopping path.
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+13
+Picker Agent
+SRP 
+Solver
+Q-Learning
+Algorithm
+Store 
+Environment
+New 
+online 
+order?
+Observation
+Action
+Reward
+Routing
+Policy
+Update
+Policy
+Yes
+No
+Figure 1
+Approach overview including a detailed breakdown of the picker agent.
+4.1.1.
+SRP Mathematical formulation To model the SRP we use binary decision variables
+xij for defining routing decisions. Let xij be the binary variables indicating whether an edge (i,j)
+is traversed. The proposed formulation reads as follows:
+SRP:
+minimize
+�
+(i,j)∈E
+wij xij
+(5)
+s.t.
+�
+i∈V
+xji = 1
+∀j ∈ Lo ∪ {0}
+(6)
+�
+i∈V
+xij = 1
+∀j ∈ Lo ∪ {v + 1}
+(7)
+�
+i∈O
+�
+j∈O
+xij ≤ |O| − 1
+∀O ⊂ V,i ̸= j
+(8)
+x ∈ {0,1}.
+(9)
+Objective function (5) minimizes the total distance (or duration) of the shopping path. The flow
+conservation at the starting position, product picking positions, and finishing position are ensured
+by constraints (6) and (7). Constraints (8) are added to eliminate sub-tours. Finally, constraints
+(9) define the type and bounds of each variable.
+
+Author: Article Short Title
+14
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+4.1.2.
+SRP Solution approach In the context of diPRPs, each SRP needs to be solved
+in fractions of a second. Furthermore, in a simulation procedure (later used for optimizing the
+picking policies), we need to solve hundreds if not thousands of SRP instances. Relying solely on a
+mathematical solver is not sufficient when the picking lists comprise a considerably large number
+of products. For that reason, we tackle this problem with an efficient exact solution approach that
+can quickly solve the mathematical formulation presented in the previous section. We employ a
+cutting planes algorithm where cuts are added to remove fractional solutions from the admissible
+region of the linear relaxation (Algorithm 1).
+Algorithm 1 Cutting Planes
+1: procedure CuttingPlanes(V, Lo)
+2:
+CutsAdded ← True;X ← ∅;Y ← ∅;Z ← 0;
+3:
+while CutsAdded do
+4:
+X ← Solve assignment model (5)-(7) considering sets V and Lo;
+5:
+Y ← Separate arcs in sets of connected components;
+6:
+Z ← Count the number of connected component sets in Y;
+7:
+if Z = 1 then
+8:
+CutsAdded ← False;
+9:
+else
+10:
+Add cuts (8) to eliminate sub-tours;
+11:
+CutsAdded ← True;
+12:
+if CutsAdded then
+13:
+X ← Convert positive variables to integer;
+14:
+else
+15:
+if All edges in the solution are integer then
+16:
+θo ← The picking sequence is obtained from the model solution X;
+17:
+return θo.
+The algorithm starts by solving a relaxed assignment model (Algorithm 1, line 4) and separating
+connected component sets (Algorithm 1, line 5). If there are more than one connected component
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+15
+set, the solution has sub-tours (Algorithm 1, line 10) and cuts are added to eliminate these sub-
+tours (Algorithm 1, line 11). When cuts are added, the edge variables with a positive value are
+converted to an integer. This process is repeated until no cuts are added after solving the relaxed
+assignment model and all the edge variables are integer, meaning that the optimal solution is found
+(Algorithm 1, line 17).
+4.2.
+Q-Learning
+The cutting plane approach presented in Section 4.1.2 provides a sequence θo for picking the list of
+products of online order o. However, an in-store picker has multiple options to go from one picking
+position to another. In the SRP formulation we assume shortest paths, that is, the steps to perform
+when traversing an arc (i,j) are the ones leading to the shortest distance between i and j (again,
+note that other decision rules can be used). Therefore, in order not to disturb in-store customers,
+it may be beneficial to perform different steps or detours while traveling from i to j, depending on
+the unknown patterns related to the movement of the customers in-store.
+To obtain a routing policy that follows a given picking sequence but tries to avoid customers, we
+devise a Q-Learning (Watkins 1989) approach to solve the MDP previously presented in Section
+3.3. This technique belongs to reinforcement learning, which is a branch of approximate dynamic
+programming. In simple terms, it is a value-based method which searches for action-value functions
+Q(sk,a) representing the future expected reward of taking action a from state sk.
+Let us denote Vπ(sk) as the value function that gives the total expected reward if we start from
+a state sk and follow policy π. This value includes the immediate reward received at decision epoch
+k+1 and the expected rewards to obtain in the future until the final decision epoch T. Our objective
+is to find the action array that maximizes the total reward obtained through the entire process.
+Considering the Bellman Equation (Powell 2007), V ∗(sk) =
+max
+ak∈A(ak){Rk+1 + E[V (sk+1)|sk,a]}, an
+optimal action ak is implemented by solving the following expression,
+ak = arg max
+ak∈A(sk)
+{Rk+1 + E[V (sk+1)|sk,a]}
+(10)
+
+Author: Article Short Title
+16
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+In the problem we are considering, agents do not control the characteristics of the state they will
+end up in due to the uncertain customer shopping paths. Therefore, they only have limited influence
+in the next state that they will visit, depending on their current state sk and a performed action
+ak. Let Qπ(sk,ak) be a function (Q-function) representing the quality of action ak when the agent
+is at a given state sk and follows policy π. We can define the optimal Q-function as Q∗(sk,ak). The
+relation between V ∗(sk) and Q∗(sk,ak) is given by the following expression,
+V ∗(sk) = max
+a∈A(sk)Q∗(sk,a)
+(11)
+Therefore, an optimal policy π∗ for the problem we aim to solve can be derived through the
+following expression,
+π∗(sk) = arg max
+a∈A(sk)
+Q∗(sk,a)
+(12)
+If we can identify Q∗, we have an optimal policy to control the actions of the picker agent in the
+store environment in which he was trained. This policy can be determined using the referred Q-
+Learning approach, using the Bellman equation to approximate Q∗. The Bellman equation based
+on a Q-function can be expressed recursively by
+Q(s,a) = R + γ max
+a′ Q(s′,a′)
+(13)
+The Q-learning equation to iteratively approximate Q∗ is given by
+Qnew(s,a) = Q(s,a) + α
+�
+R + γ max
+a
+Q(s′,a) − Q(s,a)
+�
+(14)
+Where the learning rate α adjusts how different previous and new Q values are from each other.
+As the agent explores more and more the environment, the approximated Qnew values will converge
+to Q∗. To extract an optimal picking policy for a store environment, we propose Algorithm 2.
+The algorithm starts by initializing the Q-function with random optimistic values (Algorithm 2,
+line 2). These optimistic values promote the exploration of new states. Then, a cyclic procedure
+is repeated for a certain number of episodes (Algorithm 2, line 3). First, the store environment
+initializes at the initial state s0 (Algorithm 2, line 4) and, while it does not reach the final state
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+17
+Algorithm 2 Q-Learning procedure
+1: procedure QLearning(MDP, α, γ, ϵ)
+2:
+Initialize Q(s,a) using random optimistic values;
+3:
+for each episode do
+4:
+Initialize the environment with the initial state s0 (picker at the depot);
+5:
+while sk is not the final state (the store is not closed) do
+6:
+Choose an action ak available from the current state sk following an ϵ-greedy strategy;
+7:
+Execute action ak, receive a reward Rk, and observe a new state sk+1;
+8:
+Update Q-function by doing Q(sk,ak) ← Q(sk,ak) + α
+�
+Rk + γ arg max
+a
+Q(sk+1,a) − Q(sk,ak)
+�
+9:
+if Q converges then
+10:
+Break;
+11:
+return Q
+sT (Algorithm 2, line 5), it chooses an action, executes it, and updates the Q-function (Algorithm
+2, lines 6 to 8). The choice of the action is performed using an ϵ-greedy, meaning that a random
+action is chosen with probability ϵ, and the action with the best Q value is chosen with probability
+1 − ϵ. After finishing an episode, the algorithm verifies if the Q-function converged (i.e., check if
+the maximum difference between old and new q-values is lower than a convergence threshold). If
+it did converge, the algorithm returns the Q-function, otherwise, it continues to the next episode.
+4.3.
+Illustrative example
+The solution approach developed in this work converges to optimal policies that are substantially
+different from a shortest path-based policy, depending on the reward shape assumed. Figure 2
+illustrates a comparison between two paths with similar origin and destination locations. In this
+figure, the size of the nodes measures the average number of customers that have been present at
+each location. We observe that the central aisle and the product positions in the top middle aisles
+were more crowded during the simulated episode. The SP policy (shortest path-based), in red,
+results in a shorter distance but it encounters more customers. On the other hand, the QL policy
+(Q-Learning-based), in blue, results in a large detour that, on average, encounters fewer customers.
+
+Author: Article Short Title
+18
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+Figure 2
+Illustrative example of a SP (red) and a QL path (blue).
+The blue path is obtained by simply following a policy extracted from the Q values. Figure 3
+presents an example of three steps performed using a Q-table (lookup table).
+States
+https://www.baeldung.com/cs/epsilon-greedy-q-learning
+Available	Actions
+S1
+(85,	123)
+…
+…
+A1
+A2
+A3
+S2
+(77,	123)
+S3
+(62,	123)
+Q((85,	123),	77)
+Q((85,	123),	93)
+-
+…
+Q((77,	123),	62)
+Q((77,	123),	85)
+Q((62,	123),	54)
+Q((62,	123),	63)
+-
+…
+Q((62,	123),	77)
+…
+…
+…
+…
+1st step
+2nd step
+3rd step
+Figure 3
+An example with three steps obtained from the Q-table that was used to obtain the Q-learning path
+in the illustrative example.
+Each row represents a state that is a tuple containing the current position (node 85) and the
+target position (node 123). Each column corresponds to a possible action considering the current
+positions of the picker. For instance, in the first step, the picker could walk to node 77 or node 93.
+Since the Q value of node 77 is larger, the picker moves to position 77.
+5.
+Algorithm convergence
+In this section, we present the computational experiments to explore the efficacy and efficiency of
+the proposed approach on synthetic instances. Firstly, we describe the process that was used to
+generate the synthetic instance set. Secondly, we perform a search on the training parameters to find
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+19
+good combinations of learning rate, discount rate, and ϵ-greediness and validate the convergence
+of the proposed approach on different problem sizes.
+5.1.
+Synthetic instances
+To validate the proposed approach, synthetic instances were generated by considering different
+simulation parameter combinations. We tested combinations of four store layouts with different
+sizes and with three different customer concentrations inside the store. As such, store environments
+can be tiny, small, medium, and large, and the products of customers’ shopping lists can be more
+concentrated near the entrance, the middle, or the back of the store. The number of products in
+customers’ shopping lists follows a uniform distribution with a minimum of 1 and maximum of 10
+products. These products and their picking sequence are randomly chosen and customers follow
+the shortest path when moving from one product position to another product position. The arrival
+rate of physical customers is set to 2 and the arrival rate of online orders is set to 0.2. In each
+episode, the store is open for 8 hours, and the maximum number of customers inside the store |C| is
+set to 50. The maximum number of customers per node m is 5 and each customer takes 30 seconds
+to pick a product. The walking speed is set to 1 m/s. Figure 1 summarizes the sets of parameters
+that were used to build 12 synthetic instances. Regarding rewards and penalties in the synthetic
+instances, when a picker is in the same node where a customer is, it is penalized by 3 points. If a
+picker is seen by a customer (i.e., the customer is in the nodes accessible by the picker), the picker
+is penalized by 1 point. Picker steps are penalized by 1 point and products picked are rewarded
+with an amount of 100 points.
+5.2.
+Q-learning training and results
+Despite the potential of reinforcement learning approaches, tuning them is often a non-trivial and
+time-consuming task. This is essentially because testing parameter combinations can be an expen-
+sive and noisy process. To overcome these difficulties, we tested several parameter configurations
+to determine policies for each synthetic instance. Each configuration was run for 5000 episodes
+using a single thread. All computations were run on Intel Xeon @ 2.5 GHz processing units and the
+
+Author: Article Short Title
+20
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+Table 1
+Sets of parameters used to build the generated instances.
+Store layout {Tiny,Small,Medium,Large}
+Customer concentration {Entrance,Middle,Back}
+Physical customer arrival rate {2}
+Online order arrival rate {0.2}
+Open time (h) {8}
+Maximum number of customers inside {50}
+Maximum number of customers in a location {5}
+Product picking time (s) {30}
+Walking speed (m/s) {1}
+Customer encounter penalty (p1) {3}
+Customer visible penalty (p2) {1}
+Step penalty (p3) {1}
+Picking reward (p4) {100}
+Table 2
+Sets of reinforcement learning parameters used to build training configurations.
+Learning rate (α) {0.95,0.97,0.99}
+Discount rate (γ) {0.50,0.70,0.90}
+ϵ-greediness (ϵ) {0.01}
+mathematical solver adopted to compute picking sequences of the pickers was Gurobi 9.5. Table 2
+presents the parameter sets that were used to build each training configuration.
+The cumulative rewards obtained for the training episodes are presented in Table 3. The first two
+columns, “Layout” and “Paths”, indicate the layout of the store and the type of shopping paths
+performed by the customers. The third column, “Best Parameters”, indicates the combination of
+parameters (α|γ|ϵ) that obtained the values presented in the respective row. Columns four to six
+present the minimum, average, and maximum cumulative rewards obtained across all the tested
+parameter configurations. The last column presents the coefficient of variation over the last 50
+observations.
+We observe that for the larger store layouts the difference between the minimum and maximum
+cumulative rewards obtained increases. As the problem size increases, the importance of choosing
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+21
+Table 3
+Cumulative rewards obtained in the training procedure (5000 episodes).
+Cumulative rewards
+Layout
+Paths
+Best Parameters
+(α|γ|ϵ)
+Min.
+Avg.
+Max.
+CV
+(last 50 obs.)
+tiny
+far
+0.97|0.9|0.01
+28989.54
+29262.92 29440.60
+0.11
+tiny
+middle
+0.99|0.7|0.01
+27240.37
+28563.90 29262.10
+0.11
+tiny
+near
+0.95|0.9|0.01
+28325.40
+28810.05 29073.35
+0.10
+small
+far
+0.97|0.9|0.01
+23471.06
+26828.68 28569.88
+0.11
+small
+middle
+0.97|0.9|0.01
+8571.05
+21363.16 28362.71
+0.10
+small
+near
+0.97|0.9|0.01
+11729.02
+22830.18 28236.43
+0.10
+medium far
+0.99|0.9|0.01
+-249624.48
+-74201.81 26741.11
+0.11
+medium middle
+0.99|0.9|0.01
+-280708.50 -136167.61 26352.44
+0.12
+medium near
+0.97|0.9|0.01
+-275092.86 -144711.93 25733.31
+0.08
+large
+far
+0.95|0.9|0.01
+-282733.08 -172079.58 22584.38
+0.10
+large
+middle
+0.95|0.9|0.01
+-286259.32 -152702.76 17371.42
+0.18
+large
+near
+0.97|0.9|0.01
+-285686.24 -180264.63 21524.09
+0.13
+a good combination of parameters also increases. As expected, the cumulative reward obtained in
+larger stores is smaller, as the picker spends more time traveling through a larger number of nodes
+particularly when he/she needs to return to the preparation zone. Moreover, we observe that for
+the last 50 observations the coefficient of variation of the rewards is low for all instances sizes,
+enforcing the idea that the algorithm converged in all of them. At this point, to better understand
+the algorithm convergence, we present Figure 4.
+Figure 4
+Cumulative rewards obtained by the best parameter configurations used in synthetic instances (first
+500 episodes of the training session).
+
+Author: Article Short Title
+22
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+The cumulative rewards suggest that the algorithms converge in a few episodes. As expected,
+the size of the store considered in each instance is connected to the time it takes for the cumulative
+rewards to converge. While in instances with a medium layout we observe the algorithm converging
+after a few episodes, in instances with a large layout the cumulative rewards stabilize after more
+than 200 episodes. Nonetheless, the proposed approach is successful in achieving picking policies
+that do not demand substantial computational power.
+6.
+Real-world application
+This section details a real application in the context of a large European retailer. The devised
+approach was applied using the layout of a real-world store considering real shopping paths collected
+from customers using a mobile app that allows them to keep track of products they pick in-store
+as they are shopping.
+A very important aspect of the devised approach is that it does not require substantial compu-
+tational power. The output of the Q-learning approach is a policy in the form of a Q-table (lookup
+table) that can be used in simple mobile devices (i.e., phones and PDAs). Moreover, this can be
+improved over time in an offline or in an online fashion, using new data collected from customer
+and picker shopping paths to update Q-tables. It is even possible to compute Q-tables tailored for
+different parts of the day or of the week, suited to the customer flows that are more likely to occur
+during these periods.
+6.1.
+Real-world instance
+The first step to applying the devised approach to the considered real context was to represent the
+retail store with substantial detail. This is a particularly challenging process due to the lack of data
+regarding product positions, which occurs in practice. Large retailers are now digitizing physical
+stores as the opportunities provided by the new data collected are remarkable. However, in the
+retailer considered in this study, this process is still in its infancy. In the retail business, several
+products are likely to be relocated daily for a multitude of reasons (i.e., promotional activities,
+new products introduced in the assortment), which increases the difficulties of maintaining reliable
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+23
+information on product positions. Nonetheless, we mapped products to aisles, achieving a realistic
+representation of the real store. Figure 5 presents the store layout and graph considered in the
+real-world application.
+Figure 5
+Representation of the store layout and graph considered in the real-world application.
+In this figure, the nodes represented in red are aisle intersections and the nodes represented in
+blue are product positions. We opted for considering at most two product nodes inside each aisle
+as it already provides a good indication of the position of each product. In the considered graph,
+customers are allowed to enter the store by the node represented in green and, after traversing their
+shopping path, they randomly choose one of the exit nodes represented in orange. The sequence of
+products picked by the physical customers is based on real-world data collected using the mobile
+app provided by the retailer. To compute the shortest path between each pair of products in the
+customer shopping path we resorted to an implementation of a Dijkstra algorithm. The remaining
+parameters were set according to discussions with the retail store managers. The arrival rate of
+physical customers λStore is set to 10 and the arrival rate of online orders λOnline is set to 0.3. The
+maximum number of customers allowed in the store is set to 300 and we did not set a maximum
+number of customers per node in the real application. Each customer takes 30 seconds to pick a
+product and walks at a speed of 1 m/s. The store is open for 8 hours. The picker agent is awarded
+with 50 points for each product picked, it is penalized by 3 points for each customer encountered
+in the same node, and it is penalized by 1 for each customer seen in adjacent nodes.
+
+DMSAMOE
+COULIUEULE
+YBEDEAENDNJ003SSAuthor: Article Short Title
+24
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+(a) Shopping times histogram.
+(b) Most crowded nodes.
+2,frozen
+3,dairy
+2,salty grocery store
+3,wine and spirits
+2,candy grocery store
+3,take away
+3,salty grocery store
+2,essentials
+3,house cleaning
+1,dairy
+2,house cleaning
+2,the bakery
+3,essentials
+1,beauty
+2,hygiene
+3,candy grocery store
+2,fruits and vegetables
+3,fruits and vegetables
+1,hygiene
+3,breakfast
+2,beauty
+2,dairy
+3,charcuterie
+2,butchery
+3,frozen
+2,breakfast
+2,culture
+3,juices, beers and water
+2,juices, beers and water
+3,the bakery
+2,charcuterie
+2,wine and spirits
+2,take away
+3,butchery
+3,hygiene
+3,pet food and care
+3,bio and healthy
+2,fishmonger
+3,beauty
+2,bio and healthy
+3,house - textiles and decoration
+3,fishmonger
+2,pet food and care
+3,soft drinks
+(c) Categories visited by customers picking 3 products starting from hygiene, dairy, and beauty.
+Figure 6
+Real-world store simulation environment metrics.
+To further describe the real-world store simulation environment that was used to train the
+picker agents, we present Figure 6 containing the distribution of the customer shopping times,
+a representation of the most crowded areas inside the store (average number of customers per
+decision epoch), and examples of category paths performed by customers picking only 3 products
+(for better visualization) and starting from the categories hygiene, beauty, and dairy.
+We observe that most customers stay in-store for a period of 5 to 15 minutes. The most crowded
+nodes are the ones on the left side of the image, mostly corresponding to food products, and at
+the bottom of the image corresponding to the front aisle of the store near the cashiers. The order
+preparation zone from which the pickers depart is located at the top right of the picture. This area
+is usually less crowded as it includes work tools and various materials for DIY activities. We also
+observe that customers follow very distinct shopping paths even if they share a few nodes/products
+in their shopping lists.
+
+BOMEWC
+COULIUEULE
+ACVEV 30 ARA
+J003WS
+S
+NAuthor: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+25
+6.2.
+Training in a real-world context
+The training process that was used in the real-world case was similar to the one presented in Section
+5.2. First, we set up several training sessions considering different combinations of reinforcement
+learning parameters, namely, learning rates α, discount rates γ, and exploration probabilities ϵ.
+The reinforcement learning parameter sets that were used are presented in Figure 4.
+Table 4
+Sets of reinforcement learning parameters to build training configurations for the real-world instances.
+Learning rate (α) {0.1,0.5,0.9,0.95,0.97,0.99}
+Discount rate (γ) {0.1,0.5,0.9,0.95,0.97,0.99}
+ϵ-greediness (ϵ) {0.01,0.03,0.05,0.07,0.09}
+A total of 30000 episodes were run for each configuration. Policies and the average cumulative
+rewards were saved every 1000 episodes. The 10 best policies (in terms of average rewards) obtained
+in the training results related to the real retail store environment are presented in Figure 7. Every
+Figure 7
+Average cumulative rewards for the top 10 policies among all parameter configurations in the real-world
+store environment.
+1000 episodes we restarted the computation of the average cumulative rewards. Note that after the
+first 1000 episodes some policies were already performing better but had not converged. In general
+terms, the best policies converged after 20000 episodes in the training procedure. A few policies
+could not match the performance of the best policies even after 30000 episodes.
+
+Author: Article Short Title
+26
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+In some reinforcement learning problems, average cumulative rewards obtained during the train-
+ing phase (considering random actions) can be different from the ones obtained during the testing
+phase (constantly choosing the optimal action according to the Q-table). There is a large debate
+on how to evaluate these approaches and whether the ϵ-greedy exploration should be kept during
+production or testing phase where no updates will be made to the picking policy. In some problems,
+it may make sense to keep this exploration feature as the agent may get stuck in the presence of
+unseen states. This is done successfully in Mnih et al. (2015) while playing Atari games. When
+solving path problems, non-optimal policies can be affected by cycles in case there is no randomness
+considered. For this reason, we consider that the ϵ parameter is to be maintained when the policy
+is used in practice.
+6.3.
+Managerial insights
+The results shown in the previous sections suggest that the method can learn picking policies to
+be applied in real retail store environments. Indeed, we can observe the picker learning as he/she
+achieves higher and higher cumulative rewards. Nonetheless, one may ask whether we need all this
+work to pick a set of products or if it would be sufficient to just follow simpler policies as it is the
+case of a shortest path policy uniquely based on distance metrics.
+The main questions that motivated us to solve this problem still remain. Can we improve cus-
+tomer experience? What is the impact on the number of picked orders?
+To answer these questions, we measure the performance of four different picking policies regarding
+business-related indicators such as the number of orders/products picked and customer encounters.
+Q-learning (QL) Policy trained using the devised Q-learning that aims at finding a good com-
+promise for picking orders while avoiding customer encounters.
+Shortest Path (SP) Policy based on a shortest path policy (employing a Dijkstra algorithm)
+aiming at picking the most orders and ignoring customer encounters. Each step is obtained by
+solving a shortest path problem considering arcs weighted by their length.
+Myopic Policy (MP) Policy based on a myopic behavior, which aims at following the nodes
+with least customers, without getting farther from the target picking position.
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+27
+Crowded Nodes (CN) Policy that minimizes the number of customer encounters based on a
+static measure of the average of customers in each node. Each step is obtained by solving a shortest
+path problem considering arcs weighted by the average number of customers traversing them.
+Additionally, for each policy, we considered two types of routing costs to sequence the picking
+order of the products of an order. Therefore, whenever an online order arrives, the SRP can be
+solved using a cost matrix considering distances or a crowdedness measure (average number of
+customers between the origin and destination nodes). These two routing basis are called Arc
+Distance and Arc Crowdedness, respectively. This will affect the sequence in which the products
+of an order are picked.
+Table 5 presents the indicators obtained for the four policies using each of the routing basis
+considered.
+Table 5
+Average indicators obtained for 500 episodes of the
+Shortest Path (SP), Crowded Node (CN), Myopic Policy (MP), and Q-learning (QL) policies.
+Routing Basis Policy
+Avg.
+Rewards
+Avg.
+#Orders
+Avg.
+#Products
+Avg.
+#Encounters
+Arc Distance
+QL 16582.12
+41.03
+487.01
+1025.39
+Arc Distance
+SP 15388.14
+46.56
+553.79
+2072.22
+Arc Distance
+CN 16085.83
+38.51
+456.64
+745.71
+Arc Distance
+MP 15294.34
+47.65
+564.76
+2134.58
+Avg. 15838.69
+43.45
+515.55
+1494.40
+Arc Crowdedness
+QL 16997.61
+41.49
+491.58
+1018.92
+Arc Crowdedness
+SP 14991.66
+44.45
+527.50
+1868.61
+Arc Crowdedness
+CN 16845.25
+39.84
+470.69
+768.99
+Arc Crowdedness
+MP 15244.25
+45.66
+542.28
+1891.15
+Avg. 16019.69
+42.86
+508.01
+1386.92
+
+Author: Article Short Title
+28
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+Analyzing the case where route sequences are obtained using arc distances, we observe that SP
+and MP are the most operationally efficient policies, picking an average of 46.56 and 47.65 orders,
+respectively. Policies QL and CN are not so efficient in terms of number of orders picked, picking
+41.03 and 38.51 orders, respectively. These policies are clearly focused on improving customer
+experience, lowering the average number of customer encounters by more than 50% compared to
+policies SP and MP. Policy CN achieves the lowest number of customer encounters, yet it could
+pick the lowest number of orders among all policies. Policy QL drastically reduced the number of
+customer encounters without heavily compromising the number of orders picked.
+Observing the results obtained when route sequences are obtained using the arc crowdedness
+measure, we observe a similar behavior of the policies, in relative terms, regarding operational
+efficiency and customer experience. However, the route sequences obtained suffer from a reduction
+in the average number of orders picked, reducing by 1.35%, from an average 43.45 orders to 42.86.
+The number of customer encounters slightly improved, reducing by 7.19%, from an average of
+1494.40 encounters to 1386.92. Therefore, using arc crowdedness as a routing basis helped to find
+small improvements in customer experience for all policies at a cost of a small reduction in the
+number of orders picked.
+We observe the most operationally efficient policies are the policies based on following shortest
+distance paths (SP and MP). In terms of providing the best customer experience, the CN policy
+achieves the lowest average number of customer encounters. However, these policies are totally
+focused on one objective only. While the SP and MP policies cannot avoid customers, the CN
+policy is not aware that the ultimate objective is to pick orders. The QL policy achieves a good
+compromise in picking a large average number of orders of and encountering a low average number
+of customers.
+An interesting detail on these results is that the QL had to be re-trained when the routing
+basis was changed from distance to crowdedness. Note that we are dealing with a problem that
+suffers from the “curse of dimensionality” in the number of states. Given that the route sequences
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+29
+were changed, different pairs of current position and target position (state) started to appear more
+frequently. Since the QL policy trained with arc distances had not visited these states sufficiently,
+the policy had to be trained in an environment where the route sequences considered the arc
+crowdedness.
+The interesting behavior learned by the picker agent can be observed in several origin-destination
+pairs obtained in the policy. In Figure 8 we present an illustrative example of the paths followed
+by the SP, the CN and the QL policies.
+Figure 8
+Illustrative example of a SP path (red), a CN path (orange), and a QL path (blue).
+Although the SP policy performs fewer steps (traveling a shorter distance), it ends up visiting
+the most crowded nodes. On the other hand, the QL policy follows a longer path but it tries not to
+visit crowded nodes, resulting in fewer customer encounters. The CN policy performs the longest
+path as it is trying to avoid customers at all cost.
+7.
+Conclusion
+This paper proposes a new problem in online retail, the diPRP, and solves it through a reinforce-
+ment learning approach. This problem has gained particular relevance during the pandemic context
+of 2020, in which most retailers offering online shopping services resorted to new picking operations
+in-store. These new picking operations will not be ephemeral, they will prevail as an important
+fulfillment method among omnichannel retail players.
+
+BOMEWC
+COULIUEULE
+Wr1Conrte
+ACV3V 30 A3RA
+J003SSAuthor: Article Short Title
+30
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+Our solution approach encompasses several techniques, integrating the fields of mathematical
+programming, simulation, and machine learning, to obtain an in-store picking policy that is com-
+petitive in terms of the number of picked products while minimizing the number of customer
+encounters. Our study provides evidence that reinforcement learning can be an important method
+for effectively solving new management problems.
+The devised solution approach trains picker agents to perform shopping paths that reduce the
+disturbance of in-store customers during their shopping experience. Computational experiments
+performed both on synthetic instances and a real-world instance from a large European retailer
+show that the picker agents can learn efficient picking policies in a reasonable number of episodes.
+An important characteristic inherent to our approach is the low computational power demanded
+and the very short computational time required to obtain a decision.
+In addition, based on the real-world case study, we provide interesting numeric results and
+solution visualizations to support a few managerial insights. As expected, the shopping sequences
+followed by the customers often result in crowded store areas. Due to the dynamic and unpredictable
+movement of customers in-store, these crowded store areas are difficult to predict. However, we
+observe that the devised picking policies avoid these areas to reduce the number of customer
+encounters. This is not the behavior of the shortest path-based policies, which, logically, can pick
+a few more products, but nearly double the number of customers disturbed in our simulations.
+Our approach, based on reinforcement learning, allows better control over the trade-off between
+operational efficiency and customer experience. Such result may empower retailers to further pursue
+their omnichannel fulfillment strategy.
+Many developments can be further introduced in this problem. For instance, developing col-
+laborative picking policies to coordinate picking teams in this problem would be an interesting
+exercise, particularly if the customer perception regarding the number of picker encounters can be
+assessed. Additionally, improvements to the simulation environment can be introduced to consider,
+for example, shopping cart capacities and slowdowns experienced in crowded aisles. Furthermore,
+
+Author: Article Short Title
+Transportation Science 00(0), pp. 000–000, © 0000 INFORMS
+31
+exploring richer state variables may improve the quality of the picking policies. For instance, the
+number of customers encountered in each aisle can further influence the decision of the picker in an
+online fashion (probably, the state space would suffer from the so-called “curse of dimensionality”
+more severely in this case). Lastly, we would like to stress that there are multiple applications
+for the approach proposed in this paper. The future reserves the automation of several activities,
+such as robots for cleaning public spaces, robots for picking products at warehouses and stores,
+and autonomous vehicles. These technologies will likely be placed in narrow spaces that are shared
+with humans. Since these automations are typically not pleasant to the eye, our approach can be
+an interesting alternative in the future.
+Acknowledgments
+The research leading to these results has received funding from the European Union’s Horizon 2020 - The
+EU Framework Programme for Research and Innovation 2014-2020, under grant agreement 952060.
+References
+Adhi P, Hough G, Calais S, Lange T, Lenzen C (2022) Reimagining the role of physical stores in an omnichan-
+nel
+distribution
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+

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+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+1
+Universal Domain Adaptation for Remote Sensing
+Image Scene Classification
+Qingsong Xu, Yilei Shi, Member, IEEE, Xin Yuan, Senior Member, IEEE, and Xiao Xiang Zhu, Fellow, IEEE
+Abstract—This work has been accepted by IEEE TGRS for
+publication. The domain adaptation (DA) approaches available
+to date are usually not well suited for practical DA scenarios
+of remote sensing image classification, since these methods (such
+as unsupervised DA) rely on rich prior knowledge about the
+relationship between label sets of source and target domains, and
+source data are often not accessible due to privacy or confiden-
+tiality issues. To this end, we propose a practical universal domain
+adaptation setting for remote sensing image scene classification
+that requires no prior knowledge on the label sets. Furthermore,
+a novel universal domain adaptation method without source data
+is proposed for cases when the source data is unavailable. The
+architecture of the model is divided into two parts: the source
+data generation stage and the model adaptation stage. The first
+stage estimates the conditional distribution of source data from
+the pre-trained model using the knowledge of class-separability
+in the source domain and then synthesizes the source data. With
+this synthetic source data in hand, it becomes a universal DA task
+to classify a target sample correctly if it belongs to any category
+in the source label set, or mark it as “unknown” otherwise. In
+the second stage, a novel transferable weight that distinguishes
+the shared and private label sets in each domain promotes the
+adaptation in the automatically discovered shared label set and
+recognizes the “unknown” samples successfully. Empirical results
+show that the proposed model is effective and practical for
+remote sensing image scene classification, regardless of whether
+the source data is available or not. The code is available at
+https://github.com/zhu-xlab/UniDA.
+Index Terms—Remote sensing image classification, source data
+generation, transferable weight, universal domain adaptation.
+This work is jointly supported by the German Research Foundation
+(DFG GZ: ZH 498/18-1; Project number: 519016653), by the European
+Research Council (ERC) under the European Union’s Horizon 2020 research
+and innovation programme (grant agreement No. [ERC-2016-StG-714087],
+Acronym: So2Sat), by the Helmholtz Association under the Framework of
+the Helmholtz Excellent Professorship “Data Science in Earth Observation
+- Big Data Fusion for Urban Research”(grant number: W2-W3-100), by
+the German Federal Ministry of Education and Research (BMBF) in the
+framework of the international future AI lab ”AI4EO – Artificial Intelligence
+for Earth Observation: Reasoning, Uncertainties, Ethics and Beyond” (grant
+number: 01DD20001) and by German Federal Ministry for Economic Affairs
+and Climate Action in the framework of the ”national center of excellence
+ML4Earth” (grant number: 50EE2201C). The work of X. Yuan is supported by
+the National Natural Science Foundation of China (grant number: 62271414),
+Zhejiang Provincial Natural Science Foundation of China (grant number:
+LR23F010001), Westlake Foundation (grant number: 2021B1 501-2) and the
+Research Center for Industries of the Future (RCIF) at Westlake University.
+(Qingsong Xu and Yilei Shi contributed equally.) (Corresponding author: Xiao
+Xiang Zhu.)
+Q. Xu, and X. X. Zhu are with the Chair of Data Science in Earth Ob-
+servation, Technical University of Munich (TUM), 80333 Munich, Germany.
+(e-mails: [email protected]; [email protected]).
+Y. Shi is with the Chair of Remote Sensing Technology, Technical Univer-
+sity of Munich (TUM), 80333 Munich, Germany. (e-mail: [email protected]).
+X. Yuan is with the School of Engineering, Westlake University, Hangzhou,
+Zhejiang 310030, China (e-mail: [email protected]).
+NOMENCLATURE
+M
+Pre-trained model on real source domain.
+F
+Feature extractor. θf denotes parameters of F.
+C
+Classifier. θc denotes parameters of C.
+D′
+Non-adversarial domain discriminator. θd′ de-
+notes parameters of D′.
+D
+Adversarial domain discriminator. θd denotes
+parameters of D.
+p(x)
+Real source domain distribution.
+q(x)
+Target domain distribution.
+p(ys)
+True labeled distribution of the source domain.
+y
+One-hot vector that represents a label.
+py(y)
+Estimated categorical distribution of the true
+labeled distribution p(ys).
+z
+Standard normal vector of a low-dimensional
+noise.
+pz(z)
+Multivariate Gaussian distribution describing
+the source data points.
+p(x | y, z)
+Conditional distribution of source data.
+G
+Generator that obtains the empirical distribu-
+tion p(x | y, z). θg denotes parameters of G.
+xs
+Source image.
+xf
+Synthetic source image.
+xt
+Target image.
+Ds
+Real source domain.
+Df
+Synthetic source domain. |Df| denotes the size
+of synthetic source domain.
+Dt
+Target domain.
+Yf
+Label set of the synthetic source domain.
+Yt
+Label set of the target domain.
+Yf
+Private label set of the synthetic source domain.
+Yt
+Private label set of the target domain.
+ξ
+Jaccard index.
+¯y
+Probability vector predicted by the classifier C.
+φj(x)
+Activation at the jth layer of the style loss
+network.
+Gφ
+j (x)
+Gram matrix.
+wf(x)
+Sample-level transferable weight for synthetic
+source data (scalar).
+wt(x)
+Sample-level transferable weight for target data
+(scalar).
+w0
+Decision threshold (scalar).
+d(x)
+Domain similarity of target domain samples to
+the synthetic source domain samples (scalar).
+max ¯y(x)
+Confidence of predicted probabilities (scalar).
+ℓcls
+Classifier loss.
+ℓstyle
+Style loss.
+arXiv:2301.11387v1  [cs.CV]  26 Jan 2023
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+2
+ℓadv
+Adversarial loss function for adaptation.
+ℓf
+ce
+Cross-entropy loss on the synthetic source do-
+main.
+ℓsimi
+Binary cross-entropy loss for non-adversarial
+domain discriminator.
+I. INTRODUCTION
+R
+EMOTE sensing image scene classification is a pro-
+cedure for assigning semantic labels according to the
+content of remote sensing scenes [1], which is beneficial
+to traffic analysis, urban area monitoring and planning [2],
+[3], land-use and land-cover [4], and hazard detection and
+avoidance [5], among other applications. In recent years,
+many deep learning approaches have been proposed for scene
+classification of remote sensing images [6], [7], such as
+autoencoder [8], convolutional neural networks (CNNs) [9],
+generative adversarial networks (GANs) [10], prototype-based
+memory networks [11], and transformer [12]. These methods
+usually assume that the training and testing data share the
+same distribution. However, in a real application, due to the
+influence of sensors, geographic locations, imaging conditions,
+and other factors, the distribution of training and testing data
+may be different. This phenomenon is referred to as the
+domain gap [13]. To address the domain gap problem among
+different datasets, domain adaptation (DA) algorithms have
+been proposed. DA aims to leverage a source domain to learn
+a model that performs well on a different but related target
+domain [14].
+In remote sensing scene classification, most existing DA
+approaches [15], [16] are proposed to tackle the domain gap
+between different domains by learning a domain invariant
+feature representation. Based on the knowledge of the rela-
+tionship between the source and target label space (category-
+gap), DA can be divided into closed-set DA, partial DA, and
+open-set DA. Specifically, closed-set DA usually addresses
+the domain adaptation problem by leveraging the adversarial
+learning behaviors of GANs to perform distribution alignment
+in the pixel, feature, and output spaces [5], [15], [17], [18],
+which assumes a shared label set between the source and
+target domains, as shown in Fig. 1(a). In order to relax
+this assumption, two alternatives have been proposed: partial
+DA [19], in which the target label space is considered a subset
+of the source label space, as shown in Fig. 1(b), and open-set
+DA [20], in which the source label space is considered a subset
+of the target label space, as shown in Fig. 1(c). For example,
+an open-set domain adaptation algorithm [21] is proposed
+in which transferability and discriminability are explored for
+the purpose of remote sensing image scene classification.
+However, these DA methods have two major bottlenecks in
+the domain adaptation of remote sensing scene classification
+in the wild.
+• In a general scenario, we cannot select the proper domain
+adaptation methods (closed-set DA, partial DA, or open-set
+DA) because no prior knowledge about the target domain
+label set is given.
+• The source dataset is not available in many practical ap-
+plication scenarios of remote sensing. For example, many
+satellite companies and users will only provide pre-trained
+models instead of their source data due to data privacy and
+security issues. In addition, the source datasets, like high-
+resolution remote sensing images, may be so large that it
+is not practical or convenient to transfer or retain them to
+different platforms.
+To address the first challenge, a novel scenario of univer-
+sal domain adaptation (UniDA) is proposed. As shown in
+Fig. 1(d), UniDA removes all constraints and includes all the
+above adaptation settings [22]. UniDA may contain a shared
+label set and hold a private label set for a given source label set
+and a target label set. Two challenges are exposed in a UniDA
+setting. (1) If we naively match the entire source domain
+with the entire target domain, the mismatch of different label
+sets will deteriorate the model. Thus, the samples coming
+from the shared label set between the source and target
+domains should be automatically detected and matched. (2)
+The target samples from private label sets should be marked
+as “unknown” since there are no labeled training data for
+these classes. Currently, different transferability criteria (such
+as entropy in
+[22], pseudo-margin vector in [23], and the
+mixture of entropy, confidence, and consistency in [24]) have
+been proposed to distinguish samples from shared label sets
+and those in private label sets in the field of computer vision.
+To address the second challenge, in computer vision, source-
+free domain adaptation is under continuous exploration [25]–
+[28]. For example,
+[29] proposes the universal source-free
+domain adaptation setting for natural image classification.
+However, existing UniDA methods [22]–[24], [29] in computer
+vision normally assume that the source data set is available
+when building the classifier platform. This assumption is not
+valid and practical for the second challenge. Thus, developing
+a universal domain adaptation method without source data
+(Fig. 1(e)) has a practical value and is thus desired in real
+application scenarios of remote sensing image classification.
+In UniDA without source data, pre-trained models can be
+available. Pre-trained models not only serve as strong baselines
+for the original dataset, but also contain knowledge of the
+original dataset. Therefore, generating synthetic source domain
+data from the pre-trained model is the first problem to be
+solved. There are recent works for distilling a network’s
+knowledge by a small dataset [30] or no observable data [31].
+It is worth noting that we cannot use generative adversarial
+networks to directly generate artificial data (similar to [32]),
+because the core of UniDA without source data is to restore
+the category distribution (including the shared label set and
+the private label set) from the pre-trained model.
+Bearing these concerns in mind, we propose the UniDA
+without source data in order to introduce the UniDA setting
+into remote sensing datasets. In this case, we merely have
+access to the pre-trained model from the source domain. We
+have no information about the source data distribution that
+was used to train. UniDA without source data poses two major
+technical challenges for designing the corresponding models
+in the wild. (1) Distilling the knowledge of source data from
+the pre-trained model. The knowledge is consistent with the
+source in the category distribution (including the shared label
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+3
+Fig. 1. Different domain adaptation scenarios. (a) Closed-set DA, which assumes that the source domain and the target domain have shared label sets. (b)
+Partial DA, which assumes that target label sets are considered a subset of source label sets. (c) Open-set DA, which assumes that source label sets are
+considered a subset of target label sets. (d) Universal DA, which imposes no prior knowledge on the label sets. Label sets are divided into shared and private
+label sets in each domain. (e) Universal DA without source data. The source dataset is not available in the practical universal DA scenarios of remote sensing.
+Fig. 2. Overview of the proposed UniDA without source data (SDG-MA). The model consists of a source data generation stage and a model adaptation stage.
+set and the private label set), and is as close as possible to
+the target in style. (2) Domain adaptation should be applied
+to align distributions of the synthetic source and target data in
+the technical challenges of UniDA.
+To address these two challenges, our proposed UniDA with-
+out source data for remote sensing images consists of a source
+data generation (SDG) stage and a model adaptation (MA)
+stage. In the SDG stage, we reformulate the goal as estimating
+the conditional distribution rather than the distribution of the
+source data, since the source data space is exponential with
+the dimensionality of data. After the conditional distribution
+of the source data is obtained, a well-defined criterion can be
+used to distinguish different degrees of uncertainty in order
+to separate the target samples from the shared label set and
+those from the private label. However, uncertainty is usually
+measured by entropy [22], [33], which lacks discriminability
+for uncertainty when the categorical distributions are relatively
+uniform [24]. Thus, a novel transferable weight is defined by
+considering confidence and domain similarity.
+In a nutshell, our contributions are as follows:
+• We introduce a more practical and challenging UniDA
+setting for remote sensing image scene classification.
+• We propose a new UniDA model (SDG-MA), which is
+composed of a source data generation stage and a model
+adaptation stage.
+• In order to generate reliable source domain samples, a
+novel conditional probability recovery method of the source
+
+Closed set DA
+Partial DA
+Open set DA
+(a)
+(b)
+(c)
+Resident
+Industry
+Farmland
+Resident
+Industry
+Farmland
+Resident
+Industry
+Source
+Source
+Source
+Domain
+Domain
+Domain
+Resident
+Industry
+Industry
+Farmland
+Resident
+Resident
+Industry
+Farmland
+Target
+Target
+Target
+Domain
+Domain
+Domain
+Universal DA
+Universal DA without Source Data
+(d)
+(e)
+Resident
+Industry
+Grass
+River Lake
+Source
+Source
+Domain
+Domain
+Industry
+Resident
+Airplane
+Beach
+Resident
+Airplane
+Beach
+Industry
+Target
+Target
+Domain
+Domain
+Private label sets
+Shared label sets
+Private label sets
+Shared label setsSource Data Generation
+P,(y)
+Label y
+Feature Extractor F
+Classifier C
+(Fixed)
+(Fixed)
+ Concat
+Generator G
+Noise z
+p(xly,z)
+sampling
+Style Loss Network
+t style
+pz(z)
+Xt
+Model Adaptation: Testing
+Model Adaptation: Training
+Classifier C
+ef
+xt
+Wf, Wt
+Xt
+argmax y
+Feature Extractor
+TYes
+Discriminator D'
+F
+siml
+F
+>Unknown
++M
+W+
+Xf
+No
+D'
+Discriminator D
+advIEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+4
+domain is designed to distill category knowledge.
+• A novel transferable weight is utilized to distinguish the
+shared label sets and the private label sets in each domain.
+• Experimental results on four UniDA settings for remote
+sensing image scene classification demonstrate that the
+proposed model is effective and practical, regardless of
+whether the source domain is available or not.
+II. RELATED WORK
+Most existing DA settings for remote sensing image scene
+classification can be summarized as closed-set, partial, and
+open-set DA based on the label set relationship. Closed-set
+DA is a scenario where the source and target domains share
+the same label set. The main challenge in this scenario is
+to overcome the domain gap that comes as a result of the
+samples being taken from different distributions. Among the
+recent work on closed-set DA for remote sensing, adversarial
+learning frameworks have attracted significant interest because
+of the improved quality of alignment between distributions
+by adapting representations of different domains. GANs are
+commonly used at feature maps generated from CNNs where
+a domain discriminator is trained to correctly classify the
+domain of each input feature. For example, domain-adversarial
+neural networks (DANN) [34], Siamese GAN [35], Atten-
+tion GAN [10], and domain adaptation via a task-specific
+classifier (DATSNET) framework [36] are presented for the
+classification of remote sensing images, by learning an in-
+variant representation. Recently, a multitude of closed-set DA
+algorithms for remote sensing image scene classification [37]–
+[43] is designed to reduce the global or local distribution
+differences between domains. In addition, closed-set DA with
+multiple source domains [44] is proposed for remote sensing
+image classification. However, it is difficult to ensure that the
+source domain and the target domain have common classes.
+Thus, partial DA and open-set DA are proposed to relax
+this limitation. Partial DA handles the case where the target
+classes are a subset of source classes. This task is solved by
+performing importance-weighting on source examples that are
+similar to samples in the target [19], [45], [46]. Open set DA
+is a more realistic version, where the new classes will appear
+in the target domain. In the open set DA setting, the target
+domain contains unknown classes that do not present in the
+source domain. In remote sensing image scene classification,
+an open set DA algorithm via exploring transferability and
+discriminability (OSDA-ETD) [21] is proposed to reduce
+the distribution discrepancy of the same classes in different
+domains and enlarge the distribution discrepancy of different
+classes in different domains. In addition, some open set DA
+networks based on adversarial learning [47]–[50] and graph
+convolutional networks [51], [52] are presented for remote
+sensing image scene classification. However, almost all these
+methods rely on prior knowledge about the relationship be-
+tween label sets of source and target domains and assume
+the co-existence of source and target data. Thus, in order
+to promote the development of DA methods, we propose
+a general setting (UniDA) for remote sensing image scene
+classification.
+III. METHODOLOGY
+In this section, we elaborate the problem of the UniDA
+setting without source data and address it by a novel dual-
+stage framework (SDG-MA), shown in Fig. 2.
+A. Problem Setting
+For UniDA setting without source data (SDG-MA), we
+merely have access to the pre-trained model M, including
+feature extractor F and classifier C. We have no information
+about the source data distribution p(x) that is used to train
+M. Thus, considering the MA in the second stage, our first
+goal is to generate reliable source data xf from the pre-
+trained model M. The synthetic distribution is consistent with
+the source data distribution p(x) in the category distribution
+(including the shared label set and the private label set), and
+is as close as possible to the target domain in style. However,
+it is impracticable to estimate p(x) directly since the source
+data space is exponential with the dimensionality of data.
+Thus, as shown in the source data generation stage of Fig. 2,
+we generate the set by modeling a conditional probability
+of x given two random vectors y and z. y (y ∼ py(y)) is
+a probability vector that represents a label, where py(y) is
+an estimation of the true labeled distribution p(ys) of the
+source domain. z (z ∼ pz(z)) is a low-dimensional noise,
+where pz(z) is a random distribution describing the source
+data points. Thus, we reformulate the goal as to estimate the
+conditional distribution of source data p(x | y, z) instead of
+the distribution p(x).
+After obtaining the conditional distribution of source data
+p(x | y, z) from SDG stage, it becomes a UniDA task but
+now with synthetic source domain. Our second goal is to align
+distributions of the synthetic source domain and target domain
+in the technical challenges of domain gap and category gap. A
+synthetic source domain and a target domain are represented
+by Df =
+��
+xi
+f, yi
+f
+�
+∼ p(x | y, z)
+�nf
+i=1 sampled from condi-
+tional distribution p(x | y, z) and Dt =
+��
+xi
+t
+�
+∼ q(x)
+�nt
+i=1
+sampled from target distribution q(x), respectively. We denote
+by Yf (Yt) the label set of the synthetic source (target) domain.
+The shared label set is denoted by Y = Yf ∩ Yt. The private
+label sets of the source and target domain are represented by
+Yf = Yf\Y and Yt = Yt\Y , respectively. The Jaccard index
+of the label sets of the two domains, ξ =
+|Y |
+|Yf ∪Yt|, is used to
+measure the overlap in classes.
+For UniDA setting with source data (MA), the real source
+domain Ds =
+��
+xi
+s, yi
+s
+�
+∼ p(x)
+�ns
+i=1 is available. Thus, only
+the MA stage is used to align distributions of the real source
+domain and target domain in the technical challenges of
+UniDA.
+B. Source Data Generation
+Source data generation includes two modules, conditional
+probability generation module and data diversity module.
+Specifically, firstly, conditional probability generation module
+is presented to prove that the conditional distribution of source
+data p(x | y, z) can be estimated by estimating the categorical
+likelihood p(y | x) and the property likelihood p(z | x).
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+5
+Secondly, in order to generate a reliable source domain for
+UniDA, the generated data xf must meet two conditions: 1) in
+data content, all category distributions in the pre-trained model
+M can be restored, including source-share and source-private
+category distributions, and 2) in data style, the generated data
+can remain similar to the target domain style distribution.
+Thus, to meet these two conditions, a data diversity module
+is proposed to ensure the data diversity of the generated
+source domain. In addition, different schemes of data diversity
+generation are compared in Section IV-C.
+1) Conditional Probability Generation Module: Recall that
+y (y ∼ py(y)) and z (z ∼ pz(z)) are a probability vector of a
+source distribution and a low-dimensional noise, respectively.
+The variables y and z are conditionally independent of each
+other given source data x, since they both depend on x but
+have no direct interactions. In order to generate a reliable and
+balanced source domain Df, the probability of each sampled
+point x is 1/|Df|, and the probability at any other point is
+zero. Thus, Df = {arg maxxp(x | y, z)}. Based on Bayesian
+theory [31], [53], the arg maxxp(x | y, z) can be expressed as
+follows:
+arg max
+x
+p(x | y, z)
+= arg max
+x (log p(y | x, z) + log p(x | z) − log p(y | z))
+= arg max
+x (log p(y | x) + log p(y | z)
++ log p(x | z) − log p(y | z))
+= arg max
+x (log p(y | x) + log p(x | z))
+= arg max
+x (log p(y | x) + log p(z | x)
++ log p(x) − log p(z))
+≈ arg max
+x (log p(y | x) + log p(z | x)).
+(1)
+In this way, the distribution p(x | y, z) can be estimated by
+estimating the categorical likelihood p(y | x) of the variable
+y given x and the property likelihood p(z | x) of the variable
+z given x. Thus, as shown in the ‘Source Data Generation’
+module in Fig. 2, a generator G is designed to obtain the
+empirical distribution p(x | y, z) by combining y and z
+randomly sampling from the distributions py(y) and pz(z).
+In our experiments, we set py(y) to the random categorical
+distribution of source domain that produces one-hot vectors
+as y, and pz(z) to the multivariate Gaussian distribution that
+produces standard normal vectors as z.
+2) Data Diversity Module: First, in order to recover the
+data content from the pre-trained model M, a classifier loss
+ℓcls is designed. Specifically, given a sampled class vector y
+and a sampled noise vector z as inputs, G is trained to produce
+a synthetic source domain sample that M is likely to classify
+as ¯y. The classifier loss can force the generated data to follow
+the similar class distribution from model M, by minimizing
+the distance between y and ¯y, which can be formulated as
+follows:
+ℓcls(y, ¯y) = −
+�
+i∈Yf
+yi log M(G(y, z))i.
+(2)
+Notably, y and ¯y are not scalars but probability vectors of
+length Yf. Thus, the cross-entropy between two probability
+distributions is utilized to measure the distance between y and
+¯y.
+However, the classifier loss ℓcls easily leads to generate
+similar data points for each class in the synthetic source
+domain. Furthermore, it is necessary for domain adaptation
+to transfer synthetic source images to the target style. A
+style loss ℓstyle is presented to measure differences in style
+between a synthetic source image xf and a target image xt.
+The style of remote sensing images represents colors, textures,
+edges, common patterns, and other image style descriptions.
+Concretely, we make use of a 16-layer VGG network pre-
+trained on the ImageNet [54] to measure multi-scale feature
+style differences between images, which can be described as:
+ℓstyle(xf, xt) =
+4
+�
+j=1
+���Gφ
+j (xf) − Gφ
+j (xt)
+���
+2
+F ,
+(3)
+Gφ
+j (x) =
+1
+CjHjWj
+Hj
+�
+h=1
+Wj
+�
+w=1
+φj(x)c,h,wφT
+j (x)c,h,w,
+(4)
+where φj(x) is the activation at the jth layer of the style loss
+network, and is a feature map of shape Cj ×Hj ×Wj. Gφ
+j (x)
+denotes a Gram matrix that is equal to the average value of the
+product of the feature and the transposition of the feature. The
+Gram matrix can grasp the general style of the entire image.
+The style loss ℓstyle(xf, xt) is the squared Frobenius norm of
+the difference between the Gram matrices of synthetic source
+image xf and target image xt. In addition, different layers
+have different feature styles in the VGG network. Therefore,
+we sum the Gram matrices difference for each of the four
+activation layers in the VGG-16.
+C. Model Adaptation
+The objective of MA is to update the pre-trained model M,
+which distinguishes samples from the target shared label set
+Y and those in the target private label set Yt. One important
+challenge for UniDA is detecting transferable samples. In
+order to address this challenge, the sample transferable weight
+wf(xf) or wt(xt) is utilized during the training stage to
+estimate the confidence that xf or xt is from the shared
+label set. Furthermore, during the testing stage, we use the
+transferable weight as a decision threshold w0 to decide
+whether we should predict a class or mark the sample as
+“Unknown,” a designation that represents all labels unseen
+during training. This is expressed as:
+y(xt) =
+� Class
+wt(xt) > w0
+Unknown
+otherwise.
+(5)
+1) The Transferable Weight: The transferable weight is de-
+rived from uncertainty and domain similarity. Similar to [22],
+[55], the domain similarity d(x) is obtained by the non-
+adversarial domain discriminator D′. The d(x) term can be
+seen as the quantification of the similarity of target domain
+samples to the synthetic source domain samples. In particular,
+a smaller d(xf) for a synthetic source sample and a larger
+d(xt) for a target sample mean that they are more likely to be
+in the shared label set.
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+6
+On the other hand, we adopt the assumption that the target
+data in Y have a lower uncertainty than target data in Yt.
+Thus, in order to further separate target samples from the
+shared label set and those from the private label, a well-
+defined criterion can be used to distinguish different degrees
+of uncertainty. However, uncertainty is usually measured by
+entropy [22], [33], which lacks discriminability for uncertainty
+when the categorical distributions are relatively uniform [24].
+The confidence of predicted probabilities ¯y(x) is a better
+measure when the generated categories of source samples are
+relatively uniform. Digging the confidence further, as private
+label sets of synthetic source Y f have no intersection with
+shared label sets Y , samples from p(xf, yf | yf ∈ Y f) are not
+influenced by the target data and keeps the highest certainty.
+In addition, the target samples that are more similar to the
+source domain samples are more likely to be in the shared label
+set. Different schemes of the transferable weight are further
+compared and analyzed in Section IV-C.
+With the above analysis, it is reasonable to expect that:
+E(xf ,yf )∈p|yf ∈Y f d(xf) > E(xf ,yf )∈p|yf ∈Y d(xf)
+> E(xt,yt)∈q|yt∈Y d(xt) > E(xt,yt)∈q|yt∈Y td(xt),
+(6)
+E(xf ,yf )∈p|yf ∈Y f max ¯y(xf) > E(xf ,yf )∈p|yf ∈Y max y(xf)
+> E(xt,yt)∈q|yt∈Y max ¯y(xt) > E(xt,yt)∈q|yt∈Y t max ¯y(xt).
+(7)
+Thus, the sample-level transferable weight for synthetic source
+data points and target data points can be respectively defined
+as:
+wf(x) = −d(x) − max ¯y(x),
+(8)
+wt(x) = d(x) + max ¯y(x).
+(9)
+Note that d(x) ∈ [0, 1] and max ¯y(x) ∈ [0, 1] by the max-min
+normalization. The weights are also normalized into interval
+[0, 1] during training.
+2) Domain Adaptation: To perform domain adaptation dur-
+ing the training stage, the objective function aims to move the
+target samples with higher transferable weight towards positive
+source categories Y . To achieve this, input x from either do-
+main is fed into the feature extractor F, as shown in Fig. 2. The
+extracted features F(x) is forwarded into the label classifier C
+and the non-adversarial domain discriminator D′, to obtain the
+transferable weights wf and wt. The extracted feature F(x)
+is forwarded into the adversarial domain discriminator D to
+adversarially align the feature distributions of the generated
+source and target data falling in the shared label set. Thus, the
+adversarial loss function for adaptation is defined as:
+ℓadv = − Ex∼pwf(x) log D(F(x))
+− Ex∼qwt(x) log(1 − D(F(x))).
+(10)
+Adversarially, the feature extractor F strives to confuse D.
+Thus, domain-invariant features in the shared label set are ob-
+tained. In order to train the classifier C on the synthetic source
+domain with labels, the cross-entropy loss is the following:
+ℓf
+ce = E(xf ,yf )∼pL(yf, C(F(xf))),
+(11)
+where L is the standard cross-entropy loss. Furthermore, to
+better reflect domain similarity, we predict samples from the
+synthetic source domain as 1 and samples from the target
+domain as 0. Thus, similar to [22], [55], a binary cross-entropy
+loss is used to train non-adversarial domain discriminator D′.
+ℓsimi = − E(xf ,yf )∼pL(1, D′(F(xf)))
+− E(xt,yt)∼qL(0, D′(F(xt))).
+(12)
+Algorithm 1 Optimization of UniDA without source data
+Require: Pre-trained model M on the source domain, unla-
+beled data Xt in the target domain, batch size B;
+Ensure: Classification model M of the shared classes Y and
+the unknown class in target domain;
+I. Source data generation stage:
+1: for epochSDG = 1 to epochSDG,max do
+2:
+Fix the pre-trained model M and the style loss network
+(VGG-16); Randomly sample xt of size B from Xt;
+3:
+for each mini-batch do
+4:
+Generate source data xf by G, which combines
+categorical vectors y (y
+∼ py(y)) and standard
+normal vectors z (z ∼ pz(z));
+5:
+Train G by minθg(ℓcls(y, M(xf)) + ℓstyle(xf, xt));
+6:
+end for
+7: end for
+II. Model adaptation stage:
+8: if starting adaptation then
+9:
+for epochMA = 1 to epochMA,max do
+10:
+Randomly sample xt of size B from Xt and generate
+xf of size B by G;
+11:
+for each mini-batch do
+12:
+wf
+and
+wt
+are
+obtained
+by
+C(F(x))
+and
+D′(F(x));
+13:
+Train D by maxθd(−ℓadv);
+14:
+Train F and C by minθf ,θc(ℓf
+ce − ℓadv);
+15:
+Train D′ by minθd′ (ℓsimi);
+16:
+end for
+17:
+end for
+18: end if
+D. Optimization
+Algorithm 1 depicts the optimization flow of UniDA with-
+out source data procedure, which consists of two independent
+stages. θg, θf, θc, θd, and θd′ are parameters of G, F, C,
+D, and D′, respectively. First, the SDG stage estimates the
+conditional distribution p(x | y, z) of source data from the
+pre-trained model M. Thus, we train generator G via the
+data diversity module, and combine them as a single objective
+function:
+ℓ(θg) = min
+θg (ℓcls(y, M(xf)) + ℓstyle(xf, xt)).
+(13)
+Second, the training of the MA stage can be written as a
+minimax game:
+ℓ(θd, θf, θc) = max
+θd
+min
+θf ,θc(ℓf
+ce − ℓadv),
+(14)
+ℓ(θd′) = minθd′ (ℓsimi).
+(15)
+The gradient reversal layer [56] is used to reverse the gradient
+between F and D to optimize the MA stage in an end-to-end
+training framework.
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+7
+IV. EXPERIMENTS
+A. Experimental setup
+1) Datasets:
+To verify our algorithm, we select the
+RSSCN7, UC Merced, AID, and NWPU-RESISC45 to build
+the cross-domain remote sensing image scene datasets. Specif-
+ically, the RSSCN7 dataset [57] contains 2800 remote sensing
+scene images, which are from seven typical scene categories.
+There are 400 images in each scene type, and each image has
+a size of 400×400 pixels. The UC Merced dataset [58] is
+widely used for remote sensing image scene classification. It
+consists of 2100 remote sensing images from 21 scene classes.
+Each scene class contains 100 RGB images with an image size
+of 256×256 pixels. The AID dataset [59] is a large-scale
+aerial image dataset acquired from Google Earth. It contains
+10,000 images with a size of 600×600 pixels, which are
+divided into 30 classes. The NWPU-RESISC45 dataset [60]
+consists of 31,500 remote sensing images divided into 45 scene
+classes. Each class includes 700 images with a size of 256×256
+pixels. The spatial resolution varies from about 30 m to 0.2
+m for most of the scene classes.
+As shown in Table I, four UniDA tasks for remote sensing
+scene classification are established. Specifically, the RSSCN7
+dataset is suitable as the source domain because of its small
+number of categories. Thus, three cross-domain scenarios
+are conducted: RSSCN7 → UCM, RSSCN7 → AID, and
+RSSCN7 → NWPU. For RSSCN7 → UCM, we use the five
+public categories as the shared label set—namely farmland,
+forests, dense residential areas, rivers, and parking lot—the
+remaining two as the private source label set, and the re-
+maining sixteen of UC Merced as the private target label
+set. For RSSCN7 → AID and RSSCN7 → NWPU, we use
+the six public categories as the shared label set (the five
+previously enumerated plus industries). In addition, a fourth,
+more complex UniDA task with a higher Jaccard index, AID
+→ NWPU, is carried out. In this setting, we use the twenty
+public categories as the shared label set, and the rest of the
+AID and NWPU datasets as the private target label sets. Some
+sample images of shared label sets from these four datasets
+are shown in Fig. 3.
+2) Evaluation Protocol: The model is tested only on sam-
+ples from the target domain; all the target-private classes are
+grouped into a single “Unknown” class. Specifically, during
+the testing stage of MA, if the target sample’s transferable
+weight is lower than a predetermined threshold w0, the input
+image is classified as “Unknown.” Thus, the average of per-
+class accuracy for all classes, including the shared classes and
+the “Unknown” class, is the final result. Note that we run each
+experiment three times and report the average results.
+3) Implementation Details:
+All experiments are imple-
+mented in Pytorch [61]. In the setting of the SDG stage, we use
+the standard normal vector z of length 10 in all experiments.
+The generator G is similar to that of ACGAN [62], which
+consists of two fully connected layers followed by seven trans-
+posed convolutional layers (the number of convolution kernels
+is all four) with batch normalization after each layer. The size
+of the generated image xf is 3×256×256. Adam [63] with a
+learning rate of 0.001 is used for the generator. In addition, we
+Fig. 3.
+Some sample images of five shared categories extracted from four
+datasets. Top row to the bottom row are RSSCN7 dataset, UC Merced dataset,
+AID dataset, and NWPU-RESISC45 dataset, respectively.
+compute style reconstruction loss at layers relu1 2, relu2 2,
+relu3 3, and relu4 3 of the VGG-16 style loss network. For
+the model pre-trained on source data, it consists of a feature
+extractor F and a classifier network C. A ResNet-50 model
+with initial weights trained on ImageNet [64] is used as the
+backbone of the feature extractor. The classifier network is a
+fully connected network with a single layer. The cross-entropy
+loss is utilized to pre-train the model on source data. The
+stochastic gradient descent (SGD) with a learning rate of 0.001
+and momentum of 0.9 is used for the model pre-trained on
+source data. Furthermore, the classification accuracy between
+the predictions of the generated data xf and the given label
+y is used to compute the recoverability of the categories in
+the pre-trained model. After the SDG stage, the generator G
+with the highest classification accuracy is utilized for the MA
+stage.
+In the setting of the MA stage, the pre-trained model from
+source data is used to initialize the feature extractor F and the
+classifier network C. In addition, The discriminators D and D′
+consist of three fully connected layers with ReLU between
+the first two. We train F, C, D, and D′ for 40000 iterations
+with Nesterov momentum SGD. The initial learning rate is set
+to 0.001, which is decayed using the same schedule as [56].
+During the testing stage, when the Jaccard index ξ ≥ 0.2 (AID
+→ NWPU), the decision threshold w0 = 0.6. Otherwise, w0
+is set to 0.8.
+4) Methods to Be Compared: We compare the performance
+of the proposed UniDA (with and without source data) with
+the following methods.
+• Source-only: source-only is only trained on the real source
+data, and directly tested on the target domain based on the
+trained model and the target transferable weight.
+• UDA [22]: UDA is proposed to first introduce the universal
+DA setting in computer vision, which is a method with
+using source data. To discover the shared label sets and the
+private label sets to each domain, the transferable weights
+
+RSSCN7
+UCM
+AID
+NWPU
+Dense residential
+Parking lot
+Farmland
+Forests
+RiversIEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+8
+TABLE I
+FOUR UNIDA TASKS FOR REMOTE SENSING SCENE CLASSIFICATION
+RSSCN7 → UCM
+Total label sets
+Shared label sets
+Private label sets
+ξ
+Source domain: RSSCN7 dataset
+7
+5
+2
+0.18
+Target domain: UC Merced dataset
+21
+5
+16
+RSSCN7 → AID
+Total label sets
+Shared label sets
+Private label sets
+ξ
+Source domain: RSSCN7 dataset
+7
+6
+1
+0.16
+Target domain: AID dataset
+30
+6
+24
+RSSCN7 → NWPU
+Total label sets
+Shared label sets
+Private label sets
+ξ
+Source domain: RSSCN7 dataset
+7
+6
+1
+0.12
+Target domain: NWPU-RESISC45 dataset
+45
+6
+39
+AID → NWPU
+Total label sets
+Shared label sets
+Private label sets
+ξ
+Source domain: AID dataset
+30
+20
+10
+0.27
+Target domain: NWPU-RESISC45 dataset
+45
+20
+25
+TABLE II
+CLASSIFICATION ACCURACY OF DIFFERENT METHODS ON RSSCN7 → UC MERCED (%).
+UniDA with source data
+RSSCN7 → UCM
+Category
+Methods
+Farmland
+Forests
+Dense residential
+Rivers
+Parking
+Unknown
+Avg Shared
+Avg
+Source-only
+93.00
+98.00
+42.00
+60.00
+67.00
+1.63
+72.00
+60.27
+I-UAN [23]
+92.00
+97.00
+72.00
+70.00
+30.00
+11.63
+72.20
+62.10
+UDA [22]
+87.00
+98.00
+83.00
+84.00
+68.00
+1.13
+84.00
+70.19
+CMU [24]
+89.00
+99.00
+68.00
+78.00
+100.00
+12.25
+86.80
+74.38
+MA
+85.00
+98.00
+74.00
+79.00
+100.00
+14.63
+87.20
+75.11
+UniDA without source data
+MA-only
+77.00
+13.00
+55.00
+69.00
+77.00
+0.44
+58.20
+48.57
+SDG-MA w/o d
+60.00
+55.00
+31.00
+86.00
+86.00
+4.38
+63.60
+53.73
+SDG-MA w/o y
+90.00
+56.00
+28.00
+70.00
+100.00
+0.19
+68.80
+57.36
+SDG-MA
+92.00
+64.00
+63.00
+76.00
+93.00
+17.13
+77.60
+67.52
+are defined based on domain similarity and entropy.
+• I-UAN [23]: an improved universal adaptation network (I-
+UAN) is a UniDA method with source data. In I-UAN, the
+transferable weight of the source domain is defined based
+on a pseudo-margin vector (maximum predicted probability
+minus second highest predicted probability) to distinguish
+the shared label set. The sample-wise transferable weight of
+the target domain is proposed based on the confidence to
+distinguish the shared and private label sets in target domain.
+• CMU [24]: calibrated multiple uncertainties (CMU) is pro-
+posed, with a novel approach in which transferable weights
+are estimated by a mixture of complementary uncertainty
+quantities: entropy, confidence, and consistency. CMU is a
+UniDA method with real source data.
+• MA-only: MA-only uses the initialized generator G to
+generate source data. The generator is initialized randomly.
+Then, MA is performed between the synthetic source data
+and the target data.
+B. Experimental Results
+1) Results on RSSCN7 → UCM: Our first experiment is
+conducted on RSSCN7 → UCM, including two cases using
+UniDA with source data and UniDA without source data. The
+results are listed in Table II. From Table II, in the UniDA
+setting with source data, it can be seen that the accuracy of all
+methods improves compared to source-only. This phenomenon
+illustrates that a domain shift appears in RSSCN7 and UCM
+datasets. Furthermore, in the UniDA with source data setting,
+our proposed MA achieves much better performance than
+all the other baselines, with an average accuracy of 75.11%.
+In particular, the average accuracy of shared label sets and
+all label sets improves by 0.4% and 0.73%, respectively,
+compared with the best baseline CMU [24]. These findings
+demonstrate that the proposed sample-level transferable weight
+in MA, including confidence and domain similarity, is more
+efficient than entropy in UDA [22], pseudo-margin vector
+in I-UAN [23], and the mixture of entropy, confidence, and
+consistency in CMU [24] for remote sensing image scene
+classification.
+In the second case of UniDA without source data, we
+observe that our proposed SDG-MA framework significantly
+outperforms the MA-only method by 18.95% on the average
+accuracy of all label sets. It is obvious that the proposed
+source data generation is effective and practical in the UniDA
+setting without source data of remote sensing images. Notably,
+compared with the MA, our SDG-MA maintains a more
+prominent performance in the “Unknown” class. It has been
+demonstrated that data points generated by the SDG effectively
+cover the distribution of the source data.
+2) Results on RSSCN7 → AID: Our second experiment is
+conducted on RSSCN7 → AID; results are shown in Table III.
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+9
+Fig. 4.
+Feature visualization on RSSCN7 → UC Merced. For domain, blue and green represents the real source domain and synthetic source domain,
+respectively. Red refers to the target domain. For category, yellow plots are “unknown” samples, others are “known” samples.
+Fig. 5. Feature visualization on AID → NWPU. For domain, blue and green represents the real source domain and synthetic source domain, respectively.
+Red refers to the target domain. For category, yellow plots are “unknown” samples, others are “known” samples.
+(a) Threshold parameter w0 on RSSCN7 → UC
+Merced
+(b) Threshold parameter w0 on AID → NWPU (c) Size of shared label sets on AID →
+NWPU
+Fig. 6. Decision threshold analysis for the parameters w0 and varying size of shared label sets Y .
+
+40 
+40 -
+40 -
+30
+20-
+20.
+20
+10.
+0
+Domain
+-10
+-20
+-20 
+30
+-40
+40
+-40
+-20
+20
+40
+40
+-20
+20 
+-40
+-20
+20
+30
+30
+30
+20
+20 
+20 
+10
+10-
+10 -
+Category
+F 0
+10
+-10
+-20 
+20
+-30
+20
+-30 
+30
+-20
+-10 
+10 
+20
+30
+0
+10
+20
+30
+-30
+-20
+-10
+10 
+30
+(a) Source only
+(b) UniDA with source data
+(c) UniDA without source data
+(MA)
+(SDG-MA)60 
+40
+20
+Domain
+20
+-20
+20 
+40
+30
+20
+20
+Category
+-10 
+20 
+-60
+20
+20
+(a) Source only
+(b) UniDA with source data
+(c) UniDA without source data
+(MA)
+(SDG-MA)100
+ Target domain
+90
+- Target-unkonwn
+80
+70
+(%)
+60
+Accuracy 
+50
+40
+30
+20
+10
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+0
+1
+1.4
+1.6
+1.8
+2100
+-Target domain
+90
+-Target-unkonwn
+80
+70
+(%)
+60
+Accuracy (
+50
+40
+30
+20
+10
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+0
+1
+1.4
+1.6
+1.8
+2100
+90
+Target domain
+-Target-unkonwn
+80
+70
+60
+50
+40
+30
+20
+10
+0
+0
+5
+10
+15
+20IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+10
+TABLE III
+CLASSIFICATION ACCURACY OF DIFFERENT METHODS ON RSSCN7 → AID (%).
+UniDA with source data
+RSSCN7 → AID
+Category
+Avg Shared
+Avg
+Mehods
+Farmland
+Forests
+Dense residential
+Rivers
+Parking
+Industries
+Unknown
+Source-only
+79.19
+100.00
+90.73
+63.41
+80.00
+68.46
+0.12
+80.30
+68.84
+I-UAN [23]
+93.24
+99.60
+94.63
+59.27
+81.79
+61.79
+10.84
+81.72
+71.60
+UDA [22]
+92.16
+99.60
+97.56
+47.56
+93.33
+68.21
+2.52
+83.07
+71.56
+CMU [24]
+83.51
+98.00
+89.51
+59.51
+91.79
+80.77
+10.51
+83.85
+73.37
+MA
+92.70
+100.00
+94.63
+58.78
+94.36
+70.51
+13.48
+85.16
+74.92
+UniDA without source data
+MA-only
+91.89
+16.00
+55.37
+73.41
+56.67
+59.49
+0.36
+58.81
+50.46
+SDG-MA w/o d
+92.70
+93.60
+61.22
+47.32
+99.49
+56.92
+2.92
+75.21
+64.88
+SDG-MA w/o y
+90.00
+67.20
+71.22
+77.56
+92.31
+35.38
+6.11
+72.28
+62.83
+SDG-MA
+97.84
+69.20
+78.78
+37.32
+96.92
+62.05
+17.93
+73.69
+65.72
+TABLE IV
+CLASSIFICATION ACCURACY OF DIFFERENT METHODS ON RSSCN7 → NWPU-RESISC45 (%).
+UniDA with source data
+RSSCN7 → NWPU
+Category
+Avg Shared
+Avg
+Methods
+Farmland
+Forests
+Dense residential
+Rivers
+Parking
+Industries
+Unknown
+Source-only
+70.14
+97.71
+67.71
+34.14
+74.00
+74.00
+0.23
+69.62
+59.71
+I-UAN [23]
+85.71
+98.14
+96.43
+35.57
+28.43
+89.00
+5.52
+72.21
+62.69
+UDA [22]
+77.29
+97.29
+88.14
+40.86
+90.86
+76.71
+2.93
+78.52
+67.72
+CMU [24]
+80.00
+95.57
+72.86
+47.71
+84.14
+80.43
+12.18
+76.79
+67.56
+MA
+84.57
+94.29
+88.43
+40.00
+92.00
+80.14
+14.36
+79.90
+70.54
+UniDA without source data
+MA-only
+83.29
+0.00
+10.00
+0.00
+36.57
+24.00
+22.04
+25.64
+25.13
+SDG-MA w/o d
+91.43
+37.29
+78.57
+59.71
+76.14
+76.57
+1.21
+69.95
+60.13
+SDG-MA w/o y
+93.14
+52.14
+39.43
+37.86
+81.86
+42.43
+12.94
+57.81
+51.40
+SDG-MA
+91.86
+88.00
+86.57
+28.00
+84.86
+65.86
+25.20
+74.19
+67.19
+TABLE V
+CLASSIFICATION ACCURACY OF DIFFERENT METHODS ON AID→NWPU (%). FA.: FARMLAND, FO.: FOREST, DR: DENSE RESIDENTIAL, RI.: RIVER,
+PA.: PARKING, IN.: INDUSTRIAL, BE.: BEACH, MR: MEDIUM RESIDENTIAL, SR: SPARSE RESIDENTIAL, AI.: AIRPORT, BR.: BRIDGE, BA.:
+BASEBALLFIELD, CH.: CHURCH, DE.: DESERT, ME.: MEADOW, MO.: MOUNTAIN, RS: RAILWAY STATION, ST.: STADIUM, ST: STORAGE TANKS, CO.:
+COMMERCIAL.
+UniDA with source data
+AID→NWPU
+Category
+Avg Shared
+Avg
+Methods
+Fa.
+Fo.
+DR
+Ri.
+Pa.
+In.
+Be.
+MR
+SR
+Ai.
+Br.
+Ba.
+Ch.
+De.
+Me.
+Mo.
+RS
+St.
+ST
+Co.
+Unknown
+Source-only
+89.00
+86.00
+50.71
+66.00
+78.14
+16.57
+0.00
+66.71
+78.86
+62.43
+0.00
+0.00
+78.43
+89.86
+75.43
+93.43
+0.00
+0.00
+93.57
+54.57
+0.01
+53.99
+51.42
+I-UAN [23]
+96.00
+96.29
+78.86
+74.86
+86.29
+75.86
+94.86
+83.00
+86.57
+51.00
+90.29
+78.71
+83.43
+80.71
+88.43
+81.71
+84.14
+81.00
+95.29
+57.00
+10.52
+82.21
+78.80
+UDA [22]
+91.71
+92.57
+78.29
+71.29
+81.14
+77.29
+91.57
+70.00
+83.00
+45.14
+81.29
+74.57
+73.29
+78.71
+80.14
+85.43
+82.14
+50.00
+88.43
+37.43
+2.57
+75.67
+72.19
+CMU [24]
+89.57
+83.86
+80.43
+76.43
+84.00
+71.00
+88.57
+79.29
+87.00
+39.00
+86.71
+73.29
+74.00
+72.00
+75.57
+89.71
+87.43
+60.43
+94.00
+54.14
+13.81
+77.32
+74.30
+MA
+93.86
+87.43
+64.14
+78.57
+89.00
+83.43
+95.43
+82.57
+88.00
+58.00
+94.71
+79.00
+76.29
+79.86
+91.57
+70.71
+84.86
+46.00
+92.86
+60.14
+14.11
+79.82
+76.69
+UniDA without source data
+MA-only
+0.29
+0.00
+0.00
+0.14
+6.29
+0.00
+6.71
+0.00
+0.00
+14.86
+4.14
+0.00
+0.00
+0.00
+0.00
+0.00
+1.86
+0.00
+85.14
+0.00
+30.83
+5.97
+7.16
+SDG-MA w/o d
+70.71
+63.29
+51.71
+62.57
+85.57
+81.29
+79.71
+22.57
+81.29
+22.14
+86.71
+83.29
+36.29
+88.86
+89.43
+3.43
+78.43
+80.86
+80.86
+39.43
+4.03
+64.42
+61.55
+SDG-MA w/o y
+82.57
+2.57
+29.86
+70.43
+84.57
+24.71
+95.71
+44.43
+76.14
+64.57
+92.71
+75.57
+62.71
+83.29
+76.43
+20.71
+82.14
+69.86
+90.29
+31.29
+8.07
+63.03
+60.41
+SDG-MA
+80.57
+79.43
+29.57
+77.00
+83.71
+82.29
+79.57
+20.57
+90.43
+44.86
+94.00
+81.00
+52.86
+91.71
+88.57
+12.00
+73.00
+62.71
+78.71
+41.57
+20.22
+67.21
+64.97
+A similar tendency is observed in the UniDA with source
+data setting for remote sensing images. The proposed MA
+outperforms all the compared methods. Again, the experi-
+ment demonstrates that the proposed sample-level transferable
+weight filters out data coming from shared and private label
+sets on feature alignment and provides a better criterion
+for “unknown” class detection than the existing methods.
+In addition, the MA achieves the best accuracy outcomes
+compared with the baselines, for some shared categories (such
+as “Farmland,” “Forests,” and “Parking”).
+In the UniDA setting without source data, our proposed
+method has improved by 15.26% compared with the MA-only
+method. This phenomenon once again verifies the reliability of
+the proposed SDG. For identifying unknown classes, the SDG-
+MA yields excellent performance. However, the average ac-
+curacy of all categories is lower than the source-only method.
+The reason for this finding is that the difference between
+RSSCN7 and AID in the shared label sets is relatively smaller
+than other UniDA tasks.
+3) Results on RSSCN7 → NWPU: Our third experiment
+is conducted on the RSSCN7 → NWPU, and the results
+are provided in Table IV. In the UniDA setting with source
+data for remote sensing images, our proposed source-base
+UniDA is 10.83 percentage points greater than the source-only
+method and achieves the highest average accuracy among all
+methods for recognizing all target samples, which verifies the
+effectiveness of our proposed MA.
+In the UniDA setting without source data, our proposal
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+11
+improves by 42.06%, compared with the MA-only method.
+Notably, the SDG-MA method exhibits a huge performance
+for “Unknown” category.
+4) Results on AID → NWPU: The experimental results on
+AID → NWPU are reported in Table V. In the UniDA setting
+with source data, the proposed MA improves by 25.27%
+compared with the source-only method, which confirms the
+effectiveness and practicality of the proposed MA in a complex
+UniDA task with a higher Jaccard index. In addition, our
+proposed method achieves superior performance among all
+methods on most shared categories and can achieve the highest
+classification accuracy among all methods for recognizing
+private label sets in target samples (the “Unknown” category).
+However, I-UAN [23] is better than MA in identifying shared
+categories, because the proposed MA has a significant drop in
+the accuracy of some shared categories, such as “Stadium.”
+In the UniDA setting without source data, the MA-only
+method achieves poor results (only 7.16%) due to a lack of
+reliable source domain data. Conversely, SDG-MA achieves
+superior performance (64.97%) because reliable source data
+is generated by SDG. Furthermore, SDG-MA achieves the
+highest accuracy for identifying the “Unknown” category,
+which further proves that the proposed SDG module can
+generate a uniform distribution that approximates the real
+source domain.
+C. Model Analysis
+1) Feature Distribution Analysis: To fully understand the
+proposed UniDA with source data and UniDA without source
+data, we provide the feature distributions of RSSCN7 → UCM
+and AID → NWPU in Figs. 4 and 5, respectively. The t-
+SNE [65] is used to visualize the learned source and target fea-
+tures with corresponding domain labels and category labels. As
+shown in Fig. 4(a), before adaptation (source only), there are
+domain shifts between the real source domain (blue) and the
+target domain (red) according to the domain distribution. From
+the category labels, the distributions of the shared categories
+are fragmented and most target private samples are attached
+near the shared samples. After applying MA (Fig. 4(b)) and
+SDG-MA (Fig. 4(c)), domain shifts are effectively alleviated.
+In addition, separability between shared categories is increased
+and most target private samples are separated from the shared
+samples. These phenomena demonstrate that our proposed
+MA strategy is effective for feature alignment. Furthermore,
+comparing MA (Fig. 4(b)) and SDG-MA (Fig. 4(c)), we can
+observe that the synthetic source data distribution (green) and
+the real source data distribution (blue) show a high degree of
+consistency in class distribution and data diversity. It has been
+demonstrated that the synthetic source data generated by SDG
+is effective and reliable.
+In Fig. 5(a), we can observe that the real source domain
+and the target domain have larger data shifts in the UniDA
+task AID → NWPU than the task RSSCN7 → UCM. After
+applying MA (Fig. 5(b)), it is clear that MA alleviates the
+distribution discrepancy in domain labels. Again, this finding
+demonstrates that the proposed MA is practical and effective.
+After applying SDG-MA (Fig. 5(c)), intra-class compactness
+and inter-class separability are significantly improved com-
+pared with source only. In addition, the intra-class compact-
+ness of the “Unknown” category (Fig. 5(c)) is improved
+compared with MA (Fig. 5(b)). This phenomenon further
+verifies the validity of the generated source data.
+2) Ablation Study of Model Adaptation: In order to ver-
+ify the efficacy of the proposed sample-level transferability
+weight, we perform ablation studies that evaluate variants of
+SDG-MA, which are listed in Tables II, III, IV, and V. SDG-
+MA w/o d is the variant that does not integrate the domain
+similarity into the sample-level transferability weight. SDG-
+MA w/o y is the variant that does not integrate confidence
+into the sample-level transferability criterion. As shown in
+Tables II, III, IV, and V, SDG-MA outperforms SDG-MA w/o
+d and SDG-MA w/o y, which indicates that both the domain
+similarity and the confidence in the transferability weight are
+necessary and important for UniDA tasks.
+3) Decision Threshold Analysis: The hyperparameter w0 is
+used to decide whether the model would label a sample as
+“Unknown” or use the predicted label. We analyze two cases
+of ξ < 0.2 (RSSCN7 → UCM) and ξ > 0.2 (AID → NWPU),
+which are described in Fig. 6(a) and Fig. 6(b), respectively.
+As shown in Fig. 6, “Target domain” represents the average
+accuracy of all classes, which measures the generation ability
+of the source domain space and the domain adaptation ability
+of the model. “Target-unknown” is the target accuracy of the
+“Unknown” class, which is a crucial metric for evaluating the
+vulnerability and robustness of the model. Note that there are
+large differences in the results for a threshold in a wide range
+between 0 and 2.0. When ξ is less than 0.2 (Fig. 6(a)), the
+average accuracy of the target domain maintains a high and
+stable accuracy between 0 and 0.8, and target-unknown rises
+significantly after 0.4. Thus, w0 can be set to 0.8 for the
+case of ξ < 0.2. Furthermore, when ξ is greater than 0.2
+(Fig. 6(b)), the average accuracy of the target domain exhibits
+little variance at higher values in a wide range between 0
+and 1. However, the accuracy of target-unknown increases
+significantly after exceeding 0.8. Thus, in order to ensure a
+positive comprehensive accuracy when the case of ξ > 0.2,
+w0 can be set to 1.
+4) Varying Size of Shared Label Sets: We explore the effect
+of the percentages of shared and private label sets on SDG-
+MA by varying the size of Y . This is done on AID → NWPU.
+Fig. 6(c) shows the accuracy of SGD-MA with different Y .
+When Y = 0, the source domain and target domain have no
+overlap on label sets, i.e. Yf ∩ Yt = ∅. It is observed that
+SDG-MA classifies all categories into “Unknown”. Further-
+more, when Y keeps increasing, the performance of SDG-MA
+remains stable and has high precision. It has demonstrated that
+SDG-MA is robust for different percentages of shared and
+private label sets.
+5) Ablation Study of Source Data Generation: We go
+deeper into the efficacy of the proposed SDG by performing
+an ablation study that evaluates the data diversity module.
+The results on RSSCN7 → UCM and AID→ NWPU are
+shown in Tables VI and VII, respectively. Our proposed
+SDG-MA performs better than SDG-MA without classifier
+loss and SDG-MA without style loss, indicating both the
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+12
+TABLE VI
+ANALYSIS OF SOURCE DATA GENERATION ON RSSCN7 → UC MERCED.
+Stages
+Methods
+RSSCN7 → UC Merced
+Farmland
+Forests
+Dense residential
+Rivers
+Parking
+Unknown
+Avg Shared
+Avg
+SDG
+SDG-MA w/o classifier loss
+2.00
+2.00
+19.00
+4.00
+17.00
+10.38
+8.80
+9.06
+SDG-MA w/o style loss
+85.00
+0.00
+0.00
+7.00
+81.00
+16.00
+34.60
+31.50
+Different Data Diversity
+Generation Schemes
+3C-GAN [32]
+91.00
+1.00
+1.00
+9.00
+87.00
+15.31
+37.80
+34.05
+KEGNET [31]
+31.00
+12.00
+21.00
+82.00
+72.00
+23.13
+43.60
+40.19
+Ours
+92.00
+64.00
+63.00
+76.00
+93.00
+17.13
+77.60
+67.52
+TABLE VII
+ANALYSIS OF SOURCE DATA GENERATION ON AID→NWPU. FA.: FARMLAND, FO.: FOREST, DR: DENSE RESIDENTIAL, RI.: RIVER, PA.: PARKING, IN.:
+INDUSTRIAL, BE.: BEACH, MR: MEDIUM RESIDENTIAL, SR: SPARSE RESIDENTIAL, AI.: AIRPORT, BR.: BRIDGE, BA.: BASEBALLFIELD, CH.: CHURCH,
+DE.: DESERT, ME.: MEADOW, MO.: MOUNTAIN, RS: RAILWAY STATION, ST.: STADIUM, ST: STORAGE TANKS, CO.: COMMERCIAL.
+Stages
+Methods
+AID→NWPU
+Fa.
+Fo.
+DR
+Ri.
+Pa.
+In.
+Be.
+MR
+SR
+Ai.
+Br.
+Ba.
+Ch.
+De.
+Me.
+Mo.
+RS
+St.
+ST
+Co.
+Unknown
+Avg Shared
+Avg
+SDG
+SDG-MA w/o classifier loss
+0.00
+0.00
+11.71
+0.00
+0.00
+7.71
+0.00
+19.43
+0.00
+20.00
+0.00
+1.43
+8.71
+0.00
+0.00
+0.00
+1.14
+0.00
+5.14
+1.29
+39.93
+3.83
+5.55
+SDG-MA w/o style loss
+65.57
+35.71
+7.14
+36.57
+76.14
+63.29
+55.57
+46.57
+71.14
+25.57
+93.57
+78.86
+47.14
+87.43
+82.71
+0.14
+75.86
+6.71
+53.43
+50.43
+15.81
+52.98
+51.21
+Different Data Diversity
+Generation Schemes
+3C-GAN [32]
+69.43
+71.00
+19.43
+57.43
+82.86
+71.00
+78.00
+45.14
+80.71
+30.43
+90.57
+82.14
+42.86
+93.43
+76.71
+0.00
+73.43
+61.29
+84.71
+50.57
+22.77
+63.06
+61.14
+KEGNET [31]
+53.29
+16.57
+16.57
+35.57
+87.00
+64.43
+29.86
+25.00
+35.29
+14.14
+75.29
+83.00
+39.86
+87.43
+65.43
+0.00
+16.43
+18.43
+43.71
+38.57
+19.30
+42.29
+41.20
+Ours
+80.57
+79.43
+29.57
+77.00
+83.71
+82.29
+79.57
+20.57
+90.43
+44.86
+94.00
+81.00
+52.86
+91.71
+88.57
+12.00
+73.00
+62.71
+78.71
+41.57
+20.22
+67.21
+64.97
+TABLE VIII
+ANALYSIS OF DIFFERENT PRE-TRAINED MODELS ON RSSCN7 → UC MERCED
+Methods
+RSSCN7 → UC Merced
+Farmland
+Forests
+Dense residential
+Rivers
+Parking
+Unknown
+Avg Shared
+Avg
+SDG-MA (Pre-trained model on ImageNet)
+92.00
+37.00
+51.00
+69.00
+79.00
+21.56
+65.60
+58.26
+SDG-MA (Pre-trained model on RSSCN7 without initial weights trained on ImageNet)
+46.00
+23.00
+6.00
+2.00
+13.00
+7.63
+18.00
+16.27
+Ours (Pre-trained model on RSSCN7 with initial weights trained on ImageNet)
+92.00
+64.00
+63.00
+76.00
+93.00
+17.13
+77.60
+67.52
+classifier loss and the style loss in the data diversity module
+are crucial and necessary for synthetic source data generation.
+More specifically, when the classifier loss and the style loss
+are not considered for SDG-MA, both category and overall
+accuracy are relatively poor. This phenomenon indicates that
+the restoration of data content and ensuring data diversity
+are paramount to the generation of reliable data. In addition,
+SDG-MA without style loss outperforms SDG-MA without
+classifier loss, meaning that the classifier loss (recovering the
+data content) is even more crucial.
+6) Comparison of Different Data Diversity Generation
+Schemes: In the SDG stage, data diversity is the key to the
+successful generation of source data distributions. Recently,
+two mainstream methods have been applied to ensure the
+diversity of data generation. The first, GAN-based methods
+(such as 3C-GAN [32]), are used to produce target-style
+training samples. Specifically, a discriminator is introduced
+to match the distributions between the target samples and
+the generated source samples through the use of adversar-
+ial training. The second, a decoder loss in KEGNET [31],
+produces similar data points for each class and increases the
+pairwise distance between sampled data points. We compare
+our proposed style loss in data diversity module with these
+two methods on RSSCN7 → UCM and AID→NWPU. The
+results are presented in Tables VI and VII, respectively. It
+can be seen that the generating ability of our proposed style
+loss is significantly better than that of 3C-GAN [32] and
+KEGNET [31], with respect to solving the problem of source
+domain generation in UniDA without source data. In addition,
+our style loss maintains a more prominent and uniform per-
+formance in per-class accuracy. It has been demonstrated that
+data points generated by the style loss have better intra-class
+and inter-class diversity.
+7) Ablation study of the pre-trained model in SGD: An
+ablation study of the pre-trained model is conducted to in-
+vestigate the effect of the pre-training models with different
+datasets and different initializations on the source data gener-
+ation. The results on RSSCN7 → UC Merced are presented
+in Table VIII. It is worth noting that for the pre-trained model
+on ImageNet [64], the feature extractor comes from the pre-
+trained ResNet-50 on ImageNet, and the classifier is from the
+pre-trained ResNet-50 on RSSCN7, in order to ensure the
+condition of the UniDA setting. Compared with our proposed
+SGD-MA (pre-trained model on RSSCN7 with initial weights
+trained on ImageNet), it is obvious that the overall average of
+SDG-MA by using the source data generated from the pre-
+trained model on RSSCN7 is better than that of SDG-MA by
+using the pre-trained model on ImageNet. Furthermore, we can
+observe that initial weights trained on ImageNet have a rela-
+tively large impact on the proposed source domain generation
+module, by comparing the pre-trained model without initial
+weights trained on ImageNet and the pre-trained model with
+initial weights trained on ImageNet. Thus, we can conclude
+that both the pre-trained model based on ImageNet (natural
+images) and the pre-trained model based on RSSCN7 (remote
+sensing images) have an impact on the proposed source data
+generation. The pre-trained model based on ImageNet is used
+to provide reasonable initial weights of the feature extractor,
+and the pre-trained model based on RSSCN7 provides an
+effective category distribution of remote sensing images for
+the proposed SGD.
+V. CONCLUSIONS
+We have introduced a novel Universal Domain Adaptation
+setting for remote sensing image scene classification, including
+UniDA with source data (MA) and UniDA without source data
+(SDG-MA). UniDA removes all constraints on the relationship
+
+IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
+13
+between label sets of the source and target domains, which has
+a high practical value and promotes the development of DA in
+remote sensing. To realize universal domain adaptation with
+or without source data, a dual-stage framework is proposed,
+consisting of a source data generation stage and the purpose
+of the model adaptation stage. The source data generation
+stage is to estimate the conditional distribution of the source
+data and generate reliable synthetic source images from both
+data content and data style, when the source data is not
+available. Furthermore, the model adaptation stage aims to
+detect samples from the target shared label sets and those
+in target private label sets utilizing the proposed transferable
+weight. This work can serve as a starting point in a challenging
+UniDA setting for remote sensing images. However, it is
+difficult for the transferable weight in model adaptation to
+tune an optimal threshold to apply it to all UniDA tasks of
+remote sensing images. Thus, in the future, we will focus on
+adaptively learning the threshold through the use of an open-
+set classifier.
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4NFAT4oBgHgl3EQflx07/content/tmp_files/2301.08619v1.pdf.txt

@@ -0,0 +1,655 @@
+FULL SOLUTION OF THE FACTORIALITY QUESTION FOR
+q-ARAKI-WOODS VON NEUMANN ALGEBRAS VIA CONJUGATE
+VARIABLES
+MANISH KUMAR, ADAM SKALSKI, AND MATEUSZ WASILEWSKI
+Abstract. We establish factoriality of q-Araki-Woods von Neumann algebras (with the
+number of generators at least two) in full generality, exploiting the approach via conjugate
+variables developed recently in the tracial case by Akihiro Miyagawa and Roland Speicher,
+and abstract results of Brent Nelson. We also establish non-injectivity and determine the
+type of the factors in question. The factors are solid and full when the number of generators
+is finite.
+The history of q-deformations in the context of (Gaussian) von Neumann algebras started
+with the paper [BKS], where the authors defined the so-called q-von Neumann algebras
+Γq(HR), with q ∈ (−1, 1), acting on the q-deformed Fock space associated with the com-
+plexification of a real Hilbert space HR. These algebras are tracial, and could be seen as
+natural deformations of the free group factors. Thus from the beginning it was a natural
+question whether they also have trivial centres (i.e. are factors) – naturally excluding the case
+of dim(HR) = 1, when Γq(HR) is generated by a single self-adjoint operator. The original
+paper showed that this is indeed true for infinite-dimensional HR. It took almost ten years
+to establish factoriality in full generality, that is for dim(HR) ≥ 2. This was achieved by ´E.
+Ricard in [Ric], after earlier partial results obtained in [Sni] and [Kro].
+Meanwhile in [Hia] another von Neumann algebraic q-deformation was constructed by
+F. Hiai, who combined the construction of [BKS] with the quasi-free (Araki-Woods) deforma-
+tion of D. Shlyakhtenko ([Sh1]). This time the initial data involves not only the real Hilbert
+space HR, but also a group (Ut)t∈R of orthogonal transformations of HR, and the resulting von
+Neumann algebras Γq(HR, Ut) are in general non-tracial. Also here the factoriality question
+has attracted a lot of interest, starting from the original article [Hia], and later continued
+in [BM], [SW] and [BMRW]. In the last of these P. Bikram, K. Mukherjee, ´E. Ricard and
+S. Wang establish factoriality whenever dim(HR) ≥ 3 and also for dim(HR) = 2, but only
+when the deformation generated by (Ut)t∈R is sufficiently small. It should be also noted that
+[Ne1] shows that for q small enough Γq(HR, Ut) is isomorphic to Γ0(HR, Ut), so in that case
+already the results of [Sh1] yield factoriality.
+Here we resolve the matter in full, showing that for arbitrary q ∈ (−1, 1) the algebra
+Γq(HR, Ut) is a factor if and only if dim(HR) ≥ 2. We take a very different approach to that
+taken in [BM] and [BMRW], and instead of studying ‘mixing subspaces’ of Γq(HR, Ut) we
+adopt a very recent approach to ‘dual’ or ‘conjugate variables’ developed in the tracial case of
+Γq(HR) in [MS] by A. Miyagawa and R. Speicher. It turns out that the construction of [MS]
+can be also extended to the non-tracial case, which in conjunction with the general study
+of consequences of the existence of suitable generators with finite free Fisher information
+2010 Mathematics Subject Classification. Primary: 46L36; Secondary 46L10, 46L53, 46L65.
+Key words and phrases. q-Araki-Woods von Neumann algebra, factoriality; conjugate variables; full factors.
+1
+arXiv:2301.08619v1  [math.OA]  20 Jan 2023
+
+conducted in [Ne2] allows us not only to settle completely the factoriality question, but also
+establish the non-injectivity and determine the type of the associated factor. Thus we obtain
+the following theorem, which is the main result of this paper.
+Theorem. Let q ∈ (−1, 1), let HR be a real Hilbert space of dimension at least 2, and let
+(Ut)t∈R be a group of orthogonal transformations of HR. Then the q-Araki-Woods von Neu-
+mann algebra Γq(HR, Ut) is a non-injective factor of type
+�
+�
+�
+III1
+if G = R×
+∗ ,
+IIIλ
+if G = λZ, 0 < λ < 1,
+II1
+if G = {1},
+where G < R×
+∗ is the closed subgroup generated by the eigenvalues of the generator of (Ut)t∈R.
+If dim(HR) < ∞ then these factors are solid and full.
+The plan of the paper is as follows: in the remainder of the introduction we set some
+notation. In Section 1, treating solely the case of finite-dimensional initial space, we introduce
+the notions of dual/conjugate variables and establish their existence for q-Gaussian systems
+inside q-Araki-Woods algebras. Then in Section 2 we establish the main results of the paper;
+in particular Theorem above is a combination of Theorems 2.4, 2.5 and 2.6.
+Throughout the paper we fix q ∈ (−1, 1). All scalar products are linear on the right. Given
+a real Hilbert space HR and (Ut)t∈R, a group of orthogonal transformations of HR, we denote
+the associated q-Araki-Woods von Neumann algebra by Γq(HR, Ut). For a full description of
+the construction of Γq(HR, Ut) we refer to the original article [Hia].
+1. Dual and conjugate variables for q-Araki-Woods von Neumann algebras in
+finite dimensions
+In this section we will consider only the case of finite-dimensional HR. Fix then d ∈ N and
+write H for the complexification of HR = Rd. We assume that we are also given (Ut)t∈R, a
+group of orthogonal transformations of Rd, whose generator (both on HR and on H) will be
+denoted by A. The space H is thus equipped both with the standard scalar product and with
+the deformed scalar product ⟨ξ, η⟩U := ⟨ξ, 2A
+1+Aη⟩; if we want to stress the difference we will
+sometimes use the notation HU. Further we write Fq(HU) for the associated q-Fock space,
+with Fq(HU)alg as the subspace spanned by finite tensors, and e0 = Ω the vacuum vector. We
+denote the associated q-Araki-Woods von Neumann algebra Γq(HR, Ut) simply by M and the
+canonical q-quasi free state ⟨Ω, ·Ω⟩Fq(HU) by ϕ. Finally for each ξ ∈ Fq(HU)alg, we denote by
+W(ξ) the unique element in M which satisfies W(ξ)Ω = ξ.
+Let {e1, . . . , ed} be a linearly independent set of vectors in H, and for i ∈ {1, . . . , d} let Ai =
+W(ei), i.e. Ai ∈ M and AiΩ = ei. We say that a tuple (D1, . . . , Dd) of unbounded operators
+on Fq(HU) with Fq(HU)alg contained in their domains and 1 contained in the domains of
+their adjoints is a (normalized) dual system for (A1, . . . , Ad) if for all i, j ∈ {1, . . . , d}
+[Di, Aj] = ⟨¯ej, ei⟩UPCΩ = ϕ(AjAi)PCΩ
+and DiΩ = 0.
+Here ¯ξ denotes the usual conjugate of a vector ξ in Cd and PCΩ denotes the projection onto
+the one-dimensional subspace CΩ. Before we proceed any further, let us note that existence
+of dual variables implies existence of conjugate variables. We will actually show directly that
+the existence of dual variables implies existence of the conjugate variables with respect to the
+quasi-free difference quotients (see [Ne2, Definition 3.11]), which in turn implies existence of
+the usual conjugate variables (see [Ne2, Remark 3.13]).
+2
+
+Recall that the quasi-free difference quotients ∂i are defined as unique derivations from
+C[Ai, . . . , Ad] into M⊗Mop such that ∂i(Aj) := ϕ(AjAi)1 ⊗ 1 for all i, j ∈ {1, . . . , d}. The
+conjugate variable for ∂i will be a vector ξi ∈ L2(M, ϕ) such that
+⟨ξ, x1⟩ = ⟨1 ⊗ 1, ∂i(x)(1 ⊗ 1)⟩
+for all x ∈ dom(∂i).
+Proposition 1.1 (See [MS, Theorem 2.5]). Suppose that (D1, . . . , Dd) is a normalized dual
+system for (A1, . . . , Ad). Then (D∗
+11, . . . , D∗
+d1) are conjugate variables for (A1, . . . , Ad).
+Proof. It suffices to check that for all i ∈ {1, . . . , d}, n ∈ N and j1, . . . , jn ∈ {1, . . . , d} we
+have ⟨D∗
+i 1, Aj1 . . . Ajn1⟩ = ⟨1 ⊗ 1, ∂i(Aj1 . . . Ajn)(1 ⊗ 1)⟩.
+The left-hand side is equal to
+⟨1, DiAj1 . . . Ajn1⟩. The defining property of Di says that DiAj1 = Aj1Di + ϕ(Aj1Ai)PΩ. It
+follows that
+⟨1, DiAj1 . . . Ajn1⟩ = ⟨1, Aj1Di . . . Ajn1⟩ + ϕ(Aj1Ai)⟨1, Aj2 . . . Ajn1⟩
+= ⟨1, Aj1Di . . . Ajn1⟩ + ϕ(Aj1Ai)ϕ(Aj2 . . . Ajn)
+Continuing in this way we will obtain the final formula:
+⟨1, DiAj1 . . . Ajn1⟩ =
+n
+�
+k=1
+ϕ(AjkAi)ϕ(Aj1 . . . Ajk−1)ϕ(Ajk+1 . . . Ajn) + ⟨1, Aj1 . . . AjnDi1⟩,
+where the last term vanishes as Di1 = 0. This is equal to ⟨1 ⊗ 1, ∂i(Aj1 . . . Ajn)(1 ⊗ 1)⟩,
+because the value ∂i(Aj1 . . . Ajn) can be computed exactly as for free difference quotients,
+merely replacing Kronecker deltas δijk with the covariance ϕ(AjkAi).
+□
+Lemma 1.2. Let {ei}1≤i≤d and {fi}1≤i≤d be two linearly independent sets in H such that
+for every j ∈ {1, . . . , d} we have fj = �d
+k=1 xjkek for some xjk ∈ C. If Ai = W(ei) and
+Ci = W(fi), i ∈ {1, . . . , d}, then a dual system for {Ai}1≤i≤d exists if and only if one for
+{Ci}1≤i≤d does.
+Proof. Note that the definition of Wick operators assures that Cj = �d
+k=1 xjkAk, j ∈
+{1, . . . , d}. If {Di}1≤i≤d denotes the dual system for {Ai}1≤i≤d, then {Ei}1≤i≤d is the dual
+system for {Ci}1≤i≤d, where for each i ∈ {1, . . . , d} we set Ei = �d
+k=1 xikDk. Indeed, let us
+check:
+[Ei, Cj] =
+d
+�
+k=1
+d
+�
+l=1
+xikxjl[Dk, Al] =
+d
+�
+k=1
+d
+�
+l=1
+xikxjl⟨¯el, ek⟩UPCΩ
+= ⟨
+d
+�
+l=1
+xjlel,
+d
+�
+k=1
+xikek⟩UPCΩ = ⟨ ¯fj, fi⟩UPCΩ.
+□
+Dual variables. Fix then {e1, . . . , ed}, an orthonormal set in H with respect to the unde-
+formed scalar product and as before let Ai = W(ei). For i, j ∈ {1, . . . , d} set Bij = ⟨ei, ej⟩U.
+Denote by [d]∗ the set of words in letters from the alphabet {1, . . . , d} and for any word
+w = jn . . . j1 ∈ [d]∗, define ejn...j1 = ejn ⊗ · · · ⊗ ej1.
+One notes that W(ξ) = l¯ξ + l∗
+ξ
+for any ξ ∈ H where l∗
+ξ is the creation operator; this is a very easy instance of the gen-
+eral Wick product formula (see for example [ABW, Proposition 2.12]).
+Hence we have
+Ai(ejn...j1) = eijn...j1 + �n
+k=1 qn−kBijkejn...ˆjk...j1, where ˆjk means to omit jk.
+3
+
+The aim is to exploit the results of [Ne2]; to that end we want to first define for each
+i ∈ {1, . . . , d} operators Di : Fq(HU)alg → Fq(HU)alg such that
+DiΩ = 0,
+[Di, Aj] = BjiPCΩ,
+j ∈ {1, . . . , d}.
+We use below the notation of [MS, Section 4], both in the formulation and in the proofs; in
+particular B(n+1) appearing the following lemma denotes a collection of partitions introduced
+after [MS, Example 4.3]. We will just note the places where the arguments need to be extended
+or modified.
+Lemma 1.3. The algebraic formula for the dual variables is given as follows (i ∈ {1, . . . , d},
+n ∈ N, j1, . . . , jn ∈ {1, . . . , d}):
+Di(ejn . . . ej1) =
+�
+π∈B(n+1)
+(−1)π(0)−1qcross(π)δB
+p(π)es(π),
+where δB
+p(π) := �
+(l,m)
+l>m
+∈π Bjl,jm.
+Proof. Check first that
+[Di, Aj]Ω = Diej =
+�
+π∈B(2)
+(−1)π(0)−1qcross(π)δB
+p(π)es(π) = BjiΩ.
+Then we compute
+DiAjn+1(ejn . . . ej1) =Di
+�
+ejn+1...j1 +
+n
+�
+l=1
+qn−lBjn+1,jlejn...ˆjl...j1
+�
+= Diejn+1...j1 +
+n
+�
+l=1
+�
+σ∈B(n)
+(−1)σ(0)−1qcross(σ)+n−lδB
+p(σ)Bjn+1,jles(σ)
+and
+Ajn+1Di(ejn . . . ej1) =
+�
+π∈B(n+1)
+(−1)π(0)−1qcross(π)δB
+p(π)ejn+1s(π)
++
+�
+π∈B(n+1)
+|s(π)|
+�
+k=1
+(−1)π(0)−1qcross(π)+|s(π)|−kδB
+p(π)Bjn+1,js(π)kes(π)\s(π)k.
+In the first step of the proof of [MS, Proposition 4.5] all terms in the last factor of the second
+sum are identified with some terms in the last factor of the first sum, by taking a pair (π, k)
+and setting σ ∈ B(n) by removing the singleton s(π)k and putting l = s(π)k. Then we just
+have to observe that
+δB
+p(σ)Bjn+1,jles(σ) = δB
+p(π)Bjn+1,js(π)kes(π)\s(π)k
+(and compute the crossings exactly as in [MS]).
+After the subtracting one is left in the first sum with the following terms:
+Diejn+1...j1 +
+n
+�
+l=1
+�
+σ∈B(n):σ(0)≥l
+(−1)σ(0)−1qcross(σ)+n−lδB
+p(σ)Bjn+1,jles(σ)
+4
+
+Now to each pair (σ, l) as above we associate σ′ ∈ B(n + 2) by inserting a ‘new point’ at l
+and pairing it with n + 1. Thus the last expression simplifies to (after counting the crossings
+as in [MS])
+Diejn+1...j1 −
+�
+σ′∈B(n+2):σ′(n+1) not a singleton
+(−1)σ′(0)−1qcross(σ′)δB
+p(σ′)es(σ′);
+note that singletons do not change under this procedure, and δB
+p(σ)Bjn+1,jl = δB
+p(σ′).
+The rest of the argument is just collecting the terms.
+□
+The following is the main result of this Section.
+Proposition 1.4. For each i ∈ {1, . . . , d} we have e0 := Ω ∈ Dom D∗
+i . Thus (D∗
+1e0, . . . , D∗
+de0)
+forms a set of conjugate variables for (A1, . . . , Ad).
+Proof. As in [MS, Theorem 4.6] the proof amounts to studying the expression of the form
+⟨e0, Di(
+�
+w∈[d]∗
+αwew)⟩Fq(HU),
+where the sum is finite (but arbitrary). It is easy to see that it coincides with
+(∗) :=
+∞
+�
+m=1
+�
+π∈B(2m),π(0)=m
+�
+|w|=2m−1
+αw(−1)m−1qcross(π)δB
+p(π),w
+where δB
+p(π),w is written for δB
+p(π) in order to make the dependency on w explicit. Fix for the
+moment m ≥ 1 and do two things at once: first rewrite each word w of length 2m − 1 as vjw′
+with v, w′ words of length m − 1 and j ∈ {1, . . . , d}, and second identify each π ∈ B(2m),
+π(0) = m with a permutation π′ ∈ Sm−1 (exactly as in [MS, Theorem 4.6]). We then have
+�
+π∈B(2m),π(0)=m
+�
+|w|=2m−1
+αw(−1)m−1qcross(π)δB
+p(π),w
+= (−1)m−1q
+m(m−1)
+2
+�
+π′∈Sm−1
+�
+|v|=m−1
+d
+�
+j=1
+�
+|w′|=m−1
+αvjw′qinv(π′)BjiδB
+π′(v),w,
+where δB
+ρ(v),w = �m−1
+l=1 Bvρ(l)wl = ⟨eρ(v), ew⟩H⊗|w|
+U
+and inv(π′) denotes the number of inversions
+of the permutation π′ ∈ Sm−1. Further
+(−1)m−1q
+m(m−1)
+2
+�
+π′∈Sm−1
+�
+|v|=m−1
+d
+�
+j=1
+�
+|w′|=m−1
+αvjw′qinv(π′)BjiδB
+π′(v),w
+= (−1)m−1q
+m(m−1)
+2
+�
+|w′|=m−1
+�
+|v|=m−1
+d
+�
+j=1
+Bjiαvjw′⟨ev, ew′⟩Fq(HU)
+= (−1)m−1q
+m(m−1)
+2
+�
+|v|=m−1
+⟨ev,
+�
+|w′|=m−1
+d
+�
+j=1
+Bjiαvjw′ew′⟩Fq(HU).
+So now we fix v of length m − 1 and look at the vector �
+|w′|=m−1
+�d
+j=1 Bijαvjw′ew′. We
+note that this is nothing but �d
+j=1 Bij ˜Lvj
+��
+|w′|=m−1 αvjw′evjw′
+�
+, where ˜Lvj denotes the
+5
+
+composition of the m relevant undeformed free annihilation operators (acting on Fq(HU)alg)
+of the form ˜Lek for k ∈ {1, . . . , d}, whose action on Fq(HU)alg is given simply by
+˜Lek(ξ1 ⊗ · · · ⊗ ξn) = ⟨ek, ξ1⟩ξ2 ⊗ · · · ⊗ ξn.
+Naturally we also have
+d
+�
+j=1
+Bij ˜Lvj
+�
+�
+�
+|w′|=m−1
+αvjw′evjw′
+�
+� =
+d
+�
+j=1
+Bij ˜Lvj
+�
+�
+�
+|w′′|=2m−1
+αw′′ew′′
+�
+�
+where we have used the fact that the set {e1, . . . , ed} in H is orthonormal in the undeformed
+scalar product. Set Ti,v : Fq(HU)alg → Fq(HU)alg, Ti,v := �d
+j=1 Bij ˜Lvj. We need to argue that
+Ti,v is bounded (and estimate its norm). Consider then a free left ‘undeformed’ annihilation
+operator ˜L(ξ) for ξ ∈ H. Then for any η ∈ H we have
+⟨ξ, η⟩ =
+�
+2A
+1 + A
+� 2A
+1 + A
+�−1
+ξ, η
+�
+=
+�� 2A
+1 + A
+�−1
+ξ, η
+�
+U
+,
+so that setting ˜ξ = ( 2A
+1+A)−1ξ we see that ˜Lξ = L˜ξ, where L˜ξ denotes the free left annihilation
+operator on Fq(HU)alg i.e. L˜ξ(ξ1 ⊗ · · · ⊗ ξn) = ⟨˜ξ, ξ1⟩Uξ2 ⊗ · · · ⊗ ξn. By [MS, Lemma 2.2] (and
+linearity) we have ∥L˜ξ∥B(Fq(HU)) ≤ C∥˜ξ∥HU (where C > 0 depends only on q). But
+∥˜ξ∥2
+HU = ⟨˜ξ, ˜ξ⟩U = ⟨ξ, ( 2A
+1 + A)−1ξ⟩ ≤ D2∥ξ∥2
+H,
+where D := ∥( 2A
+1+A)−1∥
+1
+2 . Thus finally for each j ∈ {1, . . . , d} we have ∥˜Lej∥B(Fq(HU)) ≤ CD,
+and setting B := maxi,j∈{1,...,d} |Bij|, we obtain for each v ∈ [d]∗, |v| = m − 1,
+∥Ti,v∥B(Fq(HU)) ≤ dB(CD)m.
+Then we obtain the following:
+(∗) ≤
+∞
+�
+m=1
+q
+m(m−1)
+2
+�
+|v|=m−1
+∥ev∥Fq(HU)∥Ti,v∥∥
+�
+w∈[d]∗
+αwew∥Fq(HU).
+It is easy to check (as in [MS]) that if we set E = maxi,j∈{1,...,d} |⟨ei, ej⟩U| then we have for
+each v ∈ [d]∗ of length k the estimate
+∥ev∥2
+Fq(HU) ≤ Ek[k]|q|!.
+The rest is just gathering the estimates:
+(∗) ≤
+� ∞
+�
+m=1
+q
+m(m−1)
+2
+dm−1E
+m−1
+2
+�
+[m − 1]|q|!dB(CD)m
+�
+∥
+�
+w∈[d]∗
+αwew∥Fq(HU),
+and noting that the series inside the brackets converges.
+□
+In terminology of [Ne2] the last proposition can be rephrased as saying that the set
+{A1, . . . , Ad}, which generates M, has finite free Fisher information. Together with Lemma
+1.2 this yields the following corollary.
+6
+
+Corollary 1.5. Let HR be a finite-dimensional real Hilbert space equipped with an orthogo-
+nal group (Ut)t∈R. The algebra Γq(HR, Ut) equipped with the canonical state ϕ is generated
+by a finite set G = G∗ of eigenoperators of the modular group of ϕ with finite free Fisher
+information.
+Proof. By [Hia, Proof of Theorem 2.2] we can choose a set of linearly independent vectors
+(ξ1, . . . , ξd) in H such that W(ξ1), . . . , W(ξd) form a self-adjoint set of eigenoperators of the
+modular group of ϕ. Lemma 1.2 and Proposition 1.4 imply that {W(ξ1), . . . , W(ξd)} has
+finite free Fisher information (see [Ne2, Remark 3.13]).
+□
+2. Consequences for structure of q-Araki-Woods von Neumann algebras
+We begin by quoting the main results of [Ne2] and some facts established in [SW] (see also
+[BM]).
+Theorem 2.1 ([Ne2], Theorem A). Let M be a von Neumann algebra with a faithful normal
+state ϕ. Suppose M is generated by a finite set G = G∗, |G| ⩾ 2 of eigenoperators of the
+modular group σϕ with finite free Fisher information. Then (Mϕ)′ ∩ M = C. In particular,
+Mϕ is a II1 factor and if H < R×
+∗ is the closed subgroup generated by the eigenvalues of G
+then M is a factor of type
+�
+�
+�
+III1
+if H = R×
+∗
+IIIλ
+if H = λZ, 0 < λ < 1
+II1
+if H = {1}.
+Theorem 2.2 ([Ne2], Theorem B). Let M be a von Neumann algebra with a faithful normal
+state ϕ. Suppose M is generated by a finite set G = G∗, |G| ⩾ 2 of eigenoperators of the
+modular group σϕ with finite free Fisher information. Then Mϕ does not have property Γ.
+Furthermore, if M is a type IIIλ factor, 0 < λ < 1, then M is full.
+Theorem 2.3 ([SW], Lemma 5(2) with its proof, and Theorem 7(1)). Let (HR, Ut) =
+(KR, U′
+t) ⊕ (LR, U′′
+t ) be the decomposition into, respectively, the almost periodic and the weakly
+mixing part.
+Denote M := Γq(HR, Ut) and write M1 and M2 for the expected subalgebras
+corresponding to, respectively, the almost periodic and the weakly mixing parts. Then
+(i) Mϕ ⊂ M1, hence if x ∈ M ∩ M′ then x ∈ M1;
+(ii) if (Ut)t∈R admits a non-zero fixed vector then M is a factor.
+With Corollary 1.5 and these tools in hand we can completely characterize factoriality of
+q-Araki-Woods algebras and establish all the other results listed in the introduction.
+Theorem 2.4. Let (HR, Ut) be given, with dim(HR) ⩾ 2. Then M := Γq(HR, Ut) is a factor.
+Moreover, if G < R×
+∗ is the closed subgroup generated by the eigenvalues of A then M is a
+factor of type
+�
+�
+�
+III1
+if G = R×
+∗
+IIIλ
+if G = λZ, 0 < λ < 1
+II1
+if G = {1}.
+Proof. By Theorem 2.3 the center of M is contained in the almost periodic part, so we may
+assume it is nontrivial. If it is one dimensional, then it necessarily contains a non-zero Ut-
+invariant vector, so this case is covered by Theorem 2.3 as well. We can therefore assume
+that we are in the almost periodic case with dim(HR) ⩾ 2. If HR is infinite dimensional then
+7
+
+factoriality has been obtained by Hiai ([Hia, Theorem 3.2]). In the finite dimensional case we
+can use Corollary 1.5 and Theorem 2.1.
+If (Ut)t∈R is almost periodic then the centralizer Mϕ is irreducible in M (as follows from
+Corollary 1.5 and Theorem 2.1 in finite dimensions and [Hia, Theorem 3.2] in the infinite
+dimensional case). Therefore in this case the type classification can be simply obtained from
+the spectral data of A as in the statement (see [Hia, Section 1]). On the other hand, if there
+is a non-trivial weakly mixing part, [BM, Theorem 8.1] implies that M is a III1 factor (see
+also [Hia, Theorem 3.4] for an earlier result in the purely weakly mixing case).
+□
+Theorem 2.5. The factor Γq(HR, Ut) is not injective as soon as dim(HR) ⩾ 2.
+Proof. To prove non-injectivity, we will find an expected non-injective subalgebra; note that
+the case where the weakly mixing part is non-trivial has already been covered in [Hia, Theorem
+3.4], where it was proved that in the purely weakly mixing case Γq(HR, Ut) is a non-injective
+factor.
+We therefore assume that we are in the almost periodic case. It means that either we will
+find a two dimensional subspace on which (Ut)t∈R is trivial, or a two dimensional subspace
+on which (Ut)t∈R is ergodic. In both cases the corresponding q-Araki-Woods algebra will be
+non-injective and with expectation, from which we will be able to conclude. In the former
+we are just dealing with a q-Gaussian algebra, which was covered in [Nou, Theorem 2]. In
+the latter we can conclude from Corollary 1.5 and Theorem 2.2 that we have a type IIIλ full
+subfactor, and fullness implies non-injectivity.
+□
+We saw above that the q-Araki Woods factor is full when it is of type IIIλ, 0 < λ < 1
+and dimension of HR is finite. We now establish fullness in the remaining type III1 finite-
+dimensional case as well. We also establish solidity of such factors (see [Oza] for the original
+definition for finite von Neumann algebras and [HR] for the modification needed in the general
+case).
+Theorem 2.6. Let (HR, Ut) be given with 2 ≤ dim HR < ∞. Then M := Γq(HR, Ut) is solid
+and full.
+Proof. If (Ut)t∈R is trivial, then M is a q-Gaussian algebra and its fullness is proved in [MS].
+If M is of type IIIλ, 0 < λ < 1, the statement about fullness follows from Corollary 1.5 and
+Theorem 2.2.
+It follows from [Kuz] that the Cuntz-Toeplitz algebra Tq(H) ⊂ B(Fq(HU)), i.e. the C∗-
+algebra generated by the left creation operators {l∗
+ξ : ξ ∈ H}, is nuclear, as an extension of
+the Cuntz algebra by compacts. Arguing exactly as in [Sh2, Section4] we can thus deduce
+that M satisfies the Akemann-Ostrand (AO) property. This further implies by [HR, Theorem
+A] that M is ω-solid, where ω denotes a fixed non-principal ultrafilter and hence is solid (see
+the definition of ω-solidity in [HR, Section 1]). Theorems 2.1 and 2.2 together with Corollary
+1.5 imply that the centralizer of M with respect to the canonical state is a non-injective II1
+factor. We can thus invoke [HR, Proposition 3.10] to conclude fullness when M is a type III1
+factor.
+□
+Remark 2.7. By [HI, Theorem 6.2] q-Araki-Woods factors are full if (Ut)t∈R has a weakly
+mixing part. The same theorem also covers some almost periodic examples, but if the eigen-
+values of the generator of (Ut)t∈R grow sufficiently fast then fullness remains an open problem.
+Acknowledgments.
+A.S. was partially supported by the National Science Center (NCN)
+grant no.
+2020/39/I/ST1/01566.
+M.W. was partially supported by the National Science
+8
+
+Center (NCN) grant no. 2021/43/D/ST1/01446. The project is co-financed by the Polish
+National Agency for Academic Exchange within Polish Returns Programme.
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+stanta, 2001),” Theta, Bucharest, 2003, pp. 169–202.
+[HR]
+C. Houdayer and S. Raum, Asymptotic structure of free Araki-Woods factors, Math. Ann. 363
+(2015), no. 1-2, 237–267.
+[HI]
+C. Houdayer and Y. Isono, Connes’ bicentralizer problem for q-deformed Araki-Woods algebras,
+Bull. Lond. Math. Soc. 52 (2020). no. 6, 1010–1023.
+[Kro]
+I. Kr´olak, Factoriality of von Neumann algebras connected with general commutation relations –
+finite dimensional case, in “Quantum probability”, Banach Center Publ. 73 (2006), pp. 277–284.
+[Kuz]
+A. Kuzmin, CCR and CAR Algebras are Connected Via a Path of Cuntz–Toeplitz Algebras,
+Comm. Math. Phys., to appear, https://doi.org/10.1007/s00220-022-04580-x.
+[MS]
+A. Miyagawa and R. Speicher, A dual and conjugate system for q-Gaussians for all q, Adv. Math.
+413 (2023), 108834.
+[Ne1]
+B. Nelson, Free monotone transport without a trace, Comm. Math. Phys. 334 (2015), no. 3, 1245–
+1298.
+[Ne2]
+B. Nelson, On finite free Fisher information for eigenvectors of a modular operator, J. Funct. Anal.
+273 (2017), no. 7, 2292–2352.
+[Nou]
+A. Nou,
+Asymptotic matricial models and QWEP property for q-Araki–Woods algebras,
+J. Funct. Anal. 232 (2006), no. 2, 295–327.
+[Oza]
+N. Ozawa, Solid von Neumann algebras, Acta Math. 192 (2004), no. 1, 111–117.
+[Ric]
+´E. Ricard, Factoriality of q-Gaussian von Neumann algebras, Comm. Math. Phys. 257 (2005), no.
+3, 659–665.
+[Sh1]
+D. Shlyakhtenko, Free quasi-free states, Pacific J. Math. 177 (2) (1997), 329–368.
+[Sh2]
+D. Shlyakhtenko, Some estimates for non-microstates free entropy dimension with applications to
+q-semicircular families, Int. Math. Res. Not. 51 (2004), 2757–2772.
+[SW]
+A. Skalski and S. Wang, Remarks on factoriality and q-deformations, Proc. Amer. Math. Soc. 146
+(2018), no. 9, 3813–3823.
+[Sni]
+P. ´Sniady, Factoriality of Bo˙zejko-Speicher von Neumann algebras, Comm. Math. Phys. 246 (2004),
+no. 3, 561–567.
+Institute of Mathematics of the Polish Academy of Sciences, ul. ´Sniadeckich 8, 00–656
+Warszawa, Poland
+Email address: [email protected]
+Email address: [email protected]
+Email address: [email protected]
+9
+

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@@ -0,0 +1,1382 @@
+SciPost Physics
+Submission
+Spin polarization induced by decoherence in a tunneling
+one-dimensional Rashba model
+S. Varela1, M. Peralta1, V. Mujica2, B. Berche3, E. Medina4*
+1 Institute of Materials Science and Nanotechnology, Technische Universit¨at Dresden,
+Dresden, Germany
+2 School of Molecular Sciences, Arizona State University, Tempe, AZ 85281, United States
+3 Laboratoire de Physique et Chimie Th´eoriques, Universit´e de Lorraine, Nancy, France
+4 Departamento de F´ısica, Colegio de Ciencias e Ingenier´ıa, Universidad San Francisco de
+Quito, Quito, Ecuador
+*[email protected]
+January 6, 2023
+Abstract
+Basic questions on the nature of spin polarization in two terminal systems and the
+way in which decoherence breaks Time-Reversal Symmetry (TRS) are analyzed.
+We exactly solve several one-dimensional models of tunneling electrons and show
+the interplay of spin precession and decay of the wavefunction in either a U(1)
+magnetic field or an effective Spin-Orbit (SO) magnetic field. Spin polarization is
+clearly identified as the emergence of a spin component parallel to either magnetic
+field. We show that Onsager’s reciprocity is fulfilled when time reversal symmetry
+is present and no spin polarization arises, no matter the barrier parameters or
+the SO strength.
+Introducing a B¨uttiker’s decoherence probe, that preserves
+unitarity of time evolution, we show that breaking of TRS results in a strong spin
+polarization for realistic SO, and barrier strengths. We discuss the significance of
+these results as a very general scenario for the onset of the Chiral-Induced Spin
+Selectivity effect (CISS), now possibly matching experiments in a quantitative
+manner.
+Contents
+1
+Introduction
+2
+2
+Barrier model with a magnetic field
+3
+2.1
+Spectrum, eigenfunctions, and wavevectors
+3
+2.2
+Spin precession under the barrier with magnetic field
+5
+3
+Barrier model with a Rashba term
+7
+3.1
+Spectrum, eigenfunctions, and wavevectors
+7
+3.2
+Spin precession under the barrier for Rashba
+10
+4
+Other helicity Hamiltonians
+11
+1
+arXiv:2301.02156v1  [cond-mat.mes-hall]  5 Jan 2023
+
+SciPost Physics
+Submission
+5
+Decoherence with B¨uttiker’s probe
+12
+6
+Summary and Discussion
+16
+References
+18
+1
+Introduction
+The Spin-Orbit (SO) coupling is many times neglected in electron transport because of the
+energy scale of the coupling, meV for C, N, O in chiral molecules and e.g. Si, Ga, and Ge
+semiconductors in the bulk. Although the source of SO coupling in many technologically rel-
+evant materials is atomic, how this coupling translates to transport depends on the geometry
+of the connection between spin active atoms. This way in flat graphene, while atomic SO
+coupling is meV, the effective transport SO coupling is µeV. The SO coupling cancels from
+the nearest neighbour contribution due to interference effects and it is only when the second
+neighbours coupling is introduced one gets a meV interaction. On the other hand, bending
+the graphene sheet and producing e.g. nanotubes, increases the SO coupling by three orders
+of magnitude, as it becomes a first neighbours interaction[1]. The same enhancement is seen
+in silicene which has a corrugated surface structure that breaks the orthogonality of π orbitals
+and the σ structure in graphene[2].
+In the case of electron transport in molecules, the SO coupling has been largely disregarded
+but has come into light due to large spin activity reported as Chiral-Induced Spin Selectivity
+(CISS) effect[3]. CISS effect is observed for both point chiral[4] in amino acids, helical chirality
+such as DNA[5], and helicene[6, 7]. Biological molecules generally combine both as in e.g.
+oligopeptides[8, 9].
+There exists an enormous gap in understanding between the size of the SO coupling
+in molecules, in the meV range, and the magnitude of the spin polarization effect in CISS
+effect experiments, above 40%[5]. This percentage exceeds the polarization strength produced
+by transmission through ferromagnets. The SO coupling is almost universally regarded as
+the spin active ingredient in CISS effect and theoretical estimates have yielded the correct
+qualitative behavior i.e. helicity states with the propagation axis as the quantization axis of
+the electron spin. On the other hand, the prediction for the magnitude of the spin polarization
+is at least ten times smaller, when correct atomic SO coupling strengths are contemplated
+[10, 11, 12, 13].
+An important issue on the symmetries involved in electron transmission with two terminals
+with SO coupling was pointed out by Yang, van der Wal and van Wees [14, 15]. As was clearly
+argued, Onsager’s reciprocity precludes the possibility of spin polarization in the two terminal
+setting in the linear regime, in contrast with the results of many works in the literature, both
+experimentally and numerically. It is then important to observe how symmetry arguments
+play out in specific calculations as reference results[14, 16]. Symmetry arguments alone cannot
+say how sensitive these results will be in the face of weak symmetry-breaking perturbations,
+in this case, of Time-Reversal Symmetry (TRS).
+In this work we will discuss the simplest transmission model through a SO active barrier,
+2
+
+SciPost Physics
+Submission
+as tunneling is a very common electron transfer mechanism in large molecules [17]. We will
+explore the possibilities of spin-polarized electron transmission in a one-dimensional two-probe
+setting for an exactly solved model. The action of a U(1) field on tunneling electrons[18] will
+be contrasted with the effective momentum-dependent magnetic field arising from the SO
+coupling. A very important issue in the latter case is the velocity operator’s non-diagonal
+nature that secures the flux’s continuity through boundaries[19]. The strong result is that
+while a U(1) magnetic field polarizes spin along its direction, under the action of the barrier,
+no spin polarization results under the action of a spin-orbit magnetic field. Following, we
+solve the model for the spin-orbit magnetic field under the effects of weak TRS breaking
+decoherence effects, enacted through B¨uttiker’s probe. Large spin polarization highlights a
+high sensitivity to TRS breaking with realistic SO couplings. These findings address the core
+issue of CISS effect.
+2
+Barrier model with a magnetic field
+2.1
+Spectrum, eigenfunctions, and wavevectors
+We will first fully solve analytically for the emblematic problem of a magnetic field under
+a barrier for spinful particles[18]. The correct solution to this problem allowed for properly
+addressing the tunneling time problem. B¨uttiker realised that spin precession in the field is
+modulated by the spin-dependent decay of the wavefunctions under the barrier, generating
+polarization in the direction of the magnetic field. The Hamiltonian in this case is given by
+H =
+�
+( p2
+x
+2m + V0)1σ − Γσz,
+if 0 < x < a
+( p2
+x
+2m)1σ,
+otherwise,
+(1)
+where 1σ is the unit matrix in spin space and σi are the Pauli spin matrices, with i = x, y, z.
+Γ = ℏωL/2 where ωL is the Larmor frequency, and V0 is the barrier height. This Hamiltonian
+has the dispersion relation depicted in Fig. 1. The choice of coordinates is slightly different
+from that of B¨uttiker so we can discuss all one-dimensional models with the same notation.
+The input wavefunction we choose to be
+ψ =
+1
+�
+1 + |s|2
+� 1
+is
+�
+.
+(2)
+The values of s = ±1 correspond to the two eigenfunctions of the σy matrix, and s =
+±i correspond to the two eigenfunctions of the σx matrix, appropriately normalized. The
+eigenvalues of the Hamiltonian are
+E = p2
+x
+2m − σΓ + V0,
+(3)
+where σ = ±1 is the spin degree of freedom. Using E = ℏ2k2/2m, we define the wavevector
+outside the barrier k. This Hamiltonian is not time-reversal invariant since inverting time
+flips px and σ, and these flips change the energy. From the eigenvalue equation, one can then
+distinguish between the different wavevectors under the barrier
+κλ
+σ = λ
+�
+k2 − k2
+0 + σk2
+B
+�1/2 ,
+(4)
+3
+
+SciPost Physics
+Submission
+Figure 1: The dispersion relation for the barrier with a magnetic field. The figure depicts the
+degenerate κλ
+σ vectors that occur in the barrier range.
+where k2
+B = 2mΓ/ℏ2 and k2
+0 = 2mV0/ℏ2. As can be seen, κλ
+σ can only be real or imaginary.
+Thus we have either exponentially decaying solutions for k2 < k2
+0 − σk2
+B or plane waves
+otherwise. As the Hamiltonian commutes with σz in the barrier region we can superpose
+eigenfunctions of σz as
+ψ1
+=
+1
+�
+1 + |s|2
+� 1
+is
+�
+eikx +
+�A+
+A−
+�
+e−ikx,
+(5)
+ψ2
+=
+ϵ
+�1
+0
+�
+eiκ+
++x + ζ
+�0
+1
+�
+eiκ+
+−x + η
+�1
+0
+�
+eiκ−
++x + θ
+�0
+1
+�
+eiκ−
+−x,
+(6)
+ψ3
+=
+�D+
+D−
+�
+eikx.
+(7)
+The boundary conditions are
+ψi(xb)
+=
+ψi+1(xb),
+(8)
+ˆvxψi(xb)
+=
+ˆvxψi+1(xb),
+(9)
+where ˆvx = (∂H/∂x)/m is the velocity operator and xb the boundary between the different
+space regions. We match the wavefunction and the amplitude flux. This latter boundary
+condition is very important to realize and has often been confused in the literature. Any
+dependence of the mass on position (effective mass) should be carefully considered to yield
+the appropriate hermitian velocity operator[20]. If the mass is a constant, then the boundary
+conditions amount to matching the wavefunctions and the first derivatives thereof. Although
+this problem can be separated into two spinless tunneling problems with different barrier
+heights[18], we have decided to phrase somewhat more cumbersomely to make a few important
+points when considering spin-orbit coupling.
+4
+
+E
+K_
++
+21
+kSciPost Physics
+Submission
+The system of equations above can be solved to yield
+D+ =
+t+
+�
+1 + |s|2
+=
+2k(κ−
++ − κ+
++)e−ia(−κ−
++−κ+
+++k)
+�
+|s|2 + 1
+�
+eiaκ−
++(κ−
++ − k)(κ+
++ + k) − eiaκ+
++(κ−
++ + k)(κ+
++ − k)
+�,
+D− =
+ist−
+�
+1 + |s|2
+=
+2iks(κ−
+− − ��+
+−)e−ia(−κ−
+−−κ+
+−+k)
+�
+|s|2 + 1
+�
+eiaκ−
+−(κ−
+− − k)(κ+
+− + k) − eiaκ+
+−(κ−
+− + k)(κ+
+− − k)
+�,
+A+ =
+r+
+�
+1 + |s|2
+=
+(κ−
++ − k)(k − κ+
++)
+�
+eiaκ−
++ − eiaκ+
++
+�
+�
+|s|2 + 1
+�
+eiaκ−
++(κ−
++ − k)(κ+
++ + k) − eiaκ+
++(κ−
++ + k)(κ+
++ − k)
+�,
+A− =
+isr−
+�
+1 + |s|2
+=
+is(κ−
+− − k)(k − κ+
+−)
+�
+eiaκ−
+− − eiaκ+
+−
+�
+�
+|s|2 + 1
+�
+eiaκ−
+−(κ−
+− − k)(κ+
+− + k) − eiaκ+
+−(κ−
+− + k)(κ+
+− − k)
+�. (10)
+where the t± and r± denote the transmission and reflection amplitudes.
+We recall κλ
+σ =
+λ
+��
+k2 − k2
+0 + σk2
+B
+�
+. In the next section, we obtain the behavior of the spin as a function
+of the magnetic field strength consistent with tunneling and we see both regular Larmor
+precession with V0 = 0, and precession combined with spin alignment in the field direction
+when tunneling occurs.
+2.2
+Spin precession under the barrier with magnetic field
+One readily verifies the differential decay of the transmission with the length of the barrier as
+T+ ∼ e−2κ+
++a, and T− ∼ e−2κ+
+−a as long as E < V0 ∓ ℏωL/2. The following relations quantify
+the polarization of the electron, the transmitted (T) wave is
+ψT =
+1
+�
+|D+|2 + |D−|2
+�D+
+D−
+�
+eikx,
+(11)
+and the spin averages are defined by
+⟨sz⟩
+=
+ℏ
+2⟨ψT |σz|ψT ⟩ = ℏ
+2
+|D+|2 − |D−|2
+|D+|2 + |D−|2 ,
+⟨sy⟩
+=
+ℏ
+2⟨ψT |σy|ψT ⟩ = iℏ
+2
+D+D∗
+− − D∗
++D−
+|D+|2 + |D−|2 ,
+⟨sx⟩
+=
+ℏ
+2⟨ψT |σx|ψT ⟩ = ℏ
+2
+D+D∗
+− + D∗
++D−
+|D+|2 + |D−|2 .
+(12)
+Analogous relations can be written for the reflected wave. A spin oriented in the y direction
+(corresponding to s = −1, see Eq. (2) will Larmor precess around the magnetic field (in z
+direction) when V0 = 0 as shown in Fig. 2.
+On the other hand, for V0 > E ± ℏωL/2, spin precession around the magnetic field is only
+part of the average spin motion, since each spin component decays at a different rate under
+the barrier. This gives rise to a z-component that aligns with the direction of the field[18].
+Figure 3 depicts the qualitative motion for the latter case. For V0 < E ± ℏωL/2 only Larmor
+precession follows.
+5
+
+SciPost Physics
+Submission
+Figure 2: Precession of the spin as it goes through an increasing barrier length a starting from
+the 1/
+√
+2 (1
+− i) state, for V0 = 0. This is the simple Larmor precession initially surmised
+for tunneling times[18].
+Figure 3: Relaxation of spin toward magnetic field direction due to tunneling when k2 <
+k2
+0 ∓ k2
+B[18]. For long enough barriers the spin becomes completely polarized in the direction
+of the field. The reference values taken for the plot are k = 2/a, k0 = 3/a, and kB = 1/a,
+where a is the barrier length.
+6
+
+1.0
+0.5
+0.0
+<Sx)
+2
+-0.5
+B
+0
+a
+-1.0
+0
+2
+4
+6
+8
+10
+a1.0
+2
+(Sz)
+0.5
+B
+a
+0.0
+<sx)
+-0.5
+-1.0
+0
+2
+4
+6
+8
+10
+aSciPost Physics
+Submission
+Finally, Fig. 4 shows the transmitted probability in z quantization axis. The input spin
+orientation is along the negative y axis and the transmitted wave selects the up spin orientation
+due to the slower decay of the lower energy state under the barrier.
+This produces spin
+alignment with the magnetic field.
+Figure 4: Transmission contrast between spin components for a magnetic field under the
+barrier. One can see the preferred spin polarization due to the slower decay of the lower
+energy spin configuration under the barrier, leading to an alignment of the entering spin to
+the magnetic field.
+A very important relationship to check is that of the conservation of angular momentum.
+As proven in reference [18], the conservation can be stated exactly as
+(R+ + R−)⟨sz⟩R = −⟨sz⟩(T+ + T−),
+(13)
+where T+ = |t+|2 and R+ = |r+|2 and ⟨sz⟩R is the reflected (R) spin component in the z
+direction.
+Such a relationship is verified in Fig. 5 where the changed angular momentum transmitted
+is compensated by the opposite angular momentum reflected. We have thus verified B¨uttiker’s
+scenario for tunneling with a magnetic field under the barrier. Before addressing the case of
+the SO coupling in one dimension, some useful gauge concepts will be introduced.
+3
+Barrier model with a Rashba term
+3.1
+Spectrum, eigenfunctions, and wavevectors
+We solve the scattering problem for the following model
+H =
+�
+( p2
+x
+2m + V0)1σ + Λpxσy,
+if 0 < x < a
+( p2
+x
+2m)1σ,
+otherwise,
+(14)
+where 1σ is the unit matrix in spin space and σi are the Pauli spin matrices. H acts on the
+spinors ψ = (ψ+(x) ψ−(x)) where |ψ±|2dx is the probability of find a particle between x and
+x + dx with spin ±ℏ/2. This Hamiltonian can be obtained from a helical model of a molecule
+7
+
+1.0
+1
+x 10-4
+V2
+(-i
+(0）
+0.8
+2
+0.6
+T+
+B
+0.4
+0
+a
+(0)
+0.2
+T
+(i）
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+kBSciPost Physics
+Submission
+Figure 5: Angular momentum conservation balancing transmitted spin up (z-component)
+and reflected spin down. As there is no incident spin-up current the two previous components
+(Eq. (13)) must balance.
+with p wave overlaps and SO active Carbon/Nitrogen atoms[21, 22, 23]. The magnitude of
+the SO coupling considered is in fact derived from the overlaps that are only feasible for the
+chiral structure considered in those models.
+We take the incident beam to have an amplitude
+ψ =
+√
+2
+√
+1 + s2
+� 1+s
+2
+1−s
+2
+�
+,
+(15)
+where s = 1 corresponds to the up-spin normalized eigenstate of the σz matrix and s = −1 to
+the down-spin state. The normalization also allows access to all spin states in the x−z plane of
+the Bloch sphere. Here we will illustrate how the SU(2) gauge vector for the one-dimensional
+Rashba Hamiltonian becomes crucial in the barrier boundary conditions[19] which has been
+missed in previous treatments.
+We can faithfully rewrite the Hamiltonian in the following form
+H =
+1
+2m (ˆpx1σ + mΛ(x)σy)2 + V0 − mΛ2
+2
+,
+(16)
+where we can identify the SU(2) gauge field Ax = Ay
+xσy = mΛ(x)σy. The velocity opera-
+tor defined by vx = ∂H/∂px = ((px/m)1σ + Λσy), where no effective mass differences are
+considered[20] for the different scattering regions. Solving for the eigenvalues of this Hamil-
+tonian, we arrive at
+E =
+1
+2m (px + mσΛ)2 − mΛ2
+2
++ V0,
+(17)
+where σ = ±1 is the spin quantum number (eigenvalue label of SU(2) Hamiltonian). Equating
+E = ℏ2k2/2m we define the wavevector outside the barrier region as k. Starting from the
+eigenvalue, we can solve for the possible values of px = ℏq. A new quantum number arises that
+distinguishes right and left propagating waves. The resulting possible values of the wavevector
+under the barrier are
+qλ
+σ = λ
+�
+k2 + k2so − k2
+0 − σkso,
+(18)
+8
+
+1.5
+X 10-2
+4
+1.0
+2
+0
+(T++T-)<sz)
+-2
+0.5
+-4
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+-0.5
+(R++ R_)<sz)R
+-1.0
+0
+2
+4
+6
+8
+10
+kSciPost Physics
+Submission
+where kso = mΛ/ℏ and k2
+0 = 2mV0/ℏ2. The meaning of the quantum numbers is depicted in
+Fig. 6, where the degeneracy of two Kramer’s pairs is evident. Note that for each direction of
+propagation, there are two distinct wavevectors with opposite spin labels and that the previous
+wave vector can be real or complex depending on the values of the incoming wavevector (with
+energy E = ℏ2k2/2m) and the height of the potential barrier.
+As is easily derived from
+Figure 6: Dispersion for Hamiltonian in Eq. (14) The labels correspond to the wavevectors in
+the barrier region with qλ
+σ
+Eq. (14), the Hamiltonian commutes with σy, and ˆpx so it has common eigenstates with
+σy and the ˆpx eigenstates.
+In the σz basis the wavefunctions in the different regions are
+parameterized as follows
+ψ1
+=
+� 1+s
+2
+1−s
+2
+�
+eikx +
+�A+
+A−
+�
+e−ikx,
+ψ2
+=
+α
+√
+2
+�1
+i
+�
+eiq+
++x + β
+√
+2
+� 1
+−i
+�
+eiq+
+−x + γ
+√
+2
+�1
+i
+�
+eiq−
++x + δ
+√
+2
+� 1
+−i
+�
+eiq−
+−x,
+(19)
+ψ3
+=
+�D+
+D−
+�
+eikx,
+(20)
+where the coupling between the direction of propagation and spin orientation has been im-
+plemented by the appropriate qλ
+σ wavevectors. The boundary conditions at the barriers limits
+xb are
+ψi(xb)
+=
+ψi+1(xb),
+ˆvxψi(xb)
+=
+ˆvxψi+1(xb),
+(21)
+where i = 1, 2, the index of the regions, and ˆvx = (ˆpx + mΛσy) /m. The second condition guar-
+antees the continuity of the probability flux and not just the derivative of the wavefunction[19].
+The linear system of eight unknowns can be explicitly solved for the transmission and reflec-
+9
+
+E
+qt
+q+
+9.
+bSciPost Physics
+Submission
+tion amplitudes
+t+
+=
+(1 + i)∆ke−iak �
+(1 − is)eiksoa + (s − 1)e−iksoa�
+(e−ia∆(∆ + k)2 − eia∆(k − ∆)2)
+,
+t−
+=
+−(1 + i)∆ke−iak �
+(s + i)eiksoa − (1 + is)e−iksoa�
+(e−ia∆(∆ + k)2 − eia∆(k − ∆)2)
+,
+r+
+=
+(k − ∆)(∆ + k)(s + 1)
+�
+(k − ∆)2e2ia∆ + (∆ + k)2e−2ia∆ − 2(k2 + ∆2)
+�
+2 (e−ia∆(∆ + k)2 − eia∆(k − ∆)2)2
+,
+r−
+=
+−(k − ∆)(∆ + k)(s − 1)
+�
+(k − ∆)2e2ia∆ + (∆ + k)2e−2ia∆ − 2
+�
+k2 + ∆2��
+2 (e−ia∆(∆ + k)2 − eia∆(k − ∆)2)2
+, (22)
+where ∆ =
+�
+k2 + k2so − k2
+0.
+Such amplitudes will be very important to understand how
+decoherence effects generate spin polarization. The barrier region amplitudes are
+α
+=
+(−1)1/4eiaq−
++k(s − i)(k + ∆)
+−eiaq+
++(k − ∆)2 + eiaq−
++(k + ∆)2 ,
+β
+=
+(−1)1/4k(1 − is)eiaq−
+−(∆ + k)
+eiaq−
+−(∆ + k)2 − eiaq+
+−(k − ∆)2 ,
+(23)
+and
+γ
+=
+(−1)1/4k(s − i)eiaq+
++(k − ∆)
+eiaq+
++(k − ∆)2 − eiaq−
++(∆ + k)2 ,
+δ
+=
+(−1)1/4k(1 − is)eiaq+
+−(k − ∆)
+eiaq+
+−(k − ∆)2 − eiaq−
+−(∆ + k)2 .
+(24)
+We recall that qλ
+σ = λ
+�
+k2 + k2so − k2
+0 − σkso.
+3.2
+Spin precession under the barrier for Rashba
+The transmission of up-spin as a function of the entry spin polarization is
+|t+|2
+=
+8|∆|2k2[1 + s cos(2ksoa)]
+|e−ia∆(∆ + k)2 − eia∆(k − ∆)2|2 ,
+|t−|2
+=
+8|∆|2k2[1 − s cos(2ksoa)]
+|e−ia∆(∆ + k)2 − eia∆(k − ∆)2|2
+(25)
+from where we can see that |ttotal|2 = |t+|2 + |t−|2 so the total conductance is
+Gtotal
+=
+G+ + G−
+=
+e2
+h |ttotal|2 =
+16e2|∆|2k2
+h |e−ia∆(∆ + k)2 − eia∆(k − ∆)2|2 ,
+(26)
+where G± = (e2/h)|t±|2, which is spin independent[19, 24] even in the presence of a barrier
+and an open system at either end of the barrier.
+10
+
+SciPost Physics
+Submission
+Figure 7: Precession of spin around the spin-orbit magnetic field. Note that as the k vector
+inside the barrier has always a real part, we have pure precession with no tilting toward the
+magnetic field as with the magnetic field in the previous section. This happens below and
+above the barrier.
+Following the definitions for average spin components in Eq. (.12), we can now see the
+behavior of the injected spin into the spin-orbit active barrier. Fig. 7 shows how the average
+spin traverses the spin-active barrier region with SO coupling. In analogy with B¨uttiker’s
+U(1) magnetic field, we can define a new momentum-dependent magnetic field BSO given
+the mapping λpxσy = −γBSO · σ that results in BSO = −(Λ/γ)pxuy. BSO lies in the nega-
+tive y direction for the model Hamiltonian. Precession follows correctly the torque equation
+d⟨s⟩/dt = γ⟨s⟩×BSO. Note that, as even under the barrier, the wavevector is complex, unlike
+the magnetic field case, precession proceeds with no generation of a spin component along the
+BSO direction. Also, both spin components suffer the same decay within the barrier (although
+dependent on SO) independent of their spin orientation (see Eq. (18)).
+We can see from comparing the two cases (magnetic field and SO) that one superposes
+different k vectors corresponding with the same energy when traversing the spin active region.
+The two wavevectors, having different real parts, cause precession due to a torque around the
+direction of BSO. In the case of a real magnetic field, under the barrier, the k vector is purely
+imaginary, and only a spin-dependent decay ensues (see Eq. (4)), producing an alignment of
+the spin in the magnetic field direction. In the case of the SO, there is always a real part to the
+k vector (even under the barrier) so that precession occurs for energies above and below the
+barrier. No spin polarization along BSO follows from this scenario in tune with time-reversal
+symmetry.
+4
+Other helicity Hamiltonians
+Varying the Hamiltonian under the barrier to the case where the eigenstates are projected
+along the direction of propagation (helicity states), can be interpreted directly from the pre-
+11
+
+1.0
+(Sx)
+0.5
+0.0
+(Sz)
+2
+-0.5
+0
+y
+0
+a
+-1.0
+Bso
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+aSciPost Physics
+Submission
+vious results. The Hamiltonian in this case is
+H =
+�
+( p2
+x
+2m + V0)1σ + Λpxσx,
+if 0 < x < a
+( p2
+x
+2m)1σ,
+otherwise.
+(27)
+The SO magnetic field is now in the x direction as BSO = −Λpx/γ ux. Working out the
+eigenstates in the SO active region, the eigenstates will be those of σx matrix, and the possible
+k vectors will be κλ
+σ = λ
+�
+k2 + k2so − k2
+0−σkso as before. The k-vector under the barrier always
+has a real part that results in a spin precession. If we start from a spin orientation in the
+z-axis, then the spin will precess around the x axis without generating a spin component in
+the x direction. So no changes from the conclusion in the previous section follow in this case.
+5
+Decoherence with B¨uttiker’s probe
+The spin-orbit coupling does not contrast between spin species, so it cannot, alone, account
+for polarised spin polarization as expected in CISS effect. Nevertheless, the perfect conditions
+under which these results are valid i.e., no coupling to a TRS breaking probe beyond the
+two terminals, are not met, especially at room temperature conditions.
+A thermalization
+of electron transport to the environment through the electron-phonon or electron-electron
+interactions is inevitable. This environment can be modeled as a lumped probe that disrupts
+the delicate coherences that yield Bardarson’s theorem that translates into transport as the
+Onsager reciprocity relations in the linear regime. This turns our attention to a tunneling
+molecular system to a three-probe scenario.
+Figure 8: B¨uttiker’s probe under the spin-orbit active barrier. The probe absorbs each eigen-
+state spin species under the barrier with the same scattering matrix, so no spurious spin
+selection is induced. Flux conditions are imposed on building the S matrix for a wideband
+B¨uttiker probe.
+The B¨uttiker’s voltage probe[25] is an ingenious way to introduce decoherence processes
+through the scattering matrix for an exactly solved model. Here we introduce a generalization
+of the probe used previously in the context of persistent currents[26, 27, 28] (see Fig.
+8).
+The probe is spin insensitive, so we do not introduce extraneous sources of spin selection.
+This is achieved by introducing two probes, one for each spin species at the same point
+connected to a third reservoir thermalized to a Fermi distribution at temperature T. The
+probe is wide-band, supporting the wavevectors injected by the barrier channels. The probe
+scattering matrix returns an amplitude consistent with a simple electron reservoir unrelated
+12
+
+(1)
+ikx
+ikx
+0
+D
+I(α, β, , )
+T(α',β', Y', S"
+S
+A.
+(.
+a+Kma
+N(C_eiky +e-ixy)SciPost Physics
+Submission
+Figure 9: Precession when the decoherence probe couples to a particular point under the
+barrier. A noticeable disruption of spin precession is observed, generating a spin polarization
+analogous to an actual magnetic field (see Fig. 3, realizing broken time-reversal symmetry).
+to the input amplitude (while preserving unitarity) so that a disruption to the interferences
+occurs according to the local fluxes of each spin orientation. Such a probe introduces TRS
+breaking that generates the B¨uttiker tilting of the spin in the BSO direction producing net
+spin polarization.
+The behavior of the B¨uttiker probe follows the combination of an ideal lead with v = ℏκ/m
+that supports a current dI = ev(dN/dE)f(E)dE in the energy interval dE, where f(E) is the
+Fermi distribution, dN/dE = 1/2πℏv is the density of states. This model can then induce
+level broadening[26, 27], ([28] for Hamiltonian version) and level shifts under the barrier, and
+also depends on where the decoherence event occurs. Besides the coupling of the probe to the
+barrier, we can also control the temperature through the Fermi distribution of the attached
+reservoir. A somewhat different model is discussed in [29], where they consider inelastic events
+and thus probability leakage. Here S†S = 1 so that only decoherence is contemplated.
+Figure 8 shows the four regions that must be matched for continuity and flux. Under
+the barrier, the matching occurs at position (x0, y0) = (x0, 0) where y describes the coordi-
+nate of the third probe. The Scattering (S) matrix can then emulate a generic dephasing
+process[26]. Matching flux conditions at x0 yields the following S matrix between input and
+output amplitudes for each spin species (spin eigenstates under the barrier).
+Ψout =
+�
+�
+√
+Nζ−
+β′
+δ
+�
+� = S−Ψin =
+�
+�
+�
+−(A + B)
+−√εeiq−
+−x0
+√εeiq+
+−x0
+√εe−iq+
+−x0
+−Ae−2i∆x0
+B
+−√εe−iq−
+−x0
+B
+Ae2i∆x0
+�
+�
+�
+�
+�
+√
+N
+δ′
+β
+�
+� ,
+(28)
+where Ψin,out represent the input/output amplitudes to the junction and S− is the scatter-
+ing matrix for the spin-down label. The labels follow the usage previously introduced where
+qλ
+σ, with σ the spin label and λ the sense of propagation label. A = (√1 − 2ε − 1)/2 and
+B = (√1 − 2ε + 1)/2, while N = ef(E)dE/2πℏv with f(E) the Fermi distribution, e the
+electron charge, E the energy and v the velocity of the carriers in the lead[25]. 0 < ε < 0.5
+describes the coupling of the probe to the barrier from uncoupled to fully coupled.
+13
+
+1.0
+0.5
+0.0
+2
+<Sz)
+-0.5
+0
+a
+Bso
+-1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+a(nmSciPost Physics
+Submission
+The results regarding the influence of decoherence, barrier length, and SO coupling is
+depicted in Figs. 9 and 11. In Fig. 9, one can see that the SO coupled to the third probe
+produces a smooth disruption of the spin precession which is no longer in the (x, z) plane but
+achieves a large polarization in the direction of the SO magnetic field (see inset). Thus, the
+BSO acts as a regular symmetry-breaking interaction such as a real magnetic field in Fig. 3.
+Figure 10: Transmission contrast of spin orientations along the y quantization axis (BSO field
+orientation). The input spin wavefunction in the (1 0) orientation, which has equal amplitudes
+in the latter basis, acquires a preferred orientation aligned with the BSO field analogously to
+the U(1) magnetic field case.
+Figure 11: Spin polarization generated by decoherence analogous to that caused by a real
+external magnetic field. The contour plot shows the effect of a spin-orbit and decoherence
+coupling, consistent with estimates of ref.[21] for tunneling. The appearance of alignment of
+the spin to the BSO is very sensitive to the coupling to the B¨uttiker probe, producing up to
+16% polarization with realistic values of SO coupling.
+14
+
+5
+X 10~4
+4
+1-
+3
+T
+2
+1
+0
+0.00
+0.05
+0.10
+0.15
+0.200.20
+0.15
+0
+-0.05
+-0.10
+-0.15
+0.10
+-0.20
+-0.25
+-0.30
+-0.35
+0.05
+-0.40
+0.00
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+kso(nm-1)SciPost Physics
+Submission
+Figure 10 depicts the transmission contrast between spin components along the y axis in
+which the BSO field is oriented. As can be seen, the spin acquires a preferred direction along
+the −y direction aligning with the SO magnetic field. This is analogous to the case of the TRS
+breaking U(1) magnetic field considered at the outset. When the SO strength is zero, the two
+components merge, showing no spin activity, and only express the transmission through the
+barrier.
+Figure 11 shows the dependence of the new polarization as a function kso, and the coupling
+to the reservoir. In contrast to previous figures depicting qualitative precession features, here
+we have set the parameters to realistic ranges for spin-orbit strength, barrier height, and
+barrier-probe coupling[21]. It is evident that even for weak coupling (ε) and ESO ∼ 10 meV,
+one can achieve polarizations of 40%. The polarization effects can yield positive and negative
+polarizations depending on the length of the barrier a and exhibit a non-trivial temperature
+dependence. No spin polarization is produced without the SO interaction, no matter the
+coupling to the third probe.
+Figure 12: Spin polarised component for k = 0.6 nm−1, k0 = 2 nm−1, a = 4 nm, x0 = 1.5 nm.
+Manifest interference effects in the spin polarization through the coupling to the third probe.
+Small couplings to the probe can produce large polarizations while larger couplings degrade
+the polarization. Note the high sensitivity to the probe coupling that cannot be surmised
+solely based on symmetry arguments.
+Figure 12 shows the non-monotone/interference effects of coupling to the third probe and
+the sensitivity to the coupling to the third probe. Sufficiently large values of SO wavevec-
+tor kso, can produce a large polarization at low coupling, while large couplings degrade the
+polarization. Figure 13 depicts the temperature dependence of the polarization as a func-
+tion of the barrier length. The temperature dependence is expressed through the parameter
+N = ef(E)dE/2πℏv proportional to the thermal occupation of the probe. As temperature
+rises N is reduced so that as the temperature is increased, the polarization increases for fixed
+barrier length. Non-monotone effects with the probe coupling can change this temperature
+dependence. They will be determined by the specific nature of how the electron spin current
+15
+
+0.10
+0.08
+0
+-0.1
+0.06
+-0.2
+-0.3
+0.04
+-0.4
+-0.5
+-0.6
+0.02
+0.00
+0.00
+0.05
+0.10
+0.15
+0.20
+kso(nm-1)SciPost Physics
+Submission
+Figure 13: Temperature dependence of spin polarization as a function of the barrier length
+and occupation of the B¨uttiker probe for k = 0.4 nm−1, kso = 0.1 nm−1, k0 = 4 nm−1,
+x0 = 0.8 nm and ε = 0.2.
+The polarization increases with temperature for the range of
+parameters chosen.
+thermalizes to the environment e.g., electron-phonon, electron-electron interactions.
+6
+Summary and Discussion
+We have discussed a one-dimensional system with SO interaction which can be derived from
+a three-dimensional model ignoring the orbital degree of freedom[21, 22].
+The spin-orbit
+coupling arises from the geometrical arrangement of the p orbitals of the helical model’s
+chiral structure. Without it, the SO coupling would be orders of magnitude smaller[22], as
+happens comparing SO coupling of planar graphene and carbon nanotubes[1]. Due to the
+atomic origin of the SO coupling in this model, the helix’s spin and orbital degrees of freedom
+are uncoupled at the lowest order and in the half-filling model[22, 23]. Thus the orbital degree
+of freedom only modulates the kinetic energy and adds orbital angular momentum, which can
+also enhance spin-orbit effects, as shown in ref.[30]. So chirality has a role in the present CISS
+model in generating its ∼10 meV strength.
+In the succession of models presented for transmission through a spin-active barrier, we
+have expressed spin polarization as the manifestation of a lack of TRS that selects a spin
+orientation. We first discussed B¨uttiker’s model of a U(1) magnetic field under a barrier.
+The differential decay of the amplitudes for spin-up and spin-down produces a reorientation
+of the spin along the magnetic field.
+It serves as the trademark of TRS breaking in the
+spin polarization.
+In the following model we assessed the SO coupling where an effective
+magnetic field can also be identified BSO, mapping the SO coupling to a Zeeman-like term.
+Nevertheless, this effective field depends on the propagation direction and does not break TRS.
+The exact solution of the tunneling problem, which has not been satisfactorily solved before
+in the literature in this form, yields nevertheless the expected result, implied by Onsager’s
+reciprocity for two terminal devices, i.e. no spin polarization independent of the magnitude
+of the SO coupling. Spin precession around BSO with no differential decay of different spin
+components is shown. Thus, in the two terminal setup, at T = 0, the chiral structure and
+16
+
+0.0
+N= 0.1
+-0.1
+N= 0.01
+-0.2
+N= 0.001
+0.3
+-0.4
+E= 0.2
+1
+2
+3
+4
+5
+a(nm)SciPost Physics
+Submission
+SO will then not be enough for spin selectivity since, as we have shown above, the spin-orbit
+coupling only makes for spin precession due to the spin torque of the SO magnetic field BSO
+with no asymmetric treatment of both spin species.
+One of the most emblematic features of the CISS effect is that it is measured at room tem-
+perature (although see[31, 32]) in molecular systems that are strongly coupled to the thermal
+environment. Strictly coherent quantum models can only be part of the story. The final model
+addresses minimal coupling to the environment through a third probe, making for decoher-
+ent/dephasing albeit unitary processes. Symmetry arguments cannot assess beforehand, how
+sensitive the system will be to weak-TRS-breaking events. These events are incorporated in
+the last model as a time reversal symmetric interaction (SO coupling) under a barrier coupled
+to a B¨uttiker probe. Our exact results show a high sensitivity of spin polarization, where the
+combination of SO coupling and decoherence acts analogously to a U(1) magnetic field. The
+input spin reorients to the effective magnetic field producing a net spin polarization. This is a
+very different mechanism from the first model considered, where differential decay of each spin
+orientation gives polarization. Here delicate interferences that guarantee time-reversal sym-
+metry are disrupted, producing a large effect of even 40% for small couplings to the B¨uttiker
+probe.
+The proposed scenario, in the context of an exactly solved model, addresses many issues
+that have related to the CISS effect: i) The size of the SO coupling is set to realistic values
+in agreement with theoretical estimates in the meV range[22]; ii) The models reproduce the
+Onsager relations for two terminal devices for TRS interactions; iii) Coupling weakly to the
+environment through a third probe (other than terminals) produces highly sensitive effects of
+the polarization capacity of chiral molecules, that can match the order of magnitude found in
+experiments. Additionally we note that other interactions such as electron- and spin-phonon
+have been modeled[33] in order to investigate their effect on electron transport and spin
+polarization through chiral molecules. These works show that non-zero coupling to a thermal
+reservoir is necessary to have spin selectivity[34, 35, 36].
+Of course, this is the bare-bones model for the spintronics of chiral molecules. The nature
+of the environment coupling to electron transport should be developed in much more detail to
+assess its quantitative correctness. The incorporation of the interplay between orbital and spin
+degrees of freedom, not yet addressed to our knowledge, should enrich further the possibilities
+of the theory to describe, more completely, the CISS effect.
+In summary, one of our main conclusions in that decoherence effects, even those associated
+with small coupling constants, can translate into significant changes in the spin polarization.
+This important result allows for a B¨uttiker-probe representation of several mechanisms, in-
+cluding electron-electron and electron-phonon interactions, that can be related to the CISS
+effect. There is an important connection between our treatment and a Liouville equation
+description of electron transfer in molecules [37], that we will explore extensively in a forth-
+coming article.
+Acknowledgements
+Author contributions
+17
+
+SciPost Physics
+Submission
+Funding information
+E.M. acknowledges support from Project 17617 Spin Active Molec-
+ular electronics by the Universidad San Francisco de Quito. M.P. and S.V. acknowledge the
+support given by the Dresden Junior Fellow by the Chair of Materials Science and Nanotech-
+nology at the Technische Universit¨at Dresden. V.M acknowledges the support of Ikerbasque,
+the Basque Foundation for Science, and the W.M. Keck Foundation through the grant ”Chi-
+rality, spin coherence and entanglement in quantum biology”.
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+Optirank: classification for RNA-Seq data with optimal ranking
+reference genes
+Paola Malsot1,∗, Filipe Martins1, Didier Trono1, Guillaume Obozinski2,†
+1Ecole Polytechnique Fédérale de Lausanne
+2Swiss Data Science Center, EPFL & ETH Zürich
+January 13, 2023
+Abstract
+Classification algorithms using RNA-Sequencing (RNA-Seq) data as input are used in a variety of
+biological applications. By nature, RNA-Seq data is subject to uncontrolled fluctuations both within and
+especially across datasets, which presents a major difficulty for a trained classifier to generalize to an
+external dataset.
+Replacing raw gene counts with the rank of gene counts inside an observation has proven effective to
+mitigate this problem. However, the rank of a feature is by definition relative to all other features,
+including highly variable features that introduce noise in the ranking.
+To address this problem and obtain more robust ranks, we propose a logistic regression model, optirank,
+which learns simultaneously the parameters of the model and the genes to use as a reference set in the
+ranking.
+We show the effectiveness of this method on simulated data. We also consider real classification tasks,
+which present different kinds of distribution shifts between train and test data. Those tasks concern
+a variety of applications, such as cancer of unknown primary classification, identification of specific
+gene signatures, and determination of cell type in single-cell RNA-Seq datasets. On those real tasks,
+optirank performs at least as well as the vanilla logistic regression on classical ranks, while producing
+sparser solutions.
+In addition, to increase the robustness against dataset shifts, we propose a multi-source learning scheme
+and demonstrate its effectiveness when used in combination with rank-based classifiers.
+1
+Introduction
+RNA-Sequencing provides a way to probe the state of cells and tissues, by measuring the level of expression
+of thousands of genes. Since its introduction, RNA-Seq data has been used in differential expression analysis
+to highlight genes that are differentially expressed in two contrasting conditions (stereotypically healthy
+versus diseased), pointing towards potentially actionable drug targets and molecular mechanisms. In this
+context, normalization of RNA-Seq data has been extensively studied: we provide in the subsequent section
+an overview of common methods. However there is still a lack of consensus on normalization for classification
+tasks, which is crucial given the recent emergence of machine-learning assisted diagnosis based on RNA-Seq
+data (for instance Cascianelli et al., 2020; Shen et al., 2020; Tan and Cahan, 2019).
+In practice, among other normalization techniques also used in differential expression analysis, ranking
+normalization seems to have had particular success in combination with classification algorithms (Shen et al.,
+2020; Scialdone et al., 2015). Ranking normalization consists simply in replacing the raw read count of genes
+∗[email protected]
+†[email protected]
+1
+arXiv:2301.04653v1  [q-bio.GN]  11 Jan 2023
+
+by their ranks amongst the read count of other genes for the same observation. Lausser et al. (2016) show a
+consistent improvement of score when ranking normalization is used.
+A potential weakness of ranking normalization is that the rank of an otherwise informative gene could
+be perturbed by genes whose expression fluctuate independently from the variable of interest. An obvious
+solution is to rank gene expressions only relative to a set of stable genes, which we call reference set. The
+difficulty is, however, in choosing this set. With this motivation, and to solve binary classification problems
+based on robust and adaptive ranks, we propose optirank, a logistic regression model based on ranks relative
+to a reference set, where the latter is learned at the same time as the weights of the logistic model.
+1.1
+Overview of Normalization Techniques
+Multiple factors alter the number of reads obtained for a gene beyond the number of corresponding RNA
+molecules in the biological sample, the quantity of interest. For instance, the preservation technique of
+a biological sample and its temperature influence the natural degradation process of RNA; the length
+and the GC-content of an RNA molecule will affect its reading rate. Normalization aims at obtaining a
+representation of the data invariant to those aforementioned nuisance factors. Ideally, the remaining variation
+after normalization should be attributable solely to the phenomenon of interest (stereotypically, disease versus
+normal). In this way, the analyst avoids the risk of confounding the effect of the variable of interest with the
+effect of the nuisance variables, and the learned model is applicable to an external dataset obtained with a
+different configuration of nuisance factors.
+The simplest normalization, total count normalization, divides the raw read count on each gene by the total
+number of reads in the corresponding observation. However, highly expressed genes can induce a misleadingly
+low proportion of reads on other genes which are normally expressed. More refined techniques, such as TMM
+and DESeq counteract this artifact by using a more robust normalization factor. Quantile normalization
+matches the distribution of gene expressions to a reference distribution. Other methods correct the expression
+thanks to controls, such as housekeeping genes hypothesized to be constant across conditions or RNA spike-ins
+whose quantities are controlled during library preparation.
+For the interested reader, Dillies et al. (2012) and Evans et al. (2016) evaluate these methods in differential
+expression analysis with real and simulated data. To our knowledge, there is no similar review on the use of
+RNA-Seq normalization as a pre-processing step prior to classification.
+2
+Learning from ranks
+2.1
+Ranking with respect to a reference set
+Given a vector of gene expression levels x ∈ Rd, the rank of gene j, as understood in the common sense, can
+be obtained by counting the number of genes k (among all d genes) that are less expressed than gene j (i.e.
+xk < xj).
+We introduce a more general notion of rank, the rank rΓ
+j of xj relative to a reference set Γ, by:
+rΓ
+j =
+d
+�
+k=1
+1[k ∈ Γ] 1[xj > xk]
+(1)
+Note that we clearly recover the classical rank in the particular case where Γ = {1, . . . , d}.
+In this paper, we will encode the set Γ by a vector γγγ ∈ {0, 1}d with γj = 1[j ∈ Γ] . We can thus express
+rΓ
+j as rΓ
+j = �d
+k=1 γk 1[xj > xk]. To simplify notations we will also drop the exponent Γ, which will be defined
+from the context.
+Now, to define simultaneously all the ranks associated with the elements of a vector xi we introduce the
+matrix of binary comparisons Ci ∈ Rd×d, with Ci
+jk = 1[xij > xik] . The vector of ranks ri associated with a
+reference set encoded by the indicator vector γγγ is computed as ri = Ciγγγ.
+2
+
+gene A
+1.7
+gene B
+7.3
+gene C
+10.0
+gene D
+5.1
+gene E
+3.4
+Expression 
+Data
+gene A
+1.7
+gene E
+3.4
+gene D
+5.1
+gene B
+7.3
+gene C
+10.0
+Ordered 
+Expression Data
+gene A
+0
+gene E
+1
+gene D
+2
+gene B
+3
+gene C
+4
+Usual 
+Ranks
+gene A
+0
+gene E
+1
+gene D
+1
+gene B
+1
+gene C
+2
+Ranks 
+with respect to 
+reference set  
+(A, B)
+gene A
+0
+gene E
+1
+gene D
+2
+gene B
+2
+gene C
+3
+Ranks 
+with respect to 
+reference set  
+(A, B, E)
+Figure 1: Example of ranking gene expression data with respect to a reference set.
+We will use the index i ∈ {1, . . . , n} to index the n observations in the data.1
+2.2
+Classification: learning the reference set along with a linear model on adap-
+tive ranks
+We consider a classical supervised learning problem in which input data is encoded as ranks ri of the form
+above, and the output data is a label yi ∈ Y. For a loss function ℓ : Y × R → R, we consider a regularized
+empirical risk minimization of the form
+min
+w∈Rd, b∈R
+1
+n
+n
+�
+i=1
+ℓ(yi, w⊤ri + b) + Ω(w),
+(2)
+where Ω is a (typically convex) regularizer. By expressing ri as a function of γγγ and of Ci and minimizing the
+empirical risk with respect to γγγ as well, we propose to learn the reference set Γ.
+This leads to the following optimization problem:
+min
+γγγ∈{0,1}d, w∈Rd, b∈R
+1
+n
+n
+�
+i=1
+ℓ(yi, w⊤Ciγγγ + b) + Ω(w).
+(3)
+Given that the integrality constraint on γγγ makes the optimization problem combinatorial, we propose to
+relax the constraint γγγ ∈ {0, 1}d to γγγ ∈ [0, 1]d. We empirically found that adding a cardinality constraint on
+the reference set, γγγ⊤1 = s instead of penalizing with a sparsity inducing regularization, such as the ℓ1-norm,
+was useful to obtain fast convergence. We suppose that this constraint removes an indeterminacy of scale
+between w and γγγ which appears once the integrality constraint is removed, given that (α w, 1
+α γγγ) yields
+identical losses values for any scaling factor α; it would be implicitely removed as well by regularizers on w
+and γγγ but only at convergence.
+Given the relaxed constraint γγγ ∈ [0, 1]d, and in order to nonetheless obtain solutions with γγγ ∈ {0, 1}d, we
+propose to solve a sequence of problems of the form
+min
+γγγ∈[0,1]d, w∈Rd, b∈R
+1
+n
+n
+�
+i=1
+ℓ(yi, w⊤Ciγγγ + b) + Ω(w) + λp ρ(γγγ)
+s.t.
+γγγ⊤1 = s,
+(4)
+1Note that the introduced notations adopt the convention of using a strict inequality in the definition of the rank. It is
+possible to obtain two other similar definitions of ranks by replacing 1[xj > xk] by 1[xj ≥ xk] or by 1[xj > xk] + 0.5 1[xj = xk] .
+All definitions are obviously equivalent when there are no ties and when Γ = {1, . . . , d}. The case where there are ties and where
+these other notions of ranks become relevant is discussed in Appendix A.
+3
+
+for an increasing sequence of regularization coefficients λp, where ρ is the concave “push” penalty defined by
+ρ(γγγ) =
+d
+�
+j=1
+γj(1 − γj),
+(5)
+which effectively “pushes” the entries of γγγ towards the extreme points of the hypercube [0, 1]d. More precisely,
+starting from λp = 0, the solution of each problem in the sequence is used as initialization to warm-start the
+next one, and the sequence is terminated when the solution satisfies the constraint γγγ ∈ {0, 1}d.
+It is obviously possible to only solve the above problem for λp = 0 and renounce the integrality constraints.
+Actually, the presence of the capped-simplex constraints γγγ ∈ [0, 1]d and γγγ⊤1 = s are themselves sufficient
+to obtain that, at the optimum, γγγ∗ tends to lie on a lower dimensional face of the capped-simplex, so that
+a significant fraction of its entries are exactly equal to 0 or 1. In preliminary experiments, we also did
+not observe significant differences whether integrality constraints are strictly enforced or not. The main
+motivations to nonetheless enforce them, are that (a) the additional computational effort is small compared
+to the cost of solving the problem with λp = 0, (b) the interpretability of the obtained ranks is otherwise lost,
+and (c) that it tends to produce slightly sparser solutions.
+3
+Optimization procedure
+Block proximal coordinate descent.
+When λp = 0, problem (4) is bi-convex. More precisely, the
+objective function to minimize, which we denote by O(λp;γγγ, w, b), is convex w.r.t. (w, b) when γγγ is fixed
+and convex w.r.t. (γγγ, b) when w is fixed. This suggests that a form of alternating descent algorithm can be
+used, such as block coordinate descent, in which blocks of variables, here (γγγ, b) and (w, b), are alternatively
+updated (see for example Tseng and Yun, 2009; Xu and Yin, 2013).
+In addition, since the regularizer Ω is convex and potentially non-differentiable (e.g., elastic net regularization),
+descent w.r.t. w can be suitably realized with proximal gradient steps, provided that the proximal operator
+for Ω can be computed efficiently. For γ, the optimization step satisfying the constraint γγγ ∈ [0, 1]d | γγγ⊤1 = s
+also involves a proximal operator: the projection on this constraint set called capped-simplex. We derive this
+proximal operator in Appendix B.1.
+Therefore, to solve each instance of problem (4), we use a block proximal coordinate descent algorithm
+(BPCD). Shi et al. (2014) propose a BPCD algorithm to solve bilinear logistic regression problems with
+convex regularizers. Our implementation is similar to theirs, except that we use different blocks and a simpler
+stopping criterion, which is better suited to the non-convexity of the push-penalty and to the implementation
+of the path-following algorithm described next. We detail our implementation in Appendix B.3.
+Initialization.
+Given that the optimization problem is non-convex, the initialization matters: for reasons
+of symmetry we set w = 0 and γγγ = s
+d1, i.e., the center of the capped-simplex.
+Path-following algorithm.
+Concerning the sequence of values of λp used for the problems of the form (4),
+given that the term λpρ eventually creates local minima at all vertices of the capped-simplex, it is important
+not to increase λp too quickly, which could produce suboptimal solutions. We use the approach proposed
+by Zaslavskiy et al. (2009). In essence, we adjust the next λp to ensure a sufficiently small increase of the
+objective value O for the previously found solution. The strategy is detailed in Appendix B.4.
+To summarize, we propose to solve each instance of problem (4) with a block proximal coordinate descent
+algorithm (BPCD), and to increase λp according to a rule inspired by the path-following algorithm in
+Zaslavskiy et al. (2009). This scheme is summarized in Algorithm 1.
+4
+
+Algorithm 1 Optimization Procedure
+γ ← s
+d1, w ← 0,
+λp = 0
+▷ Initialization
+while γγγ /∈ {0, 1}d do
+▷ Iterating on λp
+(γγγ, w, b) ← argminγγγ∈[0,1]d,γγγ⊤1=s, w∈Rd, b∈R O(λp;γγγ, w, b)
+▷ solved with BPCD
+λp ← λ′
+p , with O(λ′
+p;γγγ, w, b) − O(λp;γγγ, w, b) = ϵ.
+end while
+return γγγ, w, b
+Note.
+When λp > 0, although ρ is non-convex, problem (4) can still be formulated as a multi-convex
+problem, suitable to the block coordinate descent (see Appendix B.2 for a derivation).
+3.1
+Computing the product Ciγγγ with complexity O(d).
+A priori, the computation of the matrix vector product Ciγγγ involves d2 multiplications. But it is clear that
+to compute classical ranks it is sufficient to sort the data, which can be done with a complexity of O(d log d).
+Since Ciγγγ is none else than the vector of ranks with respect to the reference set Γ, it seems reasonable to
+think that the same complexity can be achieved, and this is indeed the case. Assuming that there are no ties,
+and if σi is a permutation sorting the entries of xi, i.e. such that xi,σi(1) < · · · < xi,σi(d), the inner-sum can
+be calculated recursively by applying the same permutation to γγγ and applying, from j = 1 to d,
+ri,σi(j) ← ri,σi(j−1) + γσi(j−1),
+(6)
+with, by convention, ri,σi(0) = 0.
+The complexity is therefore dominated by the sorting operation and is thus O(d log d). Moreover, sorting
+the data needs to be done only once at the beginning of the optimization, so that effectively the number of
+operation needed to compute Ciγγγ each time γγγ is updated is O(d). The exact same reasoning applies to the
+computation of w⊤Ci which is therefore also O(d), once the inverse of σi is computed. With the alternative
+definitions of rank and in the presence of ties, the calculations are more subtle, but the same complexity can
+be obtained. They are detailed in Appendix A.2.
+4
+Benchmark: competing classification algorithms
+We will apply the proposed methodology to solve a number of binary classification problems on first synthetic
+and then real RNA-Seq data. To serve as a basis of comparison, we choose standard logistic regression or
+random forest classifiers that rely either on a rank representation or not.
+Optirank: a sparse rank-based logistic regression with learnable reference set.
+To solve binary
+classification tasks, we propose optirank, a logistic regression model on rank-transformed data, with ranks
+computed with respect to a learnable reference set. Our model optirank is fitted within the framework
+introduced in section 2.2, by solving the optimization problem (3) with a logistic loss ℓ(y, a) = −y log (S(a))−
+(1 − y) log (1 − S(a)), where S(x) denotes the sigmoid function S(x) = 1/(1 + e−x), y ∈ Y = {0, 1} being the
+binary label, and with an elastic net regularization
+Ω(w) = λ1 ∥w∥1 + λ2 ∥w∥2
+2 ,
+to induce sparsity in the set of features whose rank is relevant to the classification task. In fact, given that
+we consider RNA-Seq data, and that there are potentially significant correlations between genes, the use of
+the elastic net, with a Euclidean regularization on top of the Lasso terms, aims at stabilizing feature selection
+(see Zou and Hastie, 2005).
+5
+
+Competing algorithms.
+We will compare our optirank
+algorithm with classical logistic regression
+(lr) equipped with the same elastic net regularization Ω, and logistic regression on rank-transformed data
+(rank-lr), still with the same regularization. In addition, in tasks involving real data, we will also compare
+our method to the random forest (rf) and to the SingleCellNet algorithm (SCN) proposed by Tan and Cahan
+(2019) for cell-typing tasks that we consider in our benchmark (see Section 6). The SCN algorithm consists of a
+pre-processing pipeline which identifies gene pairs with informative differential expression and transforms the
+RNA-Seq data into a binary matrix indicating the order of gene pairs followed by a random forest classifier.
+Implementation details.
+The stopping criterion in the scikit-learn (Pedregosa et al., 2011) implementation
+of logistic regression being different from the one in optirank, to ensure that this discrepancy does not
+affect the comparison on real datasets, we re-optimize the weights w learned by optirank with the logistic
+regression of scikit-learn. Additional details about the classifiers can be found in Appendix C.
+5
+A synthetic data distribution model with unstable ranks
+In order to illustrate the potential and limitations of optirank, we present in the following a synthetic
+example in which the robustness of the rank normalization is challenged. We test whether optirank is
+effectively able to overcome the difficulty of the task.
+5.1
+Model
+As mentioned in Lausser et al. (2016), the strength of rank-normalization can be linked to the fact that ranks
+are invariant to observation-wise monotone perturbations of the gene expressions. Those perturbations can be
+easily envisioned, for example by considering that counts depend in a quadratic (and observation-dependent)
+fashion on the RNA content in the observation. By contrast, the following example focuses on a weakness of
+rank normalization that arises in the presence of a non-monotone, nonetheless simple, perturbation. Those
+perturbations occur in real data; for instance, Leek et al. (2010) report a case in which a group of genes shifts
+between different batches of samples. In our example, we consider a similar perturbation: we suppose that
+there is a group of perturbed genes, uninformative for the classification task at hand, that introduces noise in
+the ranks of relevant features by fluctuating in a coordinated and observation-wise manner.
+More precisely, the expression levels of the genes in this group, called P, are all assumed to shift by an
+additive amount close to ∆i (unique to the observation i). This introduces noise in the ranking of the other
+stable genes: indeed, since the perturbed genes in P shift in a coordinated fashion, the rank of a stable gene
+is increased or decreased by an amount proportional to the number of genes in P that cross it.
+We propose the following synthetic model: the non-perturbed expression of a gene j in observation i, �
+Xij,
+follows a Gaussian distribution centered on the typical expression value of gene j, µj:
+�
+Xij ∼ N(µj, σ2) ,
+(7)
+where σ defines the magnitude of the baseline noise in the data. We generate values for µj by sampling
+uniformly on the expression interval, which we set to [0, 1].
+The expression of a perturbed gene in P is generated by adding to the unperturbed expression �
+Xij an
+observation-wise shift ∆i that we sample from a centered Gaussian distribution N(0, τ 2), with τ defining the
+typical magnitude of the perturbation. Summing all contributions, the expression of a gene j in observation i,
+Xij, is generated as:
+Xij =
+� �
+Xij + ∆i
+if j ∈ P
+�
+Xij
+otherwise.
+(8)
+6
+
+Finally, we assign to each observation i a label generated from a simple logistic model on the ranks within
+the stable (non-perturbed) genes (forming the set S):
+P(Y = 1|X = x) = σ(w⊤rΓ + b),
+with
+wj = 0, ∀j /∈ S
+and
+Γ = S.
+(9)
+The generation of the parameters w, γγγ, and b is detailed in Appendix D.
+5.2
+Results
+We benchmarked on this synthetic data three different classifiers based on logistic regression: the simple
+logistic regression (lr), the logistic regression on ranked data (rank-lr), and optirank, which can produce
+the rank-lr model as a particular case but offers the additional flexibility of restraining the reference set.
+Concerning the choice of regularization hyperparameters, note that we tuned only the ridge regularization
+coefficient λ2 (and s for optirank), setting the lasso penalty λ1 to zero for all three classifiers, given that we
+consider sample sizes n that are large compared to the number of variables d.
+With default simulation parameters, optirank outperforms both rank-lr and lr (see Table 1). Moreover,
+optirank is empirically able to recover the true reference set. Indeed, the cosine similarity SC between the
+vector γγγ used to generate the data (see equation 9) and the one found by optirank is high: SC = 0.95 ± 0.03.
+Test Balanced Accuracy (%)
+Logistic Regression (lr)
+78 ± 2
+Logistic Regression on Full Ranks (rank-lr)
+80 ± 3
+optirank (our model)
+96 ± 0.4
+Table 1: Test balanced accuracy (in %) for classifiers on the synthetic example of Section 5. Default simulation
+parameters were set to d = 50, dP = 40, n = 1000, τ = 0.2, and σ = 0.05.
+We investigated how this comparison evolves when we change the number of perturbed genes dP or the
+dimension of the gene expression profile d, while maintaining the ratio between the number of observations
+and the dimension (n/d) fixed. Not surprisingly, the superiority of optirank over lr and rank-lr fades
+when the number of perturbed genes dP becomes small relative to the dimension of the gene expression
+profile. Indeed, Figure 2 shows that when d increases while keeping the number of perturbed genes, dP ,
+equal to 40 , rank-lr and lr scores rise to the level of optirank (whose performance degrades slightly).
+In accordance, when the number of perturbed genes is increased while keeping the dimension of the gene
+expression to 50, the performance of rank-lr and lr degrades, while the score of optirank remains high.
+This outlines the fact that the perturbation on the usual ranks of informative genes becomes smaller as the
+ratio dP /d decreases.
+Concerning the cosine similarity between the ground truth reference set and the reference set found by
+optirank, its dependence on the simulation parameters dP and d is small: Figure 4 in Appendix D.3 shows
+that the overlap remains high.
+For the dependence of scores on τ, which defines the magnitude of the observation-wise shift ∆i, see
+Figure 3 in Appendix D.3.
+In summary, this synthetic task exemplifies (a) how a non-monotone perturbation can effectively degrade
+the performance of the rank normalization, and (b) that optirank is robust to the kind of perturbation
+introduced thanks to its learnable ranking reference set.
+6
+Experiments on real RNA-Seq data
+In this section, we benchmark optirank on multiple biologically relevant classification tasks with real
+RNA-Seq data. These tasks present qualitatively different dataset shifts between train and test data. In
+Subsection 6.1, we evaluate how robust different algorithms are to these dataset shifts. To enhance robustness,
+7
+
+50
+75
+100
+125
+150
+175
+200
+d
+0.75
+0.80
+0.85
+0.90
+0.95
+test balanced accuracy (%)
+lr
+rank-lr
+optirank
+0
+10
+20
+30
+40
+dP
+0.75
+0.80
+0.85
+0.90
+0.95
+test balanced accuracy (%)
+lr
+rank-lr
+optirank
+Figure 2: Dependence of the test-accuracy with respect to the dimension of the gene expression profile d
+when dP = 40, and with respect to the number of perturbed genes dP for d = 50. The shaded area shows the
+standard error across runs.
+we then investigate in Subsection 6.2, an alternative learning scenario in which multiple source datasets are
+merged in the training data. In this manner, we hope that the algorithm learns better to be robust to the
+kind of perturbation it will encounter in the test data.
+6.1
+Classification with different dataset shifts
+6.1.1
+Classification tasks
+CUP-related tasks.
+Cancer of unknown primary (CUP) occurs when a patient has a metastatic tumor
+whose organ of origin (where the primary tumor was located) cannot be determined. A lot of effort has
+been dedicated to develop classifiers to predict the organ of the primary tumor based on RNA-Seq data
+of the metastatic tissue, in the hope of personalizing and enhancing the treatment given to CUP patients
+(Laprovitera et al., 2021). However, an obstacle in building efficiently such classifiers is the scarsity of
+RNA-Seq data of metastatic tumors. As a result, classifiers are often trained and tuned on datasets of
+primary tumors, which are biologically different from metastasis, and metastatic samples are reserved to
+the external classifier validation. This is precisely what we do in the three tasks TCGA, PCAWG, met500.
+In the task TCGA, classifiers are trained on the TCGA dataset comprising primary tumors (The Cancer
+Genome Atlas Research Network et al., 2008), and tested on a held out portion of the same dataset. In the
+task PCAWG, those same classifiers trained on TCGA were tested on an external dataset PCAWG also with
+primary tumors (The ICGC/TCGA Pan-Cancer Analysis of Whole Genomes Consortium, 2020). Finally, the
+task met500 evaluates those classifiers on the met500 dataset (published by Leiserson et al., 2015), which
+comprises metastatic tumors of various origins. The two latter tasks represent different challenges in terms
+of dataset-shift between train and test. The task PCAWG is subject to technical variation between two
+separately obtained datasets, which we call batch-effect. An additional difficulty in the met500 task is that a
+metastasis differs biologically from its primary tumor, resulting in a so-called biologically induced dataset shift.
+Cell-typing single-cell data.
+Single-cell RNA-Seq (scRNA-Seq) provides a way to probe gene expression
+at the cell-level resolution. A preliminary step in single cell data analysis resides in the cell-type identification
+of each observation/cell, i.e. cell typing. To achieve this automatically, Tan and Cahan (2019) propose to
+train a classifier on a labeled dataset comprising many cell types —commonly known as a cell-atlas, and use
+it to infer the cell-types in an unlabelled dataset. One difficulty is that the train and test scRNA-Seq data
+8
+
+are potentially generated by different sequencing platforms. Tan and Cahan (2019) evaluate the robustness
+of their classifier SCN for various cross-platform train and test data combinations (Tasks Baron-Murano,
+Baron-Segerstolpe, MWS-TM10x, MWS-TMfacs, TM10x-MWS, TM10x-TMfacs and TMfacs-MWS). We use
+the same tasks to investigate the usefulness of optirank in counteracting dataset shifts induced by different
+sequencing platforms and compare with the original method SCN.
+BRCA task.
+The task BRCA consists in predicting the presence of the BRCA mutation from the RNA-Seq
+data in breast primary tumors from the TCGA dataset. This task does not directly answer a real-life
+classification problem. However, the classifier coefficients could be used to obtain a (sparse) transcriptional
+signature for BRCA cancer.
+We list all tasks while grouping them by type of dataset shift (if any) between the train and test data in
+Table 2.
+Dataset-shift
+Tasks
+None (same distribution)
+BRCA, TCGA
+Batch-effects
+PCAWG
+Technical dataset-shift (Different
+sequencing platforms)
+Baron-Murano, Baron-Segerstolpe, MWS-TM10x,
+MWS-TMfacs, TM10x-MWS, TM10x-TMfacs and
+TMfacs-MWS
+Biologically induced dataset-shift
+met500
+Table 2: Classification tasks by type of dataset-shift between train and test set.
+Additional details about the data sources are in Appendix E.1.
+6.1.2
+Data pre-processing
+For fairness of comparison and as first step of dimensionality reduction, the data was reduced to include the
+1000 genes occuring in informative pairs identified by SCN. Logged raw-cpm values were used as input for
+each classifier (see Appendix E.2 for additional details).
+6.1.3
+Hyperparameter selection
+The ℓ1 and ℓ2 regularization coefficients of optirank, lr and rank-lr , and the s/d ratio for optirank were
+tuned via internal cross-validation (i.e., using held out data from the same source as the training data; see
+Appendix E.3.1, E.3.2 and E.4 for details). SingleCellNet was used with parameters suggested by Tan and
+Cahan (2019), and random forest (rf) was trained with 300 trees. Optimal hyperparameters were chosen
+with the one-standard error rule (Hastie et al., 2015) which selects the sparsest model with a score within one
+standard error of the best one.
+6.1.4
+Results and discussion
+The performance in terms of balanced accuracy of all classifiers on the different classification tasks presented
+in Section 6.1.1 are summarized in the following table (Table 6.1.4). We highlighted in bold the scores of
+classifiers that did not score significantly worse than the winning method, according to a paired Student’s
+t-test.
+Interestingly, the advantage of logistic regression-based classifiers relying on a rank-representation
+(optirank, rank-lr) over their non-ranked counterpart (lr) is not consistent, but rather depends on
+the task considered. Indeed, on the tasks TM10x-MWS and TMfacs-MWS, lr clearly surpasses its ranked
+counterparts, while on the tasks met500, Baron-Murano and TM10x-TMfacs we notice the opposite trend.
+9
+
+SCN
+lr
+optirank
+rank-lr
+rf
+BRCA
+50.1 ± 0.3 (4)
+58 ± 2 (1)
+52 ± 2 (3)
+53 ± 1 (2)
+50.0 ± 0 (5)
+TCGA
+92 ± 1 (5)
+99.14 ± 0.23 (2) 99.06 ± 0.26 (3) 99.3 ± 0.3 (1)
+95.7 ± 0.5 (4)
+PCAWG
+76.0 ± 4.5 (2)
+76.2 ± 5.4 (1)
+74.3 ± 5.4 (4)
+74.4 ± 5.8 (3)
+60 ± 4 (5)
+met-500
+67 ± 4 (4)
+71 ± 4 (3)
+77 ± 4 (1)
+75 ± 4 (2)
+60 ± 4 (5)
+Baron-Murano
+89 ± 2 (3)
+87 ± 3 (4)
+93.0 ± 1.9 (2)
+93.2 ± 1.9 (1)
+62 ± 3 (5)
+Baron-Segerstolpe 93.5 ± 1.6 (3)
+93.4 ± 2.2 (4)
+93.6 ± 2.2 (2)
+94.0 ± 1.8 (1)
+60 ± 3 (5)
+MWS-TM10x
+72 ± 2 (4)
+86 ± 2 (1)
+84 ± 2 (3)
+85 ± 2 (2)
+53 ± 1 (5)
+MWS-TMfacs
+70 ± 2 (4)
+86 ± 1 (3)
+87.3 ± 1.6 (2)
+87.4 ± 1.7 (1) 50.01 ± 0.01 (5)
+TM10x-MWS
+51.5 ± 0.3 (4)
+72 ± 2 (1)
+65 ± 2 (2)
+63 ± 2 (3)
+51.0 ± 0.2 (5)
+TM10x-TMfacs
+80 ± 1 (4)
+91 ± 1 (3)
+92.3 ± 0.9 (2)
+92.7 ± 0.9 (1)
+58 ± 1 (5)
+TMfacs-MWS
+51.0 ± 0.2 (4)
+71 ± 2 (1)
+64 ± 1 (3)
+66 ± 2 (2)
+50.2 ± 0.1 (5)
+Table 3: Average balanced accuracies in % (across folds and classes) of competing classifiers on the different
+tasks detailed in Section 4. Horizontal lines separate tasks with different types of dataset shift, from top to
+bottom: generalization to the same distribution, robustness to batch-effects, robustness to biologically-induced
+dataset shifts and robustness across sequencing platforms. The integer in parenthesis denotes the rank of the
+classifiers in terms of average balanced accuracy (lower is better). Classifiers which did not score significantly
+worse than the best classifier according to a paired Student’s t-test (with a level of 5 %) are highlighted in
+bold (see Appendix E.5 for additional details).
+This indicates that the rank representation confers additional robustness against dataset shifts only in some
+instances.
+A burning question is whether there is an advantage of ranking relative to a subset of genes compared
+to ranking among all. At first sight, this doesn’t seem to be the case: the performances of optirank and
+rank-lr are similar. In a more thorough analysis in which we carried paired Student’s t-tests for every task
+and every pair of classifiers (see Appendix E.5), only the task TM10x-MWS showed a significant difference
+between rank-lr and optirank, in favor of optirank.
+In summary, in these tasks, the ranking reference set found by optirank is not more robust than the
+classical full reference set.
+A possible explanation is that an optimal restricted reference set does not
+necessarily exist. Contrarily to the synthetic example in Section 5 and to certain observations made on real
+data (see Leek et al., 2010), where a group of genes shift in one direction and perturb the ranking, in the
+tasks we consider, the dataset-shift could either be a monotone transformation or could shift genes in opposite
+directions. In both these scenarios, the ranks of certain stable genes would not be affected by the dataset
+shift. Alternatively, one could argue that even if such an optimal restricted reference set existed, the only way
+to discover it would be by inspecting the test dataset. We address this question in an additional experiment
+presented in the next section.
+Aside from the performance aspect, it is important to note that by definition, the classical ranking
+normalization is computed with the measurement of all (reference) genes. In contrast, optirank can find
+models that require only a small number of genes to be sequenced, which can be a decisive advantage in some
+medical applications. In the tasks we consider, solutions found by optirank require around 500 genes, half of
+the thousand used by rank-lr. However, it is worth noting that when the logistic regression performs well,
+there is no advantage of using optirank, as the latter tends to produce less sparse solutions (see Appendix
+E.5.6).
+It is worth noting that in general, the random forest rf performs worse than other classifiers, and that
+SCN does not provide a competitive advantage on single-cell typing tasks.
+10
+
+6.2
+Enhancing robustness with a multi-source learning scenario
+In this experiment, we investigate whether in the presence of dataset shifts, merging two source datasets in
+the training set increases the classification accuracy on the third external target dataset. The rationale is that
+the algorithm could learn to be robust to the kind of perturbation it will encounter in the target dataset2.
+To achieve this, we constructed three tasks: TCGA-PCAWG-met500, Baron-Segerstolpe-Murano and MWS-
+TMfacs-TM10x. The tasks are named after the datasets that compose them: the first is the main source
+dataset, the second the auxiliary source dataset and the third is the target dataset. We compare the
+performance in the multi-source scenario (where the main and auxiliary source datasets are merged into a
+training set) to a baseline scenario in which the auxiliary source dataset is not used (single-source scenario).
+Appendix E.3.2 provides additional details about the construction of those tasks.
+ANrank-lr.
+One could ask if, with the help of the auxiliary source dataset, robust ranking reference genes
+can be identified in a simpler manner than in optirank, in particular, with a selection step decoupled from
+the fitting process. To answer this question, we constructed an additional logistic regression classifier based
+on adaptive ranks, ANrank-lr, that selects the ranking reference genes based on a simple ANOVA test.
+For each gene, the vulnerability to dataset-shift is assessed with a two-way ANOVA which determines the
+effect of label and dataset jointly. The s most robust genes are selected as ranking reference (See Appendix
+C.6 for additional details). For completeness, we evaluate ANrank-lr both in the multi-source and in the
+single-source scenario. In the single-source setting, ANrank-lr uses the auxiliary source dataset only for the
+ANOVA test (and not during fitting nor validation).
+Data preprocessing and hyperparameter selection.
+Data preprocessing was done as described in
+the previous section. Appendix E.3.2 details the procedure to obtain the cross-validation splits: special
+care was taken to have similar training dataset sizes in the multi-source and single-source scenarios. The
+hyperparameter grids used for cross-validation are the same as in the previous experiment. Concerning
+ANrank-lr, the number of ranking reference genes s and the elastic net regularization coefficients are tuned
+over the same grid as for optirank.
+6.2.1
+Results and Discussion.
+There is a clear benefit brought by merging two source datasets in the training phase (multi-source scenario).
+Indeed, for nearly all classifiers and tasks, the average balanced accuracy is greatly increased in the multi-
+source scenario (Table 6.2.1) compared to the single-source scenario in which the auxiliary source dataset is
+not used (see Table E.5.8 in Appendix E.5.8). Accordingly, for both single-cell typing tasks, the leading score
+is greater in the multi-source scenario than in the single-source scenario. Moreover, the regression classifiers
+based on ranks (optirank, ANrank-lr, and rank-lr) outperform the simple logistic regression (lr).
+However, as in the previous section, the performances of optirank and rank-lr seem comparable: in the
+single-cell tasks, the paired t-tests do not reveal any significant difference between the two classifiers (see
+Appendix E.5).
+It is worth noting that despite the simplicity of its method for restricting the reference set, ANrank-lr reaches
+a level of performance comparable with the other ranked-based algorithms, in particular optirank, and likewise
+outperforms the simple logistic regression. This is particularly interesting since, by definition, ANrank-lr can
+produce sparse solutions. Indeed, in Appendix E.5.7, we note that optirank and ANrank-lr find solutions
+involving a similar number of genes. In accordance with the results of the previous section, the simple logistic
+regression produces substantially sparser solutions.
+Runtime comparison.
+The runtime for competing classifiers was measured in the previous single-source
+scenario. Table 26 in Appendix E.5.9 attests that the fitting time of both optirank and ANrank-lr are
+reasonable and in some instances lower than the one of their competitors.
+2The art of combining multiple labeled source datasets in order to classify a target dataset under a dataset-shift is referred as
+multi-source domain adaptation in the literature (See for example the review by Sun et al. (2015)).
+11
+
+ANrank-lr
+SCN
+lr
+optirank
+rank-lr
+rf
+TCGA-PCAWG-met500
+81 ± 4 (1)
+69 ± 5 (4)
+80 ± 4 (2)
+68 ± 4 (5)
+71 ± 4 (3)
+65 ± 4 (6)
+Baron-Segerstolpe-Murano
+98.0 ± 0.3 (3)
+93.2 ± 0.8 (5) 97.2 ± 0.4 (4)
+98.1 ± 0.3 (2)
+98.2 ± 0.3 (1) 67 ± 3 (6)
+MWS-TMfacs-TM10x
+95.63 ± 0.40 (3)
+83 ± 1 (5)
+92.5 ± 0.9 (4) 95.64 ± 0.41 (2) 95.9 ± 0.4 (1) 72 ± 2 (6)
+Table 4: Multi-source scenario. Average balanced accuracies in % (across folds and classes) of competing
+classifiers on the tasks detailed in section 6.2 in the case in which the first two source datasets are merged in
+the training phase. The integer in parenthesis denotes the rank of the classifiers in terms of average balanced
+accuracy (lower is better). Classifiers which did not score significantly worse than the best classifier according
+to a paired Student’s t-test are highlighted in bold (see App. E.5 for additional details).
+Conclusion
+According to the literature, rank normalization confers increased robustness against distribution shifts that
+occur in RNA-Seq data. This success is linked to the fact that rank normalization is invariant to all perturbing
+monotone transformations that occur between different datasets and/or samples. However, a potential
+weakness of using rank normalization is that the rank of genes that might be biologically relevant can be
+perturbed by fluctuations of irrelevant ones.
+To counteract this problem, we proposed optirank, an algorithm that learns a ranking relative to an optimal
+reference genes set while learning a classification or regression model. We showed on a synthetic example,
+inspired by observations on real data, how rank-normalization can suffer from collective fluctuations of an
+ensemble of genes that perturb the ranks, and demonstrated the ability of optirank to eliminate those genes
+from the ranking reference set, thereby allowing it to solve successfully the classification task.
+We then assessed the performance of optirank on 11 real classification tasks, presenting different challenges
+in terms of distribution shifts occurring between train and test data. Indeed, we hypothesize that our model
+is able in some instances to remove from the ranking reference set genes that have the propensity to shift in
+the test distribution, thereby perturbing the ranks learned on the training data. Firstly, we observed that the
+advantage of the rank transformation is not systematic. Moreover, contrary to our hypothesis, restricting the
+reference set, as is done by optirank, does not seem to provide increased robustness compared to ranking
+relative to the full set of genes.
+As an additional way to tackle distribution shifts occurring between train and test data, we propose a
+multi-source learning scheme. In this scheme, we train a classifier on a union of two different datasets in
+which a dataset shift occurs, hoping to make it more robust and efficient on a third external dataset. We
+show that this scenario is particularly useful in the cell-typing tasks, in particular when used in synergy with
+rank-based classifiers. We also explored an alternative way of restricting the reference set, with a simple
+ANOVA test that exploits the multiple sources in the training data. Despite its simplicity, the resulting
+classifier, which we call ANrank-lr, achieves a level of performance similar to optirank.
+Finally, it is important to mention that restricting the reference set reduces the number of genes needed to
+be sequenced, while maintaining the level of robustness and accuracy of the rank normalization. Therefore, in
+certain medical applications where sparsity is desired, it can be worth considering the classifiers optirank and
+ANrank-lr.
+Acknowledgements
+This work was funded under the Swiss Data Science Center collaborative project grant C19-02.
+12
+
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+Code and Data Availability
+The code and the data necessary to reproduce the results are available on the Github repository https:
+//github.com/paolamalsot/optirank.
+14
+
+A
+Notions of rank in the presence of ties
+For the RNA-Seq data we consider in several experiments, the read count of a few genes are equal to 0. This
+leads to ties between these genes, which motivated us to extend the formulation proposed to that case.
+A.1
+Different rank definitions
+Ever since ranks were introduced in statistics, there have been discussions on how to correctly treat
+ties (Kendall, 1945).
+For more references and discussion on recent related work, we refer the reader
+to Amerise and Tarsitano (2015).
+The three simplest approaches, and the ones which could be relevant in our setting, consist in assigning
+to a group of tied genes whose ties are initially broken arbitrarily, respectively their minimum, maximum, or
+average rank.
+For simplicity, and given that the problem of ties is not central to the set of ideas that we are presenting
+and might be irrelevant in many cases, the generalization of classical ranks to ranks with respect to reference
+set that we introduce in the paper stems from the minimum rank definition. However, our implementation of
+the optirank algorithm uses the better behaved average rank.
+We propose in the following generalizations of these different ranks to the case where ranks are defined
+with respect to a reference set Γ.
+Minimum rank.
+In Section 2.1 we introduced the following definition of rank rΓ
+j of xj relative to a reference
+set Γ:
+rΓ
+j =
+d
+�
+k=1
+1[k ∈ Γ] 1[xj > xk] .
+(10)
+The above definition of ranks assigns to tied values the minimum rank they would have if they were arbitrarily
+ordered, hence the name of minimum rank. This rank is also referred to as the standard competition rank for
+obvious reasons. Note that with the mathematical definition above, the smallest rank equals 0; to obtain
+ranks that starts at 1 it suffices to add 1 to all rank values.
+Average rank.
+A way of handling ties which has better properties, in particular which keeps constant the
+sum of the ranks, is to assign to them the average value of these ranks, hence the name average ranking. In
+mathematical terms, this is:
+rΓ
+i =
+d
+�
+j=1
+1[j ∈ Γ] (1[xi > xj] + 0.5 1[xi = xj]) − 0.5
+(11)
+=
+d
+�
+j=1
+γj (1[xi > xj] + 0.5 1[xi = xj]) − 0.5 ,
+(12)
+where the offset of 0.5 sets again the lowest rank to the value of 0. (The fact that this offset appears here
+while it did not appear for the minimum rank could seem surprising, but it is necessary because we sum over
+all values of j including j = i.) Again to obtain ranks that starts at 1 we can add 1 to all ranks. The average
+rank is also sometimes called the fractional rank.
+Maximum rank.
+Yet another possible definition, which is symmetric with the minimum rank, consists
+in replacing the strict inequality in (10) by an inclusive inequality and adding an offset of 1 (to keep our
+convention that the ranks start at 0). This results in:
+15
+
+rΓ
+j =
+d
+�
+k=1
+1[k ∈ Γ] 1[xj ≥ xk] − 1 .
+(13)
+The maximum rank is also called modified competition rank.
+In our implementations and experiments, we systematically used the average rank for two main reasons.
+First, if two variables (i.e. genes in our case) have systematically close values (i.e. that are either equal or
+tend to be very close and in arbitrary order), we would expect the coefficients wj associated with their ranks
+in a classification model to be close. In that case, it is natural to request that the linear score does not change
+much whether they are exactly equal or not. Given that the sum of the ranks of tied values is constant for
+the average rank, it satisfies this property, which fails for the min and the max rank. Second, for a non-trivial
+reference set, and when there are no numerical ties, the average rank only assigns integer valued ranks for the
+elements of the reference set Γ, and any element falling exactly in-between two consecutive reference elements
+has a half-integer value, which is a nice property to have.
+A.2
+Recursion to compute average/maximum/minimum ranks in O(d) from
+sorted data
+In this section, we derive the recursion used to compute the rank with respect to a reference gene set in the
+presence of ties for all definitions of ranking mentioned previously.
+Let x ∈ Rd represent the vector of gene expressions and Π = (A1, . . . , AK) its ordered partition, such that
+�
+∀k ∈ {1, . . . , K}, ∀i, i′ ∈ Ak,
+xi = xi′,
+∀j < k, ∀i ∈ Aj, i′ ∈ Ak,
+xi < xi′.
+(14)
+A.2.1
+Recursion for the average rank
+The recursive relationship for the rank of gene i ∈ Aki is derived as follows, from the definition in Eq. (12):
+ri =
+d
+�
+j=1
+γj (1[xi > xj] + 0.5 1[xi = xj]) − 0.5 =
+K
+�
+k=1
+�
+j∈Ak
+γj
+�
+1[ki > k] + 0.5 1[ki = k]
+�
+− 0.5
+(15)
+=
+ki
+�
+k=1
+|Γ ∩ Ak|
+�
+1[ki > k] + 0.5 1[ki = k]
+�
+− 0.5,
+(16)
+where ki denotes the index of the partition such that i ∈ Aki.
+Obviously, ri = ri′ if ki = ki′. If we call �rki this common value, with a little work, we obtain the following
+recursive relationship:
+�rk = �rk−1 + 1
+2
+�
+|Γ ∩ Ak−1| + |Γ ∩ Ak|
+�
++ 0.5.
+(17)
+A.2.2
+Recursion for the maximum rank
+The recursion for the maximum rank strategy defined by Eq. (13) can be obtained similarly. With the same
+notations, we have
+�rk = �rk−1 + |Γ ∩ Ak| .
+(18)
+16
+
+A.2.3
+Recursion for minimum rank
+The recursive relation for the minimum rank accounting for the presence of ties can be obtained similarly.
+With the same notations, we have:
+�rk = �rk−1 + |Γ ∩ Ak−1| .
+(19)
+B
+Optimization algorithms
+B.1
+Projection on the capped-simplex
+To be self-contained, we rederive in this section an efficient algorithm to compute the projection on the
+capped-simplex. Note that it is a classical result that this projection can be calculated in O(dlogd) operations,
+as demonstrated in Duchi et al. (2008).
+Suppose that for a given x in Rd, we wish to solve:
+min
+z∈Rd
+1
+2 ∥z − x∥2
+2 ,
+such that
+0 ≤ zi ≤ 1
+and
+d
+�
+i=1
+zi = k.
+(20)
+Because of the convexity of Problem (20), any point satisfying the KKT conditions is primal optimal (and
+vice versa).
+The Lagrangian is:
+L (z, λ, ν, µ) = 1
+2 ∥z − x∥2
+2 + λ
+�
+d
+�
+i=1
+zi − k
+�
+−
+d
+�
+i=1
+νizi +
+d
+�
+i=1
+µi(zi − 1).
+(21)
+We enforce the KKT conditions, namely complementary slackness (CS), primal feasibility (PF), dual
+feasibility (DF), and primal stationarity (PS), to get
+∀i ∈ {1, . . . d},
+�
+νizi = 0,
+µi(zi − 1) = 0,
+(CS)
+d
+�
+i=0
+zi = k,
+(PF)
+∀i ∈ {1, . . . d},
+νi, µi > 0,
+(DF)
+∇zL = 0.
+(PS)
+With a little work, we get that:
+zi = clip(xi − λ, 0, 1),
+(22)
+with clip(x, a, b) = max(a, min(x, b)) (provided a < b) and where λ is found by solving ψ(λ) = k, with:
+ψ(λ) =
+d
+�
+i=0
+clip(xi − λ, 0, 1).
+(23)
+In order to find the solution of (23), one can follow this procedure:
+1. Order xi and xi − 1 values (yielding 2d ordered values or knots). As a result, there are 2d − 1 intervals
+between subsequent knots.
+2. Calculate the slope of ψ(λ) on each interval.
+17
+
+3. Calculate iteratively the value of ψ(λ) at each knot, starting from the greatest xi where ψ(λ) = 0. One
+can stop when ψ(λ) > k.
+4. Determine the interval on which ψ(λ) = k.
+5. Solve the linear equation on this interval to find λ.
+B.2
+Adaptive step sizes and a multi-convex formulation
+In this section, we detail the exact problem formulation to which a block proximal coordinate descent
+algorithm (BPCD, Xu and Yin, 2013) can be applied. Our formulation and algorithm is in particular very
+close to the ones considered in Shi et al. (2014).
+As stated in Section 3, the goal is to find the solution of a minimization problem of the form:
+min
+γγγ∈[0,1]d, w∈Rd, b∈R
+1
+n
+n
+�
+i=1
+ℓ(yi, w⊤Ciγγγ + b) + Ω(w) + λp ρ(γγγ)
+s.t.
+γγγ⊤1 = s,
+(24)
+for an increasing sequence of coefficients λp ≥ 0, where a �→ ℓ(y, a) is assumed to be a convex and differentiable
+loss function, with Lipschitz gradients, where Ω(w) is a convex regularizer, and where ρ is the concave “push”
+penalty defined by
+ρ(γγγ) =
+d
+�
+j=1
+γj(1 − γj).
+(25)
+When λp > 0, the previous problem is clearly non convex w.r.t. γγγ. It is obviously possible to use block-
+coordinate proximal coordinate descent on functions that are not even convex with respects to each of the
+individual blocks considered (Razaviyayn et al., 2013; Csiba and Richtárik, 2017). But, with among others
+the motivation of being able to use adaptive step-sizes that can be defined in a principled way, we propose
+to exploit the fact that the optimization problem can be reformulated as another bi-convex problem, by
+introducing a variational form for the concave penalty ρ.
+Indeed, thanks to the Fenchel-Legendre transform it is possible to express the concave push-penalty as an
+infimum over a set of linear functions parameterized by the dual variable υυυ, as follows
+ρ(γγγ) = inf
+υυυ∈Rd
+�
+ρ∗(υυυ) − γγγ⊤υυυ
+�
+,
+with
+ρ∗(υυυ) =
+d
+�
+j=1
+1
+4(1 + υj)2.
+(26)
+Therefore, the minimization problem (24) can be written in a multi-convex form:
+min
+γγγ∈[0,1]d, b∈R,
+w∈Rd,υυυ∈Rd
+1
+n
+n
+�
+i=1
+ℓ(yi, w⊤Ciγγγ + b) + Ω(w) + λp ρ∗(υυυ) − λpγγγ⊤υυυ
+s.t.
+γγγ⊤1 = s.
+(27)
+Problem (27) is convex w.r.t. (w,b, υυυ) with γγγ fixed, and with respect to (γγγ, b) with (w, υυυ) fixed. We
+therefore apply a BPCD scheme to minimize problem (27), except that, for the variable υυυ, the exact
+minimization is immediate, which we therefore use instead of a gradient update. Note that the minimization
+with respect to υj yields υj = 2γj −1 = ∂ρ
+γj (γγγ) so that effectively the update on υ amounts to replacing ρ by its
+tangent, as typically done in convex-concave optimization algorithms. The advantage of using a multi-convex
+formulation is that we can use Armijo type adaptive stepsizes for each variable (In particular, the proofs of
+convergence in Csiba and Richtárik (2017) generalize immediately to the case where Armijo-type linesearches
+are used). Intuitively, using the tangent approximation to ρ when performing a step on γ prevents from
+increasing the step-size too quickly, which might help to prevent that γ is projected “too early” on vertices of
+the capped-simplex that are suboptimal local minima.
+The algorithm that we obtain is similar to the algorithm proposed by Shi et al. (2014).
+18
+
+B.3
+BPCD algorithm with adaptive step sizes
+In this section, we detail the BPCD algorithm (Xu and Yin, 2013; Shi et al., 2014) that we use to solve the
+minimization problem in equation (4).
+Note:
+The objective function to minimize in problem (27), which we call O, can be decomposed in two
+parts, a differentiable part h and a potentially non-differentiable part ψ whose proximal operator can be
+computed efficiently:
+h(w,γγγ, b,υυυ) = ℓ(yi, w⊤Ciγγγ + b) − λpγγγ⊤υυυ
+(28)
+ψ(w,γγγ, b,υυυ) = Ω(w) + λpρ∗(υυυ) + ι{γγγ∈∆s},
+(29)
+with ∆s = {γγγ ∈ [0, 1]d | γγγ⊤1 = s} the capped-simplex and ι{γγγ∈∆s} = 0 if γγγ ∈ ∆s and ι{γγγ∈∆s} = +∞ else.
+The main part of the BPCD algorithm is detailed in Algorithm 2. It consists of a main loop that involves
+alternative updates in w, γγγ, b and υυυ. We refer the reader to the previous section for an explanation on the
+dual variable υυυ. The loop is terminated when a convergence criterion is met.
+Note.
+In the following, we denote by p a set of variables (w,γγγ,υυυ, b). We adopt the convention that the
+same index k indexes the set and the enclosed variables, such that: pk = (wk,γγγk,υυυk, bk).
+Algorithm 2 Block Proximal Coordinate Descent
+Input: {xi, yi} for i = 1 . . . n
+k ← 0
+p0 ← Initialization() a
+Lw, Lγγγ, Lb ← InitializationStepsize(p0)
+p ← p0
+k ← 0
+repeat
+k ← k + 1
+(w, Lw) ← ProxStep_w(p, Lw)
+▷ Updating w
+(b, Lb) ← ProxStep_b(p, Lb)
+▷ Updating b
+(γγγ, Lγγγ) ← ProxStep_γ(p, Lγγγ)
+▷ Updating γγγ
+υυυ ← 2γγγ − 1
+pk ← p
+▷ Storing solution
+until ConvCrit(pk, pk−1) or k > max_iter
+return pk
+aNote that when λp > 0, (w0, γγγ0, υυυ0, b0) are initialized from the previous solution.
+Algorithm 3 details the proximal update step with adaptive stepsizes for the variable w. It consists of a
+loop that searches over logarithmic decreasing stepsizes until a criterion that ensures a sufficient decrease in
+the objective is met. For each stepsize, the next step proposed is computed with a proximal operator.
+The updates for the variables γγγ and b are conceptually identical and can be obtained by permuting
+appropriately the role of the different variables.
+19
+
+Algorithm 3 Proximal update step for w
+function ProxStep_w(p−, Linit)
+m ← −1
+L ← Linit
+repeat
+L ← max(Lmin, L ηm) a
+w∗ ← argmin
+w∈Rd ⟨∇wh(p−), w − w−⟩ + L
+2 ∥w − w−∥2
+2 + ψ(w,γγγ−, b−, υ−)
+p ← (w∗,γγγ−, b−, υ−)
+m ← m + 1
+until CritProx_w(p, p−, L)
+return w∗, L
+end function
+aLmin was set to 10−10, and η to 1.5.
+The criterion for accepting the inverse stepsize L is:
+Algorithm 4 Criterion stepsize for w
+function CritProx_w(p, p−, L)
+return
+�
+h(p) < h(p−) + ⟨∇wh(p−), w − w−⟩ + L
+2 ∥w − w−∥2
+2
+�
+end function
+The BPCD algorithm terminates either when progress in the objective becomes inappreciable, or when
+the update in the variables are very small. The convergence criterion is formulated as:
+Algorithm 5 Convergence Criterion
+function ConvCrit(pk, pk−1)
+ϵ ← 10−5, ϵ2 ← 10−10
+if O(pk−1) − O(pk) < ϵD then
+return true
+else if max(
+��wk − wk−1��2 , (bk − bk−1)2,
+��γγγk − γγγk−1��2) < ϵ2 then
+return true
+else
+return false
+end if
+end function
+In practice, we set the denominator D to O(p2) at the first relaxation iteration with λp = 0.
+Algorithm 6 Initialization
+function Initialization( )
+b0 ← 0
+w0 ← 0
+γγγ0 ← s
+d1
+υυυ0 ← 2γγγ − 1
+return p0
+end function
+20
+
+The initial stepsizes are calculated with the projection of the derivative of h on the Hessian.
+Algorithm 7 Initialization of stepsize
+function InitializationStepsize(p0)
+Lw ← ∇wh(p0)T ∇2
+wh(p0)∇wh(p0)
+∥∇wh(p0)∥2
+Lγγγ ← ∇γγγh(p0)T ∇2
+γγγh(p0)∇γγγh(p0)
+∥∇γγγh(p0)∥2
+Lb ← ∇bh(p0)T ∇2
+bh(p0)∇bh(p0)
+∥∇bh(p0)∥2
+return (L0
+w, L0
+γγγ, L0
+b)
+end function
+B.4
+Path following algorithm and stopping criteria
+Motivation.
+The objective function in (4) must be solved for an increasing sequence of λp > 0, until
+γγγ∗ ∈ {0, 1}d. As we noted in Section 3, one must be careful not to increase λp too quickly. In the following,
+we detail the path-following algorithm used to determine the sequence of λp.
+Rationale.
+The general procedure of the path following algorithm consists in increasing λp "progressively",
+each time solving problem (4) starting from the previous solution. The criterion used to determine the next
+λp is based on the increase in the objective O at the current solution. The detailed algorithm is laid out in
+the following paragraph.
+Path-following algorithm.
+First, problem (4) is solved without push-penalty (λp = 0), yielding a solution
+p(0) = {w, b,γγγ}. Then, λ(1)
+p
+is chosen with the procedure detailed below (see equation (30)) and the BPCD
+algorithm is run again starting from p(0) to minimize the objective with λ(1)
+p , yielding the solution p(1). This
+procedure is repeated until γγγ is very close to the vertices of the capped-simplex (see criterion (31) below), or
+after reaching the maximum number of so-called relaxation iterations (10000).
+Chosing the next λp.
+The rule for choosing λ(i+1)
+p
+, starting from a solution p(i) is:
+O(p(i), λ(i+1)
+p
+) − O(p(i), λ(i)
+p ) = Mϵ
+(30)
+M controls the tradeoff between speed and accuracy and was set to 100. ϵ is the tolerance on the objective
+value decrease used in the stopping criterion of the BPCD algorithm (see Alg. 5).
+Stopping criterion.
+The algorithm is stopped when the solution is sufficiently close to the vertices of the
+capped-simplex, more precisely, when:
+�
+j
+|γj − ˜γj| < δ,
+(31)
+with ˜γj = sign(γj − 0.5) the rounded version of γj, and where δ was set to 10−10.
+Note.
+For a better coordination with the path following algorithm, the stopping criterion presented in the
+BPCD algorithm Shi et al. (2014) was changed to an absolute criterion on the loss change at the last iteration
+(see Algorithm 5).
+21
+
+References.
+The above path-following algorithm was developed in Zaslavskiy et al. (2009) to solve a
+graph-matching problem. Here we have used a slightly simpler version than the one detailed in the paper.
+B.5
+Normalization
+To limit the scaling of the inverse stepsize Lw with s (in algorithms 2 and 3), it is useful to normalize the
+bilinear product w⊤Ciγγγ by an appropriately chosen constant κ which we set to s for reasons now explained.
+It is a classical result that a proximal step with any constant L larger than the Lipschitz constant of the
+gradient of the function produces a valid update. In other words, any step-size of the form 1
+L where L is
+larger than the Lipschitz constant is a valid stepsize. To obtain stepsizes that have a reasonable scaling with
+respect to different hyperparameters such as s or d, it can be useful to normalize the data or the loss function
+such that the Lipschitz constant of the gradient is controlled.
+In this section, we thus bound
+�����
+∂2h
+∂w2
+j
+�����
+2
+and use the fact that any continuous function with bounded
+derivative is also Lipschitz continuous. More precisely, the Lipschitz constant equals 2M, M being the bound
+on the derivative.
+The bound on
+�����
+∂2h
+∂w2
+j
+�����
+2
+, where we introduce the normalization factor κ to determine, is calculated as
+follows:
+∂2h
+∂w2
+j
+= 1
+n
+n
+�
+i=1
+∂2σ
+∂z2
+i
+�
+∂zi
+∂wj
+�2
+, with zi = w⊤Ciγγγ
+κ
++ b .
+(32)
+Using the fact that γγγ lies in [0, 1]d, that |Ci
+jk| ≤ 1, ∀i, j, k and that �
+j γj = s, we get:
+�
+∂zi
+∂wj
+�2
+=
+� �
+k Ci
+jkγk
+κ
+�2
+≤
+�
+∥γγγ∥1
+κ
+�2
+≤
+�
+s
+κ
+�2
+.
+(33)
+For reasons now obvious, if we set κ to s, and combine the previous equations, using the fact that the
+second derivative of the sigmoid function is bounded by Msig =
+1
+6
+√
+3 we get:
+�����
+∂2h
+∂w2
+j
+�����
+2
+< Msig.
+(34)
+C
+Classifiers
+In this section, we describe the implementation of all classifiers used in the results section. Each classifier
+implemented the "balanced" class-weight setting: each observation is re-weighted, such that the weight of
+each class is equal in the objective.
+C.1
+SingleCellNet
+The code for SingleCellNet was downloaded from the repository at https://github.com/pcahan1/singleCellNet.
+In our code for the comparison of classifiers, we wrapped the original code into a scikit-learn compatible
+classifier. In the first section, we detail the steps of the pipeline of SingleCellNet to select informative genes.
+In the second, we detail the classification algorithm, in the case where the output variable is binary (which is
+our case).
+22
+
+C.1.1
+Gene selection
+The first step of the pipeline consists in down-sampling the expression values to 1500 counts per observation
+and scaling it up such that the total expression per observation is 10000. Secondly, a log-transformation is
+applied to the expression data, and each gene is scaled to unit variance and zero mean.
+Then, a first skimming step is applied: it retains genes which are expressed in more than α1 observations, or
+for which the average expression among observations where it is expressed (at least α2) is above a threshold
+µ. After this, the correlation coefficient between the gene expression and the output variable is calculated for
+each gene, and the nnTopGenes with lowest and highest (signed) correlation coefficients are retained.
+We use the default values of SingleCellNet, namely α1 = .05, α2 = .001, µ = 2 and nTopGenes = 100.
+C.1.2
+Classification algorithm
+The expression matrix is transformed into a binary nobs × n2
+TopGenes matrix, where each column encodes
+the orientation of the corresponding gene pair. The correlation coefficients between each column and the
+output are calculated and the nTopPairs top columns are retained. We use the default value of nTopPairs = 100.
+The resulting matrix, supplemented by 100 random observations obtained by shuffling, is used as input to a
+random forest (implemented with scikit-learn) consisting of 1000 trees, trained to optimized the balanced
+accuracy. The other parameters of the random forest are set to default values (see Appendix C.2 for a
+description of the default values).
+C.2
+Random Forest
+The classifier rf was implemented with the scikit-learn library, setting the number of trees to 300 and the
+class-weight to "balanced". We set the number of trees to 300 after verifying empirically that a larger number
+of trees wouldn’t produce better solutions. The other hyperparameters were set to default values. Namely,
+each split minimizes the "Gini" impurity of resulting leaves, each leaf is split until it is pure,
+√
+d randomly
+chosen variables are considered at each split, and nobs observations are drawn (with repetition) to train each
+tree.
+C.3
+Logistic Regression
+The classifier lr was implemented with the scikit-learn library, specifying the maximum number of iterations
+to 10 000, the tolerance to 10−3, choosing the saga solver (Defazio et al., 2014), and setting the class-weight
+to "balanced". Regularization was not applied if not specified (Note that to achieve zero ℓ2-regularization,
+one must set the C parameter to a high value).
+C.4
+Logistic Regression on ranks
+A rank-transformation (with average ranking, see Appendix A) was applied to the data prior to the logistic
+regression classifier detailed above in Appendix C.3.
+C.5
+Optirank
+Implementation. The source code of our implementation of optirank
+is available on the repository
+https://github.com/paolamalsot/optirank. The algorithm is detailed in Appendices B.3 and B.4.
+Note.
+For the classification tasks on real datasets (in Section 6), lr was used to refit w and b with frozen
+γγγ. The goal is to prevent the incompatible stopping criteria between lr and optirank to interfere with the
+comparison between both classifiers.
+Balanced setting. As all the other classifiers, optirank
+was trained with data points reweighted
+inversely proportionally to the size of each class, to balance the classes.
+23
+
+C.6
+ANrank-lr
+The classifier is similar to rank-lr (detailed previously in app. C.4), except that the ranking transformation
+is applied with a previously chosen ranking reference set. The ranking reference set is chosen with a simple
+ANOVA that eliminates from the reference set genes whose expression is subject to dataset shift. Note that
+ANrank-lr uses the auxiliary source dataset only for the ranking reference set selection. For each gene, a
+two-way ANOVA with factors label and dataset is carried out. We then select as ranking reference the genes
+which have the smallest F-value corresponding to the marginal effect of dataset - in other words, the ones less
+susceptible to dataset-shift.
+D
+Synthetic example: Supplementals
+D.1
+Generation of w, γγγ and b
+In this section, we describe the generation of parameters w, γγγ and b for the simulation of the task described
+in Section 5.
+Let us first recall from equation 9 that the label of each sample follows a simple logistic model on the
+ranks within the stable genes:
+P(Y = 1|X = xi) = σ(w⊤ri + b)
+with
+wj = 0 ∀j /∈ S, Γ = S.
+(35)
+= σ(w⊤Ciγγγ + b).
+(36)
+In practice, once w, γγγ and b are generated, it suffices to draw the label from a Bernoulli distribution
+whose probability of success is given by the above equation.
+Simulation procedure.
+First, the design matrix X is generated following the model detailed in Section 5.
+From the synthetic model, γγγ is fixed to γγγi = 1 ∀i ∈ S, 0 otherwise. A non-scaled version of w is sampled as:
+wi ∼ (−1)sgn(N(0, 1) + 1), with sgn ∼ B(0.5). b is then adjusted to ensure that the labels generated are
+balanced (i.e. that the average of the probability of generating a positive class is 0.5 across observations).
+Next, w and b are scaled by the same factor to control the noise of the generated data. The noise is such
+that in expectation, the balanced accuracy between the most probable label and the label drawn is 98 %.
+D.2
+Classifiers comparison
+Simulation parameters.
+The default parameters for the simulation task were set to: d = 50, dP = 40,
+n = 1000, τ = 0.2, and σ = 0.05. Figures 2 and 3 indicate the score for varying values of simulation
+parameters (d ∈ {50, 100, 150, 200}, dP ∈ {0, 10, 20, 30, 35, 40}, and τ ∈ {0, 0.125, 0.25, 0.375, 0.5}). Each
+simulation experiment was repeated 4 times, resulting in the error-bars displayed on the figures.
+Note.
+In the experiment with increasing values for d, we also increased the size of the training set n to
+keep the ratio n/d constant.
+Cross-validation.
+The generated dataset was split in a stratified fashion, keeping 30% for the test. Each
+classifier was trained and tuned with 5-fold internal cross-validation on the train set.
+Hyperparameter selection.
+The parameter grid for classifiers lr and rank-lr consisted in 5 log-spaced
+values for the ℓ2 regularization (0, 0.0001, 0.001, 0.01, 0.1), corresponding to the parameter λ2 in Ω(w) (see
+section 4). In addition, the grid for optirank included 5 values for the hyperparameter s/d (0.2, 0.4, 0.6, 0.8, 1).
+The hyperparameter λ1 was set to 0. For each classifier, we selected the model with highest validation
+balanced accuracy.
+24
+
+D.3
+Supplementary figures
+D.3.1
+Simulation results for varying τ
+Motivation.
+We investigated how the outcome of the comparison between the competing classifiers (lr,
+rank-lr and optirank) is affected by the simulation parameter τ. τ defines the magnitude of the coordinated
+shift of the perturbed genes in P from one observation to the other.
+Results.
+Figure 3 shows that the superiority of optirank over rank-lr and lr is maintained throughout
+the simulation parameter range that we explored.
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.75
+0.80
+0.85
+0.90
+0.95
+test balanced accuracy (%)
+lr
+rank-lr
+optirank
+Figure 3: Dependence of test-accuracy with respect to the simulation parameter τ. Default simulation
+parameters were set to d = 50, dP = 40, n = 1000, and σ = 0.05.
+D.3.2
+Overlap between the true γ and the one fitted by optirank
+25
+
+50
+75
+100
+125
+150
+175
+200
+d
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+overlap
+0
+10
+20
+30
+40
+dP
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+overlap
+Figure 4: Cosine similarity CS between the true γ and the one fitted by optirank, as a function of the
+dimension d and the number of perturbed genes dP . The shaded area shows the standard error across runs.
+Default simulation parameters were set to d = 50, dP = 40, n = 1000, τ = 0.2, and σ = 0.05.
+26
+
+E
+Results on real data
+E.1
+Datasets
+In this section we detail the data sources and the selection of examples used for each of the classification
+tasks.
+TCGA.
+The public count-table for the RNA-Seq data (with dbGab accession number phs000178.v10.p8) and
+the metadata was downloaded from the GDC data portal (https://portal.gdc.cancer.gov). We selected
+primary tumors with RNA-Seq data, with disease type in (’Adenomas and Adenocarcinomas’, ’Squamous
+Cell Neoplasms’, ’Cystic, Mucinous and Serous Neoplasms’, ’Ductal and Lobular Neoplasms’) and tissue
+of primary in (’Breast’,’Bronchus and lung’,’Esophagus’,’Stomach’,’Colon’,’Stomach’,’Pancreas’,’Rectum’,
+’Rectosigmoid junction’, ’Prostate’). The label for the classification task was taken from the tissue of primary,
+by agglomerating "Colon", "Rectum" and "Rectosigmoid junction" in one class called "Colorectal".
+Note.
+In the task TCGA-PCAWG-met500, we ensured the binary problems One-vs-Rest involved a positive
+class present in the auxiliary source dataset PCAWG. For this reason, we had to remove the binary problems
+"Bronchus and lung" VS Rest and "Prostate" VS Rest.
+PCAWG.
+The PCAWG dataset comprises among other the TCGA dataset. Care was taken to exclude
+instances from the PCAWG that belong to the TCGA dataset during the selection of examples. The RNA-
+Seq data was downloaded from https://dcc.icgc.org/releases/PCAWG/transcriptome/transcript_
+expression and the metadata from https://dcc.icgc.org/releases/PCAWG/transcriptome/metadata.
+The same disease types and tissues of primary as in TCGA were selected, the class labels were processed in
+the same way.
+BRCA.
+The mutation status of the BRCA1 and BRCA2 genes for the examples selected in the TCGA
+was obtained on the GDC data portal, and aggregated into one binary class label indicating the presence of
+at least one mutation among BRCA1 and BRCA2. We selected instances whose tissue of primary was the
+Breast. The expression data is the same as the one in the TCGA.
+met-500.
+The expression data and the metadata was downloaded from https://xenabrowser.net/
+datapages/?cohort=MET500%20(expression%20centric). Only metastatic tumors were selected, of the
+same primary tissue as in the TCGA. The class-label was generated with the tissue of primary.
+Single-Cell datasets: Baron, Murano, Segerstolpe, MWS, TM10x, TMfacs
+The datasets were
+downloaded from https://github.com/pcahan1/singleCellNet#trainsets. For each task consisting of a
+pair/triplet of datasets, only common genes and common cell types were selected. Here is a short description
+per dataset:
+Murano. Adult human pancreatic cells, sequenced with CEL-Seq2 (2120 cells)
+Baron. Adult human pancreatic cells, sequenced with inDrop (8596 cells)
+Segerstolpe. Adult human pancreatic cells, sequenced with Smart-Seq2 (2209 cells)
+MWS. Adult mouse cells, across 125 cell types, sequenced with Microwell-seq (6477 cells)
+TM10x. Adult mouse cells, across 32 cell types, sequenced by 10x (1599 cells)
+TMfacs. Adult mouse cells across 69 cell types, sequenced by smartseq2 (3182 cells)
+27
+
+E.2
+Pre-processing
+All datasets were pre-processed with the pre-processing pipeline of SingleCellNet, detailed in Appendix C.1.1.
+E.3
+Cross-validation splits
+In this appendix, we detail, per experiment and for each classification task, which dataset(s) and which
+training-testing scheme were used. All splits were realized in a stratified fashion.
+E.3.1
+Cross-validation splits for the tasks of section 6.1.
+TCGA.
+10% of the TCGA dataset was held out as a test set. The remaining 90% was used for 5-fold
+internal cross-validation.
+PCAWG.
+The same training set and cross-validation splits as in TCGA were used. The PCAWG dataset
+was used as a test set.
+met-500.
+The same training set and cross-validation splits as in TCGA were used. The met-500 dataset
+was used as a test set.
+BRCA.
+The dataset was split in four equal parts (each containing the same percentage of BRCA1 and
+BRCA2 mutated instances). A nested-cross validation scheme was applied in which for every possible
+combination, 2 parts are used for training, the third for validation, and the fourth for testing.
+Cell-typing single-cell data.
+For each task called X-Y (for instance Baron-Murano), dataset X was used
+as a training set and dataset Y as a test set. The training data was generated by randomly picking (maximum)
+100 cells for each cell type. The rest of the training data is used for validation. This process is done 5 times.
+The test dataset is used as a whole.
+E.3.2
+Cross-validation splits for the tasks of section 6.2 in the multi-source and single-source
+scenario
+We now describe the cross-validation splits in the tasks involving three datasets: one main source dataset,
+a auxiliary source dataset and a target dataset. These tasks are named with the pattern X-Y-Z, where X
+is the main source dataset, Y the auxiliary source dataset and Z the target dataset (for instance TCGA-
+PCAWG-met500). we distinguish two training scenarios: the multi-source and single-source scenarios. In the
+multi-source scenario, datasets X and Y are merged, both in the train and validation splits.
+In the single-source scenario dataset Y is not used (except by ANrank-lr which uses dataset Y in its entirety,
+for the sole purpose of selecting the reference genes during the training phase). Indeed, in the single-source
+scenario, ANrank-lr does not use dataset Y for fitting the coefficients of the logistic regression nor for
+validation.
+TCGA-PCAWG-met500.
+We used 5-fold cross-validation to divide the TCGA and PCAWG datasets
+into train/validation splits, for each dataset separately. In the single-source scenario, only the TCGA is
+used. In the multi-source scenario, we compose each train and validation split by doing the union of the
+corresponding splits in the TCGA and PCAWG cross-validation splits.
+Cell-typing single-cell data.
+In the single-source setting, we pick randomly from dataset X 100 cells
+for each cell-type, the rest of dataset X goes in the validation split. We repeat this 5 times to generate 5
+validation folds. ANrank-lr uses a small(er) version of dataset Y, which comprise 100 cells for each cell-type.
+In the multi-source scenario, we took care not to augment the size of the training set. To achieve this, we
+used the same CV splits as in the single-source setting, except that we downsampled to 50 the number of
+28
+
+examples from dataset X for each cell-type, to which we added 50 examples of dataset Y for each cell-type.
+We added to the validation splits generated in the single-source setting the portion of dataset Y which was
+not used in the training set.
+E.4
+Hyperparameter grid
+In table 5, we list the hyperparameters tuned for each classifier, and the corresponding values which were
+tried in the parameter grid.
+Classifier
+Parameter Name
+Values
+optirank
+λ1
+{0, 0.0001, 0.001, 0.01, 0.1}
+λ2
+{0, 0.0001, 0.001, 0.01, 0.1}
+s/d
+{0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0}
+ANrank-lr
+λ1
+{0, 0.0001, 0.001, 0.01, 0.1}
+λ2
+{0, 0.0001, 0.001, 0.01, 0.1}
+s/d
+{0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0}
+lr
+λ1
+{0, 0.0001, 0.001, 0.01, 0.1)
+λ2
+{0, 0.0001, 0.001, 0.01, 0.1}
+rank-lr
+λ1
+{0, 0.0001, 0.001, 0.01, 0.1}
+λ2
+{0, 0.0001, 0.001, 0.01, 0.1}
+rf
+-
+-
+SCN
+-
+-
+Table 5: Hyperparameters for classifiers in Section 6.1.4. Note that the dimension d is 1000 for all classifiers.
+Note that in the scikit-learn implementation of the logistic regression, the elastic net regularization is
+prescribed by the parameters C and ℓ1_ratio. Therefore, these parameters were set according to the values
+of λ1 and λ2.
+E.5
+Comparison between classifiers
+Scoring function: Balanced Accuracy.
+The balanced accuracy scoring function was used as a a scoring
+metric both for hyperparameter selection and for model evaluation.
+The balanced accuracy equals the average of sensitivity (true positive rate) and specificity (true negative
+rate), and is defined as such:
+balanced accuracy = 1/2
+�
+TP
+TP + FN +
+TN
+TN + FP
+�
+,
+(37)
+where TP stands for the number of true positives, FN for the false negatives, etc..
+Note that in a balanced setting, the above formula reduces to the conventional accuracy. By contrast, in
+an unbalanced setting, a classifier which predicts all the time the predominant class will achieve a score of 1/2.
+E.5.1
+Hyperparameter selection and model evaluation
+In this subsection, we describe how hyperparameters were chosen and how the resulting model was evaluated
+and compared to its competitors.
+Each multi-class classification task was separated in nclasses One-VS-Rest binary classification tasks, the
+binary classification tasks were left as is. Each classifier was trained for each parameter combination, in
+each binary problem, for each (internal) training split, and tested in the corresponding validation set and
+on the test set. In each binary task, for every test set (there is a unique one apart in the task BRCA), we
+29
+
+selected the hyperparameters with the one-standard-error rule applied to the balanced accuracy averaged
+over validation splits. More precisely, we selected the sparsest model whose averaged balanced accuracy was
+within one standard error of the highest averaged balanced accuracy. We did not refit each classifier with the
+optimal hyper-parameter combination on the whole training set. Instead, for computational reasons, we kept
+as many models as the number of validation splits, and, as a result, the test-score, as displayed in table 6.1.4,
+was calculated with the average test score among those classifiers.
+E.5.2
+Pairwise comparisons
+The following section details how each pair of classifier was compared, in order to determine if one performs
+significantly better than the other. In order to do so, we performed the comparison per dataset. We computed
+the difference in balanced accuracy between the two classifiers for each validation fold and each binary
+classification task (in such a way that no average is involved). Every one of those differences was accumulated
+as an independent observation, and a paired Student’s t-test was used to determine, based on the differences,
+if one classifier outperformed the other on the dataset at hand. (We used a two-sided test with a significance
+level of 5 %).
+Table 6 through 19 show the result of the pairwise comparisons in each task. On the lower left part of
+each table, the sign at position (i,j) indicates if the classifier corresponding to row i is significantly better (+)
+or worse (-) than the classifier in column j. The percentages in the upper-right part indicate the fraction
+of instances where classifier i performed better than classifier j. In addition, whenever this percentage is
+associated with the algorithm on line i significantly outperforming the algorithm in column j the number is
+highlighted in bold green, and conversely, if this number correspond to a case where the algorithm on row i
+performs significantly worse then algorithm in column j, it is highlighted in red italic.
+E.5.3
+Pairwise comparisons for the classification tasks of section 6.1
+Table 6: Pairwise comparisons for the task BRCA
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+50
+83
+67
+50
+SCN
+75
+92
+58
+rank-lr
++
+75
+17
+lr
++
++
+8
+rf
+-
+-
+Table 7: Pairwise comparisons for the task TCGA
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+0
+40
+44
+0
+SCN
+-
+96
+96
+72
+rank-lr
++
+24
+0
+lr
++
+0
+rf
+-
++
+-
+-
+30
+
+Table 8: Pairwise comparisons for the task PCAWG
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+44
+40
+32
+20
+SCN
+52
+40
+0
+rank-lr
+36
+20
+lr
+16
+rf
+-
+-
+-
+-
+Table 9: Pairwise comparisons for the task met-500
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+24
+40
+36
+24
+SCN
+-
+72
+56
+12
+rank-lr
++
+24
+4
+lr
+-
++
+20
+rf
+-
+-
+-
+-
+Table 10: Pairwise comparisons for the task Baron-Murano
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+15
+62
+18
+25
+SCN
+-
+85
+62
+25
+rank-lr
++
+12
+25
+lr
+-
+-
+22
+rf
+-
+-
+-
+-
+Table 11: Pairwise comparisons for the task Baron-Segerstolpe
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+29
+49
+24
+0
+SCN
+62
+58
+7
+rank-lr
+27
+2
+lr
+0
+rf
+-
+-
+-
+-
+Table 12: Pairwise comparisons for the task MWS-TM10x
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+19
+49
+60
+7
+SCN
+-
+76
+84
+7
+rank-lr
++
+56
+7
+lr
++
+0
+rf
+-
+-
+-
+-
+31
+
+Table 13: Pairwise comparisons for the task MWS-TMfacs
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+14
+39
+32
+8
+SCN
+-
+84
+88
+7
+rank-lr
++
+33
+6
+lr
++
+5
+rf
+-
+-
+-
+-
+Table 14: Pairwise comparisons for the task TM10x-MWS
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+20
+47
+77
+16
+SCN
+-
+81
+86
+30
+rank-lr
+-
++
+70
+13
+lr
++
++
++
+14
+rf
+-
+-
+-
+-
+Table 15: Pairwise comparisons for the task TM10x-TMfacs
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+7
+53
+50
+4
+SCN
+-
+96
+89
+10
+rank-lr
++
+45
+1
+lr
+-
++
+-
+7
+rf
+-
+-
+-
+-
+Table 16: Pairwise comparisons for the task TMfacs-MWS
+optirank
+SCN
+rank-lr
+lr
+rf
+optirank
+18
+55
+78
+19
+SCN
+-
+80
+88
+29
+rank-lr
++
++
+71
+11
+lr
++
++
++
+12
+rf
+-
+-
+-
+-
+32
+
+E.5.4
+Pairwise comparisons for the classification tasks of section 6.2, in the single-source
+scenario
+Table 17: Pairwise comparisons for the task Baron-Segerstolpe-Murano
+optirank
+ANrank-lr
+SCN
+rank-lr
+lr
+rf
+optirank
+28
+12
+50
+15
+22
+ANrank-lr
+38
+78
+50
+25
+SCN
+-
+-
+90
+60
+25
+rank-lr
++
+2
+22
+lr
+-
+-
+-
+20
+rf
+-
+-
+-
+-
+-
+Table 18: Pairwise comparisons for the task MWS-TMfacs-TM10x
+optirank
+ANrank-lr
+SCN
+rank-lr
+lr
+rf
+optirank
+33
+20
+32
+47
+8
+ANrank-lr
+32
+63
+53
+8
+SCN
+-
+-
+73
+72
+8
+rank-lr
+-
++
+50
+8
+lr
++
+10
+rf
+-
+-
+-
+-
+-
+Table 19: Pairwise comparisons for the task TCGA-PCAWG-met500
+optirank
+ANrank-lr
+SCN
+rank-lr
+lr
+rf
+optirank
+52
+12
+40
+24
+12
+ANrank-lr
+4
+12
+8
+8
+SCN
+-
+-
+72
+68
+16
+rank-lr
++
+32
+12
+lr
+-
+-
++
+16
+rf
+-
+-
+-
+-
+-
+33
+
+E.5.5
+Pairwise comparisons for the classification tasks of section 6.2, in the multi-source
+scenario
+Table 20: Pairwise comparisons for the task Baron-Segerstolpe-Murano
+optirank
+ANrank-lr
+SCN
+rank-lr
+lr
+rf
+optirank
+28
+0
+45
+10
+10
+ANrank-lr
+5
+62
+38
+10
+SCN
+-
+-
+98
+78
+20
+rank-lr
++
++
+22
+12
+lr
+-
+-
++
+-
+12
+rf
+-
+-
+-
+-
+-
+Table 21: Pairwise comparisons for the task MWS-TMfacs-TM10x
+optirank
+ANrank-lr
+SCN
+rank-lr
+lr
+rf
+optirank
+43
+18
+55
+22
+5
+ANrank-lr
+15
+58
+27
+7
+SCN
+-
+-
+83
+85
+15
+rank-lr
++
+20
+3
+lr
+-
+-
++
+-
+8
+rf
+-
+-
+-
+-
+-
+Table 22: Pairwise comparisons for the task TCGA-PCAWG-met500
+optirank
+ANrank-lr
+SCN
+rank-lr
+lr
+rf
+optirank
+68
+36
+56
+80
+48
+ANrank-lr
++
+20
+36
+28
+12
+SCN
+-
+56
+56
+16
+rank-lr
++
+-
+56
+20
+lr
++
++
++
+20
+rf
+-
+-
+-
+-
+34
+
+E.5.6
+Comparisons in terms of sparsity
+In Section 6.1.4, we highlighted the fact that optirank, compared to rank-lr has the added beneficial
+potential to produce sparse solutions. In fact, by definition, rank-lr necessitates to know the value of all
+genes to perform the ranking.
+However, it was mentioned that on instances where lr performed well, there was no advantage in using
+optirank, as the solutions of lr are usually sparser than the ones produced by optirank. In the following,
+we support this claim.
+As a first investigation, we computed per dataset the average number of genes of the solution found
+by optirank and lr. Note that for optirank, the effective number of genes is the number of genes which
+are either in the reference set indicated by γγγ or whose corresponding coefficient wj is non-zero. Table 23
+summarizes the results and it is clear that lr produces sparser solutions than optirank.
+sor
+slr
+100 ˆP(sor ≤slr)
+100 ˆP(slr ≤sor)
+BRCA
+454
+43
+8
+92
+TCGA
+598
+588
+24
+76
+PCAWG
+598
+588
+24
+76
+met-500
+598
+588
+24
+76
+Baron-Murano
+462
+406
+57
+42
+Baron-Segerstolpe
+511
+451
+44
+67
+MWS-TM10x
+629
+365
+21
+79
+MWS-TMfacs
+666
+226
+9
+91
+TM10x-MWS
+514
+242
+27
+73
+TM10x-TMfacs
+616
+305
+36
+64
+TMfacs-MWS
+530
+204
+25
+75
+Table 23: Number of genes sor and slr in the solutions produced by optirank and lr (2 outermost left
+columns), averaged across folds and classes. The third column shows the percentage 100 ˆP(sor ≤ slr) of
+instances where the solution of optirank is sparser than the solution found by lr. The last column shows
+the percentage 100 ˆP(slr ≤sor) of instances with the opposite case, i.e where the solution of optirank is less
+sparse than the solution found by lr.
+In fact, in the task BRCA, optirank rarely produces a sparser solution, and on other tasks, apart from
+the task Baron-Murano and TMfacs-MWS, lr produces sparser solutions in a majority of cases. It should
+be clear however that in all cases where some robustness can be obtained by using a rank representation,
+the classical rank representation requires to measure all genes and has thus no sparsity whatsoever, while
+optirank attempts by construction to use a reduced set of genes, and so, even if the level of sparsity obtained
+is not comparable to that of an lr model the gain in performance might be worth it.
+E.5.7
+Comparisons in terms of sparsity in the multi-source scenario
+We carried the same analysis than in the previous section for the tasks in the multi-source scenario (see Section
+6.2). As in previous section, lr produces sparser solutions than optirank and ANrank-lr in a majority of
+cases. It is worth noting that solutions found by ANrank-lr and optirank require a similar number of genes,
+with optirank producing slightly sparser solutions.
+35
+
+sor
+sAr
+slr
+100 ˆP(sor ≤sothers)
+100 ˆP(sAr ≤sothers)
+100 ˆP(slr ≤sothers)
+TCGA-PCAWG-met500
+844
+849
+273
+20
+0
+80
+Baron-Segerstolpe-Murano
+606
+658
+364
+12
+12
+75
+MWS-TMfacs-TM10x
+604
+796
+300
+15
+2
+83
+Table 24: Number of genes sor, sAr, and slr in the solutions produced by optirank, ANrank-lr and lr (3
+outermost left columns), averaged across folds and classes. The last three columns shows the percentage of
+instances where the solution of the indicated classifier is sparser than the ones found by the other classifiers.
+E.5.8
+Supplementary table for single-source scenario
+ANrank-lr
+SCN
+lr
+optirank
+rank-lr
+rf
+TCGA-PCAWG-met500
+80 ± 3 (2)
+66 ± 4 (5)
+74 ± 4 (4)
+81 ± 4 (1)
+76 ± 3 (3)
+61 ± 4 (6)
+Baron-Segerstolpe-Murano 93 ± 1 (3) 89.4 ± 1.5 (5) 89.5 ± 2.4 (4) 93.7 ± 1.5 (2) 93.8 ± 1.7 (1) 62 ± 3 (6)
+MWS-TMfacs-TM10x
+82 ± 2 (4)
+72 ± 2 (5)
+83.3 ± 2.2 (2)
+84 ± 2 (1)
+83.0 ± 2.4 (3) 53 ± 1 (6)
+Table 25: Single-source scenario. Average balanced accuracies in % (across folds and classes) of competing
+classifiers on the tasks detailed in section 6.2 in the case in which the auxiliary source dataset is not used
+(except for ANrank-lr which uses it for ranking reference genes selection). The integer in parenthesis denotes
+the rank of the classifiers in terms of average balanced accuracy (lower is better). Classifiers which did not
+score significantly worse than the best classifier according to a paired Student’s t-test (with a 5% significance
+level) are highlighted in bold (see Appendix E.5 for additional details).
+E.5.9
+Runtime comparisons
+Table 26 shows the average runtime (in seconds) across validation splits and binary problems of competing
+classifiers with the hyperparameters set to their optimal values according to the one-standard-error-rule. We
+applied our comparison on the tasks detailed in 6.2, with classifiers trained in the single-source scenario. The
+results attest that the fitting time of optirank is reasonable, and on some tasks, it is even lower than the
+fitting time of its competitors using rank features such as ANrank-lr and rank-lr.
+ANrank-lr
+SCN
+lr
+optirank
+rank-lr
+rf
+TCGA-PCAWG-met500
+135 ± 38
+205.6 ± 0.8
+140 ± 47
+70 ± 4
+150 ± 46
+8.6 ± 0.5
+Baron-Segerstolpe-Murano
+10 ± 1
+50.8 ± 0.1
+2.0 ± 0.3
+7.7 ± 0.7
+3.6 ± 0.3
+0.75 ± 0.01
+MWS-TMfacs-TM10x
+8.8 ± 0.5
+33.82 ± 0.09
+2.5 ± 0.2
+16 ± 2
+36 ± 9
+0.92 ± 0.02
+Table 26: Runtime estimates. Average runtime (in seconds) across validation splits and binary problems of
+competing classifiers for the hyperparameter set which was selected as optimal according to the one-standard-
+error-rule. All classifiers were trained in the single-source scenario.
+F
+Computational resources
+The computational results in this work were produced with an internal computing cluster, in a batched
+fashion, amounting to a total of 2500 CPU hours. Per batch, the maximal capacity used was 30 GB of RAM
+with 2 CPUs.
+36
+

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+On the homogeneity of SnIa absolute magnitude in the Pantheon+ sample
+Leandros Perivolaropoulos1, ∗ and Foteini Skara1, †
+1Department of Physics, University of Ioannina, GR-45110, Ioannina, Greece
+(Dated: January 4, 2023)
+We test the homogeneity of the Pantheon+ sample with respect to the intrinsic absolute luminos-
+ity M = mBi − µi of the type Ia supernovae (SnIa) in Cepheid hosts and in the Hubble flow. Here,
+mBi is the corrected/standardized SnIa apparent magnitude and µi is the ith SnIa distance modulus
+obtained either from Cepheids (for SnIa in Cepheid hosts) or from the parametrized Hubble expan-
+sion rate H(z) (for the rest of the SnIa). When M is allowed to take a single value in the context of
+flat ΛCDM cosmological background H(z), we find the expected best fit values M = −19.25 ± 0.03,
+Ω0m = 0.33 ± 0.02, H0 = (73.4 ± 1) km s−1 Mpc−1 consistent with the original analysis of Brout et.
+al. When we introduce a new degree of freedom allowing M to take two values, one (M<) for nearby
+SnIa (distance di < dcrit, µi < 5 log10(dcrit/Mpc)+25) and one (M>) for more distant SnIa, we find
+a 2 − 3σ tension between the two best fit values of M> = −19.215 ± 0.03 and M< = −19.362 ± 0.05
+for dcrit ≃ 20Mpc. However, in contrast to the pure SH0ES data, this degree of freedom does not
+affect significantly the best fit values for the cosmological parameters H0 and Ω0m obtained from
+Pantheon+, for any value of dcrit, due to the dominant effects of the covariance matrix. When M is
+allowed to take distinct values Mi for each SnIa in Cepheid hosts we find using a KS test, that the
+Mi of nearby SnIa (di < 20Mpc) have less than 2.5% probability to have been drawn from the same
+probability distribution as the Mi of more distant SnIa (d > 20Mpc). These results constitute hints
+of inhomogeneities in the Pantheon+ sample which could be due to large statistical fluctuations,
+unaccounted systematic effects or new physics.
+I.
+INTRODUCTION
+The value of the Hubble constant H0 measured by di-
+rect local measurements based mainly on Type Ia super-
+novae (SnIa) standard candles calibrated using distance
+ladder methods is at 5σ tension with the corresponding
+value of H0 measured indirectly using the sound horizon
+at last scattering as a standard ruler. This discrepancy
+constitutes one of the main challenges for the standard
+cosmological model ΛCDM known as the Hubble tension.
+The most precise direct method for measuring the
+Hubble constant is based on observations of SnIa cali-
+brated with Cepheid variable stars in galaxies that host
+both Cepheid variable stars and SnIa. Cepheids in turn
+are calibrated using geometric methods (e.g. parallax) in
+the Milky Way and other nearby anchor galaxies. This is
+the distance ladder method for the direct measurement
+of H0.
+Such a distance ladder approach has been implemented
+recently by the SH0ES team (Supernovae and H0 for
+the Equation of State of dark energy) and has lead to
+a best fit value HR21
+0
+= 73.04 ± 1.04 km s−1 Mpc−1 [1].
+The corresponding indirect measurement of H0 using the
+sound horizon at recombination as a standard ruler mea-
+sured by the CMB perturbations angular power spec-
+trum under the assumption of the validity of the stan-
+dard cosmological model ΛCDM (inverse distance lad-
+der approach) has lead to an even more precise value of
+HP 18
+0
+= 67.36 ± 0.54 km s−1 Mpc−1 [2] (see also Refs.
+[3–12] for relevant recent reviews). The 5σ discrepancy
+∗ [email protected]
+† [email protected]
+(tension) between these two very precise measurements
+of H0 indicates that most probably at least one of them
+is not accurate because the assumptions on which it is
+based are not valid.
+The local direct measurement of SH0ES is consistent
+with a wide range of other less precise local measure-
+ments of H0 using alternative SnIa calibrators [13–16],
+gravitational lensing [17–20], gravitational waves [21–
+25], gamma-ray bursts as standardizable candles [26–30],
+quasars as distant standard candles [31], type II super-
+novae [32, 33], γ−ray attenuation [34] etc. (for recent
+reviews see Refs. [3, 5]).
+The SH0ES measurement relies on the following as-
+sumptions:
+• The measurements of the properties (period, metal-
+licity) and luminosities of Cepheid calibrators and
+SnIa are accurate and free of unaccounted system-
+atic errors.
+• The modeling and physical laws involved in the cal-
+ibration of Cepheids and SnIa in the three rungs of
+the distance ladder are accurate and well under-
+stood.
+A recent analysis by the authors [35] has indicated
+that a simple variation of the Cepheid/SnIa modeling
+in the SH0ES analysis introducing a single new degree
+of freedom can potentially modify the best fit value of
+H0 in such a way that it may become consistent with
+the corresponding inverse distance ladder measurement.
+This new degree of freedom allows for a transition of the
+SnIa calibrated and corrected intrinsic luminosity (ab-
+solute magnitude M) at some distance or redshift.
+It
+would therefore be interesting to introduce this new de-
+gree of freedom in the new extended Pantheon+ sample
+arXiv:2301.01024v1  [astro-ph.CO]  3 Jan 2023
+
+2
+[36–38] which includes many more SnIa than the local
+SH0ES Cepheid+SnIa sample, to investigate if this de-
+gree of freedom is excited by the data.
+The Pantheon+ SnIa luminosity sample [36–38] pro-
+vides distance moduli derived from 1701 light curves of
+1550 SnIa in a redshift range z ∈ [0.001, 2.26] compiled
+across 18 different surveys. This sample is significantly
+improved over the first Pantheon sample of 1048 SnIa,
+especially at low redshift (z).
+During the past few months when the Pantheon+ sam-
+ple has been publicly available, a wide range of studies
+have investigated various aspects of it. In particular, the
+following aspects of Pantheon+ have been investigated:
+its consistency with the cosmological principle [39, 40],
+the self-consistency level of its covariance [41], its consis-
+tency with standard electromagnetism and gravity [42],
+the constraints it can provide on modified gravity and
+generalized dark energy [36, 43–46], the constraints it
+can provide on early dark energy [47, 48], the constraints
+it can provide on the start of cosmic acceleration [49],
+the constraints on possible modification of physics at re-
+combination (e.g. electron mass variation) [50], the iden-
+tification of possible change of the best fit value of H0
+when different redshift bins are considered [51, 52], the
+effects of binning on its data [53, 54], its consistency with
+BAO+BBN data [55], the constraints it implies on gen-
+eralization of the Hubble law [56] etc.
+One novel feature of Pantheon+ is that it may be used
+to infer H0 in addition to cosmological parameters. This
+is due to the fact that it includes the distance moduli
+of SnIa in Cepheid hosts as obtained directly from the
+distance ladder analysis of the SH0ES. It also includes
+the covariance of these SnIa with the SnIa in the Hubble
+flow. The estimate of H0 was not possible in the first
+Pantheon sample because of the degeneracy between H0
+and SnIa absolute magnitude M. The inclusion of both
+the apparent magnitude mB and the distance modulus
+from Cepheids µCeph for SnIa in Cepheid hosts allows
+the independent determination of the absolute magnitude
+M = mB − µCeph which breaks the degeneracy between
+M and H0 thus allowing the independent determination
+of H0 through the Pantheon+ sample.
+Due to the new features and data included in the Pan-
+theon+ sample the following questions may be addressed:
+• Is the best fit value of the SnIa absolute magnitude
+M consistent among various subsamples of the Pan-
+theon+ sample?
+• What is the effect of the introduction of new de-
+grees of freedom (e.g. allowing for a change of M)
+on the quality of fit and on the best fit values of H0
+and cosmological parameters (e.g. matter density
+Ω0m)?
+The goal of the present analysis is to address these
+questions focusing on the possible inhomogeneities of
+the standardized/corrected intrinsic luminosity (absolute
+magnitude) of the SnIa of the Pantheon+ sample. Inves-
+tigations of possible inhomogeneities of other properties
+of the SnIa (e.g.
+color or stretch parameters [57]) are
+also interesting but are beyond the scope of the present
+study.
+The structure of this paper is the following: In the
+next section II we describe the data of the Pantheon+
+sample that are relevant for our analysis and describe
+the method used for the fit of the cosmological param-
+eters, the Hubble parameter H0 and the SnIa absolute
+magnitude M. We then implement this method and ob-
+tain the corresponding best fit parameter values for Ω0m,
+H0 and M in the context of a ΛCDM background thus
+confirming the results of the original analysis of Brout et.
+al. [36] and verifying our implementation of the method
+described there. In section III, we generalize the model
+and the fitting method by allowing for a transition of
+the SnIa intrinsic luminosity M at some distance dcrit
+from a value M< at distances d < dcrit to a value M>
+at distances d > dcrit. We find the best fit parameter
+values for Ω0m, H0 M< and M> with their uncertain-
+ties and test the consistency between the best fit values
+of M< and M>. In section IV we discuss the statisti-
+cal properties of the intrinsic luminosities Mi of SnIa in
+Cepheid hosts as obtained from the SnIa apparent mag-
+nitudes mBi and the Cepheid distance moduli µCeph
+i
+. We
+check in particular the consistency of the statistical prop-
+erties of the luminosities among different subsamples of
+the Pantheon+ sample. Finally in Section V we review
+our main results, discuss their implications and point out
+possible future extensions of our analysis.
+II.
+THE STANDARD ANALYSIS OF THE
+PANTHEON+ SAMPLE FOR ΛCDM
+The Pantheon+ sample is presented through a table
+(Pantheon+SH0ES.dat) with 1701 rows (plus a header)
+which includes the data relevant to 1701 SnIa light curves
+in 47 columns which are described at this url. It also con-
+sists of a 1701 × 1701 covariance matrix Cstat+syst which
+represents the covariance between SnIa due to systematic
+and statistical distance moluli uncertainties as described
+below.
+The relevant columns for our analysis are the
+following:
+• Column 3: Hubble Diagram Redshift (with CMB
+and peculiar velocity corrections).
+• Columns 9-10: mB corrected/standardized SnIa
+apparent magnitude and its uncertainty as ob-
+tained from the diagonal of the covariance matrix.
+• Columns
+11-12:
+µ
+=
+mB − MCeph
+cor-
+rected/standardized distance moduli where the ab-
+solute SnIa magnitude MCeph = −19.253 has been
+determined from SH0ES 2021 Cepheid host dis-
+tances.
+Its uncertainty as obtained from the di-
+agonal of the covariance matrix is included in col-
+umn 12. Column 11 is superfluous as it is trivially
+obtained from column 9 by subtracting MCeph.
+
+3
+• Column 13:
+µCeph corrected/standardized dis-
+tance moduli of the SnIa host as obtained from the
+SH0ES distance ladder analysis [1] in the context of
+the H0 distance ladder measurement. The uncer-
+tainty of µCeph is not included in this Table but it
+is incorporated in the covariance matrix. This col-
+umn has entries only in the rows which correspond
+to SnIa in Cepheid hosts. The rest of the rows have
+an entry ’-9’ in this column.
+• Column 14: Takes the value 1 if the SnIa of the
+row is in Cepheid host and 0 otherwise.
+In this section we follow [36] and use the above de-
+scribed Pantheon+ data to constrain the Hubble param-
+eter H0 = 100 h km s−1 Mpc−1, the SnIa absolute mag-
+nitude M and the matter density parameter Ω0m by min-
+imizing a χ2 likelihood:
+χ2 = ⃗QT · (Cstat+syst)−1 · ⃗Q,
+(2.1)
+where ⃗Q is a vector with dimension 1701 and components
+which are usually defined as
+Qi = mBi − M − µmodel(zi),
+(2.2)
+where mBi − M = µi is the distance molulus of the ith
+SnIa and µmodel(zi) is the corresponding distance modu-
+lus as predicted by the assumed background cosmologi-
+cal model parametrization which in the present analysis
+is assumed to be ΛCDM . Thus we have
+µmodel(zi) = 5 log(dL(zi)/Mpc) + 25,
+(2.3)
+where the luminosity distance dL(z) is
+dL(z) = (1 + z)c
+� z
+0
+dz′
+H(z′),
+(2.4)
+where c is the speed of light and in a ΛCDM background
+H(z) = H0
+�
+ΩM(1 + z)3 + ΩΛ.
+(2.5)
+The parameters M and H0 appear in Eqs. (2.1), (2.2)
+only through the combination M ≡ M−5log(H0·Mpc/c)
+and therefore they are degenerate and can not be es-
+timated separately. In order to break this degeneracy,
+M can be estimated separately using the distance lad-
+der approach by calibrating SnIa using Cepheids as was
+done with previous Pantheon sample. In the Pantheon+
+sample this degeneracy is broken within the analysis by
+modifying the definition of (2.2) to include the distance
+moduli of SnIa in Cepheid hosts which can constrain M
+independently. Thus the vector ⃗Q in the likelihood defi-
+nition (2.2) is modified as follows [36]
+Q′
+i =
+�
+mBi − M − µCeph
+i
+i ∈ Cepheid hosts
+mBi − M − µmodel(zi)
+otherwise,
+(2.6)
+where µCeph
+i
+is the distance modulus of the Cepheid host
+of the ith SnIa which is measured independently in the
+context of the distance ladder with Cepheid calibrators
+[1]. The novel feature of Pantheon+ is that the compo-
+nents Q′
+i that correspond to SnIa in Cepheid hosts are
+now fully incorporated in the sample and correlated with
+the rest of the SnIa though the provided covariance ma-
+trix. Thus, the degeneracy between M and H0 is broken
+and the three parameters M, H0 and Ω0m can be fit in
+the context of a ΛCDM background by minimizing
+χ′2(M, H0, Ω0m) = ⃗Q′T · (Cstat+syst)−1 · ⃗Q′,
+(2.7)
+where Cstat+syst denotes the covariance matrix provided
+with the Pantheon+ data including both statistical and
+systematic uncertainties. We have obtained the best fit
+parameter values for M, H0 and Ω0m and constructed the
+1σ−3σ likelihood contours by minimizing χ′2 of Eq. (2.7)
+using a simple Mathematica v12 code which is publicly
+available.
+The uncertainties for each one of the three best fit pa-
+rameters were obtained using the square roots of the diag-
+onal elements of the parameter covariance matrix which
+is the inverse of the Fisher matrix defined as
+Fij = 1
+2
+∂2χ′2(p1, p2, p3)
+∂pi∂pj
+,
+(2.8)
+where i, j = 1, 2, 3 and the parameters p1, p2, p3 corre-
+spond to M, h and Ω0m. We thus find
+M = − 19.25 ± 0.03,
+(2.9)
+h = 0.734 ± 0.01,
+(2.10)
+Ω0m = 0.333 ± 0.018,
+(2.11)
+which is in excellent agreement with the best fit values
+for h and Ω0m reported in [36]. The corresponding pa-
+rameter likelihood contours are the blue contours shown
+in Fig. 1.
+At the minimum we also find χ′2
+min = 1522.98 which
+corresponds to a χ′2 per degree of freedom of about 0.9.
+This is less than 1 and may indicate a possible overesti-
+mation of the uncertainties in the covariance matrix as
+pointed out recently in Ref. [41].
+III.
+GENERALIZED ANALYSIS ALLOWING
+TRANSITION OF SNIA LUMINOSITY.
+In order to test the homogeneity of the calibrated SnIa
+intrinsic luminosity, we now generalize the model of the
+previous section not by allowing more cosmological pa-
+rameters but by allowing a change of the absolute mag-
+nitude at a distance dcrit such that the SnIa absolute
+magnitude is of the form
+M =
+�
+M<
+d < dcrit
+M>
+d > dcrit,
+(3.1)
+
+4
+M>
+M<
+Mtot
+0.25
+0.30
+0.35
+0.40
+-19.5
+-19.4
+-19.3
+-19.2
+-19.1
+Ω0 m
+M
+M>
+M<
+Mtot
+0.70
+0.72
+0.74
+0.76
+0.78
+0.80
+h
+Figure 1. Blue contours: The 1−3σ likelihood contours for the parameters M, h and Ω0m in the context of a ΛCDM background.
+Red contours: The 1 − 3σ likelihood contours for the parameters M<, M>, h and Ω0m in the context of a ΛCDM background
+in a model that allows for a transition of the SnIa absolute magnitude from a value M< at distances d < 20Mpc to a value M>
+at distances d > 20Mpc.
+The magnitude transition critical distance dcrit may be
+associated with a critical distance modulus through the
+relation µcrit = 5log(dcrit/Mpc) + 25. By introducing
+this degree of freedom in χ′2 we obtain a generalized
+χ′′2(M<, M>, h, Ω0m) defined by using a vector ⃗Q′′ of
+the form
+Q′′
+i =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+mBi − M< − µCepheid
+i
+iff µCepheid
+i
+< µcrit, and i ∈ Cepheid hosts
+mBi − M> − µCepheid
+i
+iff µCepheid
+i
+> µcrit, and i ∈ Cepheid hosts
+mBi − M< − µmodel(zi)
+iff µCepheid
+i
+< µcrit, and i /∈ Cepheid hosts
+mBi − M> − µmodel(zi)
+iff µCepheid
+i
+> µcrit, and i /∈ Cepheid hosts,
+(3.2)
+in the expression (2.1) for χ2.
+Previous studies [35, 58–60] have found hints of in-
+homogeneities of astrophysical properties including the
+parameters of the Tully-Fisher relation at a distance of
+about 20Mpc.
+In Ref. [35] it was also pointed out that if the SH0ES
+data are reanalyzed by allowing for a change of the SnIa
+absolute magnitude at dcrit = 50Mpc then the best fit
+value of the Hubble parameter shifts to a value almost
+identical with the inverse distance ladder best fit value al-
+beit with significantly increased uncertainties. Motivated
+by these studies we first set dcrit = 50Mpc (µcrit = 31.5)
+and minimize the generalized χ′′2(M<, M>, h, Ω0m). We
+thus find the following best fit parameter values with the
+corresponding 1σ uncertainties
+M< = − 19.25 ± 0.03,
+(3.3)
+M< = − 19.23 ± 0.05,
+(3.4)
+h = 0.742 ± 0.02,
+(3.5)
+Ω0m = 0.332 ± 0.018,
+(3.6)
+with no change of the quality of fit since χ′′2
+min = 1522.61
+(∆χ′′2
+min = −0.3). Thus for dcrit = 50Mpc we find no
+hint of discrepancy between M< and M> and thus no
+inhomogeneity with respect to the SnIa intrinsic lumi-
+nosities. In addition no significant change is observed in
+the best fit value of h in contrast to the corresponding re-
+sult for the SH0ES data analysis where the same degree
+
+5
+d=20Mpc
+M=MCeph=-19.253
+29
+30
+31
+32
+33
+34
+35
+-20.5
+-20.0
+-19.5
+-19.0
+-18.5
+-18.0
+Distance Modulus μPanth+=mB-MCeph=mB+19.253
+M=mB-μCeph
+Figure 2. The absolute magnitudes of SnIa residing in Cepheid hosts. The best fit SnIa standardized and corrected absolute
+magnitude based on the SH0ES analysis is also shown (blue dashed line).
+of freedom induced a shift in the best fit value of h to
+h = 0.67±0.04. This may be due to the small number of
+SnIa in Cepheid hosts for the M> bin (4 SnIa) combined
+with the much larger number of SnIa in the Hubble flow
+bin for the Pantheon+ sample and the much more exten-
+sive covariance matrix. Similarly, for other values of dcrit
+no significant change is found for the best fit values of the
+parameters h and Ω0m. However, for dcrit ≃ 20Mpc we
+find a mild discrepancy between the best fit values of M<
+and M>.
+In particular, for dcrit = 20Mpc we find the follow-
+ing best fit parameter values with the corresponding 1σ
+uncertainties
+M< = − 19.362 ± 0.05,
+(3.7)
+M< = − 19.215 ± 0.03,
+(3.8)
+h =
+0.745 ± 0.01,
+(3.9)
+Ω0m = 0.331 ± 0.018,
+(3.10)
+with a significant improvement of the quality of fit since
+χ′′
+min
+2 = 1513.3 (∆χ′′
+min
+2 = −9.7) corresponding to re-
+duction of the Akaike Information Criterion (AIC) by
+∆AIC = −7.7 (for a single additional parameter). The
+corresponding likelihood contours for these parameters
+are shown in Fig. 2 (red contours for M< and M>).
+Thus for dcrit = 20Mpc we find hints of discrepancy
+between M< and M> at a 2 − 3σ level and thus inhomo-
+geneity with respect to the SnIa intrinsic luminosities.
+However, no significant change is observed in the best fit
+value of h. This was also the case for the SH0ES data
+analysis for the same value of dcrit, where the same de-
+gree of freedom induced no significant shift in the best fit
+value of h [35] even though hints of discrepancy between
+M< and M> were observed at similar dcrit albeit at lower
+statistical significance (see e.g. Figs. 8, 9 of Ref. [35]).
+IV.
+STATISTICAL PROPERTIES OF SNIA
+INTRINSIC LUMINOSITIES.
+In this section we further investigate the hints for in-
+homogeneity derived in the previous section from the full
+Pantheon+ sample. We thus focus on the particular sub-
+set of the Pantheon+ sample that corresponds to SnIa in
+Cepheid hosts and investigate the statistical properties of
+their individual absolute magnitudes. The measured ab-
+solute magnitude Mi of individual SnIa in Cepheid hosts
+can be directly obtained from the Pantheon+ data as
+Mi = mBi − µCeph
+i
+(4.1)
+
+6
+-19.6
+-19.4
+-19.2
+-19.0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+SnIa Absolute Magnitude
+Probability
+d<20Mpc
+d>20Mpc
+All
+Figure 3. The probability distribution histogram of the ab-
+solute magnitudes each SnIa subsample (Nearby subsam-
+ple M <
+s : dark blue columns, Distant subsample M >
+s : red
+columns, Full sample of SnIa in Cepheid hosts:
+green
+columns)
+i.e. by subtracting column 13 from column 9 for those
+entries where column 14 is 1 1. In Fig. 2 we show a plot
+of the measured Mi for SnIa in Cepheid hosts vs the SnIa
+distance moduli µP anth+,i = mBi −MCeph (column 9) as
+obtained from the measured apparent magnitudes and
+the best fit value of M from SH0ES (MCeph = −19.253,
+blue dashed line). As shown in Fig. 2, SnIa at distances
+d < 20Mpc appear to be systematically more luminus
+(lower Mi) than the rest of the SnIa since almost all are
+below the blue dashed line corresponding to MCeph2.
+In order to compare the statistical properties of the Mi
+subsample with d < 20Mpc (M <
+s ) with the correspond-
+ing distant subsample with d > 20Mpc (M >
+s ) we show
+in Fig. 3 the probability distribution histogram of each
+subsample (M <
+s : dark blue columns, M >
+s : red columns,
+Full sample: green columns). As expected from Fig. 2
+the probability distributions for the two subsamples differ
+significantly with M <
+s being significantly skewed towards
+lower M values (brighter SnIa).
+The statistical difference between the two subsam-
+ples may be quantified using a Kolmogorov-Smirnov test.
+Based on this test, the null hypothesis that the two sub-
+samples have been drawn from the same probability dis-
+tribution is rejected at the 2.5% level. In fact the prob-
+ability that the two subsamples have been drawn from
+the same probability distribution is 2.2%.
+This result
+further amplifies the evidence for inhomogeneities in the
+SnIa corrected intrinsic luminosities of the Pantheon+
+sample.
+This inhomogeneity could be due to either a
+large statistical fluctuation in the context of the stan-
+dard model, or to an unaccounted systematic effect or to
+1 The uncertainty of each Mi is obtained from the corresponding
+entries of columns 10 and 13 only for plotting purposes as it will
+not be used in the statistical analysis of this section.
+2 A similar trend appears for the four furthest SnIa with d >
+50Mpc even though this is much less significant statistically. This
+is the origin of the effects observed in [35].
+a physics transition that has occurred at a distance of
+about 20Mpc (about 70Myrs ago) [61, 62].
+V.
+CONCLUSION-DISCUSSION
+We have tested the internal consistency of the Pan-
+theon+ sample with respect to the SnIa standardized
+and corrected intrinsic luminosity. We have allowed for a
+change of the SnIa absolute magnitude at dcrit = 20Mpc
+from M< at low distances (late times) to M> at high
+distances (early times).
+We found that such a change
+is favored by the Pantheon+ data leading to an reduc-
+tion of χ2 by ∆χ2
+min = −9.7. This corresponds to re-
+duction of the Akaike Information Criterion (AIC) by
+∆AIC = −7.7 (for a single additional parameter). Such
+a reduction provides strong evidence that the model
+where a change of M is preferred by the Pantheon+
+data over the baseline model with a single value for M.
+This conclusion is further amplified by the fact that the
+best fit value of M< is in discrepancy with the best fit
+value of M> at a level of more than 2σ. In addition, a
+Kolmogorov-Smirnov test has indicated that the proba-
+bility that the absolute magnitudes Mi of SnIa in Cepheid
+hosts at distances d < 20Mpc are drawn from the same
+distribution as the Mi of SnIa at hosts with d > 20Mpc
+is less than 2.5%. These results constitute evidence for a
+possible inhomogeneity of the SnIa intrinsic luminosities
+of Pantheon+ sample with probability more than 95%.
+However, even if this inhomogeneity manifests itself by
+the introduction of a new degree of freedom in the anal-
+ysis (replacing M by M< and M>), the best fit values of
+the cosmological parameters Ω0m and H0 are not signif-
+icantly affected. Notice however, that the best fit value
+M< = −19.362 ± 0.05 of low the distance SnIa absolute
+magnitude is fully consistent with the inverse distance
+ladder best fit value M = −19.4 ± 0.027 [61].
+The
+demonstrated
+presence
+of
+possible
+inhomo-
+geneities in the Pantheon+ sample may have implica-
+tions for other calibration parameters like the color and
+stretch parameters. It would therefore be of interest to
+extend the present analysis in such directions by testing
+the homogeneity of the Pantheon+ sample with respect
+to possible differences of the best fit values of such pa-
+rameters when these are allowed to change among dif-
+ferent subsamples of the full Pantheon+ sample. Hints
+for such inhomogeneities from the first Pantheon sam-
+ple have been already reported [57]. The corresponding
+effect on the best fit values of cosmological parameters
+when such new degrees of freedom are allowed would also
+be an interesting extension of the present analysis.
+NUMERICAL ANALYSIS FILES
+The numerical files for the reproduction of the figures
+can be found this Github repository under the MIT
+
+7
+license.
+ACKNOWLEDGMENTS
+This project was supported by the Hellenic Foundation
+for Research and Innovation (H.F.R.I.), under the "First
+call for H.F.R.I. Research Projects to support Faculty
+members and Researchers and the procurement of
+high-cost research equipment Grant" (Project Number:
+789).
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+

BtE0T4oBgHgl3EQfyAJM/content/tmp_files/2301.02653v1.pdf.txt

@@ -0,0 +1,2025 @@
+Single electron-spin-resonance detection by microwave photon counting
+Z. Wang1,2,∗, L. Balembois1,∗, M. Rancic1, E. Billaud1, M. Le Dantec1, A. Ferrier3, P.
+Goldner3, S. Bertaina4, T. Chaneliere5, D. Esteve1, D. Vion1, P. Bertet1, E. Flurin1,†
+1Quantronics group, Université Paris-Saclay, CEA,
+CNRS, SPEC, 91191 Gif-sur-Yvette Cedex, France
+2Département de Physique et Institut Quantique,
+Université de Sherbrooke, Sherbrooke, Québec, Canada
+3Chimie ParisTech, PSL University, CNRS,
+Institut de Recherche de Chimie Paris, 75005 Paris, France
+4CNRS, Aix-Marseille Université, IM2NP (UMR 7334),
+Institut Matériaux Microélectronique et Nanosciences de Provence, Marseille, France
+5Univ.
+Grenoble Alpes, CNRS, Grenoble INP,
+Institut Néel, 38000 Grenoble, France
+(Dated: January 9, 2023)
+Electron spin resonance (ESR) spectroscopy is
+the method of choice for characterizing para-
+magnetic
+impurities,
+with
+applications
+rang-
+ing from chemistry to quantum computing [1],
+but it gives access only to ensemble-averaged
+quantities due to its limited signal-to-noise ra-
+tio.
+Single-electron-spin sensitivity has how-
+ever been reached using spin-dependent photo-
+luminescence [2–4], transport measurements [5–
+8], and scanning-probe techniques [9–11]. These
+methods are system-specific or sensitive only in
+a small detection volume, so that practical single
+spin detection remains an open challenge. Here,
+we demonstrate single electron magnetic reso-
+nance by spin fluorescence detection [12], using
+a microwave photon counter at cryogenic tem-
+peratures [13].
+We detect individual paramag-
+netic erbium ions in a scheelite crystal coupled
+to a high-quality factor planar superconducting
+resonator to enhance their radiative decay rate,
+with a signal-to-noise ratio of 1.9 in one second
+integration time.
+The fluorescence signal shows
+anti-bunching, proving that it comes from indi-
+vidual emitters. Coherence times up to 3 ms are
+measured, limited by the spin radiative lifetime.
+The method has the potential to apply to ar-
+bitrary paramagnetic species with long enough
+non-radiative relaxation time, and allows single-
+spin detection in a volume as large as the res-
+onator magnetic mode volume (∼ 10µm3 in the
+present experiment), orders of magnitude larger
+than other single-spin detection techniques.
+As
+such, it may find applications in magnetic reso-
+nance and quantum computing.
+∗these authors contributed equally
+†corresponding author: emmanuel.fl[email protected]
+In ESR spectroscopy, the linewidth of an ensemble of
+paramagnetic centers is usually dominated by the fre-
+quency shifts that each center undergoes under the action
+of its local environment. This inhomogeneous broadening
+can reach large values (up to several GHz) and imposes a
+limitation to the achievable spectral resolution [1]. One
+radical way to overcome the inhomogeneous broadening
+is to perform ESR spectroscopy on individual paramag-
+netic centers, thus gaining several orders of magnitude in
+spectral resolution since single spin linewidths are typ-
+ically in the kHz-MHz range [3, 7, 14]. Besides the in-
+terest for magnetic resonance spectroscopy, single spin
+addressing is also a necessity for most spin-based quan-
+tum computing applications.
+Practical single-electron-spin-resonance should enable
+the detection and spectroscopy of a wide range of para-
+magnetic centers buried in an insulating matrix, with
+a sufficiently large detection volume and signal-to-noise
+ratio. So far, none of the approaches that achieve single-
+spin detection satisfy all of these requirements. Optically
+Detected Magnetic Resonance (ODMR) can detect indi-
+vidual paramagnetic centers only when suitable energy
+levels and cycling optical transitions are present [2–4].
+ODMR-detected individual NV centers can be used to
+measure the spectrum of neighboring single electron spins
+in ambient conditions [11, 15, 16], but the detection vol-
+ume is limited to ∼ 103−104nm3 by the 1/r3 dependence
+of the dipolar interaction, which makes the detection of
+spins far outside of the diamond host challenging. Spin-
+dependent transport can detect individual spins when a
+spin-to-charge conversion pathway is present [5–8, 10],
+but this is lacking in most paramagnetic centers. Single
+electron-spin imaging was also achieved using Magnetic
+Resonance Force Microscopy [9], but spectroscopy has
+not yet been demonstrated with this platform.
+Here, we perform single electron spin resonance spec-
+troscopy by transposing fluorescence detection, a well-
+established method to detect individual emitters in the
+arXiv:2301.02653v1  [quant-ph]  6 Jan 2023
+
+2
+Figure 1.
+Principle of single spin spectroscopy by
+microwave photon counting. An individual electron spin
+(red arrow) embedded in a crystal is excited by a microwave
+pulse (in black); it then relaxes back to its ground state by
+emitting a microwave photon (green arrow), which is routed
+via a circulator towards a microwave photon counter based
+on a superconducting transmon qubit. To enhance its radia-
+tive rate ΓR, the spin is coupled magnetically to the mode of
+a high-quality-factor superconducting planar microwave LC
+resonator (in orange). The spin frequency is tuned to the res-
+onator by application of a magnetic field B0 parallel to the
+resonator plane.
+optical domain at room-temperature, to microwave fre-
+quencies and millikelvin temperatures. In optical fluo-
+rescence, an emitter is excited by a short light pulse,
+and detected by counting the emitted photons during
+the radiative relaxation [3, 17].
+Similarly, we excite a
+spin by a short microwave pulse, and detect it by count-
+ing the microwave photons it emits when returning to its
+ground state. Spin relaxation by spontaneous emission
+of microwave photons is exceedingly slow in free space;
+we thus enhance its rate ΓR resonantly by coupling the
+spin to a high-quality-factor superconducting microwave
+resonator of frequency ω0 [18], and we detect the fluo-
+rescence photon with a single-microwave-photon detector
+(SMPD) based on a superconducting qubit (see Fig. 1 for
+a schematic description). The maximum signal-to-noise
+ratio (SNR) reached by this method with a one second in-
+tegration time scales as ∼ ηΓR/
+�
+α + η(1 − η)ΓR, where
+0 ≤ η ≤ 1 is the average number of counts generated
+by the radiative decay of one spin, and α the SMPD
+dark count rate (see Methods). It is noteworthy that this
+SNR is only limited by technical imperfections and has
+no upper bound for an ideal experiment where α = 0 and
+η = 1, in contrast with earlier proposals and experiments
+of circuit-QED-enhanced magnetic resonance where the
+SNR is ultimately limited by vacuum microwave fluctu-
+ations
+[19–25]. In a recent experiment demonstrating
+the detection of ∼ 104 impurity spins by microwave flu-
+orescence, this single-spin SNR was ∼ 5 · 10−4 [12], thus
+insufficient for single-spin detection.
+Here, we reach a
+single-spin SNR of ∼ 1 by improving the resonator de-
+sign, the SMPD performance, and by using spins with
+a larger gyromagnetic ratio. Our method could be ap-
+plicable to a broad class of paramagnetic impurities and
+offers a detection volume that can be large (∼ 10 µm3
+in the present experiment). It is therefore promising for
+operational single electron spin resonance at cryogenic
+temperatures.
+We demonstrate this method with rare-earth ions in
+a crystal, specifically Er3+ ions in a scheelite crystal of
+CaWO4, which has tetragonal symmetry around its c-
+axis. The crystal used in the experiment was grown un-
+doped, but has a residual erbium concentration 3.1 ± 0.2
+ppb (see Methods), which corresponds to a ∼ 300 nm av-
+erage distance between neighboring Er3+ ions. At low
+temperatures, only the ground state Kramers doublet
+of Er3+ : CaWO4 is populated; it behaves as an effec-
+tive spin S = 1/2 with frequency ωs = γ · B0, where
+B0 is the applied magnetic field and γ the ion gyromag-
+netic tensor. The ensemble-averaged gyromagnetic ten-
+sor γ0 determines the center of the ensemble resonance
+line ωs0; it is diagonal in the (a, b, c) tetragonal frame,
+with elements γa = γb ≡ γ⊥ = 2π × 117.3 GHz/T, and
+γc ≡ γ|| = 2π×17.45 GHz/T [26]. Due to inhomogeneous
+broadening however, each individual ion has a gyromag-
+netic tensor γ = γ0 + δγ (with |δγ| ≪ |γ0|) that slightly
+deviates from γ0 [27].
+The planar resonator is patterned on top of the crystal,
+out of a superconducting niobium thin-film. The heart
+of the device is a 600 nm-wide, 100µm-long wire, which
+acts as a lumped inductance, shunted by a finger capac-
+itor (see Fig. 1 and Methods) that sets the resonance
+frequency ω0/2π = 7.335 GHz.
+The wire (z direction)
+is oriented approximately along the crystal c-axis, and
+the magnetic field B0 is applied along the sample surface
+(z − y plane), at a small adjustable angle θ with respect
+to z (see Methods). The resonator is coupled to a trans-
+mission line for exciting the spins and collecting their
+fluorescence, at a rate κc, whereas the total resonator
+damping rate κ = κc + κi also includes internal losses κi.
+A circulator routes the excitation pulses from the input
+line towards the sample, and the reflected pulses together
+with the subsequent spin fluorescence signal towards the
+input of a transmon-qubit-based SMPD. This detector is
+similar to the ones described in [12, 13], but has a much
+lower dark count rate α = 102 s−1 (see Methods).
+By coupling to the resonator, a spin at frequency ωs
+and position r demonstrates a Purcell-enhanced radia-
+tive relaxation rate ΓR = κg2
+0/(δ2 + κ2/4) that depends
+on its detuning to the resonator δ ≡ ωs − ω0 and on
+
+click3
+414
+416
+418
+420
+422
+424
+426
+0.2
+0.4
+0.6
+30
+60
+90
+-20
+-10
+0
+7.334
+7.336
+〈C〉(counts)
+ 
+B0 (mT)
+0.0
+0.5
+1.00
+100
+200
+ time td (ms)
+〈C〉(counts / ms)
+0
+2
+4
+6
+8
+0.0
+0.1
+0.2
+ time td (ms)
+〈C〉(counts / ms)
+ 
+〈C〉(counts)
+ frequency (GHz)
+ 
+ reflection (dB)
+s0
+s1
+s2 s3 s4
+s5
+s6
+.
+c
+f
+b
+e
+a
+d
+.
+-600
+-300
+0
+x (nm)
+y  (nm)
+0
+-300
+-600
+600
+300
+�R= 125 s-1
+250
+500
+1000
+6
+5
+4
+3
+2
+1
+g0/2� (kHz)
+td 
+Figure 2.
+Spin spectroscopy. (a) Simulation of the spin-resonator coupling constant g0(x, y) and relaxation rate ΓR(x, y)
+as a function of the spin position (x, y) with respect to the wire (shown as a green rectangle). (b) Magnitude of resonator
+reflection (green dots) as a function of probe frequency at single-photon level input power. A fit yields the total resonator
+linewidth κ/2π = 470 kHz, with a coupling rate κc = 1.7 · 106 s−1 and an internal loss rate κi = 1.3 · 106 s−1 (c-d) Microwave
+fluorescence spectroscopy (c) at high excitation power (∼ −97 dBm at sample input) and typical fluorescence signal (d). At
+each magnetic field B0, the average number of counts ⟨C⟩ is integrated over a ∼ 200 ms duration following the excitation pulse
+(light blue window in panel d). Blue open circles are data, red line is a Lorentzian fit with FWHM 0.45 mT. Note that the
+angle θ varies linearly between −0.006◦ and 0.006◦ over the scan. Fluorescence histograms are shown at the center (red) and
+tail (grey) of the spin ensemble line (see stars in panel c). (e) Spin spectroscopy at low power (∼ −107 dBm at sample input),
+with an integration window of 2 ms. Blue line is measured data, red line is a Lorentzian fit. The inset shows an expanded view
+of 7 peaks (labelled s0 to s6). Note that the angle θ varies linearly between −0.016◦ and 0.016◦ over the scan. (f) Fluorescence
+histograms of spin s0 (red) and background (grey) averaged over the range of B0 shown in the inset of panel e. The light blue
+window is the integration window for the data in e).
+the magnetic field vacuum fluctuations δB1(r) through
+the spin-resonator coupling strength g0(r) = ⟨↓ |S| ↑
+⟩ · γ · δB1(r) [18]. To a good approximation, δB1(r) does
+not depend on the position z along the wire and is orthog-
+onal to the latter, so that the coupling strength can be
+re-written as g0(x, y) = (1/2)γ⊥|δB1(x, y)|. The g0(x, y)
+map is shown in Fig. 2a for our resonator design, and
+shows that g0/2π is larger than 3 kHz, and thus ΓR is
+larger than ∼ 500s−1, for spins located below the wire at
+a depth smaller than ∼ 150 nm, corresponding to a vol-
+ume of ∼ 10 µm3. This implies that ΓR/√α ≥ 50/
+√
+Hz,
+and therefore suggests that single-spin sensitivity may be
+reached over this whole volume.
+The properties of the fluorescence signal, which is the
+sum of the contributions of all the spins excited by the
+pulse, strongly depend on the excitation power. We first
+record the spectrum of the Er3+ : CaWO4 resonance with
+a high input power (∼ −97 dBm), thus exciting many
+weakly coupled ions that have low ΓR. The average count
+rate as a function of time following the pulse shows an
+excess compared to the dark count level (see Fig. 2d) and
+decays non-exponentially over a time scale of ∼ 100 ms.
+We plot the average number of counts integrated over
+200 ms ⟨C⟩ as a function of magnetic field B0 applied
+along the z direction in Fig. 2c.
+A smooth, approxi-
+mately Lorentzian, peak is observed at B0 = 419.5 mT,
+close to ω0/γ||, proving it is the Er3+ spin resonance.
+Its inhomogeneous Full-Width-Half-Maximum linewidth
+0.5 mT corresponds to a ∼ 8 MHz-wide distribution.
+We then record the line with ∼ 20 dB lower excita-
+tion power while simultaneously reducing the integra-
+tion time to 2 ms, thus exciting and detecting only the
+most strongly coupled and fastest relaxing spins.
+The
+integrated count ⟨C⟩(B0) now shows qualitatively differ-
+ent behavior and appears as a sum of narrow, unevenly
+distributed peaks, with typical amplitude ∼ 0.1 excess
+count over the noise floor (see Fig.2e). The fluorescence
+curve when tuned to one of these peaks shows an ex-
+ponential decay (see Fig. 2f), with a time constant of
+∼ 2 ms.
+These features suggest that each peak corre-
+sponds to the microwave fluorescence signal originating
+from a single Er3+ ion spin; analogous to the optical fluo-
+rescence spectrum of a collection of individual solid-state
+emitters [17, 28, 29]. Note that while we observe a large
+fluorescence signal at the centre of the inhomogeneous
+absorption line, some individual peaks are still found far
+from the centre; a common observation in low-density
+spectra of optical emitters, and a natural consequence of
+
+4
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+θ (deg)
+421
+422
+B0 (mT)
+0.0
+0.1
+〈C〉(counts)
+s0
+s1
+s2
+s3
+s4
+s5
+s6
+�
+Figure 3.
+Single-spin-resolved rotation pattern Average
+number of excess count ⟨ �C⟩ as a function of the magnetic field
+amplitude B0 and its angle θ with respect to the projection
+of the crystal c axis on the sample surface. The range is the
+same as in the inset of Fig. 2, and the same labeling of the spin
+lines is used. The data acquisition time was approximately
+one week.
+the random nature of inhomogeneous broadening. This
+is also possibly supplemented in our particular device by
+the strain imparted by the thermal contractions of the
+metallic wire on the substrate just below [30, 31].
+To demonstrate the stability and reproducibility of the
+peaks, we perform a two-dimensional magnetic field scan
+by recording a background-corrected average number of
+counts (see Methods), named ⟨ ˜C⟩ hereafter, as a func-
+tion of B0 and θ (see Fig. 3). Eight different spin peaks
+are resolved, and their spectrum is readily followed in
+magnetic field. It appears that each ion has its own gy-
+romagnetic tensor γ, close to γ0 but with different values
+for the principal axes and also a symmetry axis that can
+slightly deviate from the c-axis, vividly illustrating the
+concept of inhomogeneous broadening. The lines are so
+narrow that each ion γ could, in principle, be determined
+to better than 10−6 accuracy (using a suitably calibrated
+magnetic field). Because the deviation δγ of the gyro-
+magnetic tensor from the ensemble-averaged γ0 is due to
+the local electrostatic and strain environment, its accu-
+rate measurement can also be turned into a sensitive way
+to probe it (as done with NV centers in diamond in par-
+ticular [32]). Note that our measurements also call for a
+better modeling of the response of rare-earth ion spins to
+applied electric or strain fields.
+We now select one of the peaks (s0) and bring fur-
+ther proof of its single-spin nature. We first measure the
+excess counts ⟨ ˜C⟩ as a function of the microwave pulse
+duration, and observe sinusoidal oscillations with a fre-
+0
+5
+10
+Pulse duration  (µs)
+T
+0.0
+0.1
+0.2
+〈C〉(counts)
+a
+0
+1
+A (arb. unit)
+0.0
+0.2
+0.4
+0.6
+Rabi frequency (MHz)
+b
+−20
+−10
+0
+10
+20
+Offset  (# excitation pulses)
+k
+0.0
+0.5
+1.0
+Corrected g(2)
+c
+0
+50
+C (counts)
+0.00
+0.02
+0.04
+0.06
+p C
+( )
+d
+0
+20
+40
+measurement time 
+ (s)
+tm
+0
+5
+10
+SNR
+e
+no
+pulse
+� pulse
+tm = 1 s
+A
+T
+~
+Figure 4.
+Characterization of spin s0. (a) Rabi oscil-
+lation: measured average excess count ⟨ �C⟩ (red dots) as a
+function of excitation pulse duration T (see inset), and cor-
+responding fit (solid line) by a sine function with linearly in-
+creasing offset. (b) Extracted Rabi frequency (magenta dots)
+as a function of excitation pulse amplitude A, and correspond-
+ing linear fit through origin (solid line).
+(c) Background-
+corrected auto-correlation function g(2) (blue columns) and
+corresponding ±1-standard deviation error bars (red) mea-
+sured as a function of the offset k between excitation pulses.
+(d) Measured probability distribution p(C) of the total count
+C integrated over the first 2 ms of 7.5 ms-long sequences, ei-
+ther with no excitation pulse applied (grey) or with a π ex-
+citation pulse (red). Sequences are repeated and counts are
+summed during a measurement time tm = 1 s. Solid lines are
+Poissonian fits, yielding the spin signal Cspin = 12.4 (differ-
+ence between the mean values of the two distributions) and
+the standard deviations δC0 = 5.5 and δCπ = 6.5. (e) Mea-
+sured signal-to noise ratio Cspin/δCπ (magenta dots) as a
+function of the measurement time tm, and fit with the func-
+tion A√tm (solid line). Data taken at B0 = 421.042 mT and
+θ = −0.024◦.
+quency that depends linearly on the pulse amplitude (see
+Fig. 4a and b), as expected for the Rabi oscillation of a
+single spin. Superposed on these oscillations is a grad-
+ual increase in counts, which we attribute to heating of
+the bath of defects that are responsible for the resonator
+internal losses upon microwave excitation, as observed
+
+5
+in [12] (see Methods). We then use the SMPD to measure
+the photon statistics of the fluorescence signal and reveal
+its single emitter nature. For this task, we acquire a large
+number of fluorescence traces following a π pulse, label
+them by index j, and compute the dark-count-corrected
+intensity-intensity correlation function g(2)(k) between
+traces whose index differs by k (see Methods). For N
+emitters, g(2)(0) should be equal to (N − 1)/N; in par-
+ticular, g(2)(0) should be equal to 0 for a single-emitter
+since it can emit only one photon in each sequence. We
+measure g(2)(0) = 0.23 ± 0.06, and g(2)(k) = 1 ± 0.04
+for k ̸= 0 (see Fig. 4c), thus showing clear anti-bunching
+in each sequence, whereas the emission from different se-
+quences is uncorrelated. The non-zero value of g(2)(0)
+may be due to heating; in any case, the fact that its
+value is well below 0.5 further suggests that the spectral
+peak under study is a single microwave photon emitter,
+in the form of an individual Er3+ electron-spin.
+We use the same dataset to quantify the single-spin
+SNR for a certain measurement time tm. For that, we
+sum the counts obtained over sequences played during a
+tm time window, integrated over the first 2 ms following
+the excitation pulse, yielding the number of counts C.
+Figure 4d shows the count probability histogram p(C)
+for tm = 1 s, with and without π pulses applied. Both his-
+tograms are well reproduced by a Poissonian distribution
+(see Fig. 4d). The single-spin SNR defined as Cspin/δCπ
+has a value of 1.91. Here, Cspin is the difference between
+the mean number of counts and δCπ the half-width of the
+distribution with π pulse applied. A comparison with the
+expected SNR requires measuring the overall efficiency η,
+which we find to be equal to η = 0.12±0.01 by integrating
+the fluorescence signal after the π pulse with subtracted
+background. This value of η results from the SMPD finite
+efficiency, resonator internal losses, and microwave losses
+in-between the spin resonator and the SMPD (see Meth-
+ods). We deduce an optimal theoretical SNR of ∼ 2.5
+(see Methods), close to our measured value.
+We also
+verify that the SNR scales as the square root of the mea-
+surement time tm up to at least 1 minute (see Fig. 4e),
+indicative of good measurement stability.
+The ability to address individual spins with microwaves
+opens the way to using them as spin qubits for quan-
+tum computing, and it is thus interesting to characterize
+their coherence properties. The longitudinal relaxation
+time T1 is obtained simply from the fluorescence curve
+decay; we select a spin (s6) with T1 = 1.46 ± 0.05 ms
+(see Fig. 5a) at resonance (δ = 0).
+We then measure
+the free-induction-decay time using a Ramsey sequence
+π/2X − τ − π/2Φ, with the relative inter-pulse phase
+Φ = 2π∆τ, where ∆ = 0.025 MHz.
+The excess count
+⟨ ˜C⟩ shows oscillations at frequency ∆ + δ, damped with
+an approximately Gaussian shape and a characteristic
+relaxation time T ∗
+2 = 170 ± 33 µs, corresponding to a
+∼ 2 kHz single-spin linewidth (see Fig. 5c). We use the
+Ramsey sequence to accurately measure δ, making it pos-
+0
+2
+4
+6
+0.12
+0.14
+0.16
+0.18
+0.0
+0.1
+0.2
+0.0
+0.1
+0.2
+0
+1
+2
+3
+4
+0.0
+0.1
+0
+1
+2
+3
+4
+5
+6
+0.0
+0.1
+-500
+0
+500
+0
+1
+2
+3
+4
+5
+6
+〈C〉(counts / ms)
+.
+.
+td (ms)
+〈C〉(counts)
+� (ms)
+〈C〉(counts)
+2 � (ms)
+〈C〉(counts)
+4 � (ms)
+T1 (ms)
+� / 2� (kHz)
+.
+T2 
+PDD = 2.99 ± 0.33 ms 
+T2 = 2.47 ± 0.31 ms 
+T2* = 0.17 ± 0.03 ms 
+a
+b
+T1=1.42±0.07 ms
+c
+d
+e
+�
+td
+�
+2X
+�
+�
+2����
+�
+�
+2X
+�
+2����
+�X
+�
+�
+2X
+�
+2����
+�Y �Y �Y
+�
+�
+�
+Figure 5.
+Coherence time measurements of spin
+s6.
+(a) Energy relaxation:
+measured average count rate
+⟨ ˙C⟩ (blue dots) as a function of delay td after a resonant
+π excitation pulse. Exponential fit (solid orange line) yields
+the energy relaxation time T1. (b) Purcell effect: measured
+T1 as a function of spin-resonator frequency detuning δ (or-
+ange dots). A fit to Γ−1
+R (δ) (solid black line) yields the spin-
+resonator coupling constant g0/2π = 3.54 ± 0.15 kHz.
+(c)
+Ramsey sequence (see inset):
+measured excess counts ⟨ �C⟩
+versus delay time τ between two resonant π/2 pulses with
+relative phase ϕ(τ) = 2π∆τ and ∆ = 0.025 MHz (dots).
+The corresponding fit (solid line) by a sine function with a
+Gaussian-decaying envelope (dash lines) yields a coherence
+time T ∗
+2 = 0.17 ± 0.03 ms. (d) Hahn-echo sequence (see in-
+set): measured excess counts ⟨ �C⟩ versus delay τ between sub-
+sequent pulses with a linearly increasing phase ϕ(τ) = 2π∆τ
+with ∆ = 0.001 MHz on the last pulse (red dots). The cor-
+responding fit and its envelope (solid and dash lines) yield a
+coherence time T2 = 2.47 ± 0.31 ms. (e) Periodic Dynami-
+cal Decoupling sequence (see inset): measured excess counts
+⟨ �C⟩ versus inter-pulse delay time τ (red dots).
+A linearly
+increasing phase ϕ(τ) = 2π∆τ with ∆ = 0.001 MHz is im-
+parted on the last pulse. Corresponding fit and its envelope
+(solid and dash lines) are shown, yielding the coherence time
+T P DD
+2
+= 2.99±0.03 ms. Data taken at B0 = 422.085 mT and
+θ = −0.003◦.
+
+6
+sible to determine the dependence of the spin longitu-
+dinal relaxation time T1 on δ (Fig. 5b).
+It is seen to
+increase with δ quadratically, in agreement with the ex-
+pected Γ−1
+R
+dependence [18]; a fit yields a coupling con-
+stant g0/2π = 3.56 kHz (see Fig. 5b).
+This confirms
+that non-radiative relaxation is negligible in our mea-
+surements (see Methods), and that T1 ≃ Γ−1
+R
+for the
+most strongly coupled spins.
+The Hahn echo coherence time is measured by apply-
+ing the sequence π/2X − τ − πY − τ − π/2Φ [33], with
+Φ = 2π∆τ. An oscillation at frequency ∆ is observed
+in ⟨ ˜C⟩, exponentially relaxing with a characteristic time
+T2 = 2.47 ± 0.31 ms. This is close to the radiative de-
+cay limit 2T1, indicating that the pure dephasing echo
+contribution is ∼ 16 ± 5 ms, in line with measurements
+on ensembles of Er3+ : CaWO4 electron spins [34]. This
+dephasing can be suppressed further by a 3-π-pulse Pe-
+riodic Dynamical Decoupling sequence, yielding a trans-
+verse relaxation time T P DD
+2
+= 2.99 ± 0.33 ms, which is
+equal to 2T1 to the accuracy of the measurement. These
+coherence times were also measured on a set of five Er3+
+electron spins; T ∗
+2 varies strongly among these ions (be-
+tween 5µs and 300µs), whereas T2 and T P DD
+2
+are consis-
+tently close to 2T1 (see Methods). The variation of co-
+herence time among different spins can be explained by
+the varying nuclear spin or paramagnetic environment of
+each ion, and also possibly their degree of exposure to
+surface magnetic noise given their approximate depth of
+∼ 100 − 150 nm according to Fig. 2 [31, 35]. It is also
+noteworthy that the coherence times measured here are
+on par with the longest reported for individual electron
+spins in solid-state [14], in a platform which gives access
+to several tens of these spin qubits by simply tuning the
+magnetic field.
+We now discuss the significance of our results for prac-
+tical single electron spin resonance spectroscopy.
+One
+particularly interesting aspect of our method is its appli-
+cability to a broad range of paramagnetic species, pro-
+vided their radiative relaxation rate ΓR can be enhanced
+up to ∼ 103s−1 or higher by the Purcell effect, and their
+non-radiative relaxation rate is smaller than ΓR. Note
+that no requirement on the coherence time applies, as
+the fluorescence signal is entirely incoherent.
+Indeed,
+many paramagnetic impurities have non-radiative relax-
+ation rates in the range of 10−3−103 s−1 at ∼ 1−4 K [36–
+38], and thus also likely at millikelvin temperatures.
+Although reaching the desired radiative relaxation rate
+of ΓR > 103s−1 was made easier in this work by the
+large transverse g-factor of 8.3 in Er3+ : CaWO4, this
+large relaxation rate was also demonstrated for donor
+spins in silicon with g-factors of only 2, using a simi-
+lar resonator geometry but with a narrower and shorter
+wire [25]. Whereas in our experiment the spins are lo-
+cated in the sample supporting the resonator, it is also
+possible to deposit a small volume of a spin-containing
+insulating material, such as a powder or micro-crystal,
+onto a pre-fabricated resonator device.
+Such an ap-
+proach could be suitable for measuring individual rare-
+earth-ion-containing molecules [6], nanocrystals [39], or
+proteins whose active center contains a transition-metal-
+ion [40, 41].
+Based on the
+10 µm3 detection volume
+demonstrated here using Er3+ : CaWO4, we extrapolate
+that a 0.5 µm3 detection volume would be achievable for
+an electron-spin with a g-factor of two, under the same
+experimental conditions. All these metrics could be im-
+proved with better SMPD performance, in particular re-
+duced dark count rates, highlighting a strong motivation
+for the continued development of SMPD devices.
+In conclusion, we report spectroscopic measurements
+of single rare-earth-ion electron spins by detecting their
+microwave fluorescence, gaining four orders of magnitude
+in spectral resolution by resolving the ensemble inhomo-
+geneous linewidth. In our experiment, tens of individ-
+ual spins with coherence times in excess of 1 millisecond
+are interfaced with the same microwave resonator, which
+opens new perspectives for hybrid quantum computing.
+Because of its broad applicability, large detection vol-
+ume, and spectroscopic capability, our detection method
+comes close to practical single electron spin resonance at
+cryogenic temperatures, and may thus open new appli-
+cations in ESR spectroscopy.
+Acknowledgements
+We acknowledge technical support from P. Sénat, D.
+Duet, P.-F. Orfila and S. Delprat, and are grateful for
+fruitful discussions within the Quantronics group.
+We
+acknowledge support from the Agence Nationale de la
+Recherche (ANR) through the Chaire Industrielle NAS-
+NIQ under contract ANR-17-CHIN-0001 cofunded by
+Atos, and through the MIRESPIN (ANR-19-CE47-0011)
+and DARKWADOR (ANR-19-CE47-0004) projects. We
+acknowledge support of the Région Ile-de-France through
+the DIM SIRTEQ (REIMIC project), from CEA through
+the DRF-Impulsion porgram (grant RPENANO), from
+the AIDAS virtual joint laboratory, and from the France
+2030 plan under the ANR-22-PETQ-0003 grant.
+This
+project has received funding from the European Union
+Horizon 2020 research and innovation program under
+ERC-2021-STG grant agreement no. 101042315 (INGE-
+NIOUS) and Marie Sklodowska-Curie grant agreement
+no. 792727 (SMERC). Z.W. acknowledges financial sup-
+port from the Sherbrooke Quantum Institute, from the
+International Doctoral Action of Paris-Saclay IDEX, and
+from the IRL-Quantum Frontiers Lab. We acknowledge
+IARPA and Lincoln Labs for providing the Josephson
+Traveling-Wave Parametric Amplifier.
+
+7
+Author contributions
+A.F. and P.G. grew the crystal, which M.L.D., Z.W.
+and S.B. characterized through CW and pulse EPR mea-
+surements. Z.W., D.V., P.B., E.F. designed the spin res-
+onator. Z.W. fabricated the spin resonator. L.B., E.F.
+designed the SMPD. L.B. fabricated the SMPD. M.R.
+designed and installed the magnetic field stabilization.
+Z.W., L.B., E.F. took the measurements.
+Z.W., P.B.,
+E.F. analyzed the data. Z.W., P.B., D.V., E.F. wrote
+the article, with contributions from all the authors. P.B.
+and E.F. supervised the project.
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+
+Method
+(Dated: January 9, 2023)
+Sample
+The CaWO4 crystal used in this experiment origi-
+nates from a boule grown by the Czochralski method
+from CaCO3 (99.95% purity) and WO3 (99.9 % pu-
+rity).
+A sample was cut in a rectangular slab shape
+(7 mm × 4 mm × 0.5 mm), with the surface approx-
+imately in the (ac) crystallographic plane, and the c-
+axis parallel to its short edge. The crystal structure is
+tetragonal with unit cell constants a = b = 0.524 nm
+and c = 1.137 nm, as shown in Extended Data Fig. 1.
+The erbium ions Er3+ substitute to the calcium ions
+Ca2+ (with long-range charge compensation in the crys-
+tal). These sites have a S4 symmetry, leading to a gy-
+romagnetic tensor with only diagonal elements in the
+(a, b, c) plane γa = γb ≡ γ⊥ = 2π × 117.3 GHz/T, and
+γc ≡ γ|| = 2π × 17.45 GHz/T [1]. The residual doping
+concentration of erbium is 3.1 ± 0.2 ppb, measured from
+continuous-wave EPR spectroscopy [2].
+Extended
+Data
+Fig.
+1:
+Crystal
+structure
+of
+Er3+ : CaWO4 (oxygen is not shown for clarity).
+On top of this sample, a lumped-element LC resonator
+was fabricated by sputtering 50nm of niobium and pat-
+terning the thin film by electron-beam lithography and
+reactive ion etching. The sample is placed in a 3D cop-
+per cavity with a single microwave antena and SMA port
+used both for the excitation and the readout in reflec-
+tion. As shown in Extended Data Fig.2, the "bow-tie"
+shaped resonator consists of an interdigitated capacitor
+shunted by a 94 µm × 600 nm inductive wire in the mid-
+dle. From finite-element microwave simulations, we de-
+duce an impedance of 17.5 Ω. The geometric inductance
+of the inductance wire contributes 33% of the total res-
+onator inductance. The top and bottom capacitor pads
+are shaped as parallel fingers in an effort to improve the
+resonator resilience to an applied residual magnetic field
+perpendicular to the metallic film (along x).
+20 μm
+10 μm
+0.85 mm
+1.44 mm
+Inductive wire:
+- Length: 94 μm
+- Width: 600 nm
+Extended Data Fig. 2: Resonator design.
+Experimental setup
+The complete setup schematic is shown in Extended
+Data Fig. 9
+Room-temperature setup
+Its room-temperature part uses five microwave sources
+and one FPGA-based instrument (OPX platform from
+Quantum Machine) for arbitrary waveform generation,
+digitization, and real time feedback. The OPX instru-
+ment contains 10 channels of analog outputs (AO), 10
+digital outputs (DO) and 2 analog inputs (AI).
+The pulses used to drive the spins are generated by I/Q
+mixing a pair of in-phase (I) and quadrature (Q) signals
+from the OPX with a local oscillator (LO - orange) at the
+spin resonator frequency ω0. The upconverted microwave
+signal is then split over 2 branches, one of them including
+an about 40 dB amplifier, which are then recombined.
+Only one of the branch is chosen to propagate the signal,
+with two fast switches controlled by digital lines from
+the OPX. The spin excitation pulses enters the dilution
+refrigerator through line 2.
+The SMPD operation (see [3] for details) involves one
+dc-current and three microwave sources, the role of which
+are as follows: (1) A Yokogawa current source (red) pro-
+vides the necessary flux bias to bring the SQUID-tunable
+buffer resonator of the SMPD at ωb in resonance with
+arXiv:2301.02653v1  [quant-ph]  6 Jan 2023
+
+b
+WO4
+Ca2+
+Er3+
+a
+0.524 nm
+1.137 nm2
+the spin resonator at ω0, so that the fluorescence pho-
+tons emitted by the spins are at the center of the SMPD
+detection bandwidth. (2) A pump drive (purple) at fre-
+quency ωp enables a four-wave mixing process convert-
+ing an incoming photon in the buffer into an excitation
+of a superconducting transmon qubit at ωq and a pho-
+ton in a readout (waste) resonator at ωw, according to
+ωp + ωb = ωq + ωw. (3) The readout of the qubit is per-
+formed by probing by homodyne detection (green) the
+qubit-state dependent dispersive shift of the readout res-
+onator.
+(4) Control pulses of the qubit are generated
+with a sideband mixer from one OPX IF output and the
+blue LO source. They are combined with the pump pulse
+and are sent to line 7.
+A Rohde & Schwarz microwave source (yellow) at the
+input of line 5 provides the pump power for a traveling
+wave parametric amplifier (TWPA) placed at 10 mK.
+Low-temperature setup
+The spin excitation pulses (line 2) are heavily atten-
+uated (∼ 110 dB) to minimize the thermal excitation of
+the qubit and dark counts. They are directed, through a
+double- and a single-junction circulator, to the antenna
+of the cavity containing the spin resonator. The reflected
+and output signals on this antenna are routed to the in-
+put of the SMPD through a single-junction circulator. To
+pre-characterize the spin resonator as well as the SMPD,
+the signal reflected on the SMPD input is routed to room-
+temperature via the same single- and double-junction cir-
+culators and output line 1 with a first HEMT amplifier;
+isolation of the experiment from this HEMT is provided
+by a double circulator and an extra 10dB attenuation.
+Note that during all measurements reported in the main
+text, this line 1 was left open and its HEMT switched off.
+SMPD qubit readout pulses are sent via the attenu-
+ated line 4 and a double circulator. The reflected sig-
+nal is routed to a Josephson Traveling Wave Parametric
+Amplifier (TWPA) pumped from line 5 via a directional
+coupler and to a second HEMT at the 4 K stage.
+A
+double-circulator isolates the TWPA from this HEMT.
+The SMPD pump tone and qubit reset pulses are ap-
+plied via line 7 and its 20dB directional coupler. The
+other 2 ports of the coupler are connected to a 50Ω load
+at 800 mK and a 50Ω-loaded circulator at 10mK, in order
+to minimize the noise induced by the dissipation of these
+signals.
+Single microwave photon detector
+The SMPD is operated in cycles of 12.8 µs on average.
+Each cycle is composed of three steps: (i) the pumped
+conversion of an incoming photon into a qubit excitation
+during 10 µs, (ii) the qubit dispersive readout lasting
+2 µs, and (iii) the conditional reset of the qubit to its
+ground state if it was detected excited. This reset consists
+of one or several π-pulse(s) applied to the qubit until it is
+measured in its ground state. The conditional reset time
+is thus non-deterministic, and lasts from 0.7 µs (feedback
+time with the OPX) to 0.7 + (2 + 0.7)k µs, with k the
+number of π pulses applied.
+At each cycle, a count C = 1 is detected if the qubit
+is found in its excited state (before the reset), and the
+count time is recorded with sub-microsecond accuracy.
+The SMPD is characterized independently, in absence
+of spin signal, by measuring its key figures of merit in
+terms of dark count rate, efficiency, and bandwidth.
+Dark count rate. - We define dark counts as the counts
+that are not due to the spins, that is those originating
+from spurious excitation of the transmon qubit in absence
+of incoming photons, and those due to unwanted photons
+present at the SMPD input [3]. For this detector, a dark
+count rate of 106 ± 3 s−1 has been measured over 24
+hours (data from Extended Data Fig. 7). We observed
+slow darkcount rate fluctuations over week time scales
+ranging typically from 130 s−1 to 90 s−1 mainly due to
+variation in qubit T1 and a slow cooling down of the line
+and of the qubit. We can discriminate dark count contri-
+butions from the microwave line thermal occupancy, from
+the pump heating and from the qubit thermal occupancy
+by switching off and detuning the pump tone from the
+four wave mixing frequency. By detuning the pump by
+10 MHz, the detection efficiency is set close to zero but
+the pump heating load persists, we measure count rates
+of 11 s−1. When the pump is turned off, the dark count
+rate is 9 s−1, indicating that the pump heating is negligi-
+ble. From these measurement, we conclude that 92% of
+the dark counts come from thermal microwave photons
+reaching the SMPD via its input line. This corresponds
+to a thermal population of ∼ 2.4 × 10−4 photons in the
+line, and to an effective temperature of ∼ 42 mK, to be
+compared to the measured 10 mK base temperature of
+the refrigerator.
+Efficiency.
+- The detector efficiency is measured by
+shining a microwave tone of known power at the detector
+input. The average input photon flux for a given applied
+power is calibrated in-situ by measuring the transmon
+qubit a.c. Stark shift and dephasing [4]. The SMPD ef-
+ficiency is then simply taken as the ratio between the
+counts detected over 1s and the photon flux (in pho-
+ton/s). It was measured for different input powers, as
+shown in Extended Data Fig.3a: At hundreds of input
+photons per second, a value close to the fluorescence sig-
+nal obtained at high excitation power, the efficiency is
+ηSMP D = 0.32. The detector saturates and the efficiency
+drops at input fluxes above 104 photons/s. Further opti-
+mizing the lifetime of the transmon qubit as well as the
+readout and pump power for four-wave mixing, would
+probably yield a better efficiency.
+Bandwidth. - The detector bandwidth is extracted by
+
+3
+0
+50
+100
+150
+200
+Incoming photon rate (
+ 
+)
+×103
+−1
+s
+0
+10
+20
+30
+Efficiency (%)
+7.331 7.332 7.333 7.334 7.335 7.336
+Microwave frequency (GHz)
+10
+−4
+10
+−3
+10
+−2
+10
+−1
+⟨ ⟩
+C  (counts)
+0.9 MHz
+b
+MW on
+MW off
+0
+20
+40
+Count rate (
+ 
+)
+×103
+−1
+s
+a
+Extended Data Fig. 3: SMPD characteristics.
+(a)
+SMPD efficiency. Detected click rate (red) and efficiency
+(blue) as a function of input photon flux.
+Below 104
+s−1 (linear regime), an efficiency of 32% is obtained. (b)
+SMPD bandwidth. Average number of detected counts as
+a function of photon frequency when the input microwave
+tone is switched on (red dots) or off (gray dots). The solid
+line is a Lorentzian fit to the data yielding a FWHM
+bandwidth of 0.9 MHz.
+measuring the average detected counts ⟨C⟩ as a func-
+tion of the microwave frequency. Each ∼ 10µs-long pulse
+contains 0.5 photon on average to mimic single spin de-
+tection. The full width at half maximum (FWHM) of
+a Lorentzian fit gives a bandwidth of 0.9 MHz for the
+detector, as shown in Extended Data Fig. 3b.
+Average number of counts
+In the case of Fig 2, We calculate the average number
+of counts
+⟨C⟩ = 1
+N
+N
+�
+n=1
+T
+�
+0
+cn(td)
+(1)
+by summing the counts from td = 0 to T, with N the
+number of repetitions of the experiment and cn(td) the 0
+or 1 SMPD outcome at time td.
+For the other figures (Figs. 3, 4 and 5) involving single
+spins, the fluorescence signal is measured as a function
+of time up to ∼ 5T1 after the excitation pulse, and its
+second half (close to the background level) is subtracted
+from the first part, leading to a background-corrected
+average number of counts
+⟨ ˜C⟩ = 1
+N
+N
+�
+n=1
+�
+�
+T/2
+�
+0
+cn(td) −
+T
+�
+T/2
+cn(td)
+�
+� .
+(2)
+Magnetic field alignment and stabilization
+The magnetic field B0 is generated by a 1T/1T/1T
+3-axis superconducting vector magnet.
+Magnetic field
+alignment proceeds in two steps. We first align the field
+in the sample plane (y − z) by applying a small field
+of 50 mT, and minimizing the resonator losses and fre-
+quency shift with respect to the zero-field values [2]. We
+then determine the direction of the projection of the crys-
+tallographic c-axis on the sample plane, defined as θ = 0◦,
+by measuring the erbium ensemble line (as shown in Fig.2
+of the main text) for various angles in the (y − z) plane.
+Due to the anisotropy of the gyromagnetic tensor γ0, B0
+can be expressed with angle θ and β as
+Bpeak
+0
+= ℏω0/
+�
+γ2
+∥cos2θcos2β + γ2
+⊥(1 − cos2θcos2β),
+(3)
+where θ = 0◦ corresponds to the maximum of B0 and
+β is the angle between the c-axis and sample plane. As
+shown in Extended Data Fig. 4, the fitting (solid line)
+with eq. 3 to the data (dots) yields β = 0.5◦.
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+ (deg)
+417.0
+417.5
+418.0
+418.5
+419.0
+419.5
+Bpeak
+0
+(mT)
+Extended Data Fig. 4: Magnetic field alignement.
+Measured (dots) magnetic field Bpeak
+0
+at which the center
+of the spin ensemble line is found, as a function of the
+angle θ that the field makes with the c axis projection.
+The fit with eq. 3 (line) to the data yields the θ = 0
+origin (see text), as well as the angle between the c axis
+and the sample plane, β = 0.5◦.
+Note that this procedure does not guarantee that the
+resonator nanowire exactly coincides with the θ = 0◦ di-
+rection determined by our alignment procedure. In fact,
+a small residual angle (possibly of order 1◦) likely ex-
+ists between the two directions. Since we have no way
+
+4
+to determine this angle, and since its non-zero value has
+negligible impact on any of the results found in the arti-
+cle, we used a zero value by simplicity for plotting Fig.
+2a.
+The stability of the magnetic field is determined by
+the mode of operation (current supplied mode or per-
+sistent mode) of the three superconducting coils of the
+vector magnet and by their current sources. For the spec-
+troscopy data of the main text (Figs 2 and 3), the current
+supplied mode is used with a commercial current source
+(Four-Quadrant Power Supply Model 4Q06125PS from
+AMI). On the contrary, the data of Figs. 4 and 5 re-
+quire to tune the spin-resonator frequency difference δ
+and to keep it stable (less than 10kHz variation) over
+long periods of time. To achieve this goal, we use the
+fact that one of our coils is nearly aligned with the z-
+axis and thus provides the largest component of the B0
+field; we thus minimize the noise by placing it in persis-
+tent mode. Then, the coil closest to the y axis is used
+to fine-tune δ. The (much smaller) current through that
+coil is moreover further stabilized using a custom-made
+feedback loop based on a current meter (Keithley 2700
+model).
+Microwave induced heating and corresponding
+spurious signal
+We now discuss heating effects observed (as in [3]) af-
+ter a microwave pulse is applied to the spin resonator. To
+evidence and clarify this point, we first measure in three
+different cases the transcient signal recorded after an ex-
+citation pulse resonant with the SMPD buffer resonator:
+1. Normal operation on a single spin: single spin, spin
+resonator and SMPD buffer are all in resonance.
+2. Normal operation in absence of spins: All spins are
+far off-resonance, and spin resonator and SMPD
+buffer are on resonance.
+3. Complementary diagnosis: All spins, spin resonator
+and SMPD buffer are detuned from one another.
+A 6µs-long excitation pulse is applied at time t = 0; the
+photon counting sequence starts 1 ms before the pulse,
+is interrupted (SMPD switched off) during the pulse du-
+ration, and is restarded during several ms.
+In normal operation (case 1 and 2 - red and blue in
+Fig. 5), a count rate spike is observed in the bin imme-
+diately following the excitation pulse; it corresponds to
+the decay (at rate κ−1) of the microwave energy stored
+by the pulse in the spin resonator.
+This spike disap-
+pears when the spin resonator is detuned from the signal
+(case 3), as expected.
+After the spike, even when no
+spin signal is present (case 2 - blue), extra counts above
+the background (grey) are however observed over a time
+window of about 0.3 ms after the pulse, with a decay
+time of ∼ 100µs. This extra signal is reminiscent of the
+one observed over ∼ 10 ms in [3], possibly shorter in the
+present work due to lower excitation pulse powers. For
+comparison, when detuning the spin resonator from the
+excitation (case 3 - orange in Fig. 5), the extra count rate
+is lower and reaches the background steady-state much
+faster.
+All these measurements with no spins indicate
+that the spurious extra counts decaying over ∼ 100µs in
+normal operation originate from the excitation and sub-
+sequent radiative decay of systems that are resonantly
+coupled to the spin resonator. It is tempting to identify
+them with the two-level-system bath that causes field de-
+cay and phase noise in superconducting circuits. In nor-
+mal operation with a spin (case1 - red in Fig. 5), these
+spurious extra counts of course adds to the relevant signal
+coming from the spin. To lower the impact of this tran-
+sient heating effect, the results of Fig. 4c (resp. Figs. 5a
+and b) were obtained by discarding the counts detected
+in the first 100µs (resp. 50µs) time window following the
+excitation.
+0
+2
+4
+6
+Time (ms)
+0
+1
+2
+3
+4
+5
+⟨ ̇⟩
+C (counts / ms)
+0.0
+0.2
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+t
+0
+6µs
+Detection 
+cycle13µs
+Extended Data Fig. 5: Transient response of the sys-
+tem after microwave excitation. Measured average
+click rate versus time before and after a 6 µs-long mi-
+crowave excitation pulse is applied at time 0, for cases
+1 (red), 2 (blue) and 3 (orange) - see text.
+SMPD is
+switched off during excitation. Dark count background
+is indicated in grey. Inset is a zoomed-in view around
+time 0.
+We finally study the dependence of this heating effect
+in absence of resonant spins on the excitation pulse du-
+ration and amplitude. For that we integrate over 8 ms
+the number of counts ⟨C⟩ after an excitation pulse, and
+repeat the sequence every 8 ms. The results in Extended
+
+5
+Fig.4 show an increase of ⟨C⟩ as a function of pulse du-
+ration and amplitude, as well as a characteristic time for
+this increase with pulse duration that decreases as ex-
+citation amplitude increases. We also verified that this
+increase of ⟨C⟩ is not due to microwave heating of the
+line attenuators, by repeating the same measurements
+with the spin resonator detuned from the SMPD buffer:
+a much smaller effect is observed, indicating that the ex-
+cess counts do come from the spin resonator.
+0
+5
+10
+15
+20
+25
+Pulse duration (�s)
+1.0
+1.5
+2.0
+2.5
+C  (counts)
+× 1
+× 2
+× 4
+× 7
+Pulse
+amplitude
+SMPD in resonance
+SMPD detuned
+Extended Data Fig. 6:
+Heating versus spin exci-
+tation duration and amplitude. Measured average
+counts integrated over a 8ms-long window after an exci-
+tation pulse as a function of pulse duration and ampli-
+tude A, obtained when the spin resonator is detuned from
+(dash line) or in resonance with (solid line) the SMPD
+buffer. The excitation pulse frequency is always tuned to
+the SMPD buffer one.
+Intensity-intensity correlation measurements
+We now provide more details on the intensity-intensity
+correlation measurements used to prove the single spin
+character of our experiment.
+The dataset to be analyzed corresponds to two series
+of 4363635 sequences labelled from i=0 to i=4363634 re-
+peated every tr = 7.5 ms, where one serie includes a π
+pulse at time t = 0 and the other has no excitation pulse.
+Time t = 0 is followed by 600 SMPD cycles. As explained
+in the Heating section, the first 100µs window after the
+excitation pulse is excluded from the analysis in order to
+minimize the impact of the heating effect. The count data
+in the rest of the sequence are then grouped in subsequent
+350µs-long timebins indexed by j (with j running from 0
+to 20), and centered at time τj = 100+(2j+1)×350/2 µs.
+The corresponding number of counts in the bin j of se-
+quence i is denoted as n(i)
+j .
+We first provide in Extended Data Fig. 7a a direct vi-
+sualization of the anti-bunching found on a single-spin
+peak.
+The count rate ⟨ ˙C⟩(t) is plotted as a function
+of time, first, averaged over all recorded sequences, and
+second, averaged over sequences with a count 1 in the
+first bin (conditioned curve).
+When measured on the
+background signal, the two curves are identical, whereas
+when measured on the single-spin peak (s0), the condi-
+tioned fluorescence rate is reduced at short times after
+the first count, compared to the average unconditioned
+one.
+In order to quantify this anti-bunching, we then com-
+pute the intensity-intensity correlation functions inside a
+sequence,
+g(2)(τ = τj)
+⟨n(i)
+0 n(i)
+j ⟩i
+⟨n(i)
+0 ⟩i⟨n(i)
+j ⟩i
+,
+(4)
+as well as between two sequences separated by k excita-
+tion pulses,
+g(2)(k) = ⟨n(i)
+0 n(i+k)
+1
++ n(i)
+1 n(i+k)
+0
+⟩i/2
+⟨n(i)
+0 ⟩i⟨n(i+k)
+1
+⟩i
+,
+(5)
+where we keep only the first and second bin of the two
+sequences, symmetrize the function about k=0, and av-
+erage over all pairs of sequences with same separation
+k ∈ Z.
+The intra-sequence g(2)(τ) and inter-sequence g(2)(k)
+are shown in Extended Data Fig. 7c and d. The latter
+is then corrected from the background counts (leading to
+Fig. 4c in the main text) as explained below.
+Background correction
+We now describe how we subtract from g(2) the dark
+count rate contribution, in order to obtain a background-
+corrected correlation function.
+We assume that the clicks from the detector have two
+independent origins: emission sj from the spins, and Poi-
+sonnian background noise dj due to independent dark
+count events, such that nj = sj + dj, ⟨nj⟩ = ⟨sj⟩ + ⟨dj⟩,
+and ⟨sjdj⟩ = ⟨sj⟩⟨dj⟩.
+In addition, we assume that
+the instruments during the measurement time are sta-
+ble enough so that the dark count rate is time-invariant:
+⟨dj⟩ = ⟨d⟩.
+We thus define the background-corrected autocorrela-
+tion function
+g(2)
+corr(k) = ⟨s(i)
+0 s(i+k)
+1
++ s(i)
+1 s(i+k)
+0
+⟩i/2
+⟨s(i)
+0 ⟩i⟨s(i+k)
+1
+⟩i
+(6)
+
+6
+0
+5
+time (ms)
+0.00
+0.05
+0.10
+0.15
+C  (clicks / ms)
+a
+0
+5
+time (ms)
+b
+2.5
+5.0
+7.5
+ (ms)
+0.90
+0.95
+1.00
+1.05
+g(2)( )
+c
+Background
+ pulse on spins
+Dark count
+Single emitter 
+with background
+Extended Data Fig. 7:
+Photon
+intensity
+auto-
+correlation function g(2) within one sequence. (a)
+Average count rate ⟨ ˙C⟩ as a function of delay time after
+a π excitation pulse, for all recorded sequences (red) and
+for sequences with a first click detected before 0.45ms
+(dark red). The reduction of ⟨ ˙C⟩ in the second case in-
+dicates the anti-bunching of spin fluorescence photons.
+(b) Average count rate ⟨ ˙C⟩ as a function of delay time,
+for all recorded background traces (gray) and for traces
+with a first click detected before 0.45ms (dark gray). The
+unchanged ⟨ ˙C⟩ in the second case indicates a Poissonian
+background made of independent dark count events. (c)
+Extracted g(2) fucntion for dark counts (gray dots) and
+spin fluorescence signal (red dots) as a function of delay
+time τ. Expected g(2) functions for the Poissonnian back-
+ground (black solid line) and for an ideal single emitter
+in presence of the same background (g(2)
+se - red solid line)
+fit well the experimental data. (d) Uncorrected g(2)(k)
+(blue columns) and corresponding ±1-standard deviation
+error bars (red) as a function of inter-sequence offset k.
+and express it explicitly as a function of the uncorrected
+g(2)(k) of Eq. 5 and of the measurement outcomes Aj ≡
+(⟨n(i)
+j ⟩i − ⟨d⟩)/⟨d⟩:
+g(2)
+corr(k) = (1 + A0)(1 + A1)g(2)(k) − A0 − A1 − 1
+A0A1
+. (7)
+In addition, it is interesting to compare the measured
+g(2)(τ) inside a sequence with the expected g(2)
+se (τ) that
+an ideal single emitter would give in presence of back-
+ground noise. In this case, all terms s(i)
+0 s(i)
+j
+are 0 due to
+the single emitter character, and
+g(2)
+se (τ = τj) =
+⟨n(i)
+0 ⟩i⟨d⟩ + ⟨n(i)
+j ⟩i⟨d⟩ − ⟨d⟩2
+⟨n(i)
+0 ⟩i⟨n(i)
+j ⟩i
+.
+(8)
+This function is plotted as a red solid line in Extended
+Data Fig. 7(c) and shows a good match with the mea-
+sured g(2)(τ).
+Single-spin signal-to-noise ratio
+We derive in this section the theoretical single-spin
+signal-to-noise ratio of our fluorescence-detection proto-
+col. This protocol involves identical sequences repeated
+every tr times during a total measurement time tm. In
+each sequence, a π pulse is applied at the beginning, and
+the number of counts is measured during a time td fol-
+lowing the pulse. The spin relaxation rate is ΓR.
+In our model, it is readily shown that the steady-
+state spin polarization at the beginning of each se-
+quence is Sz0 = − 1
+2 tanh(ΓRtr/2), yielding an average
+number of counts per sequence −2ηSz0(1 − exp−ΓRtd),
+and an average total number of counts Cspin
+=
+η(tm/tr) tanh(ΓRtr/2)(1 − exp−ΓRtd).
+The noise has two contributions: one from the dark
+count fluctuations, whose variance is αtdtm/tr, and one
+from the partition noise of the detected photons, with
+variance (1 − η)Cspin. Therefore, the width of the his-
+togram with π pulse is δCπ =
+�
+αtdtm/tr + (1 − η)Cspin.
+The
+signal-to-noise
+ratio
+is
+defined
+as
+SNR
+=
+Cspin/δCπ = Cspin/
+�
+αtdtm/tr + (1 − η)Cspin.
+For the parameters of our experiment (ΓR = 700s−1,
+α = 102s−1, η = 0.12), numerical optimization indicates
+a maximum SNR of 2.5 is obtained for td = 2 ms and
+tr = 3 ms. In the experiment, we use a larger repetition
+time to minimize the effect of heating; for the parameters
+used (td = 2 ms and tr = 7.5 ms), the formula yields a
+SNR of 1.95, in agreement with the measured value of
+1.91.
+A simpler approximate scaling formula is obtained
+considering that the repetition time is tr ∼ Γ−1
+R , and
+that the spin polarization Sz0 ∼ −1/2.
+Taking more-
+over td = tr, one obtains the scaling formula provided
+in the introduction for the tm = 1 s integration time,
+SNR ∼ ηΓR/
+�
+α + (1 − η)ηΓR.
+Non-radiative relaxation
+The spin-lattice relaxation time of Er3+: CaWO4 with
+B0 oriented along the c axis was measured using the tra-
+ditional inductive detection, using a spin-echo-detected
+inversion-recovery sequence.
+A relaxation time T1 =
+210 ms was found at a frequency of 7.853 GHz [2] (see
+Fig. 8). At 10 mK, the relaxation is dominated by the
+
+7
+direct phonon process, with a non-radiative relaxation
+rate ΓNR scaling as B2
+0ω3
+0 [5].
+Therefore, we estimate
+that in our conditions (ω0/2π = 7.335 GHz, and B0 ap-
+plied along the c axis), the non-radiative relaxation rate
+should be ΓNR ≃ 3.3 s−1.
+T
+T(s)
+0
+2
+4
+0.0
+-0.1
+Echo amplitude (a.u.)
+A
+A
+A/2
+echo
+T1 = 213 ± 1 ms
+Extended Data Fig. 8: Spin-Lattice relaxation time mea-
+sured with inversion-recovery sequence at 10 mK, with
+B0 along the c axis, and ω0/2π = 7.853 GHz. Green dots
+are data, solid line is a fit yielding T1 = 0.213 ± 0.001 s.
+Efficiency
+We now discuss the value measured for the overall ef-
+ficiency η = 0.12.
+Losses of counts can occur due to
+non-radiative spin relaxation, internal losses of the spin
+resonator, microwave losses between the spin device and
+the SMPD ηloss, and finite SMPD efficiency, so that η =
+[ΓR/(ΓR +ΓNR)][κc/κ]ηlossηSMP D. Given the measured
+ηSMP D = 0.32, κc/κ = 0.57, and ΓR/(ΓR+ΓNR) = 0.995
+we deduce ηloss = 0.66, which is a reasonable value for the
+microwave losses encountered upon propagation along a
+50-cm-long coaxial cable, a circulator, and the filters at
+the SMPD input.
+Summary of different spins
+Apart from the spins discussed in the main text, we
+have also measured other single spin peaks found in the
+spectroscopy measurement.
+s0 and s6 are the labelled
+spins while s7 and s8 are not indicated in the spectrum
+in Fig. 2. Here we summarize their measured coherence
+time in the table below.
+Spin T1(ms) T∗
+2(µs) Techo
+2
+(ms)
+s0
+1.26
+79
+1.38
+s6
+1.42
+170
+2.47
+s7
+2.21
+7.5
+2.1
+s8
+1.36
+315
+1.53
+[1] Antipin, A., Katyshev, A., Kurkin, I. & Shekun, L. Para-
+magnetic resonance and spin-lattice relaxation of Er3+
+and Tb3+ ions in CaWO4 crystal lattice. Sov. Phys. Solid
+State 10, 468 (1968).
+[2] Le Dantec, M. Electron spin dynamics of erbium ions in
+scheelite crystals, probed with superconducting resonators
+at millikelvin temperatures.
+PhD Thesis, Univ. Paris-
+Saclay (2022).
+URL https://tel.archives-ouvertes.
+fr/tel-03579857.
+[3] Albertinale, E. et al.
+Detecting spins by their fluores-
+cence with a microwave photon counter. Nature 600, 434–
+438 (2021).
+URL https://www.nature.com/articles/
+s41586-021-04076-z.
+[4] Gambetta, J. et al. Qubit-photon interactions in a cav-
+ity: Measurement-induced dephasing and number split-
+ting. Physical Review A 74, 042318 (2006). URL https:
+//link.aps.org/doi/10.1103/PhysRevA.74.042318.
+[5] Larson, G. H. & Jeffries, C. D.
+Spin-Lattice Relax-
+ation in Some Rare-Earth Salts. I. Temperature Depen-
+dence. Physical Review 141, 461–478 (1966). URL https:
+//link.aps.org/doi/10.1103/PhysRev.141.461.
+Pub-
+lisher: American Physical Society.
+
+8
+4K
+50K
+100 mK
+10 mK
+spin resonator
+SMPD
+50 Ohm
+(800mK)
+-10
+IR
+-20 -20
+TWPA
+IR
+HEMT
+IF
+LO
+RF
+LO
+1         2                                                                 3                  4      5      6         7 
+4. qubit reset
+2. Photon
+    Detection
+3. Readout
+1. Spin pulse
+A1: MiniCirc ZVE8G+
+A2: HVA-500M-20-B
+A-tekH60 circulator
+Marki I/Q mixer
+or SSB mixer
+Clear Mw splitter
+50 Ohm load
+Waveline S11330 
+fast switch
+3
+1
+-20
+-20
+IR
+HEMT
+-10
+-20
+-10
+IR
+-20 -20
+IR
+IR
+-10
+-20
+-10
+-10
+-20
+-20
+RF
+I
+AnaPico MWG
+Q
++
+I
+Q
+LO
+RF
+I
+Q
+LO
+RF
+I
+Q
+A1
+A1
+A2
+AnaPico MWG
+AnaPico MWG
+AnaPico MWG
+R&S MWG
+-
+910 Ohm
+230 uF
+A2
+DC block
++
+-
+Yokogawa
+Directional coupler
+Bandpass filter
+AO
+AI
+AO
+DO
+Quantum Machine
+OPX
+AO
+2
+4
+�����
+��������
+����
+��
+�����
+����
+DO
+�������� ���������
+��������
+IR
+IR
+Extended Data Fig. 9: Schematic of the setup. Wiring and all the components used in this experiment at room
+temperature and cryogenic temperature are shown.
+

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+Is Federated Learning a Practical PET Yet?
+Franziska Boenisch∗, Adam Dziedzic∗†§, Roei Schuster∗§,
+Ali Shahin Shamsabadi∗‡§, Ilia Shumailov∗§, and Nicolas Papernot∗†
+∗ Vector Institute
+† University of Toronto
+‡ The Alan Turing Institute
+Abstract—Federated learning (FL) is a framework for users
+to jointly train a machine learning model. FL is promoted
+as a privacy-enhancing technology (PET) that provides data
+minimization: data never “leaves” personal devices and users
+share only model updates with a server (e.g., a company)
+coordinating the distributed training. We assess the realistic
+(i.e., worst-case) privacy guarantees that are provided to
+users who are unable to trust the server. To this end, we
+propose an attack against FL protected with distributed
+differential privacy (DDP) and secure aggregation (SA).
+The attack method is based on the introduction of Sybil
+devices that deviate from the protocol to expose individual
+users’ data for reconstruction by the server. The underlying
+root cause for the vulnerability to our attack is the power
+imbalance. The server orchestrates the whole protocol and
+users are given little guarantees about the selection of other
+users participating in the protocol. Moving forward, we
+discuss requirements for an FL protocol to guarantee DDP
+without asking users to trust the server. We conclude that
+such systems are not yet practical.
+1. Introduction
+Federated Learning (FL) [30] is a widely deployed
+protocol for collaborative machine learning (ML). It al-
+lows a server to train an ML model on data of different
+users without requiring direct access to that data. For this
+reason, FL is often promoted as data minimizing [31]:
+instead of sharing their data, users calculate model up-
+dates, usually gradients, on a shared model obtained from
+the server. These model updates are then aggregated and
+applied iteratively to train this shared model.
+Prior work has demonstrated that the model updates
+leak sensitive information on the users’ local training
+data [24], [39]. This is to be expected since nothing in
+the design of FL prevents information leakage. Indeed,
+the root cause of why FL is inherently vulnerable to data
+reconstruction attacks is that it is designed to provide
+confidentiality (data does not leave user devices) rather
+than privacy (outputs of the computation do not leak sen-
+sitive attributes from the users’ input). Without additional
+privacy measures, FL cannot protect users from the server
+reconstructing their data.
+Fortunately, if the server is trusted to follow the pro-
+tocol as prescribed, a relatively cost-effective mitigation
+exists: the server can add noise to the model updates and,
+§. Equal contribution.
+thereby, implement differential privacy (DP) [12] during
+the aggregation [32]. This degrades the performance of
+learned models [50], but adds strong privacy guarantees.
+Unfortunately, the server cannot always be trusted. Worse
+yet, by observing local users’ model updates, an untrusted
+server can mount powerful attacks to reconstruct users’
+training data points [7], [14], [16], [49], [54], [56], [57].
+Contemporary attacks exploit the fact that neurons com-
+pute a weighted sum of their input; thus the corresponding
+gradients contain rescaled versions of the input.
+In this vein, this work raises a simple pragmatic ques-
+tion that has far-reaching consequences: can modern FL
+deployments offer meaningful privacy guarantees while ef-
+ficiently and effectively learning from thousands of users’
+data, if the server is not trusted?
+We investigate this by mounting an attack (see Fig-
+ure 1) that exerts the capabilities of an untrusted server
+in FL to extract individual user data points. We show
+that the attack is even successful when FL is combined
+with secure aggregation (SA) [9] and distributed differ-
+ential privacy (DDP) [46]. In SA, gradient aggregation
+is performed via a decentralized multiparty computation
+protocol. In DDP, each user adds a small amount of noise
+to their gradient updates. Through the aggregation of user
+updates, the cumulative noise of DDP provides sufficiently
+high privacy guarantees to protect user data from leaking
+sensitive information. The two techniques were designed
+to decrease the amount of trust placed by users in the
+central party in FL [2], [17], [27].
+To be clear, we are, of course, not claiming that
+the cryptographic primitives behind SA and DDP are
+broken. We merely notice that the trust model that they
+assume, where in each round enough honest users con-
+tribute noised gradient updates, is not necessarily realized
+in practice within FL. In reality, the server is entrusted
+with provisioning users and sampling them in each round,
+and can easily inject an arbitrary number of malicious
+sybils under their control into any round, as demonstrated
+by industry actors actively deploying FL [40].
+By sampling a target user together with a group of
+sybils that act maliciously, the server can acquire direct
+access to the target user’s non-aggregated update. This
+allows the server to eliminate any effect of SA, and
+effectively reduces the protection of DDP to a minimum
+because the noise added by a single user is not designed
+to offer the claimed privacy guarantees.
+Next, we observe that the server is also typically en-
+trusted with controlling the shared model. Prior work [7],
+[14], [51] has shown that this ability enables the server
+1
+arXiv:2301.04017v1  [cs.CR]  9 Jan 2023
+
+𝐺𝑡𝑎𝑟𝑔𝑒𝑡
+[𝑡]
+Sybil
+Users
+Target 
+User 
+M Selected 
+Users at 𝑡
+Legitimate 
+Users 
+𝑓𝑤
+𝑓𝑤
+[𝑡]
+Server
+0
+0
+Secure
+Aggregation
+ҧ𝐺𝑡𝑎𝑟𝑔𝑒𝑡
+[𝑡]
+0
+0
+ҧ𝐺𝑡𝑎𝑟𝑔𝑒𝑡
+[𝑡]
+Noise 
+Addition
+ҧ𝐺𝑡𝑎𝑟𝑔𝑒𝑡
+[𝑡]
+Figure 1: Course of our attack against FL protected by SA and DDP. 1 The server introduces a small fraction of
+sybil users into the FL application. 2 The server selects M users for participation in training round t: one target user
+and M − 1 sybils. 3 The server manipulates the shared model with trap weights [7] and sends it out to the selected
+users. 4 The target user locally calculates its gradients on the manipulated model while the sybil users return zero or
+constant value gradients that are known by the server. 5 Only the target user locally applies a small amount of noise to
+its gradients to implement DDP. 6 The target user’s local noised gradients are aggregated with the sybil users’ values.
+7 The resulting aggregate which effectively contains solely the target user’s gradients is sent to the server. 8 The
+server extracts the target user’s training data from the received gradients.
+to extract large amounts of the users’ individual training
+data points from gradients in vanilla FL. By integrating
+the trap weight approach from [7] into our attack, we
+show that an untrusted server can extract high-fidelity user
+data points for common learning tasks despite DDP and
+SA. This highlights that even elaborate combinations of
+techniques like DDP and SA with FL fail to prevent data
+reconstruction when the server’s real-world capabilities
+are taken into account.
+While exploring data reconstruction under DP, we
+make another observation that advances our understanding
+of which users are more vulnerable to data reconstruc-
+tion attacks: the noise addition does not guarantee equal
+protection over all model gradients. As we visualize in
+Figure 3, some data points can be extracted with higher
+fidelity than others. This is despite the fact that all corre-
+sponding gradients were protected with the same clipping
+and noising operations. We identify the gradient norm as
+the reason behind disparate protection. For small-norm
+gradients, noise dominates the signal of the extracted data
+more than for gradients with a large norm. In principle,
+clipping in DP for ML is supposed to bound the gradient
+norm to control for this. However, the gradient norm is
+calculated globally whereas data is extracted locally from
+the components of the gradient that correspond to the
+weighted input of a single neuron. When the gradient
+corresponding to a neuron is large but all other neurons
+in that layer have small gradients, the overall gradient can
+still be below the clipping norm. Therefore, no clipping is
+performed, the same amount of noise is added to neurons
+with large and small gradients, and data from the neurons
+with larger gradients can be extracted with higher fidelity.
+We also sketch a construction that amplifies this effect
+and makes a few individual data points nearly perfectly
+extractable, even under noise addition.
+We then discuss the centralization and resulting power
+imbalance between the server and users as the root cause
+of FL’s vulnerability against attacks like the one proposed
+in this work. This motivates us to explore the requirements
+for building a variant of FL that practically prevents at-
+tacks by a malicious server. We consider three approaches
+based on (1) decentralization, (2) user verification, and
+(3) the support of specialized hardware. We find that one
+promising direction is to add adequate amounts of noise
+to users’ gradients via a cryptographic protocol such as
+secure multiparty computation (SMCP). However as of
+yet, due to the gradients’ high dimensionality, known
+constructions’ communication costs are prohibitive.
+As an alternative direction, users in FL can take re-
+sponsibility for implementing their full privacy protection
+locally, for example, by adding enough noise to individu-
+ally implement strong privacy guarantees. This approach is
+commonly referred to as local DP (LDP). As a last resort,
+users can decide not to participate in FL protocols if they
+do not trust the server. However, the last two options are
+only available if users have the required control over their
+participation in the protocol, which is not always the case.
+In summary, we make the following contributions:
+•
+We design an attack that enables a malicious server
+to reconstruct individual training data points from
+users when FL is protected by SA and DDP. See
+Figure 1 for an illustration of our attack flow.
+•
+We experimentally validate the attack’s ability to
+reconstruct image and textual data with high fi-
+delity in different DDP setups. We show how to
+increase the fidelity of the reconstruction of users’
+data points by reducing the effect of additive noise
+at the level of individual neurons. We thus observe
+disparate leakage over gradients under DDP.
+•
+We discuss centralization in FL and its resulting
+power-imbalance between the server and the users
+as the root cause of FL’s vulnerability and analyze
+potential mitigations.
+2. Background
+This section provides background on FL, describes
+user-data extraction from gradients in the vanilla version
+2
+
+6x046678of the protocol and introduces SA and DP—extensions
+implementing dedicated defenses against privacy leakage.
+2.1. Federated Learning
+FL [30] is a communication protocol that allows a
+group of N users to jointly train an ML model f, such
+that data never leaves the respective users’ devices. FL in-
+volves a server, who coordinates the training in an iterative
+process, as follows: at round t = 0, the server initializes
+the shared model W[0] at random, typically following
+common weight-initialization practice [19], [20], [23]. At
+every round t, the server chooses a subset of M ≤ N
+users to contribute to the learning round. The server then
+sends each of these users the model fW[t]. Each user
+chooses a subsample of their training data termed a mini-
+batch, computes the gradients of an objective function
+over fW[t] for each of these samples, and returns these
+gradients, which we call a (local) update, to the server. To
+conclude the round, the server updates the shared model
+by aggregating the received gradients, multiplying them
+by a learning-rate parameter and applying the change to
+the shared model.1
+It follows from the description above that users’ train-
+ing data significantly affects the values of local updates.
+This enables a variety of privacy attacks that extract indi-
+vidual user-data points directly from the model updates,
+as highlighted in the following section.
+2.2. Data Extraction from Vanilla FL
+Prior work [7], [14], [51] has shown that an untrusted
+server can directly extract user-data from the model gradi-
+ents. In these attacks, the server leverages its control over
+the shared model.2 In [14], the server exploits this ability
+to insert a fully-connected model layer as an extraction
+module where individual data points can be directly ex-
+tracted. [7] manipulates the model weights with an attack
+they call trap weights. This attack increases natural data
+leakage from fully-connected model layers. In [51], the
+server instead modifies model parameters to extract single
+data points by increasing their gradient contribution. The
+attack operates in several iterations of the protocol to
+collect multiple gradient updates from the same user.
+The presence of such data extraction attacks highlights
+the vulnerability of vanilla FL to privacy-leakage. To pre-
+vent training data from leaking onto updates and straight
+to the hands of the server, various extensions have been
+proposed, as we now briefly describe.
+2.3. Secure Aggregation
+In SA, due to Bonawitz et al. [9], users do not send
+their individual updates to the server. Instead, they per-
+form, along with the server, a multiparty computation
+1. For readers unfamiliar with gradient optimization: such gradient-
+based weight updates are intended to prescribe which direction W[t]
+needs to move towards, for the objective to be minimized. Typically,
+the objective is a value measuring the model’s level of prediction error
+across a given mini-batch.
+2. There also exist optimization-based data reconstruction attacks
+operating on model updates. These attacks can be conducted by a
+passive attacker solely observing the gradients. However, computation is
+expensive and the reconstructed data is not necessarily of high-fidelity.
+We provide a brief overview of this type of attack in Section 7.
+(MPC) protocol that ensures the server only receives the
+average of all updates in the round. Various improvements
+of the original protocol were suggested, for example, to
+allow the server to prove the correctness of the aggregate
+computation [52], increase robustness against malicious
+users’ manipulated gradients [10], or improve communi-
+cation efficiency [5], [22].
+2.4. Differential Privacy in FL
+Nothing in the design of FL prevents information
+leakage: FL is designed to provide confidentiality (data
+does not leave user devices) rather than privacy (outputs
+of the computation do not leak sensitive attributes from
+the users’ input). As discussed in Section 2.2, this leaves
+vanilla FL vulnerable to data reconstruction attacks. To
+ensure the privacy of users’ sensitive training data, it is
+natural to consider DP approaches that work by adding
+noise to user updates. DP is a gold standard in privacy
+technology because proper application of it comes with
+a theoretical bound on the probability of an adversary
+being able to distinguish adjacent datasets, i.e. datasets
+that differ solely in one data point. This implies a bound
+on the probability of data point extraction. In other words,
+a DP approach properly applied to FL updates could,
+in principle, ensure that individual user data points are
+not revealed to whoever observes the noised updates.
+See Appendix A for more background on DP and its
+integration to ML.
+One possible approach to integrate DP into FL is
+centralized DP (CDP) [4], [28], [40], where the server
+adds noise to the mini-batch gradients received by users.
+CDP assumes that the server is trusted to add noise, which
+is not true in the threat model of this work, see Section 3.
+To address this, local DP (LDP) [47] was proposed, where
+each user adds noise to its local update before sending
+it out for aggregation, in a way that ensures the user’s
+own dataset is protected from extraction. Unfortunately,
+LDP generally results in degraded model utility due to
+the addition of large amounts of noise to every user’s
+update [50]. Distributed DP (DDP) was proposed as a
+popular middle ground between CDP and LDP, where
+multiple users independently add noise to their update,
+that is sufficient to ensure their datasets are protected
+from an extraction adversary, but only as long as their
+updates are aggregated before the adversary observes
+them. Through combination with SA [2], [11], [27] or
+similar aggregation methods [6], where the server can
+only view aggregated updates, DDP ostensibly ensures
+that the server cannot extract individual data points. But,
+importantly, this assumes that a large fraction of users
+participating in the FL round are honest and add their
+share of the noise. We challenge the applicability of this
+assumption in the real world.
+3. Threat Model and Assumptions
+We characterize our threat model and assumptions in
+terms of our adversary and the considered FL setup.
+3.1. Adversary
+Our adversary is the server, and their goal is to infer
+individual users’ local sensitive data points. Note that the
+3
+
+background in Section 2.1 implies that the server can—
+whenever they choose to—(1) control the weights of the
+shared model, (2) select which of the N users participate
+in each round, and (3) provision new users into the pool
+(including sybils controlled by the server).3 We assume
+that the total fraction of sybils out of the N users is small.
+We, furthermore, assume the server is occasionally
+malicious (OM), meaning they behave maliciously in only
+a few rounds of the protocol. When this happens, they can
+exert the above capabilities (1-3) adversarially. Do note
+that the server here does not deviate from the protocol and
+restricts themselves to only use valid operations (1-3) in
+the FL protocol. An OM server ensures the attack remains
+stealthy, and also allows the server to train a model that
+has high utility over the non-malicious rounds, which is
+an expected product of FL.
+3.2. FL Setup
+Our departure point are FL protocols deployed in real-
+world applications, such as the one described in [8]. These
+protocols initially focused on data minimization only. We
+extend them with SA and DDP, two defenses dedicated
+to additionally providing privacy protection for the FL
+protocol. We note that we chose to study an instantiation
+of FL with SA and DDP because it is the combination of
+published techniques that holds the strongest promise in
+the presence of an untrusted server.4 Our goal in studying
+FL with SA and DDP is to show that, even if servers
+(e.g., companies) adopted currently available best prac-
+tices from the academic community, end users might not
+get the promised privacy protection from FL. In addition
+to these challenges, we note that FL with SA and DDP
+is not as widely deployed as vanilla FL is. This is mainly
+due to some increased communication costs [5] and the
+computational overhead of adapted mechanisms [2], [27].
+Thus, our work conservatively characterizes the risk of
+privacy leakage from many instantiations of FL.
+4. Attacking FL under SA and DDP
+In this section, we study FL extended by DDP and
+SA—considered as a strongly privacy-protective instanti-
+ation of the protocol—and show that the server can still
+reconstruct sensitive information about the users’ training
+data. We also forge an intuition of what factors contribute
+most to the leakage. Based on our findings, in Section 8,
+we discuss future research and implementation towards
+private FL.
+For successful data reconstruction under DDP and
+SA, the server has to make use of the following three
+capabilities which it naturally holds in FL:
+1)
+Introducing sybil devices: The server needs to
+be able to introduce a fraction of manipulated
+devices in the FL protocol. These devices can
+3. Capabilities (2) and (3) have been demonstrated in the real world
+as Google researchers introduced 189 sybils devices into the Gboard
+FL system and made them participate in the protocol along with real
+users [40].
+4. Indeed, the key alternative to FL with SA and DDP would be FL
+with LDP (local DP guarantees) but this alternative is not appealing
+because it comes at a significant utility cost for the server.
+return arbitrary gradients, chosen by the server.
+In particular, they can contribute zero gradients
+to the SA.
+2)
+Controlling the user sampling: To ensure that the
+sybil devices are sampled for SA together with a
+target user, the server needs to control the user
+sampling.
+3)
+Manipulating the model weights: For improved
+data reconstruction performance, the server can
+manipulate the shared model’s weights, for ex-
+ample, relying on one of the methods discussed
+in Section 2.2.
+While the first two capabilities enable the server to
+circumvent SA and to leave the gradients of a target user
+with insufficient amount of noise for privacy protection
+under DDP, the third capability increases the amount of
+individual training data that can be reconstructed and
+extends the attack to other model architecture types.
+Attack flow. Our attack aims at reconstructing the private
+data of one target user per malicious round in the FL
+training. To do so, the attack needs to circumvent the SA
+(Section 4.1), then exploit the weak privacy guarantees of
+DDP from a user’s perspective (Section 4.2), and finally
+reconstruct the target user’s individual training data points
+(Section 4.3). See Figure 1 for the course of our attack.
+4.1. Circumventing SA
+In our attack, the server circumvents SA by sampling
+the target user together with maliciously controlled sybil
+devices for the given training round. Since for each round,
+M users are sampled for participation, the server needs to
+control at least M − 1 sybil devices. It has been shown in
+previous work [40] that inserting an arbitrary number of
+sybil devices into real-world FL deployments is practically
+feasible. Moreover, the assumption that the server controls
+all-but-one participants in a specific protocol round is not
+a strong one. The fraction of controlled sybil devices
+(thousands) in comparison to the number of total users
+(millions to billions) is negligibly small.
+Since SA-protocols provide their guarantees under the
+assumption that a certain fraction of users is honest [5],
+[9], it follows naturally that in the presence of the sybil
+devices, no guarantees can be provided to the target user.
+This is because when the gradients are aggregated over
+multiple users, and all but the target user contribute arbi-
+trary gradients, known to the server, the server can extract
+the target user’s gradients perfectly. In the easiest case,
+the sybil devices contribute all zero-gradients, such that
+the final aggregate directly only contains the target user’s
+gradients.
+Note that we do not claim that the SA or any of
+the underlying cryptographic primitives are broken. We
+solely observe that SA relies on the assumption that the
+clients participating in the execution are real clients and
+not maliciously controlled by the server [5]. However,
+in FL with an untrusted server, the users cannot verify
+this assumption. We show through our attack that this has
+severe implications on their privacy guarantees.
+Pasquini et al. [37] describe a different way to cir-
+cumvent SA based on the server sending out different
+4
+
+models to different users. While the models for non-
+target users produce zero-gradients, the target user’s model
+produces non-zero gradients which can be exploited for
+data reconstruction. An advantage of this method is that
+it does not require the server to control the user sampling,
+or to manipulate a fraction of users. Note however that
+in their scenario, DDP can still be efficiently applied if
+every user adds some noise to their (potentially zero)
+gradients. As a consequence, the total amount of noise
+can be sufficient to protect the gradients of the target user.
+Therefore, in our attack, we rely on the controlled sybil
+devices to circumvent the SA.
+Note that also alternative mechanisms to implement
+DDP, such as shuffling [6], [13] which can be put into
+place instead of SA, can be circumvented by our approach
+of inserting sybil devices with server-controlled gradients
+into the protocol.
+4.2. Exploiting DDP Guarantees
+If DDP is in place, the gradients of the target user will
+be slightly noisy—even with successful circumvention of
+SA. However, by design of DDP, the amount of noise
+added by each user is typically insufficient to provide
+a meaningful privacy guarantee from the user’s perspec-
+tive [27]. By meaningful privacy guarantees we mean,
+guarantees equivalent to what one would obtain in the
+LDP definition. This is in fact how DDP obtains a utility
+gain over LDP, which would have inserted sufficient noise
+to obtain per-user privacy guarantees that are independent
+of other users: DDP assumes all users will add enough
+noise so that the aggregate is sufficiently noised whereas
+LDP assumes each user adds enough noise to obtain
+privacy in isolation. As a consequence, in DDP, each user
+can add less noise locally than required for the desired
+total privacy level, resulting in more utility. In contrast,
+the guarantee provided by LDP allows the user to not
+trust the server or other users.
+Concretely, in an LDP version of FL, the noise added
+by each user depends solely on the noise scale σ and the
+clipping parameter c of the application. As a consequence,
+the local noise is sampled from a Gaussian distribution
+according to
+N(0, σ2c2).
+(1)
+In contrast, in DDP, the amount of noise added by each
+individual user additionally takes the number of users who
+participate in the round into account [46]. Assuming that
+M users are sampled for participation, this results in a
+local addition of Gaussian noise sampled from
+N
+�
+0,
+σ2
+M − 1c2
+�
+[46].
+(2)
+In Figure 2, we present the privacy-utility trade-offs re-
+sulting from training models on the CIFAR10 [29] dataset
+as a function of the total noise scale σ and the resulting
+models’ accuracy on a test set. We train the private
+models with a state-of-the-art framework for DP-training5
+5. https://github.com/ftramer/Handcrafted-DP. Note, however, that our
+reported accuracy and achieved privacy levels ϵ cannot directly be
+compared with the values reported in the repository. This is because
+we use different noise scales than they do and train the model for 100
+epochs while they only train for 30 epochs.
+in which all hyperparameters and model architecture are
+tuned for the task.
+Figure 2 provides two main insights. First, unsurpris-
+ingly, given the privacy-utility trade-offs mentioned above,
+the model utility decreases when the total noise scale σ
+increases. Second, the figure shows that the more users
+participate in a given training round, the less noise each
+user needs to add locally. This results from Equation (2)
+which relies on the total noise being aggregated over all
+participating users before sharing the aggregated gradients
+with the server.
+DDP assumes that each user is honest and adds the
+required noise to their gradients. However, if even one of
+the users adds less than the amount of noise it should add,
+the desired total privacy guarantees cannot be reached.
+Even worse, if, as described in Section 4, a target user in
+FL is sampled for participation solely with controlled sybil
+devices that do not provide any noise for aggregation, the
+local noise added by the target user represents the only
+protection for its gradients.
+These results mean that there is a tension between (1)
+the guarantee claimed by the server (and other users) in
+DDP and (2) the guarantee that a user who does not trust
+this server can rely on. This will lead the server optimizing
+for model utility to request that users add less noise to
+their gradients than what is needed for individual users to
+protect their data from leaking to an untrusted server.
+4.3. Reconstructing Data
+In Section 2.2, we presented different attacks that rely
+on manipulations of the shared model to extract individual
+users’ training data points. In principle, each of these
+attacks can be included to perform data reconstruction
+in our FL+SA+DDP setup. However, the attack by [51]
+extracts individual data points over several rounds of
+the FL protocol. In our setup, due to the server’s OM
+nature, single-round attacks are preferable. These allow
+the adversary to stay more inconspicuous and to train
+a more meaningful shared model throughout the benign
+rounds. The attack by [14] relies on manipulations of the
+model architecture, which are more detectable than manip-
+ulations of model parameters, such as in [7]. Finally, our
+experimental evaluation highlighted a significant advan-
+tage of [7] in comparison to [14] for data reconstruction
+under noise. [7]’s trap weights yield redundancy in the
+extracted data, i.e. the same data point can be extracted
+multiple times from gradients of different weight rows.
+We thoroughly investigate this effect in Section 5.3. The
+redundancy of extracted noisy data can be exploited to
+average out the effect of the noise and yield higher-fidelity
+data reconstruction, as we will show in Section 5.4. In
+contrast, due to the nature of their attack, in [14], each
+data point is only extractable once.
+5. Evaluation of the Attack against SA+DDP
+In this section, we present a practical evaluation of
+our attack against FL protected with SA and DDP. We
+first present our experimental setup. Then, we evaluate
+direct extraction of noisy gradients under DDP. We move
+on to experimentally evaluate the redundancy of extracted
+data with [7]’s trap weight approach, and discover how
+5
+
+Figure 2: Privacy vs. utility trade-offs under DDP/LDP. Each point on the blue line corresponds to a model trained on
+CIFAR10 with a clipping parameter c = 1, and the total noise scale σ indicated on the x-axis. Training was conducted
+over 100 epochs, the resulting privacy guarantees ϵ are reported. For the non-private baseline ϵ = ∞ (red line). We
+report accuracy loss with respect to this non-private baseline. The images depicted below the line plot display the
+rescaled noisy gradients of one data point of an individual user with noise calculated according to Equation (2) as a
+function of σ, c, and M in the training round. The more users participate, the less noise every user needs to add because,
+during aggregation, the total noise is determined by the sum of the individual noises. If, however, other users do not
+add noise, the locally added noise is the only privacy protection every individual user has (DDP reduces to LDP with
+weak privacy guarantees). As a consequence, there is a discrepancy between the privacy the user believes to get and
+the privacy they actually get: The images above the black line visualize the user’s belief on what the server can extract
+from their gradient, the images below the black line visualize what the server can actually extract under our attack.
+this redundancy can be exploited for higher-fidelity data
+reconstruction under noise. We illustrate this with empir-
+ical results reconstructing image and text data.
+5.1. Experimental Setup
+We operate in a cross-device FL setup and perform
+training on the CIFAR10 [29] dataset. We evaluate extrac-
+tion on different mini-batch sizes B ∈ {10, 20, 100}. We
+split the CIFAR10 training data at random and iid between
+the users. [7] shows that their trap weights’ extraction
+success is equal for iid and non-iid distribution, even for
+the most extreme scenario where every data point in a
+given mini-batch stems from same class. Following [7],
+we also experiment with the IMDB dataset for sentiment
+analysis and distribute data the same way.
+To evaluate different setups for DDP, we select
+{10, 100, 1000} users for participation in a given round
+of the FL protocol. To circumvent the SA, as described in
+the previous section, we sample one target user together
+with sybil devices which all return zero gradients. Other
+than that, we follow [7]’s experimental setup, use their
+six-layer fully-connected neural network and embedding
+architecture (their Table 7 and 8), initialize the first fully-
+connected layer with their trap weights for individual data
+point extractability. To project received gradients of the
+first fully-connected layer’s weight matrix back to their
+input domain, we rely on [7]’s Equation (5) which shows
+that it is sufficient to rescale the gradient of the weights
+with the inverse of the gradient of the bias for perfect
+data extraction. Note that in our case, due to DDP, noise
+is added not only to the gradient of the weights but also to
+the gradient of the bias. Therefore, not only the extracted
+Figure 3: Directly Extracted Data under DDP. Rescaled
+clipped and noised gradients from a mini-batch with 20
+data points from CIFAR10 dataset. DDP setup: c = 1,
+σ = 0.1, and M = 100.
+gradients but also our scaling factor are noisy, resulting in
+the rescaled gradients not being a perfect reconstruction
+of the original input data.
+We report the DDP-setup per-round through three
+parameters required to determine the noise magnitude
+according to Equation (2), namely the DP clip norm c, the
+DP noise multiplier σ, and the number of selected users in
+this round M. This enables us to understand how sensitive
+the attack is to choices for these hyperparameters without
+making any assumptions about other hyperparameters of
+the training run (e.g., the number of training steps).
+5.2. Noisy Data Extraction
+We first perform direct extraction from noisy gra-
+dients. The extraction of data points works exactly the
+same way as for vanilla FL [7], by initializing the model
+with trap weights before sending it to the target user, and
+then projecting their received gradients back to the input
+domain. Figure 3 shows the full resulting extracted data
+in a setup with a mini-batch of 20 data points, c = 1,
+σ = 0.1, and M = 100. While some noisy reconstructed
+6
+
+ε = inf
+= 592
+ = 33.97
+ = 5.41
+DP Models
+ε = 2.39
+Non-Private BaselineNumber of
+% Individually
+Reconstruction
+Benign Users
+Reconstructable Data
+SNRs
+1
+0.95
+0.014
+5
+0.75
+0.011
+19
+0.15
+0.010
+49
+0.0
+0.010
+TABLE 1: Influence of Fraction of Sybil Devices. Re-
+sults for FL+SSA+DDP setup with 50 participants, each
+holding B = 20 data points. We replaced a varying
+fraction of users by sybil devices and measured number
+of individually extractable data points from the target user
+and average SNRs over all their reconstructions. The more
+benign users participate in the round, the less effective
+data reconstruction becomes.
+data points resemble the original training data, other are
+less recognizable because their are an overlay of multiple
+data points, or dominated by the added noise. DDP setup:
+c = 1, σ = 0.1.
+Effect of Fraction of Sybils. We, furthermore, conducted
+experiments to quantify the effect of not replacing all other
+M − 1 users by sybil devices, but only a fraction of the
+other users. Therefore, we conducted experiments with
+extracting data from one round of the FL protocol with 50
+participants, each holding a mini-batch of 20 data points.
+Our goal was to study what privacy gain the presence of
+other benign users incurs on the target user. We visualize
+our results in Table 1. They suggest that when the target
+user is sampled solely with sybil devices (row 1), the
+server is able to extract 95% of their individual training
+data points individually, protected solely by the noise
+added locally according to DDP. The more benign users
+participate in the protocol round, the lower SNRs of the
+reconstructed data from the target user, and the fewer of
+the target user’s individual data points can be individually
+extracted. This effect stems from the aggregation within
+the SA, which overlays gradients from all users before
+sharing them with the server. Our results are aligned with
+findings by [7] who showed that averaging over several
+mini-batches (which is precisely the effect of the SA)
+degrades extraction success.
+Sufficiently Protective Noise. Deciding at which point,
+i.e., under the influence of how much noise, the recon-
+struction of a data point is sufficiently close to the original
+data point is orthogonal to this work. In particular, it will
+depend on the specific domain, task, and user-preference.
+However, users in FL should assume that the server can
+extract individual data points such as the ones depicted in
+Figure 3 from their gradients.
+In the following, we will show how the server can
+improve the fidelity of extraction by leveraging the re-
+dundancy of extractable data due to the trap weights.
+5.3. Redundancy in Extracted Data
+This section studies redundancy in extracted data of
+the trap weights method and their effect on the fidelity of
+reconstructed data.
+Redundancy in Extractable Data Points. We first study
+direct redundancy by analyzing how often each data point
+in a mini-batch with B = 100 is individually extractable
+0
+2
+4
+6
+8
+10
+# individual activations for data points
+0
+10
+20
+30
+40
+50
+60
+70
+80
+# occurances
+(a) Random weights.
+0
+5
+10
+15
+20
+25
+30
+35
+40
+# individual activations for data points
+0
+10
+20
+30
+40
+50
+60
+70
+# occurances
+(b) Trap weights.
+Figure 4: Number of Activations per Data Point. The
+number of times each of the 100 data points is individu-
+ally extractable from the model gradients. The same data
+points are individually extractable from many more dif-
+ferent gradients when using trap weights which enable us
+to use redundancy for better data reconstruction. Results
+are averaged over five different random and trap weight
+model initializations.
+0
+20
+40
+60
+80
+100
+# neuron activated by number of datapoints
+0
+50
+100
+150
+200
+250
+# occurances
+(a) Random weights.
+0
+5
+10
+15
+20
+# neuron activated by number of datapoints
+0
+100
+200
+300
+400
+500
+# occurances
+(b) Trap weights.
+Figure 5: Number of Activations per Neuron. Number
+of data points that activate each one of the 1000 neurons.
+Individual neurons are activated by fewer data points (less
+overlay) when using trap weights which enable better data
+reconstruction. Results are averaged over five different
+random and trap weight model initializations.
+from the rescaled gradients. The results depicted in Fig-
+ure 4 suggest that the trap weights, first of all, make more
+data points individually extractable in contrast to random
+model initializations, but also cause the same data points
+to be individually extractable from many more different
+weight rows’ gradients (up to 70 times over the 1000
+neurons and their respective weight rows).
+Sparsity in Extractability. We, furthermore, evaluate by
+how many data points each neuron gets activated. This
+is equivalent to the question how many data points cause
+a positive input to each neuron. Figure 5 highlights that
+with randomly initialized weights, neurons are activated
+by many more data points than with the trap weights,
+which causes that many data points overlay in a single
+gradient and we cannot extract them individually.
+To improve fidelity of reconstruction, we can leverage
+both the redundancy of extractable data and the sparsity
+in the extracted data. By averaging redundant noisy data
+points, the signal-to-noise ratio (SNR) of reconstructed
+data increases as noise averages out. We visualize this
+effect in Figure 6. Also sparsity can be exploited. The
+fewer data points activate a neuron, the fewer data points
+contained in the overlay of the rescaled gradient. Hence,
+each individual data point’s signal is more clearly present
+and identifiable in the rescaled gradients. In the following
+section, we will show how this can be used to yield
+higher-fidelity reconstruction in the image domain through
+clustering.
+7
+
+0
+10
+20
+30
+40
+50
+# averaged noisy samples
+1.46
+1.48
+1.50
+1.52
+1.54
+1.56
+1.58
+1.60
+1.62
+SNR
+#=1
+#=5
+#=10
+#=50
+Figure 6: Averaging out Noise. Mean value over #-many
+noisy reconstructions of the same data point (bottom); cor-
+responding mean image’s SNR (top). DDP setup: c = 1,
+σ = 0.1, and M = 100. Over an increasing number
+of reconstructions, the local noise averages out, yielding
+higher-fidelity images and increased SNR.
+5.4. Improving Noisy Data Reconstruction
+The previous sections highlight that DDP reduces to
+LDP with weak privacy guarantees from an individual
+user’s perspective when other users are untrusted with
+their noise addition. In this section, we show how we
+can even improve data reconstruction in this setup, further
+amplifying the small signal in the extracted gradients. We
+evaluate improvements for data reconstruction from noisy
+gradients computed under DDP on image and textual data.
+All improvements solely rely on post-processing steps to
+reduce the effect of the noise.
+Image Data. Due to the local clipping and noise addition
+by the users, the data points extracted from the gradients
+are not perfect reconstructions of the original data points.
+We can still improve reconstruction quality by leveraging
+redundancy and sparsity in the gradients to average out the
+added noise, as highlighted in the previous section. How-
+ever, without knowledge of the users data, the server has
+no means of determining which data points activate which
+neurons a priori. Therefore, it is unclear which rescaled
+gradients need to be averaged to improve reconstruction
+fidelity.
+To overcome this limitation, we employ similarity
+clustering. In this approach, the server first filters out ex-
+tracted data points with a SNR below 1. This prevents too
+noisy instances from degrading performance. In the fol-
+lowing Section 6, we will discuss why different extracted
+data points have different SNRs. Then, the server runs
+a simple k-means clustering on the extracted data, and
+finally averages all per-cluster data points. Thereby, we
+do not only leverage redundancy in individually extracted
+data points, but also the sparsity. The signal from gradients
+that represent an overlay of very few data points can mean-
+ingfully contribute to the improved signal. We evaluate
+this approach with different noise scales and mini-batch
+sizes B. Note that the number k of clusters has to be
+chosen in accordance with the mini-batch size if we want
+to be able to reconstruct every data point. Our evaluation
+suggests that clustering works best when k ≥ 2B.
+In Figure 7, we depict the results of our clustering on
+data points from the CIFAR10 dataset with a DDP setup
+with c = 1, σ = 0.1 and M = 100. The top row depicts
+10 original data points, the mid and bottom rows show
+the closest averaged clusters for mini-batches of size 10,
+and 20 respectively. The more instances are available for
+averaging, the better the resulting per-cluster averages.
+Textual Data.
+For the text classifier on IMDB, we
+initialize the weights of the embedding layer with a ran-
+dom uniform distribution (minimum=0.0,maximum=1.0)
+to
+create
+the
+inputs
+for
+the
+fully-connected
+layer,
+following
+[7].
+We
+then
+adversarially
+initialize
+this
+fully-connected-layer’s weights with the trap weights
+to perform extraction of the embeddings and invert the
+embeddings back to tokens using a lookup dictionary. In
+the presence of noise introduced for DDP, the extracted
+embeddings are slightly noisy. To overcome this, in
+presence of noise we perform the lookup by searching for
+the token with the closest embedding measured through
+the ℓ2 distance. Figure 8 shows performance of a single
+mini-batch language extraction in presence of DP. Just as
+with image data, here an attacker is capable of extracting
+the original sentence of the users, despite the applied
+noise. We do observe however that there is stochasticity
+involved—when parametrization does well on the data
+point by default, extraction gets low performance since
+the received gradient has an extremely low magnitude
+and the corresponding signal gets dominated by the
+noise. We turn to this phenomena in the next section.
+To summarize the results on image and textual data, we
+find that:
+•
+The trap weights [7] cause input data-diversity and
+redundancy in resulting gradients, which can be
+used to cancel out some of the applied noise.
+•
+NLP is not safe from attacks described in
+this paper, despite a more sophisticated input-
+embeddings mapping.
+•
+Despite using DDP, an attacker often can recon-
+struct semantic information on the individual user
+data points. This is because in the presence of
+untrusted other users, DDP reduces to LDP with
+weak privacy guarantees from the perspective of
+an individual user.
+•
+As shown in Figure 2, having users add more noise
+locally, without additional improvements of the
+protocol [44] comes with a significant decrease
+in utility which makes the solution less practical.
+6. Disparate Leakage over Model Gradients
+Throughout our experiments, we observe that with the
+exact same scale of noise added to all gradients, some
+extracted data points have a significantly higher SNR
+than others. This effect translates into different levels of
+semantic similarity in the extracted data with respect to
+the original data as we show in Figure 3. In this section,
+we explain this observation and sketch how it can be
+leveraged by the adversary to better extract data in the
+presence of noise.
+8
+
+Figure 7: Similarity Clustering to improve noisy data extraction. Original data points and average clusters obtained
+from the rescaled gradients depicted in Figure 3. First 10 original training data points from the CIFAR10 dataset (top
+row). Averaged clusters of 10 data points reconstructed from the gradients for mini-batch size B = 10 (mid row), and
+B = 20 with the first 10 examples depicted (bottom row). The numbers above the images indicate how many noisy
+reconstructions were averaged to obtain that image.
+0.0000
+0.0002
+0.0004
+0.0006
+0.0008
+0.0010
+Noise scale
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Proportion of
+recovered tokens
+Figure 8: Textual Data Extraction under Noise. Ex-
+traction performance under noise for DDP from language
+model on the IMDB dataset. Extraction remains success-
+ful, even in presence of noise. Occasional drops in perfor-
+mance occur because of near-zero gradients resulted from
+correct data classification, i.e. data points with very low
+original loss. Error bars correspond to a single standard
+deviation.
+6.1. Impact of Gradient Norm
+We find that the SNR of an extracted data point is
+tightly bound to the magnitude, i.e. the norm, of the
+respective gradients. In Figure 9, we depict the SNR
+in the rescaled clipped and noised gradients, i.e., the
+extracted data points, against their respective gradient
+norms. The figure shows that with higher magnitude
+gradients, the same amount of noise has less impact on
+the signal, whereas, with smaller magnitude gradients,
+the same amount of noise largely dominates the signal.
+Therefore, increased magnitude of model gradients results
+in an increased data leakage.
+The norm of a weight row’s gradients in the model
+depends on the model’s loss. In general, higher loss re-
+sults in higher magnitude gradients, in particular for the
+weight rows that most contribute to the loss. Intuitively,
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+norm of clipped and noised gradient
+0
+2
+4
+6
+SNR
+Figure 9: Gradient Norm vs. SNR. Norm of the clipped
+and noised gradients of 1000 weight rows against SNR
+in corresponding extracted data point, i.e. the rescaled
+gradients. With higher gradient norms, the SNR in the
+extracted data increases. DDP setup: c = 1, σ = 0.1, and
+M = 100.
+to increase data leakage from noisy gradients, the server
+could, therefore, manipulate the shared model to produce
+higher loss. In the best case, the loss would be caused
+by all weight rows in the fully-connected layer used
+for extraction with the trap weights. This ensures high-
+magnitude gradients at all the weight rows’ gradients and,
+thereby, enables enhanced extraction at all of them.
+6.2. Global vs. Local Effect of Clipping
+However, in DDP, before noise addition, users perform
+a clipping step, bounding the maximum per-layer gradient
+norm, and hence the extractable signal from a gradient
+update. More precisely, clipping bounds the total norm
+of a model layer’s gradients to the clipping parameter
+c. If all weight rows have high gradients, their joint
+norm will exceed c, and therefore, all of them will have
+to be scaled down to reduce the total norm to c. The
+effect is visualized in the middle row of Figure 10. It
+9
+
+Figure 10: Extraction Success with Additional Model
+Manipulations. Row 1 (top): Under trap weights only
+(baseline), gradients at different weight rows have varying
+SNRs under the same amount of added noise, depending
+on their magnitudes. Row 2 (middle): When the shared
+model is further manipulated (Section 6.3) and all weight
+rows contribute equally to a high loss, their gradients will
+be clipped, which results in equal information loss for all
+of them. Row 3 (bottom): When only a few weight rows
+contribute to a high loss, their gradients preserve a high
+magnitude over clipping, which allows for higher fidelity
+extraction. DDP setup: c = 1, σ = 0.1, and M = 10.
+Trap Weight Row 1
+Trap Weight Row 2
+Trap Weight Row n
+…
+…
+Class 1
+Class m
+Added Class
+1
+0
+0
+Figure 11: Amplification of Gradient Magnitudes. The
+misclassification of an example to the added class in-
+creases the magnitude of its gradient for weight rows that
+contribute to the loss.
+shows that in this scenario, the extracted data over all
+gradients has a relatively low signal, which yields low-
+fidelity reconstruction.
+Even though with DP and clipping, it is not possible
+to have high magnitude gradients over all weight rows,
+we note that the clipping is performed globally per-model
+layer. Hence, if only a few weight rows locally have a
+high magnitude gradient but all other weight rows have
+a low magnitude gradient, then their joint norm can be
+below c. As a consequence, no clipping will be performed.
+The effect of this scenario is visualized in the bottom
+row of Figure 10. It highlights that while most gradients
+yield pure-noise reconstruction, a few gradients contain a
+very high-fidelity reconstruction of the input data. These
+gradients correspond to individual neurons whose gra-
+dients were less affected by the clipping operation due
+to the local vs. global effect we described. This higher
+vulnerability of certain neurons is desirable for improved
+data reconstruction under DDP.
+6.3. Exploiting Global Clipping for Increased Ex-
+tractability
+The previous section highlights that the global per-
+layer clipping in DP still allows local parts of the gradients
+to be large. This can be exploited for higher-fidelity data
+extraction from neurons corresponding to these gradients.
+In this section, we sketch the idea for a possible model
+manipulation that gives an attacker the control on local
+gradient magnitudes to amplify this effect.
+We base our manipulation on a fully-connected neural
+network with two layers. The first layer is initialized with
+the trap weights for extraction and has a ReLU activation
+function. The second classification layer is modified to
+yield high loss (without knowledge of the user data).
+Therefore, we add an additional neuron to the classifi-
+cation layer, i.e., an additional class that does not occur
+in the data distribution. Then, we set most of the weights
+connecting the output of the previous layers’ neurons to
+this additional class to very small values, e.g. zero, and the
+weights for a few neurons’ output to high values, e.g. one.
+Due to the ReLU activation of the first layer, this
+layer’s outputs are positive. High weights, connecting
+neurons to the added class in the second layer, ”attribute”
+the loss of the misclassification to a few trap weight rows.
+As a consequence, only the gradients of these few trap
+weight rows obtain a high magnitude (as illustrated in Fig-
+ure 11 for the Trap Weight Row 2). All other trap weight
+rows (row 1 and n in Figure 11) have low-magnitude
+gradients. The overall norm of the layer’s gradients will
+stay below the clipping parameter c, hence no clipping will
+be applied, enabling high-fidelity data extraction from the
+Trap Weight Row 2.
+We instantiated the above construction with a fully-
+connected neural network consisting of 1000 neurons in
+the first layer and eleven neurons in the second layer for
+experimental evaluation. The first layer’s weights were
+initialized with trap weights, while the second layer was
+initialized with a Glorot uniform distribution. We then ma-
+nipulated the weights connecting to the eleventh (added)
+class and set varying fractions (10%, 30%, 50%, and
+100%) of them to one and the rest to zero. Using the
+CIFAR10 dataset, we performed data reconstruction.
+In Figure 10, we visualize the extracted data. For the
+top row, all weights of the second layer are initialized
+at random. In the middle row, 100% of the weights
+connecting to the added class are set to one, and in the
+bottom row, 10% of these weights are set to one and the
+rest to zero.
+We furthermore measured the SNRs of the extracted
+data and depict the results in Figure 12. The results are
+consistent to the observations from Figure 10: When few
+weight rows contribute to the high loss, their respective
+rescaled gradients, i.e. the extracted data points, have a
+higher SNR. This allows for higher-fidelity extraction. In
+contrast, when all weight rows contribute equally to the
+high loss, all their rescaled gradients have a similar (lower)
+SNR. This is because of all their gradients being (equally)
+affected by the clipping. In general, the more weight rows
+contribute to a high loss, the lower their individual SNRs.
+Our two-layer construction naturally integrates with
+other architectures starting with convolutional and embed-
+10
+
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+SNR
+0
+20
+40
+60
+80
+100
+120
+140
+160
+count
+modified
+original
+(a) 100
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+SNR
+0
+20
+40
+60
+80
+100
+120
+140
+160
+count
+modified
+original
+(b) 300
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+SNR
+0
+20
+40
+60
+80
+100
+120
+140
+160
+count
+modified
+original
+(c) 500
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+SNR
+0
+20
+40
+60
+80
+100
+120
+140
+160
+count
+modified
+original
+(d) 1000
+Figure 12: SNRs of Rescaled Clipped and Noised
+Gradients, i.e. the extracted data points. The first fully-
+connected layer used for extraction consists of 1000 neu-
+rons. The respective weight rows are initialized with our
+trap weights. The second layer is manipulated by adding
+an additional class neuron, and setting {100, 300, 500,
+1000} of the 1000 weights that are connected to this neu-
+ron to one. The remaining weights that go to this neuron
+are set to zero. The original baseline consists in randomly
+initialized weights for second fully-connected model layer.
+Noise with scale 0.001 is added to all gradients; clipping
+parameter c = 1. When fewer weight rows (in this case
+100) contribute to the high loss, the data points extracted
+from the respective rescaled gradients have the highest
+SNR (> 1.5), which allows for higher fidelity extraction
+(compare to Row 3 (bottom) in Figure 10).
+ding layers described in this work. We leave fine-tuning
+and the extension of our construction to architectures that
+end with more than two fully-connected layers for future
+work. However, the approach highlights that even when
+DDP is in place, the server can initialize a model to
+increase the likelihood of reconstructing points with high
+fidelity.
+7. Related Work
+We survey related work on privacy attacks against
+model gradients, in particular in the setup of FL.
+Passive Attacks against Vanilla FL. Phong et al. [39]
+were the first to show how gradients leak information that
+can be used to recover training data at single neurons or
+linear layers. Recent work [16], [39], [49], [54], [56], [57]
+proposed exploiting this leakage for data reconstruction,
+for example, through Generative Adversarial Networks
+[21] (GANs), or by solving a second order optimization
+problem.
+Active Attacks against Vanilla FL. Melis et al. [33] pro-
+posed membership [42] and property inference [3] attacks
+based on periodically analyzing the model updates in an
+FL setup. They consider both passive and active attackers
+in vanilla FL. Nasr et al. [35] assume active attackers (both
+users and server) for membership inference. Similarly to
+Melis et al., their attackers do not directly alter the shared
+model, but rather manipulate it through model updates
+(users through gradients, and the server by modifying the
+aggregated model update). Recent work [7], [14], [51],
+have shown that manipulating the shared model allows
+a malicious server to extract user data perfectly from
+the model gradients. In contrast to all these attacks that
+consider vanilla FL, our work attacks FL with additional
+extension for dedicated privacy protection through DDP
+and SA. As we discussed in Section 4.3, the extraction
+attacks for vanilla FL can be integrated into our attack
+flow for improved data extraction. We show this using
+[7]’s trap weights.
+Attacks against Hardened FL. To our knowledge, the
+only prior work in this vein is due to Pasquini et al. [37].
+This work noticed that in SA the server has the ability to
+dispatch inconsistent models to different users, and used
+this to circumvent SA entirely. As we note in Section 4.1,
+their attack against SA is not able to circumvent DDP
+since even when all users’s models but the target user’s
+model produce zero gradients, benign users would still add
+their share of noise. Thereby, privacy guarantees through
+DDP can still be achieved. Finally, while Pasquini et al.
+focuses on leveraging a specific capability of the server
+to circumvent a specific mechanism, we systematically
+study the server’s capabilities arising from FL’s pervasive
+centralization, and consequently offer a rich breadth of
+contributions over this work. We suggest a simpler and
+more powerful attack that relies on sybils introduced
+by the server to circumvent SA, compose this attack
+with other attacks relying on other capabilities to extract
+individual data points (the attack by [37] only extracts
+updates, not individual user data points) and attack a
+variant that also includes DDP.
+8. Discussion: Privacy-Preserving FL
+In this section, we discuss the following three main
+questions:
+•
+Q1: What is the core reason behind FL’s vulner-
+ability to privacy attacks as the one of this work?
+•
+Q2: How can the vulnerability be fixed?
+•
+Q3: What privacy guarantees can, as of now, be
+provided to FL users?
+8.1. Q1: What is the Root Cause of Vulnerability?
+We posit that the root cause of FL’s vulnerability to
+attacks, like the one introduced in this work, is the power
+imbalance in the centralized design of FL. At its core, FL
+is a highly centralized protocol where the server makes
+the final decisions. For example, DDP+SA is commonly
+regarded as a privacy-enhancing solution in modern FL,
+adding local noise and decentralizing the aggregation step
+of the original design, however, the server can circumvent
+the defense by controlling users and manipulating the
+shared model. In this paper, we demonstrate that even
+though the protocols behind DDP+SA and their funda-
+mental cryptographic primitives are correct, the underly-
+ing assumptions are not met when the server is malicious.
+We detail the factors enabling the different aspects of
+the attack presented in this work:
+11
+
+1)
+The server is able to control a fraction of users.
+2)
+The server provisions users for participation in
+each protocol-round.
+3)
+The server holds the power to manipulate the
+shared model.
+4)
+The users have no inherent way of verifying each
+other’s correctness.
+5)
+The users cannot meaningfully validate model
+updates.
+Decentralization can indeed help balance the power
+disparity, but it should be introduced very carefully and
+consider the system as a whole [38]. Next, we elaborate
+more on decentralization and other methods that can ad-
+dress the vulnerability.
+8.2. Q2: How to Fix the Vulnerability?
+We analyze how to fix the vulnerability by considering
+the following three approaches: (1) decentralization, (2)
+verification on the user-side to decrease trust assumptions
+made about the server, and (3) application of external
+hardware and (cryptographic) protocols that implement
+guarantees under the presence of an untrusted server.
+Decentralizing FL. Several approaches were introduced
+to decentralize parts of FL, or the entire protocol. We
+survey them and discuss their practical applicability.
+Decentralized methods for selecting a protocol round’s
+participants are available [4], [18], [41], [53]. For example,
+mechanisms where the users perform a self-sampling,
+such as [18], have been put forward. These allow users to
+decide about their participation and, thereby, reduce the
+power of the server. Anarchic FL [53] goes even further
+and lets users choose an arbitrary point in time to update
+the shared model with fully-individualized training pa-
+rameterization; the server observes user updates directly,
+but each user can individually determine their required
+privacy level and act accordingly (for example, train on a
+lot of data, or add a lot of noise). This does not only
+mitigate the attack vector where the server samples a
+target user along with sybil devices. If the users implement
+enough protection locally, the server cannot extract their
+private data. Yet, such mechanisms come with a significant
+increase in operational complexity and are faced with the
+major challenge of motivating users to provide truthful
+private data, compromising overall system utility.
+An interesting decentralized-FL attempt, Biscotti [41],
+addresses the issue of sybil attacks by offering a block-
+chain based protocol that selects users in a decentralized
+fashion based on past behavior that appears to demonstrate
+honesty. This is called Proof-of-Federation (PoF), and
+honesty is indicated by contributing updates to the model
+that appear close to many other updates (and are thus
+assumed to be of high quality). However, users can try
+to appear honest and yet act maliciously in ways that
+will not affect the update-quality metric, for example,
+by not performing their role when they are supposed to
+noise or verify updates (roles for round participants in
+the Biscotti protocol which have no performance quality
+metric). Biscotti would have to be extensively audited for
+vulnerabilities to this and other attacks before it can be
+safely and widely deployed. Ultimately, the designs of
+these decentralized FL systems are very different from
+FL’s original design, and usually from each other’s de-
+signs. At the time of writing, we are not aware of any
+prominent real-world FL deployments that adopt such
+decentralization.
+User-Side Verification. Alternatively to decentralization,
+or in addition to it, we can try to reduce the trust that users
+have to place in the server by performing verification on
+the user side.
+First, we discuss the verification of the shared model.
+Giving users the ability to verify the shared model’s
+integrity can prevent malicious manipulations against it,
+such as the ones of the trap weights that we integrated
+into our attack flow. Unfortunately, without any changes
+to FL, there is no full-proof way to distinguish weight
+manipulation from model weights resulting from legit-
+imate previous training. Users have no insights on the
+data of other users, updates are affected by stochasticity
+including sampling [43], non-deterministic hardware ele-
+ments [26], and large-scale FL applications with control
+flow mechanisms [8] that do not sample every user in
+every round [40]. Ultimately, this makes it impossible to
+track how the model evolves over time since a user gets
+only intermittent views of the shared model. Given these
+difficulties, we might consider solutions that modify FL
+to make manipulation-detection easier.
+Second, we discuss the option of users verifying other
+users. Privacy guarantees in DDP result from all users
+adding their share of noise to protect the aggregate of all
+gradients. If only one user does not add their share of
+noise, the claimed theoretical overall privacy guarantees
+cannot be reached. Our attack exploits this effect and
+exposes the gradients of a target user by omitting noise
+addition of all other users sampled in the protocol round.
+To prevent this attack vector, users have to verify
+that other users calculate their gradients correctly and
+add the correct amount of noise. However, due to the
+centralization in standard FL protocols, users do not have
+direct communication channels with each other. The cen-
+tral party acts as an intermediary in their interactions. As
+an alternative, FL could be deployed within a Public Key
+Infrastructure (PKI) [5]. However, a server that behaves
+semi-honestly is required during the key collection phase
+of PKI. Hence, users rely again on their trust in the server
+while nothing in the protocol prevents this server from
+acting maliciously and registering their introduced sybil
+devices to the PKI prior to training.
+In fact, protocols like SA rely on the assumption that
+users participating in the protocol are actual users and no
+sybil devices controlled by the server [5]. Yet, nothing in
+the integration of the protocol into FL verifies or enforces
+this assumption. As a consequence, users who do not trust
+the server will have to verify that other users participat-
+ing with them in a protocol round follow the protocol,
+correctly calculate their gradients, and add their share
+of noise. Verifying correct gradient calculations without
+users having to reveal their private data to each other,
+can, in principle, be implemented through zero knowledge
+proofs (ZKPs) where users commit to their private data
+and prove correct gradient calculation. However, in our
+attack, the sybil devices are controlled by the server. As a
+consequence, even when they compute correct gradients,
+the server will be able to subtract these from the aggregate
+12
+
+gradient to obtain the target user’s gradients and conduct
+extraction as described in this work. The same argument
+can be applied to correct noise addition.
+This motivates the need for users to verify that other
+users are no sybil devices. Prior sybil device detection in
+FL relies on analyzing gradients [15] under the assumption
+of a non-malicious server. This approach fails to prevent
+our attack where sybils are controlled by the server and
+can, as mentioned above contribute meaningful gradients,
+as long as the server can subtract these from the aggregate.
+Yet, if users in FL have a way to confidently determine if
+other users are sybil devices, and if users have a chance
+to refrain from contributing to the protocol under their
+presence, our attack can be mitigated.
+Finally, it is desirable to enable users to verify the
+whole FL application and its local execution. This allows
+them to make sure that their local client handles their data
+correctly, adds enough local noise, and is not manipulated
+by the server. However, in modern FL ecosystems the FL
+client applications are proprietary software encapsulated
+in dedicated partitions [48] largely inaccessible to users.
+Additionally, the developers of verification software are
+usually the same entity acting as the server, bestowing
+it with even more power. Therefore, we recommend that
+applications of purportedly privacy-preserving protocols
+should be open-source so that they are available for audits
+and verification by the community.
+Hardware and Protocol Support. As the last approach
+to fixing the vulnerability of FL, we discuss the support
+of dedicated hardware and (cryptographic) protocols to
+implement guarantees for the users.
+By relying on protocols that are based on trusted
+execution environments (TEE), e.g. [34] the server can be
+prevented from manipulating the shared model. This re-
+duces the success of data extraction as the one underlying
+our attack. Additionally, to make sure that users always
+receive a well-controlled and no a manipulated model, we
+suggest releasing the shared model publicly, for example,
+in a block-chain. This makes it impossible to manipulate
+and change a shared model after release.
+A drawback of this solution is that it offers outside
+attackers access to several intermediate model states. We
+argue that this is not too restricting, though, since the
+shared model is also sent out to a few hundreds or
+thousands of users during each round. Hence, there exists
+the possibility of the internal model states being leaked,
+anyways. Yet, [7] has shown that even non-manipulated
+ML models, under certain conditions, leak their private
+training data. Moreover, when it comes to TEEs, they
+are prone to side-channel attacks, e.g. [25]. Hence, we
+conclude that even under such hardware protection some
+privacy risk for users remains.
+Homomorphic encryption (HE) is a candidate solu-
+tion to protect user gradients against leakage to a ma-
+licious server. However, existing instantiations [55] do
+not prevent our attack since, for efficiency in training,
+users decrypt the aggregated gradients and apply them
+to their (unencrypted) local model. In our attack, the
+server controls sybil devices, hence, it obtains access to
+the decrypted aggregate and can extract the target user’s
+gradient by subtracting the sybil devices’ contributions.
+Finally, a cryptographic protocol that always adds
+enough noise to user gradients in an aggregation step
+would mitigate the privacy leakage of the attack presented
+in this work. The protocol should offer DP, but without
+making any questionable trust assumptions on other users.
+We envision a secure multiparty computation (SMPC)
+protocol that performs update aggregation, much like SA,
+but also ensures that the output is added with a sufficient
+amount of noise to implement DP before it is dispatched
+to the server. As long as within this protocol users do
+not learn about each others’ inputs, and the server only
+learns the aggregated (and noised) output, the protocol
+may be able to offer a comparatively favorable privacy-
+utility trade-off. To the best of our knowledge, so far, no
+protocol for jointly adding sufficient amounts of noise to
+user gradients in SMPC exists and given the gradients’
+high dimensionality, the costs of any such approach will
+most likely not be practical, yet.
+8.3. Q3: What Users Can Do Today?
+Let us assume that a user wishes to attain DP guar-
+antees, and with good reason, as DP is the most viable
+and protective form of privacy guarantee in use today. Our
+study of FL and its extensions demonstrates that the mere
+inclusion of a DP mechanism does not necessarily provide
+protection if this mechanism makes assumptions that are
+inaccurate in the context of the given system.
+We currently see two promising directions for users to
+attain strong privacy guarantees.
+Local Differential Privacy. The first one consists of
+implementing their full privacy protection locally, without
+trusting any other participant of the protocol, neither the
+server nor other users to contribute to their protection. One
+way to implement such protection is LDP with conserva-
+tive parameterization, and while ensuring that data point
+reuse is accounted for and prevented when needed [45].
+While LDP comes at cost of the utility of the shared
+models, approaches to successfully improve the trade-offs
+exist for FL deployments with large numbers of partic-
+ipants [44]. We think that improving LDP to desirable
+privacy-utility trade-offs is a promising direction.
+FL Protocols with Trusted Servers. The second option
+for users to obtain privacy guarantees is to opt-out of FL
+altogether, if they do not trust the server, potentially while
+trying to hide this, for example, by providing randomized
+“garbage” updates to the server. However, this approach
+comes with a corresponding utility cost and undermines
+the purpose of collaborative learning to produce a perfor-
+mant model that fits many diverse individuals.
+Finally, these options are only available if users can
+control the local FL application software, which is usually
+not the case (unless an opt-out option is provided).
+9. Conclusion
+Truly privacy-preserving ML must defend itself from
+attackers that are malicious and hence do not follow
+protocol. In this work, we presented a highly efficient
+data extraction attack against FL in a strongly protected
+deployment, namely with SA and DDP. Based on the
+attack’s success, we analyzed the question of the mini-
+mum trust model that is required to obtain meaningful
+13
+
+privacy guarantees for the users. We showed that FL can
+provide privacy guarantees if the users trust the server, or
+if they rely on adequate cryptographic protocols, and if
+adequate additional protection methods are in place. Most
+of the methods for protection aim at shifting power from
+the server to the conglomerate of users and come with
+significant costs or overhead. Therefore, such systems are
+not yet practically in place. As a consequence, as of yet,
+we recommend that users only participate in FL protocols
+that are orchestrated by a trusted server.
+Acknowledgments
+We would like to acknowledge our sponsors, who
+support our research with financial and in-kind contri-
+butions: Alfred P. Sloan Foundation, Amazon, Apple,
+Canada Foundation for Innovation, CIFAR through the
+Canada CIFAR AI Chair program, DARPA through the
+GARD program, Intel, Meta, NFRF through an Explo-
+ration grant, and NSERC through the Discovery Grant and
+COHESA Strategic Alliance, and the Ontario Early Re-
+searcher Award. Resources used in preparing this research
+were provided, in part, by the Province of Ontario, the
+Government of Canada through CIFAR, and companies
+sponsoring the Vector Institute. We would like to thank
+members of the CleverHans Lab for their feedback.
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+DP [12] formalizes the idea that no data point should
+significantly influence the results of an analysis conducted
+on a whole dataset. Therefore, the analysis should yield
+roughly the same results whether or not any particular
+dataset is included in the dataset or not. Formally, this is
+expressed by the following definition.
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+R a randomized algorithm. A satisfies (ε, δ)-DP with ε ∈
+R+ and δ ∈ [0, 1] if for all neighboring datasets D ∼ D′,
+i.e. datasets that differ on only one element, and for all
+possible subsets R ⊆ R
+P [A(D) ∈ R] ≤ eε · P [M(D′) ∈ R] + δ .
+(3)
+In ML, DP formalizes the idea that any possible
+training data point in a training dataset cannot significantly
+impact the resulting ML model [1], [36]. One notable ap-
+proach to achieve this is Differentially Private Stochastic
+Gradient Descent (DPSGD) [1]. DPSGD alters the train-
+ing process to introduce DP guarantees for weight update
+operations, and thereby protects underlying individual data
+points. Particularly, the gradient computed for each data
+point or a mini-batch of data points is first clipped in
+their ℓ2-norm to bound influence. Clipping of the gradient
+g({xi}b) for mini-batch b of data {xi}b is performed
+according to a clipping parameter c, by replacing g({xi}b)
+with g({xi}b)/ max(1, ||g({xi}b)||2
+c
+). This ensures that if
+the ℓ2-norm of the gradients is ≤ c, the gradients stays
+unaltered, and if the norm is > c, the gradient get scaled
+down to be in norm of c. After the clipping, Gaussian
+noise with scale σ is applied to the gradients of each mini-
+batch before performing the model updates.
+15
+

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+Counterfactual Explanations for Concepts in ELH
+Leonie Nora Sieger
+Paderborn University
+Paderborn, Germany
+Stefan Heindorf
+Paderborn University
+Paderborn, Germany
+Lukas Blübaum
+Paderborn University
+Paderborn, Germany
+Axel-Cyrille Ngonga Ngomo
+Paderborn University
+Paderborn, Germany
+ABSTRACT
+Knowledge bases are widely used for information management on
+the web, enabling high-impact applications such as web search,
+question answering, and natural language processing. They also
+serve as the backbone for automatic decision systems, e.g. for medi-
+cal diagnostics and credit scoring. As stakeholders affected by these
+decisions would like to understand their situation and verify fair
+decisions, a number of explanation approaches have been proposed
+using concepts in description logics. However, the learned concepts
+can become long and difficult to fathom for non-experts, even when
+verbalized. Moreover, long concepts do not immediately provide
+a clear path of action to change one’s situation. Counterfactuals
+answering the question “How must feature values be changed to ob-
+tain a different classification?” have been proposed as short, human-
+friendly explanations for tabular data. In this paper, we transfer
+the notion of counterfactuals to description logics and propose the
+first algorithm for generating counterfactual explanations in the
+description logic ELH. Counterfactual candidates are generated
+from concepts and the candidates with fewest feature changes are
+selected as counterfactuals. In case of multiple counterfactuals, we
+rank them according to the likeliness of their feature combinations.
+For evaluation, we conduct a user survey to investigate which of the
+generated counterfactual candidates are preferred for explanation
+by participants. In a second study, we explore possible use cases
+for counterfactual explanations.
+CCS CONCEPTS
+• Information systems → Semantic web description languages; •
+Computing methodologies → Description logics; • Human-
+centered computing → Empirical studies in HCI.
+KEYWORDS
+Description Logic, Knowledge Graphs, Machine Reasoning, XAI,
+Semantic Web
+1
+INTRODUCTION
+Knowledge bases are commonly used in web applications, includ-
+ing information retrieval [30], information generation [27], web
+search [5] and question answering [12]. A great share of work is
+concerned with securing correctness and completeness of knowl-
+edge bases [9, 20, 28] which often involve machine learning-based
+classification. Concepts in description logics (DLs) can serve as
+transparent, white-box models for binary classification. Many ap-
+proaches to learn concepts from positive and negative examples
+have been proposed [7, 11, 14, 17, 18, 31]. Such DL concepts can
+Person 1
+Female
+Person 2
+hasChild
+Male
+Person 3
+married
+Female
+Person 1
+Female
+Person 2
+Male
+Person 3
+married
+Female
+Person 1
+Female
+Person 2
+Male
+hasChild
+Person 3
+married
+Female
+Figure 1: The concept Female⊓∃hasChild.⊤ classifies Person 1 as a mother in KB 1 (green), while its corresponding counterfactuals
+in KBs 2 and 3 (red) are not classified as such.
+1
+Figure 1: The concept Female ⊓ ∃hasChild.⊤ classifies Per-
+son 1 as a mother in KB 1 (green), while its corresponding
+counterfactuals in KBs 2 and 3 (red) are not classified as
+such.
+directly be mapped to expressions in the Web Ontology Language
+(OWL) used in the Semantic Web [16, 24]. This makes DL concept
+learning a convenient machine learning tool to apply on webbased
+knowledge graphs like DBpedia [3], Wikidata [35], or YAGO [34],
+or semantic website data, as proposed by schema.org [15].
+Explaining algorithmic decisions of machine learning models
+to stakeholders has become increasingly important [1, 2, 13, 25]:
+subjects affected by model decisions would like to understand their
+situation and verify fair decisions; data scientists would like to
+debug and improve the model; regulatory entities would like to
+check the compliance with laws and regulations.
+However, as concepts increase in length and complexity, their
+practical utility decreases, making it increasingly difficult for stake-
+holders to understand, contest or alter decisions [36]. To increase
+acceptance and trust of stakeholders, counterfactual explanations
+have been proposed as a form of short, actionable explanations [26].
+Counterfactual explanations focus on an antecedent that would
+have caused a different outcome (classification) had it been the
+case [32]. Given a set of features 𝐴 and a classification 𝐵, a counter-
+factual statement takes on the form “If 𝐴 had not been true, then
+the classification would not have been 𝐵”, where in the current
+classification to be explained, both 𝐴 and 𝐵 are true.
+In this paper, we transfer the notion of counterfactuals to DLs and
+propose the first algorithm for generating counterfactual explana-
+tions in the DL ELH. ELH as it is an important description logic
+for many applications including the medical ontology Snomed [6]
+arXiv:2301.05109v1  [cs.AI]  12 Jan 2023
+
+Sieger et al.
+Table 1: ELH description logic constructs. For further de-
+tails, we refer to Lehmann and Turhan [23].
+Syntax Semantics
+Construct
+⊤
+ΔI
+top concept
+𝐶, 𝐷
+𝐶I, 𝐷 I ⊆ ΔI
+atomic concepts
+𝑟,𝑠
+𝑟 I,𝑠I ⊆ ΔI × ΔI
+atomic roles
+𝐶 ⊓ 𝐷
+𝐶I ∩ 𝐷 I
+intersection
+∃𝑟.𝐶
+{𝑥 ∈ ΔI|∃𝑦 ∈ ΔI
+existential restriction
+Concepts
+with (𝑥,𝑦) ∈ 𝑟 I ∧ 𝑦 ∈ 𝐶I}
+𝐶(𝑥)
+𝑥 I ∈ 𝐶I
+concept assertion
+ABox
+𝑟 (𝑥,𝑦)
+(𝑥 I,𝑦I) ∈ 𝑟 I
+role assertion
+𝐶 ⊑ 𝐷
+𝐶I ⊆ 𝐷 I
+concept subsumption
+TBox
+𝑟 ⊑ 𝑠
+𝑟𝐼 ⊆ 𝑠I
+role subsumption
+and protein-protein interaction networks [19]. Given an individual
+𝑥 with concise bounded description CBD(𝑥) and a concept 𝐶 that
+holds for 𝑥, we generate counterfactual candidates 𝑥 ′ with concise
+bounded descriptions CBD(𝑥 ′) for which the concept does not hold.
+Figure 1 shows an example. In line with Wachter et al. [36], we
+define counterfactuals as those candidates that are most similar
+to the original individual. Next, we rank them according to the
+plausibility that they appear in the real world, i.e., their “likeliness
+of the combination of their features.” We conduct a user survey
+to investigate if the selected counterfactuals are indeed preferred
+by users. Finally, we conduct a study to explore possible uses of
+counterfactual explanations.
+In what follows, Section 2 introduces preliminares, Sec-
+tion 3 describes related work, Section 4 introduces our al-
+gorithm to generate counterfactuals, and Sections 5 and 6
+conduct user studies to investigate user preferences and
+use cases. Finally, Section 7 discusses limitations. All data,
+code and materials needed for reproducing this work can
+be found at https://anonymous.4open.science/r/Counterfactual-
+Explanations-ELH-EBDA/.
+2
+PRELIMINARIES
+We give a brief overview of the description logic ELH, knowledge
+bases, their triple representation, the concise bounded description
+of entities, and the closed-world assumption. For further details,
+we refer the reader to Baader et al. [4], Brandt [6], Lehmann and
+Turhan [23].
+2.0.1
+The Description Logic ELH. In DLs [4], knowledge is repre-
+sented by concept descriptions built from atomic concepts 𝐶, 𝐷 ∈
+𝑁𝐶 and roles 𝑟,𝑠 ∈ 𝑁𝑅, where 𝑁𝐶 and 𝑁𝑅 are finite sets of concept
+and role names. As shown in Table 1, a concept in the description
+logic ELH [6, 23] can consist of the the top-concept (⊤), inter-
+sections (𝐶 ⊓ 𝐷), and existential restrictions (∃𝑟.𝐶). Their seman-
+tics is defined in terms of the interpretation I = (ΔI, ·I) which
+consists of the non-empty set ΔI, called interpretation domain,
+and the function ·I, called interpretation function, that assigns
+each 𝐴 ∈ 𝑁𝐶 a set 𝐴I ⊆ ΔI and each 𝑟 ∈ 𝑁𝑅 a binary relation
+𝑟 I ⊆ ΔI × I [22, 23]. Following Lehmann and Hitzler [22], we
+extend the definition to also assigns each individual name 𝑥 ∈ 𝑁𝐼
+an element 𝑥 I ∈ ΔI.
+2.0.2
+Knowledge Base. A Knowledge Base (KB) K consists of a
+TBox T and ABox A where the TBox defines concepts by means
+of concept inclusion (𝐶 ⊑ 𝐷) and role inclusion axioms (𝑟 ⊑ 𝑠)
+and the ABox defines individuals 𝑥 ∈ 𝑁𝐼 and their relations. An
+interpretation I is a model of the knowledge base K iff it satisfies
+all axioms in the TBox and ABox. An individual 𝑥 ∈ 𝑁𝐼 is an
+instance of a concept 𝐶 with respect to K, written K |= 𝐶(𝑥) iff in
+all models I of K, we have that 𝑎I ∈ 𝐶I. We say 𝐶 holds for 𝑥 in
+K. Otherwise, we write K ̸|= 𝐶(𝑥) and say that 𝑥 does not hold for
+C.1
+2.0.3
+Triple Representation. The ABox can be represented in the
+form of subject-predicate-object triples as follows: For each individ-
+ual 𝑥 ∈ 𝑁𝐼 , we obtain all atomic concepts 𝐶 ∈ 𝑁𝐶 and all relations
+𝑟 ∈ 𝑁𝑅 such that K |= 𝐶(𝑥) or K |= 𝑟 (𝑥,𝑥 ′) for an individual 𝑥 ′
+holds, respectively. Then each role assertion 𝑟 (𝑥,𝑥 ′) corresponds
+to a triple (𝑥,𝑟,𝑥 ′) and each concept assertion 𝐶(𝑥) to a triple
+(𝑥, rdf:type,𝐶).
+2.0.4
+Concise Bounded Description. We define the concise bounded
+description2 CBD(𝑥) of an individual 𝑥 as the set of all triples that
+contain 𝑥 as a subject (both of the form (𝑥,𝑟,𝑥 ′) and (𝑥, rdf:type,𝐶)).
+2.0.5
+Closed-world assumption. As is common practise for con-
+cept learning, we employ closed-world semantics [7, 14, 22]: For
+each individual 𝑥 ∈ 𝑁𝐼 and concept 𝐶 ∈ 𝑁𝐶, either K |= 𝐶(𝑥) or
+K ̸|= 𝐶(𝑥) holds. In the latter case, we say that 𝐶(𝑥) does not hold.
+This approach allows to handle ontologies as a database and is com-
+patible with the state-of-the-art concept learners DL-Learner [21]
+and EvoLearner [14], whose classifications we would like to explain
+with our approach.
+3
+RELATED WORK
+In the following, we introduce related work on counterfactuals and
+similarity measures for description logics.
+Counterfactuals. Stepin et al. [32] survey research papers on
+contrastive and counterfactual explanation generation. They com-
+pare definitions, summarize methods, and discuss how methods
+are grounded in theoretical approaches. While some approaches
+are based on white-box models such as decision trees [33], none
+of them generates counterfactuals for individuals in description
+logics.
+According to Wachter et al. [36], counterfactual explanations
+are statements of the form “Score 𝑝 was returned because 𝑉 had
+values (𝑣1, 𝑣2, ...) associated with them. If 𝑉 had values (𝑣′
+1, 𝑣′
+2, . . .)
+and all other variables had remained constant, score 𝑝′ would have
+been returned.” While many such explanations are possible, an
+ideal counterfactual alters values as little as possible and represents
+the ‘closest world’ under which score 𝑝′ is returned instead of 𝑝.
+1Note that the |= operator expresses reasoning and we employ a reimplementation of
+the fast instance checker [22] which follows the closed-world assumption. For example,
+the reasoner, derives 𝐷 (𝑥) from the assertions 𝐶 (𝑥) and 𝐶 ⊑ 𝐷.
+2Originally, the concise bounded description (CBD) was defined for RDF, cf., https:
+//www.w3.org/Submission/CBD/
+
+Counterfactual Explanations for Concepts in ELH
+Dandl et al. [10] generalize this idea and take further criteria into
+account. Given a statement of the form “If 𝑋 had not occurred, 𝑌
+would not have occurred”, they solve a multi-objective optimization
+problem with four objectives: (1) the prediction 𝑦 of the counter-
+factual 𝑥 ′ should be as close as possible to the desired prediction
+𝑦′; (2) The counterfactual 𝑥 ′ should be as similar as possible to the
+instance 𝑥; (3) feature changes should be sparse; (4) the counter-
+factual should have likely feature values/combinations. We use the
+approach by Dandl et al. [10] as a basis for our approach and adapt
+it to description logics.
+Similarity Measures for Description Logics. At the core of coun-
+terfactuals is the notion of ‘closest world’ and the question of how
+to operationalize this notion for description logics arises. Lehmann
+and Turhan [23] create a framework for semantic-based similarity
+measures for ELH-concepts. They define desirable properties and
+present a framework to construct similarities measures fulfilling
+many of the properties. However, we need to define a similarity
+measure for individuals and not for concepts.
+4
+COUNTERFACTUALS IN ELH
+Following Wachter et al. [36] and Dandl et al. [10] who defined
+counterfactuals for black-box machine learning models with fixed-
+size input vectors, we transfer their definition to individuals in
+description logic. Given an individual 𝑥 ∈ 𝑁𝐼 and an (atomic or
+non-atomic) concept 𝐶, 𝑥 ′ is a counterfactual candidate of 𝑥 with
+respect to 𝐶 iff
+K |= 𝐶(𝑥) and K ̸|= 𝐶(𝑥 ′)
+or
+K ̸|= 𝐶(𝑥) and K |= 𝐶(𝑥 ′)
+(1)
+i.e., the predictions of 𝑥 and 𝑥 ′ differ with respect to 𝐶. Without
+loss of generality, we focus on the case that 𝐶 holds in the original
+KB and should not hold for the counterfactual. The individual 𝑥 ′
+does not necessarily need to exist in the original KB K and can be
+a new hypothetical individual in an alternative knowledge base K′
+defined over the same atomic concepts and roles as K.
+Let 𝛿(𝑋,𝑌) := |𝑋 \𝑌 | + |𝑌 \ 𝑋 | be the edit distance between two
+arbitrary sets counting the number of additions and removals. A
+counterfactual candidate 𝑥 ′ is called a counterfactual of 𝑥 iff the
+edit distance
+𝛿(CBD(𝑥), CBD(𝑥 ′))
+(2)
+between the concise bounded descriptions of the individuals 𝑥 and
+𝑥 ′ is minimal among all counterfactual candidates. If the concept is
+⊤, no counterfactual exists in ELH and we define the edit distance
+as infinite.
+Among multiple counterfactuals, we prefer counterfactuals
+which are likely and we experimented with two variants of mea-
+suring likeliness: Let 𝑁𝐼 be the set of all existing negative individ-
+uals 𝑥 ∈ 𝑁𝐼 with K ̸|= 𝐶(𝑥). Moreover, given an individual 𝑥, let
+𝐶𝑅(𝑥) := {𝐶 ∈ 𝑁𝐶 |K |= 𝐶(𝑥)} ∪ {𝑟 ∈ 𝑁𝑅|∃𝑧 ∈ 𝑁𝐼 : K |= 𝑟 (𝑥,𝑧)}
+be the set of all atomic concepts and roles of 𝑥. We define the min-
+likeliness, variantmin of a counterfactual 𝑥 ′ with K ̸|= 𝐶(𝑥 ′) as the
+minimal distance:
+min
+𝑥 ∈𝑁𝐼
+𝛿(CR(x′), CR(x)).
+(3)
+The rationale is that a counterfactual is likely if another negative
+individual in the KB exists that is similar to it. As an alternative,
+variantmean, we define the mean-likeliness of a counterfactual 𝑥 ′
+with K ̸|= 𝐶(𝑥 ′) as the average distance:
+1
+|𝑁𝐼 |
+∑︁
+𝑥 ∈𝑁𝐼
+𝛿(CR(x′), CR(x))
+(4)
+The rationale is that a counterfactual is likely if it, on average, is
+similar to the other negative individuals in the KB.
+We decided not to take the objects of role assertions into account
+(as would be done by the CBD) because the objects of roles can be
+rather distinct in real-world knowledge bases, so that individuals
+rarely share both roles and role objects. For example, in the family
+ontology (cf., Figure 1), it is unlikely that two individuals are married
+to the same person.
+In practice, the best choice of similarity function depends on the
+dataset and use case.
+4.1
+Comparison to Dandl et al.’s
+Counterfactual Objectives
+Our definitions contain the four objectives 𝑜1–𝑜4 by Dandl.
+(𝑜1: Prediction) “A counterfactual instance produces the prede-
+fined prediction as closely as possible.” Our predefined classification
+of a counterfactual 𝑥 ′ is the opposite of the classification of 𝑥 as
+pointed out in Equation 1.
+(𝑜2, 𝑜3: Similarity, Sparseness) “A counterfactual should be as
+similar as possible to the instance regarding feature values” and
+“change as few features as possible.” Since all our CBDs (or “fea-
+tures”) are discrete and do not contain numeric values, 𝑜2 and 𝑜3
+collapse. We define the similarity according to the edit distance in
+Equation 2.
+(𝑜4: Likeliness) “Counterfactuals should have likely feature val-
+ues/combinations”. We compute the likeliness/plausibility of feature
+values as being close to existing individuals in the knowledge base,
+see Equations 3 and 4.
+4.2
+Generation of Counterfactuals
+Let K be a knowledge base, 𝐶 an ELH concept, and 𝑥 an indi-
+vidual 𝑥 for which the concept holds, i.e., K |= 𝐶(𝑥). Let 𝐷 ∈ 𝑁𝐶
+be a named concept and 𝑟,𝑟 ′ ∈ 𝑁𝑅 be role names. We construct
+counterfactuals 𝑥 ′, i.e. individuals most similar to 𝑥 for which𝐶(𝑥 ′)
+cannot be inferred. For this, we assume K’s TBox to contain only
+subsumptions 𝐶 ⊑ 𝐷 and 𝑟 ⊑ 𝑟 ′ of atomic concepts 𝐶, 𝐷 ∈ 𝑁𝐶
+and roles 𝑟,𝑟 ′ ∈ 𝑁𝑅. This assumption holds for a vast array of
+real-world knowledge bases [37].
+Algorithms 1 and 2 describe our approach to generate counter-
+factuals. First, a reasoner infers as many concept and role assertions
+as possible taking subsumptions into account (Algorithm 1, line
+5). Next, we create a deep copy K𝑖 of K for each subconcept 𝐶𝑖
+of 𝐶 = 𝐶1 ⊓ 𝐶2 ⊓ . . . ⊓ 𝐶𝑛 (line 7). The method get_individual
+retrieves the individual 𝑥𝑖 from K𝑖 corresponding to 𝑥 in K (line 8).
+We turn the 𝑥𝑖 into a counterfactual candidate with K𝑖 ̸|= 𝐶(𝑥 ′)
+(line 9) by removing axioms such that the subconcept 𝐶𝑖 does not
+hold anymore (and hence the concept 𝐶 does not hold for 𝑥𝑖 any-
+more). Algorithm 2 describes the generation of a single counter-
+factual candidate. Having generated the candidates, the candidates
+
+Sieger et al.
+1 Input: KB K, Concept 𝐶 = 𝐶1 ⊓ 𝐶2 ⊓ . . . ⊓ 𝐶𝑛 with 𝑛 ≥ 1,
+Individual 𝑥 such that K |= 𝐶(𝑥)
+2 Output: Counterfactuals of individual 𝑥 w.r.t. 𝐶
+3 Function gen_counterfactuals(K, 𝐶, 𝑥):
+4
+𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠 ← [ ]
+5
+K ← reasoner(K)
+6
+for each 𝐶𝑖 do
+7
+K𝑖 ← deepcopy(K)
+// Get copy of 𝑥 in K𝑖
+8
+𝑥𝑖 ← get_individual(K𝑖,𝑥)
+9
+K𝑖,𝑥𝑖 ← gen_candidate(K𝑖, 𝐶𝑖, 𝑥𝑖)
+10
+𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠.append((K𝑖,𝑥𝑖))
+11
+end
+12
+cfs_min ← arg min
+(K𝑖,𝑥𝑖) ∈𝑐𝑎𝑛𝑑𝑖𝑑𝑎𝑡𝑒𝑠
+𝛿(CBD(𝑥), CBD(𝑥𝑖))
+13
+cfs_mean ← deepcopy(cfs_min)
+14
+Sort cfs_min by min-likeliness
+15
+Sort cfs_mean by mean-likeliness
+16 return cfs_min, cfs_mean
+Algorithm 1: Generates two lists of counterfactuals of indi-
+vidual 𝑥 w.r.t. concept 𝐶—sorted by min-likeliness and mean-
+likeliness.
+with the least edit distance as defined in Equation 2 are selected
+as counterfactuals. All counterfactuals are then rated according
+to the two variants of likeliness (see Equations 3 and 4) and the
+algorithm returns two sorted lists of counterfactuals—one list for
+each measure.
+4.3
+Analysis of the Generation of
+Counterfactuals
+In the following, we argue that every step carried out by the al-
+gorithms is necessary and that the steps carried out are sufficient.
+ELH allows concept intersection, existential restrictions and role
+hierarchy (see Table 1). Because the concept 𝐶 is in ELH, we can
+assume that each 𝐶𝑖 (Algorithm 1, line 9) does not contain an in-
+tersection on the outer level anymore (such as 𝐶𝑖 = 𝐴 ⊓ 𝐵) and all
+remaining intersections must be within an existential restriction
+(such as 𝐶𝑖 = ∃𝑟.(𝐴 ⊓ 𝐵)). If the concept 𝐶 is an intersection, Al-
+gorithm 1 splits it into the subconcepts 𝐶𝑖 to apply Algorithm 2
+to each subconcept 𝐶𝑖. If K ̸|= 𝐶𝑖 (𝑥) for any one 𝐶𝑖, it follows
+that K ̸|= 𝐶(𝑥) does not hold. Creating additional candidates by
+making multiple 𝐶𝑖 not hold at once is not necessary, because these
+candidates could not exceed other candidates in minimizing the
+edit distance. If 𝐶𝑖 ∈ 𝑁𝐶, i.e. it is an atomic concept, all of the
+individual’s assertions to this concept or its subsumed subconcepts
+are removed (Algorithm 2, line 7). Similarly, if 𝐶𝑖 is an existential
+restriction 𝐶𝑖 = ∃𝑟.𝐴, all role assertions 𝑟 ′(𝑥,𝑎) with K |= 𝐴(𝑎)
+and K |= 𝑟 ′ ⊑ 𝑟 are removed (Algorithm 2, line 9). This happens
+regardless of whether 𝐴 is a concept, another existential restriction,
+or an intersection. This is sufficient, because we aim only to remove
+axioms with the individual as subject to generate counterfactuals.
+1 Input: KB K, Concept 𝐶, Individual 𝑥 such that K |= 𝐶(𝑥)
+2 Output: KB K, Individual 𝑥 such that K ̸|= 𝐶(𝑥)
+3 Function gen_candidate(K, 𝐶, 𝑥):
+4
+if 𝐶 ≡ ⊤ then
+5
+K ← 𝑁𝑜𝑛𝑒
+6
+else if 𝐶 ∈ 𝑁𝐶 then
+7
+Remove assertions {𝐷(𝑥) | K |= 𝐷 ⊑ 𝐶 ∧ 𝐷 ∈ 𝑁𝐶}
+from K
+8
+else if 𝐶 = ∃𝑟.𝐴 then
+9
+Remove assertions
+{𝑟 ′(𝑥,𝑎) | K |= 𝐴(𝑎) ∧ K |= 𝑟 ′ ⊑ 𝑟} from K
+10 return (K,𝑥)
+Algorithm 2: Generates a counterfactual candidate (K,𝑥) by
+changing the CBD of 𝑥 such that K ̸|= 𝐶(𝑥).
+Table 2: Overview of the final, modified datasets in terms of
+number of instances (𝑁𝐼 ), axioms, atomic concepts and roles.
+Instances
+Axioms
+Atomic
+Roles
+Dataset
+Concepts
+Family
+202
+2,033
+18
+5
+Animals
+28
+170
+19
+4
+5
+EXPLANATIONS PREFERRED BY USERS
+We conducted a survey in which we let participants rate different
+potential counterfactual explanations against each other. We then
+compared the participants’ preferences with the decisions made by
+our approach. We used modified versions of the Family and Animals
+ontologies [37] to evaluate our approach. These ontologies were
+chosen because the concept, role and individual names therein are
+familiar and understandable to average lay users—in contrast to, for
+example, ontologies related to bio-medicine or chemistry. We used
+the DL concept learner [8] with ELTL—the EL Tree Learner [7]—to
+learn the concepts to be used for counterfactual generation, since a
+future goal is to combine these programs to reach a fully automated
+explainable AI.
+5.1
+Data Generation
+Table 2 gives an overview of the datasets used for our user survey.
+5.1.1
+Family Ontology. We added a super role hasPartner of
+married to the original family ontology because the original on-
+tology had none and we wanted it to include a role hierarchy, so
+that we had an actual example of ELH and not just EL. To ob-
+tain (complex) concepts for the generation of counterfactuals, for
+each atomic concept present in the ontology, we did the following:
+First, we removed the atomic concept from the ontology. Then, we
+let the DL concept learner [8] with ELTL learn a concept using
+10 randomly chosen individuals that formerly were instances of
+the atomic concept as positive examples, and 10 random others as
+negative examples. Thereafter, we randomly selected an individual
+which is an instance of that concept and applied our counterfac-
+tual algorithm. We manually inspected the learned concepts and
+used the correct concepts for the survey (e.g. concept Father leads
+
+Counterfactual Explanations for Concepts in ELH
+to Male ⊓ ∃hasChild.⊤). For consistency, we also queried for the
+concepts for Brother and Grandmother as the corresponding con-
+cepts to Sister and Grandfather in the survey, even if ELTL did
+not correctly recognize the concept. This way, we ended up with
+Mother/Father, Sister/Brother and Grandmother/Grandfather
+for the survey.
+5.1.2
+Animals Ontology. We added two super roles to the animals
+ontology, i.e., residence and home, for the same reasons as named
+above. Furthermore, we restructured the ontology to fit ELH se-
+mantics. The atomic concepts of species were removed and their
+roles added to the individual animals. This way we could train
+ELTL [8] to learn a concept for each species, using the instance be-
+longing to that species as positive example, and all other instances
+of animals as negative examples. Our algorithm was applied after-
+wards. We collected all concepts that could be used for explanations
+(see code at https://anonymous.4open.science/r/Counterfactual-
+Explanations-ELH-EBDA/ for details) and chose 6 of them randomly
+to use these and their counterfactual candidates for the survey.
+5.2
+Setup of User Survey
+The
+full
+survey
+material
+and
+data
+can
+be
+found
+at
+https://anonymous.4open.science/r/Counterfactual-Explanations-
+ELH-EBDA/. Using the generated concepts from the family and
+animals ontologies (see above) and their respective counterfactual
+candidates, we generated short stories of artificial intelligences
+classifying people in a family tree or animals and created a
+counterfactual explanation from each counterfactual candidate.
+We conducted an online survey via SoSciSurvey from May 1,
+2022 to May 4, 2022 in German. Participants were recruited
+through social networks and snowballing. On the first pages,
+participants were informed about the content and goal of the
+survey and what counterfactual explanations are, and we collected
+sociodemographic data about age, gender and occupation. On the
+next pages, the counterfactual explanations were presented. First, a
+scenario was described in which an AI would classify instances of
+family members or animals. Then, on every page, a classification
+made by an AI was presented in one sentence, followed by one
+or multiple sentences giving counterfactual explanations for
+the classification, e.g. “I would not have classified this animal
+as a turtle, if it did not have scales”. Within the two scenarios,
+classifications were presented in randomized order, one on each
+page. For the family ontology, where each concept had led to
+two counterfactual candidates, the participants were randomly
+shown only one of the counterfactual explanations. However,
+because many explanations were quite similar (e.g. all concepts
+included counterfactual candidates referring to gender) it was
+made sure that they were presented mixed explanation types. Each
+explanation was accompanied by one item asking to rate on a scale
+from one to seven how helpful they perceived the explanation
+for understanding the decision of the program. For the animals
+scenario, participants were shown all counterfactual candidates
+(between two and five) at the same time, in random order, and
+presented the same rating scale for each of the explanations.
+5.3
+Results of User Survey
+In the following, we present the results of our evaluation of our
+counterfactual rating algorithm through a user survey.
+Sample. 72 people took part in the survey. Age ranged between
+20 and 69 (average = 34.9, median = 32, standard deviation = 12.1,
+missing age data for one participant). 30 participants were female,
+39 male and 3 diverse. Participants had mixed professions including
+both academic and non-academic ones, technical and non-technical.
+Family Ontology. We used Wilcoxon signed-rank tests to calcu-
+late significance of deviation from the central value (4 on a scale
+from 1-7) for each item and to compare both items of each concept
+with another. Tests were chosen given the fact that we use ad-hoc
+generated items, so we assume the ratings to be ordinal. Deviation
+results are presented in Table 3. For all six concepts, users preferred
+the explanation that featured a role referring to relatives of the
+individual (hasChild, hasSibling) against the explanation that
+featured a 𝐶𝑖 referring to the individuals gender (all p<.001, ex-
+cept Sister: p<.01). These explanations (and only these) differed
+significantly from the central value (all p<.001).
+Comparison of user ratings with algorithmic decisions. For all
+the 4 individuals processed by our algorithms, the subconcepts
+𝐶𝑖 ∈ 𝑁𝑅 were found to be counterfactuals by our algorithm, while
+the subconcepts 𝐶𝑖 ∈ 𝑁𝐶 were not. Consequentially, both variants
+came to the same decision as the participants.
+Animals ontology. We used Wilcoxon signed-rank tests as above
+(but for matched samples). For concepts with more than two coun-
+terfactuals, we used the Friedmann test.
+5.3.1
+User evaluations of explanations. Table 3 presents results for
+Wilcoxon test of deviation from center and if and which algorithm
+variant chose this counterfactual. “Subconcept 𝐶𝑖” describes the
+subconcept that was used for explanation, e.g. hasLegs means the
+counterfactual explanation contained “(...) if it did not have legs”.
+5.3.2
+Differences in user evaluations. For each animal, we calcu-
+lated which differences between the counterfactual explanations’
+ratings were significant with a p value <.05. The difference be-
+tween the counterfactuals of Girl were not significant. For Snake,
+habitat was rated significantly worse than the other counterfac-
+tuals. For Penguin, only the difference between the best and worst
+counterfactual was significant. For Eagle, the lowest rated two ex-
+planations (homeothermic and hasLegs) were rated significantly
+less helpful than the others. Regarding Turtle, only the difference
+between the best (hasCovering(Scales)) and worst (hasLegs) ex-
+planation was significant. Finally, for Crocodile, differences were
+not significant.
+5.3.3
+Comparison of user ratings with algorithmic decisions. Al-
+though in the animals ontology, explanations similar to each other
+existed, too, participants’ decision about these varied. The concept
+hasEggs was rather popular, while hasLegs was not. Concepts
+habitat/residence were preferred 2 of 4 times. For hasCovering,
+results were mixed; concept homeothermic was never chosen. We
+suspect that participants preferred explanations referring to fea-
+tures they find characteristic of the animal or groups of animals.
+For the animals ontology, all 𝐶𝑖 were counterfactuals. Measuring
+
+Sieger et al.
+Table 3: Ratings of counterfactual explanations (Wilcoxon
+signed-rank test). The test statistics 𝜇, 𝑍, and 𝑝 refer to the
+user preferences of a subconcept 𝐶𝑖 of the named concept
+in the first column (e.g., Mother ≡ Female ⊓ hasChild.⊤). ES
+stands for Effect Size. Subconcepts in italic were chosen by
+algorithm with likeliness variant min. Subconcepts in bold
+were chosen with variant mean. The star(*) indicates p <.01
+and double-star(**) indicates p<.001.
+Name
+Subconcept 𝐶𝑖
+𝜇
+Z
+p
+ES
+Mother
+Female
+4.3
+-1.16
+.25
+Mother
+hasChild(⊤)
+6.1
+-4.83
+.001**
+0.84
+Father
+Male
+4.1
+-0.24
+.81
+Father
+hasChild(⊤)
+6.2
+-5.07
+<.001**
+0.81
+Sister
+Female
+4.5
+-1.73
+.08
+Sister
+hasSibling(⊤)
+5.9
+-4.11
+<.001**
+0.71
+Brother
+Male
+4.3
+-1.02
+.31
+Brother
+hasSibling(⊤)
+5.9
+-4.88
+<.001**
+0.78
+Grandmother
+Female
+4.3
+-0.87
+.38
+Grandmother
+hasChild(Parent)
+6.1
+-4.88
+<.001**
+0.78
+Grandfather
+Male
+4.3
+-0.79
+.43
+Grandfather
+hasChild(Parent)
+6.0
+-4.51
+<.001**
+0.79
+Girl
+HasLegs
+4.3
+-1.67
+.09
+Girl
+residence(⊤)
+4.6
+-2.70
+<.01*
+0.32
+Snake
+HasEggs
+4.6
+-2.99
+<.01*
+0.35
+Snake
+habitat(Land)
+3.6
+-0.04
+.97
+Snake
+hasCovering(Scales)
+5.0
+-4.36
+<.001**
+0.51
+Penguin
+HasEggs
+4.9
+-4.07
+<.001**
+0.48
+Penguin
+homeothermic
+4.0
+-0.65
+.50
+Penguin
+HasLegs
+4.4
+-2.08
+.04
+0.25
+Penguin
+habitat(Water)
+4.3
+-1.82
+.07
+Penguin
+hasCovering(Feathers)
+4.5
+-2.49
+.01
+Eagle
+HasEggs
+5.0
+-4.56
+<.001**
+0.54
+Eagle
+homeothermic
+3.9
+-0.32
+.75
+Eagle
+HasLegs
+4.1
+-0.79
+.43
+Eagle
+habitat(Air)
+5.2
+-5.21
+<.001**
+0.61
+Eagle
+hasCovering(Feathers)
+5.5
+-5.92
+<.001**
+0.70
+Turtle
+HasEggs
+4.7
+-3.90
+<.001**
+0.46
+Turtle
+HasLegs
+4.1
+-1.14
+.25
+Turtle
+hasCovering(Scales)
+3.6
+-0.04
+.96
+Crocodile
+HasEggs
+4.6
+-2.87
+<.01*
+0.34
+Crocodile
+HasLegs
+4.3
+-1.50
+.13
+Crocodile
+hasCovering(Scales)
+4.7
+-3.42
+<.001**
+0.40
+the match of our algorithms with the participants’ ratings using
+F-score (2 · precision·recall
+precision+recall ), variantmin reached a score of 0.3 while
+variantmean reached 0.52.
+Interpretation of results. Overall, the survey results over both on-
+tologies can be interpreted as participants preferring explanations
+that contain features which are rather unlikely. We assume that this
+is because such features are the most salient to distinguish between
+individuals. In contrast, features which are very common (like be-
+ing of a certain gender or having legs) seem to be deemed less
+helpful. This fits our likeliness measurement idea, since removing a
+feature that does not appear often in the population for candidate
+generation should also result in a rather high likeliness score using
+our definitions. Our algorithm partly manages to cover that, but
+could be improved. variantmean seems to be more promising for
+that.
+6
+STUDY ON USE CASES
+To assess possible use cases for counterfactual explanations in de-
+scription logics, we conducted an explorative study. We wanted
+to know about users’ preferences for concept-based and/or coun-
+terfactual explanations, and how different use cases or con-
+cept length affect these. The full study material and data can
+be found at https://anonymous.4open.science/r/Counterfactual-
+Explanations-ELH-EBDA/.
+6.1
+Method
+We conducted an online survey via SoSciSurvey in German. Par-
+ticipants were recruited from August 8, 2022 to August 15, 2022.
+Recruiting and setup of the study were similar to the survey. This
+time, participants were confronted with 4 different scenario stories.
+The first two were similar to the ones in the survey (based on the
+family and animals ontologies), the others were fictional stories
+not based on existing ontologies. One was about an AI helping
+select the right medicine for a fictional person and in which case
+another medicament would be the better choice. The other was
+about an AI deciding that the customer, called Clara, does not get a
+loan from the bank and what she could do to change this. In this
+study, participants were both presented an explanation based on
+a concept (e.g. “I classify Petra as a mother, because she is female
+and has a child”) and a counterfactual explanation to rate. For each
+scenario except the family scenario, participants were randomly
+assigned to one of two groups: For one, a long concept (size >3) was
+used for explanation. The second group was presented with shorter
+concepts. Participants were asked to rate each information on a
+scale from 1-7 using three items: helpfulness for understanding the
+AI, usefulness, and if the information enhances controllability of
+the AI.
+6.2
+Results
+In this section, we present the results of our explorative study on
+explanation use cases.
+6.2.1
+Sample. 96 people took part in the survey. Age ranged be-
+tween 20 and 70, mean = 35.0, median = 32, standard deviation
+= 11.0. 46 participants were female, 46 male, 3 diverse, one left
+the question blank. 17 had professions related to IT, 33 worked
+in unrelated fields, 31 did not state it clearly enough to tell (e.g.,
+student).
+6.2.2
+Factors related to explanation rating. For each scenario and
+item, we used the Mann-Whitney U test to test for differences in
+concept-based explanation ratings from concept length. None of
+the results was significant (p>.01). In one case, a difference between
+concept-based and counterfactual explanation was significant: In
+
+Counterfactual Explanations for Concepts in ELH
+the loan scenario, the counterfactual explanation was rated signifi-
+cantly higher on the item “This information is useful for Clara” than
+the long concept-based explanation (𝑍 = -3.29, 𝑝 = <.001, Effect size
+= .34). To explore differences in ratings over scenarios, we used the
+Friedmann test. We found that, again for the usefulness-item, the
+counterfactual explanation was significantly rated higher in the
+loan scenario than in the medicine or animal scenario (𝜒(3) = 25.10,
+𝑝 <.0001) and also higher than in the family scenario. However, the
+latter difference was not significant by itself. Concept ratings were
+not affected by scenario (Wilcoxon-signed rank test). All explana-
+tions had high ratings (average ratings ranging between 5.13 and
+6.39).
+7
+DISCUSSION
+We showed that our approach is able to generate counterfactuals
+with minimal edit distance measured by the distance of their concise
+bounded descriptions, i.e. few features changes to the individual.
+Moreover, as there can be multiple counterfactuals per individual
+with minimal edit distance, we explored two likeliness measures
+to choose among them. Regarding choosing the best counterfac-
+tual for an explanation, the results of our evaluation survey show
+room for improvement, but potential that an automated selection
+of explanations can be possible.
+Concept-based vs. counterfactual explanations. The explorative
+study gave us insight into people’s thoughts regarding concept-
+based versus counterfactual explanations. While participants rated
+all explanations rather positively, we did not find much differences
+regarding the factors we took into account. However, regarding the
+loan scenario, we found that the counterfactual was rated more use-
+ful than the concept-based explanation. The loan and the medicine
+scenario differed from the other two, that were already used in the
+first survey, in the fact that the AI’s classification directly affected a
+person. However, while the counterfactual was unclearly actionable
+(have a better cholesterol level) in the medicine scenario, the loan
+scenario’s counterfactual explanation was designed to provide a
+concrete possible action (pay off debts) to change the classifica-
+tion. We suppose that this is the reason this explanation scored
+significantly higher on usefulness than the long concept and other
+counterfactuals. Overall, our results suggest that counterfactuals
+perform at least as well as concept verbalization w.r.t. explaining
+algorithmic decisions. In addition, the study indicates that study-
+ing domains where counterfactual explanations lead to actionable
+decisions might be worthwhile.
+7.1
+Limitations and future work
+First of all, the target in our approach was to create a counterfac-
+tual that would not be classified as the chosen concept. However,
+in many applications, reaching a specific classification could be
+relevant. Furthermore, we restricted the TBox to non-complex con-
+cept subsumptions, as such subsumptions rarely occur in practical
+ontologies. Future work might relax this assumption. The concepts
+we used were rather simple. A description logic that contains dis-
+junctions could be used for applications where explanations are
+more interesting to users, so we plan to expand our algorithm to
+the more complex DL ALC.
+Structure of ontologies. A point open for discussion addresses
+different structures of ontologies. As it is the case in the family
+ontology, sometimes individuals might have or have not the same
+roles, while it is very unlikely for them to have the same objects for
+these roles, as argued before. We therefore decided to just compare
+the sets of classes and roles of the individuals. For other ontologies,
+this might not be a good decision. The same goes for the question if
+subfeatures of the changed feature should be counted into the edit
+distance, as we did here. We will take a look at how various calcu-
+lation possibilities depend on use cases and ontology structures to
+be able to make specific recommendations.
+Actionability. While we tried to check for plausibility of coun-
+terfactual instances, our scoring mechanism cannot make sure yet
+that the axioms that were changed can actually be changed in the
+real world. One argument for counterfactual explanations is that in
+many applications it might be interesting for data subjects to get to
+know how they can change their classification [36]. However, the
+family ontology clearly shows an example of cases where this is
+not really possible, since people cannot usually change their gender
+or familiar relations. Poyiadzi et al. [29] discuss the relevance of
+actionability of counterfactuals. Since this is in line with our empir-
+ical findings, our future work will put more focus on applications
+where actionability can be reached and how to do this.
+7.2
+Applications
+The structure of the family ontology resembles data about individu-
+als in web sources such as DBpedia, YAGO, and Wikidata. To enrich
+such knowledge bases, additional information can be extracted
+from the web. Concept learning allows checking the consistency of
+the extracted information and inferring new (implicit) axioms from
+the explicitly stated axioms.
+Concept learning has been effectively applied to medical on-
+tologies [22], but the learned concepts can become very long [18],
+making them hard to grasp even for experts. Counterfactuals, which
+might even be verbalized in natural language, help to steer focus to
+the most important parts of the concept.
+Our study suggests that counterfactual explanations are most
+useful in actionable settings. For example, if a website is falsely
+classified as unsafe by a phishing site detector, the website owner
+might be interested in knowing which feature of his website caused
+the decision so that they can change it.
+Ultimately, we want to develop a chatbot that, in the spirit of XAI,
+provides users with natural language explanations of automatically
+learned concepts and can be applied to various use cases in areas
+including web science, medicine and finance. Meanwhile, we will
+look further into using counterfactual explanations for offering
+possible actions to alter classifications, to provide more agency to
+users.
+8
+CONCLUSION
+We propose the first approach for generating counterfactual expla-
+nations in the description logic ELH. Our approach performs well
+on the objective to generate counterfactual candidates for which
+a designated concept does not hold and which are similar to the
+individual. We discussed possibilities to improve the likeliness mea-
+surement of counterfactual candidates in accordance with findings
+
+Sieger et al.
+from a user study. Our future work will move on to more complex
+DLs.
+ACKNOWLEDGMENTS
+Funded by the Deutsche Forschungsgemeinschaft (DFG, German
+Research Foundation): TRR 318/1 2021 – 438445824
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+

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@@ -0,0 +1,1618 @@
+arXiv:2301.00564v1  [eess.SY]  2 Jan 2023
+JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+1
+Estimating Risk-Aware Flexibility Areas for EV
+Charging Pools via Stochastic AC-OPF
+Juan S. Giraldo, Member, IEEE, Nataly Ba˜nol Arias, Member, IEEE, Pedro P. Vergara, Member, IEEE,
+Maria Vlasiou, Gerwin Hoogsteen, Member, IEEE, and Johann L. Hurink, Member, IEEE
+Abstract—This
+paper
+introduces a stochastic
+AC-OPF
+(SOPF) for the flexibility management of electric vehicle (EV)
+charging pools in distribution networks under uncertainty.
+The SOPF considers discrete utility functions from charging
+pools as a compensation mechanism for eventual energy not
+served to their charging tasks. An application of the proposed
+SOPF is described where a distribution system operator (DSO)
+requires flexibility to each charging pool in a day-ahead time
+frame, minimizing the cost for flexibility while guaranteeing
+technical limits. Flexibility areas are defined for each charging
+pool and calculated as a function of a risk parameter involving
+the solution’s uncertainty. Results show that all players can
+benefit from this approach, i.e., the DSO obtains a risk-aware
+solution, while charging pools/tasks perceive a reduction in the
+total energy payment due to flexibility services.
+Index Terms—Electric vehicles, flexibility management,
+stochastic optimal power flow, risk awareness, compensation
+mechanism.
+NOMENCLATURE
+Sets
+Ωb
+Set of nodes
+ΩS
+Set of nodes with charging pools
+Ωs
+N
+Set of charging points at charging pool s ∈ ΩS
+Ωs
+K
+Set of breaking points at charging pool s ∈ ΩS
+ΩT
+Set of time periods
+Ωω
+Set of stochastic scenarios
+Parameters
+An,ω
+Characteristics of charging task n at scenario ω
+an,ω
+Arrival time of charging task n at scenario ω
+dn,ω
+Departure time of charging task n at scenario ω
+En,ω
+Required energy of charging task n at scenario ω
+Rij, Xij Resistance and reactance of branch connecting
+nodes ij
+PD
+i,t, QD
+i,t Active and reactive demand power at node i and
+period t
+ηa
+n
+Expected arrival time of charging task n
+ηd
+n
+Expected departure time of charging task n
+βs,t
+Risk parameter at charging pool s in period t
+This work was financially supported by the Netherlands Enterprise
+Agency (RVO) – DEI+ project 120037 “Het Indi¨e terrein: Een slimme
+buurtbatterij in de oude weverij”.
+J. S. Giraldo is with the Energy Transition Studies group, Nether-
+lands Organisation for Applied Scientific Research (TNO), Amsterdam,
+1043 NT, The Netherlands (e-mail: [email protected]).
+N. B. Arias, M. Vlasiou, G. Hoogsteen, and J. L. Hurink are
+with the dept. Electrical Engineering, Mathematics and Computer Sci-
+ence, University of Twente, Enschede, 7522 NB, The Netherlands, (e-
+mail:{m.n.banolarias; m.vlasiou; g.hoogsteen; j.l.hurink}@utwente.nl).
+P. P. Vergara is with the Intelligent Electrical Power Grids group,
+EEMCS, Delft University of Technology, Delft, 2628 CD, The Nether-
+lands, (e-mail: [email protected]).
+κ
+Number of breaking points
+ps,t
+Maximum power allowed of charging pool s in
+period t
+xn
+Maximum charging power of charging task n
+hs,k, bs,k Coefficients of the utility function at charging
+pool s and break point k
+αs,k
+Break point value of energy not served at pool s
+and point k
+∆t
+Duration of the time period t
+cs,t
+Unitary cost of energy at charging pool s period
+t
+Iij
+Maximum allowed current magnitude at branch
+i-j
+V
+Maximum allowed voltage magnitude
+V
+Minimum allowed voltage magnitude
+πω
+Probability of scenario ω
+Variables
+ps,t
+Reserved power for charging pool s in period t
+xs,t,ω
+Allocated power consumption for charging pool
+s in period t and scenario ω
+ρs,t,ω
+Power mismatch for charging pool s in period t
+and scenario ω
+φn,ω
+Energy not served to task n in scenario ω
+Φs,ω
+Total energy not served at charging pool s in
+scenario ω
+λs,k,ω, λs,k,ω Weights in break point k, at pool s in sce-
+nario ω
+Pij,t,ω, Qij,t,ω Active and reactive power flowing through
+branch i-j in period t in scenario ω
+Isqr
+ij,t,ω
+Squared current magnitude flowing through
+branch i-j in period t in scenario ω
+V sqr
+i,t,ω
+Squared voltage magnitude at node i in period t
+in scenario ω
+ys,k,ω
+Binary variable representing state of segment at
+break point k, pool s in scenario ω
+Rs,t
+Flexibility area of charging pool s in period t
+Zs,ω
+Cost for energy not served at charging pool s
+and scenario ω
+I. INTRODUCTION
+B
+ESIDES being an environmentally-friendly option for
+transportation, electric vehicles (EVs) can also pro-
+vide services due to the controllable nature of their load.
+Examples of these services are, amongst others, congestion
+management, peak shaving, and frequency regulation [1].
+These services may be of increased value as technical prob-
+lems, such as voltage violations and branch overloading,
+are expected to be more likely in distribution systems if
+
+JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+2
+no actions are taken [2]. For distribution system operators
+(DSOs), which are responsible for delivering electricity to
+end customers and maintaining a reliable network operation,
+it might be interesting to assess the flexibility needs in
+their networks. In a later stage, these flexibility needs may
+also be provided by entities such as aggregators to solve
+operational issues or offer it as an ancillary service. For
+this, flexibility areas may be determined corresponding to
+the range of active power in which flexibility sources can
+be managed [3].
+In [4], it is already stated that network issues can be
+tackled through flexibility management frameworks to avoid
+common issues in distribution systems, such as congestion
+or voltage limit violations. This strategy is known as DSO’s
+flexibility procurement, and it has gained momentum during
+the last few years due to its economic advantages over
+other solutions such as grid reinforcement. However, for
+a flexibility scheme to be successful, it must guarantee that
+all participants can benefit from participating and are thus
+willing to engage in the flexibility scheme [5].
+Due to driving behaviours, penetration levels, and en-
+ergy requirements, different EVs add an intrinsic, highly
+volatile stochasticity layer to the already complex flexibility
+management problem [6]. Hence, to successfully implement
+a flexibility scheme, new management mechanisms are
+needed that incentivise EV users to offer their flexibility
+and encourage them to participate in such schemes allowing
+the DSO to guarantee a high-quality delivery service under
+uncertainty. In this context, a call for flexibility consists of
+acquiring services from EVs by the DSO to ensure the safe
+operation of the grid [3], ensuring that EV’s interests are
+respected.
+Several works have studied flexibility concepts concern-
+ing EVs in distribution systems using pricing strategies.
+For example, the authors in [7] propose a roadmap with
+key recommendations for the inclusion of EVs, where they
+define EV flexibility services in terms of power, time,
+duration, and location. Furthermore, the authors in [8]
+propose an adaptive pricing strategy that helps to mitigate
+peak demand and to reduce the need for grid reinforcement.
+Likewise, in [4], a dynamic pricing strategy for peak load
+reduction is proposed to optimize the profit of charging pool
+owners, while the uncertain preferences of customers are
+accounted for via robust optimization.
+Smart charging strategies designed in [9] are able to sat-
+isfy multiple flexibility objectives and target specific groups
+of EV users according to user profile preferences. However,
+they do not take into account different pricing schemes,
+aggregator profit, and EV user compensation. Similarly, the
+authors in [10] present a stochastic optimization model for
+cooperative control of charging stations using an aggregated
+energy storage equivalent to describing the charging tasks
+of the EVs. However, although the approaches mentioned
+above can provide local peak shaving services, they are
+not designed to consider network constraints. In [11], EV
+flexibility is provided in the form of peak shaving and valley
+filling, and pricing and charging scheduling mechanisms
+are proposed based on a linear demand-price function. The
+problem is formulated as a bilevel program in which the
+distribution market clearing is simulated in the lower level
+and the upper level solves the EV charging scheduling. Al-
+though aggregated flexibility is calculated for DSO services,
+the proposed framework is deterministic disregarding the
+uncertain nature of EV parameters.
+The concept of flexibility envelopes was introduced
+in [12] as an alternative to quantifying flexibility reserves
+considering the time evolution. This concept has been used,
+for example, in [13], to show that the flexibility reserves
+depend highly on the availability of EVs. Furthermore, in
+[14], flexibility envelopes are calculated for local energy
+communities highlighting it as an ease-of-use approach for
+managing and reserving flexibility in real-time. A similar
+concept known as flexibility areas has been used to estimate
+the flexibility of the available active and reactive power
+at the TSO-DSO boundary [3]. A bottom-up aggregation
+is commonly performed to estimate such flexibility areas
+by determining the potential of different assets at the
+boundary [15]. In [16], a risk-aware framework is proposed
+to define the aggregated flexibility from TSO-DSO inter-
+connections and a two-stage linear stochastic optimization
+model is developed to optimally define the active power
+flexibility available from DSOs to TSOs via a DC-OPF.
+Moreover, as concluded in [17], OPF-based algorithms
+allow for obtaining more reliable feasible operating regions
+compared to random sampling methods. However, none of
+the above approaches does consider uncertainty.
+Stochastic programming is a common approach for han-
+dling uncertainty in electrical power systems including net-
+work constraints [18]. For example, the authors in [19] in-
+troduce a multi-period stochastic AC-OPF (SOPF) consid-
+ering different flexibility assets for congestion management
+and voltage control. A two-stage stochastic programming
+model for managing the flexibility of EVs is proposed
+in [20] for distribution systems in which EVs have already
+been fully recharged. Similarly, the authors in [21] used a
+linearized power flow model in a stochastic optimization
+model considering network constraints focusing on the net-
+work’s reliability. Robust optimization has also been used as
+in [22] to provide flexibility of EVs to DSOs through active
+and reactive power management strategies minimizing the
+amount of non-supplied energy and considering network
+constraints. A queuing network model for electric vehicle
+charging is presented in [23], where the authors define the
+power allocation in the distribution grid while avoiding
+congestion and voltage issues. However, all these works as-
+sume that users agree to participate in the flexibility scheme
+without taking into account their particular priorities.
+The willingness of participants to engage in energy
+trading is an essential factor to be considered in a flexibility
+scheme. Different approaches have been identified in the
+literature, such as solving a global optimization problem that
+is aware of all participants’ subproblems, double auction
+schemes, and using marginal utility functions [5]. The au-
+thors in [24] quantify the EV flexibility for a group of EVs
+classified by user priorities in terms of amount, time and
+duration of availability, via a data-driven approach. Even
+though the EV flexibility is properly quantified, the work
+focuses on data analysis without explicitly proposing an EV
+flexibility scheme for practical implementations. Similarly,
+an online algorithm for charging scheduling of EVs in
+charging pools is proposed in [25], aiming to optimize the
+amount of energy, charging time and prices for EV users,
+
+JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+3
+which are able to choose their most preferable option from
+a menu-based pricing scheme. Although this work ignores
+economic profits of each individual charging pool and a
+detailed operation of the electrical grid (i.e., power flow
+equations), its online nature sets it as a promising option
+for real implementations of EV flexibility schemes.
+With this, simplified representations for utility functions
+are common since they allow for using decentralized opti-
+mization algorithms. For example, the authors in [26] intro-
+duce a decentralized flexibility market based on linear utility
+functions where the prosumers’ willingness to participate
+is explicitly considered. Furthermore, in [27] the authors
+propose using piecewise-quadratic utility functions. How-
+ever, as found in [5], utility functions are often nonlinear
+and nonconvex, and in the case they are linear, they can
+be relatively flat with occasionally significant variations,
+resulting in non-smooth utility functions.
+The reviewed studies show that flexibility services via
+EV charging have been widely studied. However, we have
+identified three main gaps in the current literature which we
+attempt to fill with this paper:
+• Most papers dealing with local EV energy management
+disregard network constraints and do not consider
+uncertainties. We propose an AC multi-period SOPF
+considering network constraints and uncertainty related
+to EV requirements.
+• Most papers consider quadratic utility functions be-
+cause of their attractive properties. We propose a
+general piecewise-linear formulation that is able to deal
+with convex and nonconvex utility functions allowing
+us to represent the interests of EV users. The proposed
+utility functions represent the participants’ willingness
+to offer flexibility services in the form of energy not
+served in return for compensation.
+• We propose a methodology to estimate risk-aware flex-
+ibility areas where the DSO can guarantee operational
+limits. This is done by introducing a risk parameter
+representing the willingness of the DSO to withstand
+operational limit violations. This methodology allows
+estimating probable costs for flexibility requirements
+and gives the charging pools more freedom to manage
+the EV load.
+II. PROBLEM DESCRIPTION
+An operator entity, namely the DSO, is responsible for
+guaranteeing reliable operational conditions in an electrical
+distribution network. In addition to constraint satisfaction
+(i.e., voltage and current magnitude limits), the DSO aims
+to achieve an economically efficient operation on a day-
+ahead time frame via flexibility procurement. In this con-
+text, we consider a distribution network with a set Ωb of
+nodes, connected by a set of distribution lines. A fixed
+number of charging pools are connected to the network,
+identified by the subset ΩS ⊂ Ωb. Hereby, a charging pool
+s ∈ ΩS consists of a fixed set ��s
+N of charging points
+(e.g., the number of EV parking spaces). A charging task
+n arriving to the charging pool s is represented as n ∈ Ωs
+N,
+and is characterized by its set of requirements An. It is
+assumed that a truthful local market mechanism [28] is
+implemented, eliminating any strategic behaviour from the
+participants, meaning that all charging tasks arriving at
+a charging pool are willing to provide demand flexibility
+services in exchange for compensation. This compensation
+must reflect the charging tasks involved in the process,
+whether by a tariff reduction, a bonus, or any other kind of
+settlement [29]. Therefore, the charging pools act as local
+flexibility aggregators characterized by a utility function us
+which are able to control the charging profiles of their tasks.
+In the implemented market mechanism, the charging
+pools agree on truthfully communicating the expected re-
+quirements of their charging tasks (An) along with their
+utility functions (us) to the DSO. Therefore, the DSO
+aims to obtain optimal demand profiles for the charging
+pools, which minimize the cost for flexibility procurement
+while guaranteeing the safe operation of the network over
+a planning horizon ΩT. In operation, it would be ideal
+that the charging pools could provide the demand profiles
+required, meaning that all operational constraints are satis-
+fied. However, in real operation, the actual delivered power
+might vary around the planned profiles since the information
+from the charging pools is intrinsically uncertain, e.g., due
+to the stochastic behaviour of their charging tasks. Hence,
+the DSO needs to plan its actions taking into account the
+operation uncertainties from the charging pools. For this
+purpose, in this paper, we propose using an AC multi-period
+stochastic optimal power flow, extending the work in [18].
+Let ω ∈ Ωω be a realization in a set of stochastic
+scenarios considering possible outcomes due to the uncer-
+tainty of the characteristics of the charging tasks. Hence,
+An,ω represents the expected requirements of charging task
+n ∈ Ωs
+N in scenario ω. The DSO receives this information
+from the charging pools and solves the SOPF minimizing
+the expected costs for flexibility Zs in a day-ahead time
+frame. The DSO needs to define a risk parameter βs,t
+based on the risk it is willing to withstand over operational
+limit violations. Using the optimal solution and the risk
+parameter, the DSO calculates and communicates a lower
+and upper power bound to each charging pool valid for each
+period of the planning horizon. These bounds compose the
+flexibility area, denoted by Rs,t. A graphical representation
+of the day-ahead planning involving the DSO, charging
+pools, and charging tasks is depicted in Fig. 1 along with
+its respective section in the paper.
+In the operation stage, each charging pool s is responsible
+for the local flexibility management of its charging tasks
+considering the flexibility area provided by the DSO. This
+can be done, for example, using profile steering as in [30].
+The actual energy not served to the charging tasks at the
+end of the day is then aggregated and mapped through the
+utility function to calculate the actual cost for flexibility.
+III. MATHEMATICAL MODELS
+In this section the different components of the considered
+setting are presented.
+A. Charging tasks
+Consider a charging task n ∈ Ωs
+N in charging pool
+s ∈ ΩS with a maximum deliverable power xn. In each
+scenario ω ∈ Ωω, a charging task is characterized by the
+tuple An,ω = (an,ω, dn,ω, En,ω). The tuple is composed
+
+JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+4
+DSO
+Charging pool
+Charging task
+Characterization
+Historical data
+Utility function
+Flexibility area
+SOPF
+Day-ahead
+Operation
+Expected cost for 
+flexibility 
+Charging management
+ realization 
+Flexibility management
+Actual cost for
+flexibility
+PSfrag replacements
+us
+An,ω
+Zs
+Rs,t
+III-A
+III-C
+IV-A
+IV-B
+V-A
+V-B
+V-C
+Fig. 1. Interaction between DSO, charging pools, and charging tasks.
+by the task’s arrival time an,ω ∈ ΩT following a Poisson
+distribution characterized by its expected value ηa
+n [23]. Its
+departure time dn,ω ∈ ΩT as a function of the charging
+duration following an exponential distribution characterized
+by the rate ηd
+n [31]. And the required charging energy En,ω
+is assumed to follow a uniform distribution over the closed
+interval [e1, e2]:
+an,ω ∼Pois(ηa
+n), dn,ω ∼Exp(ηd
+n) + an,ω, En,ω ∼U(e1, e2)
+(1)
+For feasibility we assume an,ω < dn,ω ≤ |ΩT|, and that
+within the charging period, the energy required can be
+delivered at full power, i.e., En,ω ≤
+�
+dn,ω − an,ω
+�
+xn.
+It is worth mentioning that the effectiveness of the model
+is independent of the probability distribution function used
+to model the exogenous stochastic parameters. In fact, these
+scenarios can also be mapped from real data [6] or can be
+synthetically generated [9], [20], [32].
+B. Charging pools
+A charging pool s ∈ ΩS, gets an energy reserve for its
+charging operation for a future planning horizon ΩT. The
+energy reserve is composed of averaged power slots defined
+before the actual realization ps,t ∀ t ∈ ΩT and eventual
+power mismatches ρs,t,ω due to the uncertainty of the
+realizations at each scenario. In other words, ps,t represents
+the lower power bound of the charging pool at each period,
+while ρs,t,ω represents any consumption above that bound.
+Let xn,t,ω be the average power consumption allocated to
+the charging task n ∈ Ωs
+N during timeslot t at the realization
+of scenario ω. This is a decision variable determined by the
+charging pool. Then, the power consumption profile of a
+charging pool s at each stochastic scenario is expressed as:
+ps,t+ρs,t,ω =
+�
+n∈Ωs
+N
+xn,t,ω, ∀ s ∈ ΩS, t ∈ ΩT, ω ∈ Ωω (2)
+which is limited by an upper bound ps,t representing the
+power capacity of the charging pool’s connection, e.g,. at
+the transformer
+0 ≤ ps,t + ρs,t,ω ≤ ps,t,
+∀ s ∈ ΩS, t ∈ ΩT, ω ∈ Ωω (3)
+with ps,t, ρs,t,ω ≥ 0, while the power allocation of each
+task is bounded by its maximum charging power xn:
+0 ≤ xn,t,ω ≤ xn, ∀ s ∈ ΩS, n ∈ Ωs
+N, t ∈ ΩT, ω ∈ Ωω
+: an,ω ≤ t ≤ dn,ω
+(4)
+and power cannot be allocated to task n outside the
+task’s arrival and departure times; hence, xn,t,ω = 0 for
+PSfrag replacementsus
+αs,0
+αs,1
+αs,2
+αs,3 Φs,ω
+fs,2
+fs,1
+fs,3
+us,1
+us,2
+Fig. 2.
+Representation of a utility function for a charging pool s with
+κ = 3.
+t < an,ω or dn,ω < t. Note that vehicle to grid (V2G) can
+be included by making the left-hand side of (4) smaller than
+zero, for example, to allow peer-to-peer transactions inside
+the charging pool [22].
+The charging pools also offer flexibility which may imply
+that some charging tasks end with a lower charged energy
+than initially requested. This leads to energy not served at
+task n in scenario ω, defined as φn,ω:
+En,ω =
+� �
+t∈ΩT
+xn,t,ω
+�
++φn,ω,
+∀ s ∈ ΩS, n ∈ Ωs
+N, ω ∈ Ωω.
+(5)
+For charging pool s, the total amount of energy not served
+to its charging tasks is expressed as:
+Φs,ω =
+�
+n∈Ωs
+N
+φn,ω,
+∀ s ∈ ΩS, ω ∈ Ωω. (6)
+C. Discrete utility functions
+The utility function us of a charging pool s ∈ ΩS ex-
+presses the cost for flexibility as a function of the total
+energy not served Φs,ω. It has been recognized that actual
+utility functions can be highly nonlinear [5] and also not
+necessarily convex. For this reason, a general formulation is
+needed to approximate any realistic utility function. To this
+end, we propose the use of discretized utility functions using
+a semicontinuous convex combination formulation [33].
+This formulation does not rely on the nature of the utility
+function (monotonicity or convexity) to approximate it.
+An example of a utility function us is shown in Fig. 2,
+where the dashed line represents a continuous nonlinear
+function, approximated by a linear piecewise function with
+three segments.
+In
+this
+work
+we
+consider
+a
+lower-semicontinuous
+piecewise-linear function representing the utility function
+of the charging pool s:
+us =
+
+
+
+
+
+
+
+
+
+
+
+
+
+fs,0 =0
+Φs,ω = 0
+fs,1 = hs,1Φs,ω+bs,1
+0 < Φs,ω ≤ αs,1
+...
+fs,κ =hs,κΦs,ω+bs,κ
+αs,κ−1 <Φs,ω ≤ αs,κ
+(7)
+where k ∈ Ωs
+K = {0, . . . , κ} represents the set of break
+points, while {hs,k, bs,k} and {αs,k−1, αs,k} ∀ k ≥ 1
+
+JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+5
+denote the coefficients of the functions and their lower and
+upper bounds, respectively. For the sake of simplicity, we
+take us,k−1 := fs,k
+�
+αs,k−1
+�
+and us,k := fs,k
+�
+αs,k
+�
+as the
+function of segment k : k ≥ 1 evaluated on its endpoints.
+In order to satisfy (7), notice that us,0 = us,0 = αs,0 = 0.
+It must be pointed out that (7) cannot be directly inte-
+grated into a mathematical programming model. However,
+by defining multipliers λs,k,ω λs,k,ω ≥ 0 ∀ k ∈ Ωs
+K as the
+weights at each two endpoints, and binary variables ys,k,ω,
+the utility function can be expressed as a linear combination
+of the cost of the endpoints by:
+Zs,ω =
+�
+k∈Ωs
+K: k<κ
+�
+λs,k,ωus,k + λs,k,ωus,k
+�
++ λs,κ,ωus,κ
+(8)
+where the energy not supplied is defined as:
+Φs,ω =
+�
+k∈Ωs
+K: k<κ
+�
+λs,k,ω+λs,k,ω
+�
+αs,k +λs,κ,ωαs,κ
+∀ s ∈ ΩS, ω ∈ Ωω
+(9)
+To make sure that (8) and (9) lead to a proper represen-
+tation of the utility function, the following constraints are
+added:
+1=
+�
+k∈Ωs
+K: k<κ
+�
+λs,k,ω+λs,k,ω
+�
++λs,κ,ω, ∀ s ∈ ΩS, ω ∈ Ωω (10)
+λs,k,ω+λs,k+1,ω =ys,k+1,ω
+∀s ∈ ΩS, k ∈ Ωs
+K, ω ∈ Ωω : k < κ
+(11)
+�
+k∈Ωs
+K:k≥1
+ys,k,ω ≤ 1
+∀ s ∈ ΩS, ω ∈ Ωω (12)
+ys,k,ω ∈ {0, 1}
+∀s ∈ ΩS, k ∈ Ωs
+K, ω ∈ Ωω : k ≥ 1 (13)
+Notice that (10) and (11) ensure that the multipliers are
+only different from zero in the segment where ys,k,ω is
+activated, while (12) and (13) guarantee that only one seg-
+ment can be active. Hence, considering the utility functions,
+the set of variables from the charging pools is defined as
+Ycp = {Φs,ω, Zs,ω, λs,k,ω, λs,k,ω, ys,k,ω, xn,t,ω}.
+D. Distribution Network Model
+We consider a distribution network with radial topology
+behind an electrical substation denoted by ES and a set of
+branches Ωl ⊂ Ωb×Ωb. The operational state of the network
+for a given scenario ω ∈ Ωω can be calculated based on
+the power flow equations as given in constraints (14)–(19),
+adapted from [34]. Hereby, the active power balance in the
+network is ensured by:
+�
+mi∈Ωl
+Pmi,t,ω −
+�
+ij∈Ωl
+�
+Pij,t,ω + RijIsqr
+ij,t,ω
+�
++P G
+i,t,ω = PD
+i,t+
++
+�
+s∈ΩS:s=i
+ps,t + ρs,t,ω
+∀ i ∈ Ωb, t ∈ ΩT, ω ∈ Ωω
+(14)
+and the reactive power balance is given by:
+�
+mi∈Ωl
+Qmi,t,ω −
+�
+ij∈Ωl
+�
+Qij,t,ω + XijIsqr
+ij,t,ω
+�
++QG
+i,t,ω = QD
+i,t,
+∀i ∈ Ωb, t ∈ ΩT, ω ∈ Ωω
+(15)
+where PD
+i,t and QD
+i,t denote the regular active and reac-
+tive power demands at node i and timeslot t. Regular
+power demands are assumed to be deterministic parameters
+expressing the base load of all nodes disregarding EVs.
+Doing this allows us to focus on the impact of EVs. Active
+and reactive power flows to node i from its parent node
+m are denoted by Pmi,t,ω, Qmi,t,ω, while Pij,t,ω, Qij,t,ω
+are the active/reactive power flows from node i to its
+descendant nodes j. For the purposes of this work, it is
+also assumed that the charging stations operate at a unitary
+power factor and no other controllable power sources,
+such as distributed generators are available in the network,
+hence P G
+i,t,ω = QG
+i,t,ω = 0, ∀i ∈ Ωb : i ̸= ES. Also, the volt-
+age magnitude is assumed to be known for the substation
+(V sqr
+ES,t,ω = 1.0 pu). The voltage magnitude drop between
+nodes i and j is represented by:
+V sqr
+j,t,ω = V sqr
+i,t,ω − 2
+�
+RijPij,t,ω+XijQij,t,ω
+�
++
+−
+�
+R2
+ij +X2
+ij
+�
+Isqr
+ij,t,ω,
+∀ij ∈ Ωl, t ∈ ΩT, ω ∈ Ωω
+(16)
+where V sqr
+i,t,ω := V 2
+i,t,ω and Isqr
+ij,t,ω := I2
+ij,t,ω are defined to
+obtain a convex relaxation of the problem [35], while branch
+power flows are obtained using the rotated second-order
+cone constraint:
+V sqr
+j,t,ωIsqr
+ij,t,ω ≥ P 2
+ij,t,ω + Q2
+ij,t,ω,
+∀ij ∈ Ωl, t ∈ ΩT, ω ∈ Ωω.
+(17)
+Furthermore, the upper and lower bounds for nodal
+voltage and branch current magnitudes are enforced by
+V2 ≤ V sqr
+i,t,ω ≤ V
+2
+∀i ∈ Ωb, t ∈ ΩT, ω ∈ Ωω (18)
+0 ≤ Isqr
+ij,t,ω ≤ I
+2
+ij
+∀ij ∈ Ωl, t ∈ ΩT, ω ∈ Ωω (19)
+Finally, the set of variables from the distribution network
+is denoted by Ydn = {V sqr
+i,t,ω, Isqr
+ij,t,ω, Pij,t,ω, Qij,t,ω, ρs,t,ω}.
+IV. PROPOSED SOPF MODEL AND ESTIMATION OF
+FLEXIBILITY AREAS
+A. Mathematical model
+Using the mathematical formulations given in the pre-
+vious section, the proposed SOPF is cast as a two-stage
+stochastic optimization model, formulated as:
+min
+Y
+�
+ω∈Ωω
+πω
+�
+s∈ΩS
+Zs,ω −
+�
+t∈ΩT
+�
+s∈ΩS
+cs,t ps,t
+s.t.
+(2)–(6), (8)–(19)
+(20)
+where the set Y = {Ycp ∪ Ydn ∪ ps,t} contains the decision
+variables of the model, πω stands for the probability of
+scenario ω, and cs,t represents the unit cost of electricity
+at charging pool s at time t. The first-stage variables
+(here-and-now) are ps,t, representing the decisions the DSO
+takes in advance without knowing the actual realizations,
+while the second-stage variables (wait-and-see) are Ycp
+and Ydn, representing the expected stochastic behavior of
+the system after fixing the first-stage variables. Note that
+although DSOs are not allowed to retail electricity, they
+may procure flexibility from the charging pools, which
+act as local flexibility aggregators. Hence, the objective
+function in (20) minimizes the expected value of the cost
+
+JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+6
+1
+2
+3
+4
+5
+6
+7
+8
+ES
+9
+� �
+� �
+� �
+� �
+� �
+
+ 
+ 
+
+
+ 
+
+
+ 
+ 
+ 
+ 
+ 
+ 
+ !
+" #
+$ %
+&
+'
+(
+)
+* +
+, -
+. /
+0
+:
+PSfrag replacements
+Regular demand
+Charging pool
+Charging task
+Fig. 3. 34-nodes test system including four charging pools.
+for flexibility and maximizes the energy reserved for the
+charging pools.
+It must be pointed out that the SOPF in (20) is based on a
+mixed-integer second-order cone programming (MISOCP)
+problem, which is nonconvex in principle. However, if the
+two sufficient conditions defined in [35] are satisfied, then
+the relaxed continuous equivalent is convex and exact, and
+a globally optimal solution is numerically reachable [36].
+In the presented model, both conditions are satisfied since
+the only power source in the system is the substation. Thus,
+every node only consumes power, and the upper bounds of
+the voltages are not binding as long as VES,t < V. More-
+over, a numerical solution to (20) can be obtained using
+the sample average approximation (SAA) technique under
+different scenario generation methods, e.g., Monte Carlo
+(MC), moment matching, or point estimate methods [18].
+B. Estimation of the flexibility areas
+Based on the optimal solution Y∗ of the SOPF in (20),
+the empirical cumulative density function (eCDF) of ρ∗
+s,t,ω
+can be calculated, which is denoted as Fρs,t. Hence, the
+flexibility area of a charging pool s at period t is calculated
+as
+Rs,t = p∗
+s,t + F −1
+ρs,t
+�
+βs,t
+�
+.
+(21)
+where βs,t ∈ [0, 1] represents a risk parameter defined by
+the DSO for each charging pool at each time period. Notice
+that the flexibility area Rs,t is composed of two terms,
+the power reserve serving as a lower limit (p∗
+s,t) and the
+upper limit calculated for a specified quantile. It is worth
+noting that the risk of violating the operational limits and
+the flexibility area are directly proportional. This means
+that βs,t = 0 represents the most conservative alternative
+(lowest risk/smallest area), i.e., Rs,t = p∗
+s,t, while the most
+optimistic alternative (highest risk/biggest area) is given for
+βs,t = 1, leading to Rs,t = p∗
+s,t + max
+ω {ρ∗
+s,t,ω}.
+Furthermore, from the perspective of the charging pools,
+the flexibility area can be interpreted as an accepted oper-
+ating region to fulfill its charging duties within which the
+DSO expects to guarantee operational limits. Finally, notice
+that Rs,t can only be obtained after solving (20) since it
+depends on the optimal solution to uncertain realizations.
+V. TEST SYSTEM AND SIMULATIONS
+In this section, we evaluate the proposed stochastic flexi-
+bility model. For the tests, we consider a radial distribution
+system modified from [34] with 34 nodes (see Fig. 3), which
+is an 11 kV network with a peak total nominal power
+Fig. 4. Utility functions used by the charging pools.
+of 1.86 MW, 1.23 Mvar, V = 0.95 pu, and V = 1.05 pu.
+The maximum phase current at the substation transformer
+connecting nodes 1-2 has been set to I1 2 = 88 A. Four
+charging pools are placed at nodes 16, 20, 27, and 28, with
+30, 59, 36, and 16 charging tasks spread over the planning
+horizon, respectively. The planning horizon is discretized
+in 24 one-hour intervals, resembling a day-ahead planning
+procedure.
+The shape parameters ηa
+n, ηd
+n, characterizing the ar-
+rival and duration times for the EV charging tasks,
+were obtained considering the data in [9] for weekdays.
+A Monte Carlo SAA with |Ωω| = 500 was used to
+solve the two-stage SOPF (20), considering equiprobable
+scenarios, i.e., πω
+=
+1/ |Ωω|. The arrival and depar-
+ture times for each scenario were calculated as in (1),
+while the energy required at each scenario was cal-
+culated as En,ω = min{U(e1, e2), xn
+�
+dn,ω − an,ω
+�
+} with
+e1 = 0 kWh and e2 = 100 kWh. Without loss of general-
+ity, the maximum power at each charging pool has been
+set to ps,t
+=
+200 kW, a fixed cost for electricity of
+cs,t = 0.2 C/kWh was chosen, and the maximum power
+at each charging task was set to xn = 22 kW. Finally,
+the utility functions for the four charging pools have been
+parameterized as in Fig. 4 with κ = 3.
+A. Obtaining Flexibility Areas – Day-ahead planning
+Two main tests were carried out to determine the flex-
+ibility areas. The first one corresponds to the base case,
+an instance with relaxed voltage and current magnitude
+constraints and disabled flexibility from charging pools.
+The base case corresponds to a situation where all required
+energy from charging tasks is supplied as soon as possible,
+regardless of the network status. The mean and standard
+deviation of the minimum voltage magnitude at each time
+period and the maximum branch current at each time period
+for the base case are shown in Fig. 5. In Fig. 5 (a), periods
+with undervoltage problems can be seen in around 8-10h
+and 18-20h. Similarly, periods with overloading problems
+are evident in Fig. 5 (b) around 18-20h. These results
+indicate that the DSO might have a congestion problem
+during the planning horizon and the need for flexibility.
+The second test corresponds to the opposite case, i.e.,
+operational constraints are enforced and flexibility from
+charging pools is enabled. The resulting value of the ob-
+jective function found was (-)C4,059.12 for the base case
+and (-)C3,908.78 in the flexibility enabled case. Notice that
+lower values indicate less energy not served. These results
+represent a reduction of 3.7% in the total expected payment
+
+ us [e/kWh]
+0.4
+ cost
+s = 16
+Flexibility 
+0.2
+s= 20
+s = 27
+s = 28
+0.0
+1
+.
+0
+25
+50
+75
+100
+125
+Energy not served s [kWh]JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+7
+Fig. 5. Base case results for the planning horizon indicating congestion
+problems. (a) Lowest voltage magnitude. (b) Highest current magnitude.
+due to the flexibility cost in the latter case. These results
+indicate that the charging pools (aggregators) would need
+to pay for the energy not served to some charging tasks to
+comply with the DSO’s expected flexibility requirements.
+Consequently, it is expected that the DSO settles this
+difference with the charging pools as part of a flexibility
+market [29].
+The flexibility areas proposed in Section IV-B allow the
+DSO to estimate safe operation regions for the charging
+pools. The first step to obtain the flexibility areas is cal-
+culating the empirical eCDF of ρ∗
+s,t,ω based on (21). The
+eCDF of the four charging pools at s = {16, 20, 27, 28}
+are shown in Fig. 6(a) for t = 14h and in Fig. 6(b)
+for t = 19h. It can be seen that the expected power
+areas chosen depend on the period, e.g., for β27 = 0.8
+the operational powers need to be lower than or equal to
+ρ27 = 49.61 kW at t = 14, but lower than or equal to
+ρ27 = 4.96 kW at t = 19. This difference is expected due
+to the network’s characteristics, i.e., there are some periods
+where the charging pools can have more room to supply
+their charging tasks without compromising the network’s
+operational limits than in other periods. The flexibility area,
+which finally will be communicated to the charging pools,
+has been calculated using (21) for both test cases. In (21),
+the flexibility area is composed by two terms, the power
+reserve serving as a lower limit (bold line) and the upper
+limit calculated for a specified quantile, as shown in Fig. 7
+for s = 20 and s = 27 using βs,t = 0.9. The load shifting
+is evident when comparing both test cases during the whole
+time horizon, especially during critical time intervals (8-10h
+and 18-20h).
+However, load shifting is not always sufficient to solve
+the congestion problems in this test case. Therefore, the
+charging pools must also procure flexibility from the charg-
+ing tasks in the form of energy not served to guarantee
+the operational limits of the DSO. The probability density
+Fig. 6. eCDFs of the operational power ρs,t,ω for (a) t = 14h and (b)
+t = 19h.
+b ;
+a<
+Fig. 7.
+Flexibility area for βs,t = 0.9 at charging pools s = 20 and
+s = 27. (a) Base case. (b) Flexibility enabled.
+Fig. 8.
+(a) PDFs of the total energy not served. (b) eCDFs of the total
+cost for flexibility.
+function (PDF) of the total energy not served at the four
+charging pools is displayed in Fig. 8 (a). Similarly, Fig. 8 (b)
+presents the eCDF of the cost for flexibility at each charging
+pool. It can be seen that the most procured charging pools
+are s = 20 and s = 27, which belong to the same
+network feeder (see Fig. 3). Interestingly, for this feeder,
+the most pronounced voltage drops occur; hence, the DSO
+must procure flexibility in these two charging pools to solve
+voltage problems. It is then evident that some charging
+pools can have an advantageous market position and might
+behave strategically depending on their location in the
+network (e.g., due to the radial topology of distribution
+networks). Therefore, these results reinforce the importance
+of truthful and fair market mechanisms in future flexibility
+markets [28], [29].
+Moreover, from Fig. 8 (b), the DSO can estimate the
+expected cost for flexibility at each charging pool. For
+example, using the 90th percentile for s = 27, means that
+the cost for flexibility at that charging pool is expected to
+be lower than or equal to C 48.97 in at least 90% of the
+expected scenarios.
+B. Validation of the Obtained Flexibility Areas with Prob-
+abilistic Power Flow – Operation
+The next step considers an operation scenario based on
+the flexibility areas identified for day-ahead in Sec. V-A.
+Two risk values are tested in this section to show the
+impact of βs,t on the safe operation of the system. We took
+arbitrarily risk values βs,t ∈ {0.57, 0.99} for the following
+analysis. A probabilistic power flow consisting of 5,000
+MC simulations is executed, considering the uncertainties
+of the aggregated consumed power at the charging pools. A
+sequential implementation of the power flow given in [37]
+has been used due to its convergence and computational
+characteristics. Uniform distributions are assumed to cope
+with any scenario combination within the flexibility area de-
+fined by the selected risk value of the form ∼ U(ps,t, Rs,t).
+
+a)
+b)
+1.0 -
+s = 16
+0.8
+s = 20
+0.8
+s = 27
+s = 28
+0.6
+F
+CDI
+0.4 -
+0.4
+s = 16
+s = 20
+0.2 -
+0.2
+s = 27
+s = 28
+0.0
+0.0
+0
+10
+20
+30
+0
+20
+40
+60
+80
+Energy not served s [kWh]
+Zs[@]a)
+b)
+200
+200
+s= 20
+s= 20
+MY
+S=27
+s=27
+150
+50
+100
+00
+Flexibility
+50
+50 
+0
+0
+5
+10
+15
+20
+25
+5
+10
+15
+0
+0
+20
+25
+Time period [h]
+Time period [h]a)
+b)
+1.0
+1.0 -
+0.8
+0.8
+0.6 -
+0.6 -
+BS
+0.4 -
+s = 16
+0.4 -
+s = 16
+s = 20
+s= 20
+0.2 -
+0.2 -
+s= 27
+s = 27
+s = 28
+s = 28
+0.0
+0.0
+0
+20
+40
+60
+80
+100
+0.0
+2.5
+5.0
+7.5
+10.0
+ps [kW]
+ps [kW]a)
+b)
+90
+0.970
+Mean
+ ± Standard deviation
+A
+0.965
+80
+magnitude
+0.960
+70
+0.955
+60
+Mean
+0.950
+± Standard deviation
+50
+5
+10
+15
+20
+25
+5
+10
+15
+20
+0
+0
+25
+Time period [h
+Time period lhJOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+8
+=>
+?@
+Fig. 9. CDF of operation results during . (a) Lowest voltage magnitude.
+(b) Highest current magnitude.
+Fig. 10. Operation results for the planning horizon using different flexi-
+bility areas. (a) Lowest voltage magnitude. (b) Highest current magnitude.
+It is assumed that the charging pools are able to control their
+consumption within the required flexibility area. Finally, it
+must be pointed out that voltage and current magnitude
+limits are not enforced in the power flow.
+At each MC simulation, the lowest voltage and the
+highest current magnitudes of the system per time period
+are stored. In Fig. 9(a), the average of the lowest volt-
+age magnitude among the buses using both risk values
+is the continuous line, while the shaded area indicates
+its maximum and minimum values. Similarly, Fig. 9(b)
+displays the average maximum branch current magnitude
+and its maximum and minimum values. For instance, at
+20 h the average lowest voltage magnitude for βs,t = 0.57
+is 0.9514pu with a maximum of 0.9522 pu and a minimum
+of 0.9507pu. The maximum current magnitude at the same
+time has an average of 85.37 A, a maximum of 86.27 A and
+a minimum of 84.48 A. On the other hand, for βs,t = 0.99,
+the average lowest voltage magnitude is 0.9499 pu with
+a maximum of 0.9521pu and a minimum of 0.9479pu;
+while the current magnitude has an average of 87.43 A, a
+maximum of 89.95 A and a minimum of 84.69 A.
+The eCDFs of the minimum voltage magnitude consid-
+ering all periods is depicted in Fig. 10(a) for both risk
+values. It can be seen that around 88% of the scenarios
+violate the voltage limit for βs,t = 0.99, whereas for
+βs,t = 0.57 minimum voltages are always within the
+limit. The eCDFs of the maximum current magnitudes are
+displayed in Fig. 10(b) where a similar result is obtained
+with only 10% of the scenarios respecting the maximum
+current magnitude limit when βs,t = 0.99. These results
+indicate that the DSO must determine the required flexibility
+areas based on the risk it is willing to accept since there is a
+trade-off between the chosen risk value and the probability
+of violating the operational limits.
+Fig. 11. Total payment from charging pools for different risk values.
+C. Impact of Flexibility Areas on the Total Payment of the
+Charging Pools
+A final test is performed to assess the impact of the flex-
+ibility areas on the total payment received by the charging
+pools. We considered ten risk values used by the DSO (see
+Fig. 11 for the chosen values). The obtained flexibility areas
+for the different risk values were taken as power limiters
+for the charging pools, i.e., ps,t = Rs,t. On the other hand,
+the total payment, representing the revenue of the charging
+pools, was calculated as the difference between the cost
+for the energy delivered to their charging tasks and the
+cost for energy not served. Thus, positive total payment
+values are desired to guarantee revenue adequacy [38].
+We simulated 1,000 random scenarios for each risk value,
+following the same distributions as described earlier for
+the random variables. Voltage and current magnitude limits
+were enforced and the flexibility enabled.
+The obtained results are displayed in Fig. 11 using a box
+plot where the median, the interquartile range, and the 90%
+confidence intervals are depicted. Results for βs,t = 0.57
+show that the median is C -18.48, the interquartile range
+is limited by C 1,978.87 and C -1,635.18, and the confi-
+dence interval is C 4,021.13 and C -3,826.80; whereas for
+βs,t = 0.99 all these values increased considerably. Hence,
+it can be seen that the total payment for flexibility increases
+with the risk value, meaning there is a trade-off between
+the risk the DSO is willing to stand and the revenue of the
+charging pools. Interestingly, risk values βs,t < 0.57 might
+produce revenue inadequate situations, which encourages
+the use of proper compensation mechanisms for energy
+not served [32]. Consequently, it is expected that the DSO
+settles this difference with the charging pools as part of a
+flexibility market [29].
+VI. CONCLUSIONS
+In this paper, we proposed a stochastic AC-OPF for the
+flexibility management of charging pools in distribution
+networks introducing the concept of flexibility areas. The
+SOPF considers discrete utility functions for charging pools
+as a compensation mechanism for eventual energy not
+served to their charging tasks. The utility functions are
+presented using a general piecewise-linear formulation to
+deal with convex and nonconvex prosumer preferences. The
+aim is to minimize the expected cost for energy not served
+while satisfying operational constraints. An application of
+the proposed SOPF has been described, where a DSO
+specifies the flexibility area to each charging pool in a
+day-ahead time frame under uncertainty. This methodology
+allows estimating probable costs for flexibility requirements
+
+5000
+Median
+2500
+payment
+Interquartile range
+Confidence interval
+0
+Total
+2500
+-5000
+0.05
+0.15
+0.26
+0.36
+0.47
+0.57
+0.68
+0.78
+0.89
+0.99
+Risk value βs,ta)
+b)
+1.0 -
+1.0 -
+0.8
+0.8-
+β = 0.57
+0.6 -
+0.6 -
+CDF
+D
+β = 0.99
+C
+0.4 -
+0.4 
+0.2 -
+0.2 -
+β = 0.57
+β = 0.99
+0.0 
+0.0 -
+0.948
+0.949
+0.950
+88.0
+88.5
+89.0
+89.5
+Voltage magnitude [pu]
+Current magnitude [A]a)
+b)
+90
+Mean - β = 0.57
+max/min - β = 0.57
+0.970
+Mean - β = 0.99
+80
+max/min - β = 0.99
+0.965
+70
+0.960
+60
+0.955
+Mean - β= 0.57
+max/min - β = 0.57
+0.950
+50 
+Mean - β = 0.99
+max/min - β = 0.99
+0
+5
+10
+15
+20
+25
+0
+5
+10
+15
+20
+25
+Time period [h]
+Time period [h]JOURNAL OF MODERN POWER SYSTEMS AND CLEAN ENERGY, VOL. XX, NO. XX, XXXX
+9
+and gives the charging pools more freedom to manage
+the EV load. Results show that a safe flexibility area for
+charging pools can be used to address DSO’s congestion
+problems, either by load shifting or managing the energy not
+served. Moreover, the DSO is able to calculate the flexibility
+area as a function of a risk parameter βs and estimate
+probable costs for flexibility requirements. Results showed
+a trade-off between the risk the DSO is willing to stand
+and the revenue of the charging pools. At the same time,
+charging pools and tasks perceive a total energy payment
+reduction as compensation for the energy not served, which
+might stimulate charging pool operators and EV users to
+offer flexibility services (e.g., in a local flexibility market).
+Future work has to analyze the impact of the proposed
+flexibility area considering V2G enabled EVs and reactive
+power compensation capabilities.
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+Juan S. Giraldo received the B.Sc. degree in electrical engineering
+from the Universidad Tecnol´ogica de Pereira, Pereira, Colombia, in 2012,
+and the M.Sc. and Ph.D. degrees in electrical engineering from the
+University of Campinas (UNICAMP), Campinas, Brazil, in 2015 and 2019,
+respectively. From Oct. 2019 to May 2021 he was a Postdoctoral Fellow
+at the Department of Electrical Engineering, Eindhoven University of
+Technology, Eindhoven, The Netherlands (NL). Later, from June 2021 to
+Aug. 2022 he was a postdoc with the Mathematics of Operations Research
+group at the University of Twente, Enschede, NL. He is currently a
+Researcher with the Energy Transition Studies group with the Netherlands
+Organisation for Applied Scientific Research (TNO), Amsterdam, NL. His
+current research interests include the optimization, planning, and control of
+energy systems, energy markets, and machine learning applied to energy
+systems.
+Nataly Ba˜nol Arias received the B.Sc. degree in Production Engineering
+from the Universidad Tecnol´ogica de Pereira, Colombia in 2012, and
+the M.Sc. and Ph.D. degree in Electrical Engineering from the S˜ao
+Paulo State University (UNESP), Ilha Solteira, Brazil, in 2015 and 2019,
+respectively. Currently, she is a researcher at the University of Twente,
+The Netherlands. Her current research interests include the development
+of methodologies for the optimization, planning, and control of modern
+distribution systems including electric vehicles and renewable energy
+sources, energy management systems, and flexibility markets.
+Pedro P. Vergara was born in Barranquilla, Colombia in 1990. He
+received the B.Sc. degree (with honors) in electronic engineering from the
+Universidad Industrial de Santander, Bucaramanga, Colombia, in 2012,
+and the M.Sc. degree in electrical engineering from the University of
+Campinas, UNICAMP, Campinas, Brazil, in 2015. In 2019, he received
+his Ph.D. degree from the University of Campinas, UNICAMP, Brazil, and
+the University of Southern Denmark, SDU, Denmark, funded by the Sao
+Paulo Research Foundation (FAPESP). In 2019, he joined the Eindhoven
+University of Technology, TU/e, in The Netherlands as a Postdoctoral
+Researcher. In 2020, he was appointed as Assistant Professor at the
+Intelligent Electrical Power Grids (IEPG) group at Delft University of
+Technology, also in The Netherlands. His main research interests include
+the development of methodologies for control, planning, and operation of
+electrical distribution systems with high penetration of low-carbon energy
+resources (e.g, electric vehicles, PV systems, electric heat pumps) using
+optimization and machine learning approaches. Dr. Vergara has received
+the Best Presentation Award at the Summer Optimization School in 2018
+organized by the Technical University of Denmark (DTU) and the Best
+Paper Award at the 3rd IEEE International Conference on Smart Energy
+Systems and Technologies (SEST), in Turkey, in 2020.
+Maria Vlasiou is a Professor at the University of Twente, The Netherlands,
+an Associate Professor at the Eindhoven University of Technology (TU/e),
+and Research Fellow of the European research institute EURANDOM.
+She received her B.Sc. (2002, Hons.) and Ph.D. (2006) from the Aristotle
+University of Thessaloniki and TU/e, respectively. In 2006, she moved
+to the H. Milton Stewart School of Industrial and Systems Engineering,
+at the Georgia Institute of Technology, where she first worked as a
+Research Engineer and later as a Postdoctoral Fellow. Her research interests
+centre on stochastic processes and stochastic operations research. Her
+research focuses on the performance of stochastic processing networks
+with layered architectures and on perturbation analysis for heavy-tailed
+risk models. Other interests include L´evy processes, large deviations for
+non-monotone stochastic recursions, and proportional fairness in heavy
+traffic for bandwidth-sharing networks. She has supervised six PhD theses
+on these topics. Prof. Vlasiou has been invited to more than 20 foreign
+universities for collaboration and seminars. She has been associate editor
+in four journals and has refereed for about 45 international journals,
+conferences, and national science foundations. Prof. Vlasiou’s research so
+far has been funded by grants from more than 10 science foundations,
+universities, societies, and organisations. She is the co-author of more than
+50 refereed papers, the co-recipient of the best paper award in ICORES
+2013, the Marcel Neuts student paper award in MAM8, a prize at the 8th
+conference in Actuarial Science, and the recent winner of the INFORMS
+UPS G. Smith award.
+Gerwin Hoogsteen received the PhD degree from the University of Twente
+in 2017 with his thesis “A Cyber-Physical Systems Perspective on Decen-
+tralized Energy Management”. He is currently employed as permanent
+researcher in the field of smart grids within the Computer Architecture for
+Embedded Systems chair, with a focus on applying theoretical research in
+field-tests. His research interest is in energy management for smart grids,
+and in particular where it concerns multi-disciplinary research and cyber-
+physical systems. Current research directions include the use of machine
+learning and artificial intelligence in smart grids, distributed coordination,
+and cyber-security of smart grids. Hoogsteen is the founder and maintainer
+of the DEMKit and ALPG software.
+Johann Hurink received the Ph.D. degree from University of Osnabr¨uck
+(Germany) in 1992 for a thesis on a scheduling problem occurring in
+the area of public transport. Since 2009 he is a full professor at the
+University of Twente and since 2020 also the Director of the 4TU Applied
+Mathematics Institute (AMI) in The Netherlands. He has published more
+than 190 refereed papers in international journals and conferences and has
+been involved in many European and national research projects. Current
+research mainly focuses on optimization and control problems for energy
+management and smart grids.
+

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+Draft version January 3, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+PHANGS-JWST First Results: Variations in PAH Fraction as a Function of ISM Phase and
+Metallicity
+J´er´emy Chastenet
+,1 Jessica Sutter
+,2 Karin Sandstrom
+,2 Francesco Belfiore
+,3 Oleg V. Egorov
+,4
+Kirsten L. Larson
+,5 Adam K. Leroy
+,6 Daizhong Liu
+,7 Erik Rosolowsky
+,8 David A. Thilker
+,9
+Elizabeth J. Watkins
+,4 Thomas G. Williams
+,10, 11 Ashley T. Barnes
+,12 Frank Bigiel
+,13
+M´ed´eric Boquien
+,14 M´elanie Chevance
+,15, 16 I-Da Chiang (江宜達)
+,17 Daniel A. Dale
+,18
+J. M. Diederik Kruijssen
+,16 Eric Emsellem
+,19, 20 Kathryn Grasha
+,21, 22 Brent Groves
+,23
+Hamid Hassani
+,8 Annie Hughes
+,24 Kathryn Kreckel
+,4 Sharon E. Meidt
+,1 Ryan J. Rickards Vaught
+,2
+Amy Sardone
+,25, 26 and Eva Schinnerer
+11
+1Sterrenkundig Observatorium, Ghent University, Krijgslaan 281-S9, 9000 Gent, Belgium
+2Center for Astrophysics and Space Sciences, Department of Physics, University of California, San Diego
+9500 Gilman Drive, La Jolla, CA 92093, USA
+3INAF — Arcetri Astrophysical Observatory, Largo E. Fermi 5, I-50125, Florence, Italy
+4Astronomisches Rechen-Institut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg, M¨onchhofstraße 12-14, 69120 Heidelberg, Germany
+5AURA for the European Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+6Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, Ohio 43210, USA
+7Max-Planck-Institut f¨ur Extraterrestrische Physik (MPE), Giessenbachstr. 1, D-85748 Garching, Germany
+8Department of Physics, University of Alberta, Edmonton, Alberta, T6G 2E1, Canada
+9Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
+10Sub-department of Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
+11Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117, Heidelberg, Germany
+12Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, 53121, Bonn, Germany
+13Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, 53121 Bonn, Germany
+14Centro de Astronom´ıa (CITEVA), Universidad de Antofagasta, Avenida Angamos 601, Antofagasta, Chile
+15Institut f¨ur Theoretische Astrophysik, Zentrum f¨ur Astronomie der Universit¨at Heidelberg,
+Albert-Ueberle-Strasse 2, 69120 Heidelberg, Germany
+16Cosmic Origins Of Life (COOL) Research DAO, coolresearch.io
+17Institute of Astronomy and Astrophysics, Academia Sinica, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
+18Department of Physics and Astronomy, University of Wyoming, Laramie, WY 82071, USA
+19European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching, Germany
+20Univ Lyon, Univ Lyon1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230 Saint-Genis-Laval
+France
+21Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia
+22ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia
+23International Centre for Radio Astronomy Research, University of Western Australia, 7 Fairway, Crawley, 6009 WA, Australia
+24IRAP, Universit´e de Toulouse, CNRS, CNES, UPS, (Toulouse), France
+25Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA
+26Center for Cosmology and Astroparticle Physics, 191 West Woodruff Avenue, Columbus, OH 43210, USA
+ABSTRACT
+We present maps tracing the fraction of dust in the form of polycyclic aromatic hydrocarbons
+(PAHs) in IC 5332, NGC 628, NGC 1365, and NGC 7496 from JWST/MIRI observations.
+We
+trace the PAH fraction by combining the F770W (7.7 µm) and F1130W (11.3 µm) filters to track
+ionized and neutral PAH emission, respectively, and comparing the PAH emission to F2100W which
+traces small, hot dust grains. We find average RPAH = (F770W + F1130W)/F2100W values of 3.3,
+4.7, 5.1, and 3.6 in IC 5332, NGC 628, NGC 1365, and NGC 7496, respectively.
+We find that
+H II regions traced by MUSE Hα show a systematically low PAH fraction. The PAH fraction re-
+mains relatively constant across other galactic environments, with slight variations. We use CO+Hi
++Hα to trace the interstellar gas phase and find that the PAH fraction decreases above a value of
+IHα/ΣHI+H2
+∼ 1037.5 erg s−1 kpc−2 (M⊙ pc−2)−1, in all four galaxies. Radial profiles also show a
+arXiv:2301.00578v1  [astro-ph.GA]  2 Jan 2023
+
+ID2
+Chastenet, Sutter, Sandstrom et al.
+decreasing PAH fraction with increasing radius, correlated with lower metallicity, in line with previous
+results showing a strong metallicity dependence to the PAH fraction. Our results suggest that the
+process of PAH destruction in ionized gas operates similarly across the four targets.
+Keywords: Dust physics (2229), Interstellar dust (836), Polycyclic aromatic hydrocarbons (1280)
+1. INTRODUCTION
+The mid-infrared (mid-IR) emission features at 3.3,
+6.2, 7.7, 8.6, 11.3, 12.6, and 17 µm are characteristic
+of the aromatic content of interstellar dust (see reviews
+from Draine 2003; Tielens 2008; Li 2020). The carriers
+of these features are often referred to as polycyclic aro-
+matic hydrocarbons (PAHs; Allamandola et al. 1985),
+and have been included as an extension of carbonaceous
+grains to small sizes in several physical dust models (e.g.
+Desert et al. 1990; Zubko et al. 2004; Draine & Li 2007;
+Galliano et al. 2008), or as an aromatic-rich mantle cov-
+ering aliphatic grains (e.g. Jones et al. 2017, and refer-
+ence therein). In the rest of this letter, we will refer to
+the carriers of mid-IR features as PAHs.
+The (collective) brightness of these features with re-
+spect to a dust continuum emission can be used as a
+tracer of the fraction of dust in the form of PAHs. The
+mass fraction of PAHs can be measured from fitting
+models to observed dust emission measured from mid-
+through far-IR broadband photometry (e.g. Draine et al.
+2007; Galliano et al. 2008; Galliano 2018; Chastenet
+et al. 2019; Nersesian et al. 2019; Aniano et al. 2020).
+One can also fit mid-IR spectra and derive more de-
+tailed information about the relative intensities of each
+feature, like the average charge and size of the PAH
+population (Smith et al. 2007; Lai et al. 2020; Maragk-
+oudakis et al. 2022). As very prominent features, the
+emission at 7.7 and 11.3 µm can be considered a satis-
+factory proxy to trace the total emission from PAHs in
+normal star-forming galaxies (e.g. Smith et al. 2007; Lai
+et al. 2020; Draine et al. 2021).
+PAHs also play a key role in heating the ISM within
+photodissociation regions (PDRs; Bakes & Tielens 1994;
+Weingartner & Draine 2001; Tielens 2008; Bern´e et al.
+2009; Croxall et al. 2012; Wolfire et al. 2022). As a sig-
+nificant source of photo-ejected electrons, the abundance
+of PAHs greatly influences the photoelectric heating ef-
+ficiency. The close tie between PAHs and PDR heating
+has led to the suggested use of PAH emission as a tracer
+of the star-formation rate (e.g., Peeters et al. 2004; Ship-
+ley et al. 2016).
+The variations of the PAH fraction in the interstellar
+medium (ISM) of external galaxies helps us to under-
+stand their origin and evolution, as well as the mech-
+anisms that regulate their formation and destruction.
+Several studies have found that the fraction of PAHs de-
+creases in regions of ionized gas and/or because of hard
+radiation fields (e.g. Giard et al. 1994; Dong & Draine
+2011; Verstraete 2011; Salgado et al. 2016; Chastenet
+et al. 2019; Rigopoulou et al. 2021), as theory predicts
+(Siebenmorgen et al. 2004; Groves et al. 2008; Micelotta
+et al. 2010; Bocchio et al. 2012; Zhen et al. 2016), and
+becomes very low in H II regions (e.g. Pety et al. 2005;
+Lebouteiller et al. 2007; Thilker et al. 2007; Compi`egne
+et al. 2008). There is also evidence of a correlation of
+PAH features with the CO content (see, e.g., Leroy et al.
+2022a). Studying nearby galaxies, Regan et al. (2006)
+found that the 8 µm (traced by Spitzer/IRAC) and CO
+radial profiles are closely matched. Evidence of a close
+link between PAH and CO emission has also been ob-
+served in high-redshift galaxies on galactic scales (e.g.
+Pope et al. 2013; Cortzen et al. 2019).
+In addition to trends observed across different ISM
+environments, the abundance of PAHs has been shown
+to decrease in low metallicity galaxies (e.g. Engelbracht
+et al. 2008; Draine et al. 2007; Sandstrom et al. 2012).
+This deficit in PAHs has multiple proposed causes, in-
+cluding the destruction of PAHs by hard radiation fields
+present in low-metallicity galaxies (Madden et al. 2006;
+Gordon et al. 2008) or delayed PAH formation in AGB
+star atmospheres (Galliano et al. 2008). These trends
+have important implications for future observations of
+PAH emission in high-redshift galaxies, making it essen-
+tial to study PAH variation across a range of systems.
+By further establishing how metallicity trends can lead
+to the reduction in PAH emission, we will be better pre-
+pared for using these small grains to assess ISM condi-
+tions across cosmic time.
+The recent launch of JWST opens a new window for
+exploring this question. The MIRI F770W and F1130W
+filters provide coverage of two of the most prominent
+PAH features at 7.7µm and 11.3µm, while the F2100W
+filter lends a useful comparison, tracing emission from
+larger dust grains (Draine & Li 2007). The unique abil-
+ity of the MIRI instrument to map these PAH features
+at unprecedented resolution and sensitivity in galaxies
+outside of the Local Group allows us to greatly expand
+the range of ISM conditions in which measurements of
+the PAH fraction can be made and better determine
+
+PAH Fraction and ISM in PHANGS-JWST
+3
+how the local ISM conditions can influence the relative
+amount of PAHs present.
+In this Letter, we investigate the variations of the PAH
+fraction traced by a combination of JWST/MIRI filters
+across the full disk of four nearby galaxies, IC 5332,
+NGC 628, NGC 1365, and NGC 7496. By tracking the
+PAH fraction across the multi-phase ISM, we are able
+to determine how a range of ISM properties affect the
+relative abundance of PAHs with respect to large dust
+grains. This will lay the ground work for more detailed
+studies of how PAH emission varies in specific condi-
+tions, such as those found around sites of active star
+formation.
+2. DATA
+2.1. JWST/MIRI data
+The data used in this paper are part of the PHANGS-
+JWST Treasury program #2107 (PI: J.C. Lee, Lee et al.
+2022). We use MIRI (Rieke et al. 2015) observations
+in the F770W, F1130W, and F2100W filters, from the
+latest reference files at the time of processing, as de-
+scribed by Lee et al. (2022) and Leroy et al. (2022a).
+The PHANGS-JWST team used the STScI Calibration
+pipeline 1.7.1 for NIRCam and 1.7.0 for MIRI, and Cal-
+ibration Reference Data context number 0968 for both
+instruments. Table 1 gives a few key details about the
+four targets of this Letter, which are used to construct
+r/r25 maps.
+We convolve all three filter maps to 1′′ resolution,
+which is larger than the full-width at half-maximum of
+the F2100W filter (0.67′′). The convolution kernels were
+computed using the theoretical PSFs of each filter pro-
+vided by the WebbPSF (Perrin et al. 2014), and the
+method described in Aniano et al. (2011). We choose to
+slightly degrade the data to increase the signal-to-noise
+(S/N) in the F2100W band. By doing so, we are also
+able to match the resolution of the CO and Hα data (see
+below).
+2.2. Ancillary data
+We use 12CO (2 − 1) “broad” moment 0 maps and
+corresponding error maps from the PHANGS-ALMA
+survey combining 12m + 7m + Total Power (Leroy et al.
+2021a,b), to trace molecular gas, at ∼ 1′′ (∼ 40 − 90 pc
+in our sample) resolution.
+To convert CO intensity
+to molecular gas surface density, ΣH2 in M⊙ pc−2,
+we use a constant CO-to-H2 conversion factor αCO =
+4.35 M⊙ pc−2 (K km s−1)−1 as recommended for solar
+metallicity, star-forming galaxies (Bolatto et al. 2013)1,
+1 With the assumption of a flat Hi distribution, the choice of αCO
+is not the dominant uncertainty.
+and 12CO (2 − 1) conversion factor and a 12CO(2 −
+1)/12CO(1 − 0) line ratio R21 = 0.65 (den Brok et al.
+2021; Leroy et al. 2022b).
+We use Hα maps from the PHANGS-MUSE survey
+(Emsellem et al. 2022) to trace ionized gas. We use the
+‘native’ resolution maps, with an average ∼ 0.8′′ resolu-
+tion, and a 0.2′′ pixel size. To trace the 12 + log(O/H)
+metallicity in our targets, we use the 2D maps by
+Williams et al. (2022). The maps were created by inter-
+polating H II regions-derived metallicity maps (with S-
+calibration), using a Gaussian Process Regression tech-
+nique, based on PHANGS-MUSE maps of Hα intensity.
+These maps have fixed physical-, and different angular
+resolutions (Emsellem et al. 2022), all lower than 1′′ ex-
+cept for NGC 1365 (1.15′′). We convolve all maps with
+a resolution lower than 1′′ to that value, to match the
+convolved MIRI maps.
+We use Hi measurements from MeerKAT (C. Eiben-
+steiner et al., in prep), at 15′′ resolution for NGC 7496,
+and from THINGS (Walter et al. 2008) for NGC 628
+(∼ 11′′ for the ‘natural’ weighted map). We convert the
+maps to have units of atomic gas mass surface density in-
+cluding helium assuming optically thin 21 cm emission,
+using the prescription from Leroy et al. (2012, see also
+Walter et al. 2008). We lack resolved 21 cm mapping for
+IC 5332 and NGC 1365. Based on the observed flatness
+of the atomic gas surface density over the inner parts
+of galaxy disks (e.g., Schruba et al. 2011; Bigiel & Blitz
+2012; Kennicutt & Evans 2012; Wong et al. 2013), we
+assume that the atomic gas has a flat distribution with
+ΣHI = 8 M⊙ pc−2 (including helium). The Hi data have
+the lowest resolution among the datasets for our sample.
+Since the Hi distribution is expected to be reasonably
+smooth across the disk of all our targets (as a likely re-
+sult of low resolution; Leroy et al. 2013), we choose to
+work at the MIRI F2100W resolution to retain as much
+information about the distribution of the PAH emission
+within galaxies as possible. We reproject these maps to
+the MIRI pixel grid, with pixel size ∼ 0.11′′.
+2.3. Noise properties and masks
+We
+measure
+background
+standard
+deviation
+in
+NGC 7496 as it is the only one that offers enough pixels
+off target, at 1′′ resolution. Lee et al. (2022) give details
+of the reduction of the data, including noise. The back-
+ground removal is done by anchoring the JWST data
+with Spitzer, where the background in these sources
+was better estimated.
+This is done as few off-source
+pixels are available in most of the early targets.
+We
+find that MIRI maps have similar noise, and we assume
+NGC 7496 estimates throughout the sample.
+We re-
+move pixels with signal-to-noise S/N ≤ 3 in all MIRI
+
+4
+Chastenet, Sutter, Sandstrom et al.
+bands, with background noise values of 0.05, 0.05, and
+0.1 MJy sr−1. In NGC 1365 and NGC 7496, we mask
+pixels in the center due to IR brightness of the active
+galactic nuclei (AGN), which saturate the signal mostly
+in the F2100W band. This is done using the WebbPSF
+package (Perrin et al. 2014), and centering the F2100W
+PSF on the central coordinates of each target. For this
+preliminary work, we remove additional conspicuous ar-
+tifacts from the central AGN that remain after this anal-
+ysis. These spikes are due to the central saturation, and
+although they are not flagged as ‘bad pixels’ by data
+reduction, they are an obvious artifact. We mask these
+by hand to ensure they do not bias our initial results.
+They are shown in gray scale in Figure 1, but are not
+used in the analysis.
+3. PAH FRACTION
+In Draine et al. (2021), the authors found that the
+luminosity of the 7.7 µm feature2 normalized to the to-
+tal IR luminosity can approximate the PAH fraction
+in all but the most extreme cases (Draine & Li 2007;
+Draine et al. 2021).
+Here, we use the JWST/MIRI
+RPAH = (F770W + F1130W)/F2100W ratio as a proxy
+for the PAH fraction. Over a broad range of PAH size
+(in terms of number of carbon atoms), and assuming a
+Galactic interstellar radiation field (Mathis et al. 1983,
+at 10 kpc), the 7.7 µm feature is more representative of
+the ionized PAH population, and the 11.3 µm feature
+of the neutral population (e.g., Rapacioli et al. 2005;
+Draine et al. 2021). We assume the F2100W filter is free
+from contribution from PAHs and is instead dominated
+by small, hot dust grains, as seen in the Spitzer/IRS
+spectra of local galaxies observed by the SINGS project
+(Smith et al. 2007, see also Draine & Li 2007, Dale et al.
+2009). This assumption is validated by a clear difference
+in behavior between the F2100W and PAH–dominated
+band emission, demonstrated in Leroy et al. (2022a).
+Normalizing the emission at 7.7 and 11.3 µm by the ob-
+served flux at 21 µm help us focus on the PAH-only frac-
+tion. Previous studies of the PAH population in nearby
+galaxies completed with Spitzer have shown that the
+ratio of 8 µm to 24 µm is a good tracer of the PAH
+fraction (see e.g. Smith et al. 2007; Marble et al. 2010;
+Croxall et al. 2012). In addition, models of PAH and
+dust emission have shown 7.7 µm-to-Total IR luminos-
+ity is a good tracer the PAH mass fraction (qPAH; Draine
+& Li 2007). Using the MIRI filters, we can improve on
+2 In this model, the 7.7 µm feature luminosity is determined by
+integrating between two set “clip” points, λ = 6.9 and λ = 9.7.
+This is not exactly equal to fluxes observed using the F770W
+filter, which spans λ = 6.6 − 8.6, but is similar.
+this by using the bands centered on the PAH features
+(F770W and F1130W) in place of the 8 µm data while
+the F2100W replaces the 24 µm continuum tracer. As
+both the PAH features and the 21 µm continuum are
+from stochastic heating, using these fluxes to determine
+the PAH fraction removes any significant dependence
+on the radiation field (except in extreme situations, see
+Draine & Li 2007, their Figure 13).
+3.1. Qualitative description
+Figure 1 shows RPAH in the four PHANGS-JWST
+early targets.
+The white contours show the brightest
+H II regions (Groves et al. subm.; Santoro et al. 2022). In
+all cases, it appears that RPAH shows clear depressions
+within H II regions, though NGC 628 and NGC 7496
+show this contrast most clearly.
+In the right section of Table 1, we show the mean of
+the ratio, ⟨RPAH⟩, and associated 16th − 84th percentile
+range. NGC 1365 shows the highest average for RPAH,
+and IC 5332 the lowest. All 16th−84th ranges agree rea-
+sonably well, and NGC 7496 shows the broadest range.
+The top left panel of Figure 2 shows the radial profiles of
+RPAH, in bins of r/r25. In all panels, the error bars are
+3 times the standard error of the mean (SEM). These
+errors are small because of the number of pixels in each
+bin (the standard deviation is much larger). Note that
+not all galaxies have observations extending to the same
+r/r25.
+All galaxies show an overall decreasing trend with
+radius. There is however, a slightly different trend in
+NGC 1365 and NGC 7496, which both show an increase
+in abundance ratio at small radii before exhibiting a
+steady decrease. Both are Seyfert galaxies (NGC 1365:
+1.8, NGC 7496: 2.0; e.g., Garc´ıa-Bernete et al. 2022)
+hosting an AGN as well as central bars that feed high
+density regions at their centers. Even though we mask
+the region around the AGN, this increasing trend at
+small radii could be due to the influence of the central
+AGN on the PAH population.
+The definitive impact
+of AGNs on the PAH population is not yet completely
+clear. For example, Garc´ıa-Bernete et al. (2022) found
+differences in the relative strengths of different PAH fea-
+tures in AGN host galaxies and star-forming galaxies at
+kpc scales. However, Lai et al. (2022) recently found rel-
+atively small variations in PAH size and ionization and
+a decrease in PAH emission only in the direct line-of-
+sight of the AGN, using JWST observations. Similarly,
+Viaene et al. (2020) found that the influence of the AGN
+is only relevant close to the nucleus. Additionally, there
+is evidence for PAH molecules surviving the harsh en-
+vironment surrounding AGNs (e.g., Jensen et al. 2017;
+Alonso-Herrero et al. 2014; Garc´ıa-Bernete et al. 2022).
+
+PAH Fraction and ISM in PHANGS-JWST
+5
+Target
+R. A.
+Dec
+Distance [Mpc]
+P.A. [◦]
+i [◦]
+r25 [′]
+⟨RPAH⟩
+16th − 84th perc.
+IC 5332
+23 : 34 : 27.488
+−36 : 06 : 3.89
+9.01
+74.4
+26.9
+3.03
+3.3
+1.8 − 4.8
+NGC 628
+01 : 36 : 41.745
++15 : 47 : 1.11
+9.84
+20.7
+8.9
+4.94
+4.7
+3.8 − 5.6
+NGC 1365
+03 : 33 : 36.458
+−36 : 08 : 26.37
+19.57
+201.1
+55.4
+6.01
+5.1
+3.8 − 6.3
+NGC 7496
+23 : 09 : 47.288
+−43 : 25 : 40.28
+18.72
+193.7
+35.9
+1.67
+3.6
+1.8 − 5.3
+Table 1. Right ascension (R. A.) and declination (Dec) coordinates (J2000), the r25 radius, in arc-minutes, used in this Letter
+for our four targets, from the HyperLeda database (Makarov et al. 2014). Distances are from Anand et al. (2021). Position
+angles and inclinations are from Leroy et al. (2021b). Additional information can be found in Table 1 of the survey paper (Lee
+et al. 2022). The second part of the Table shows the mean and 16th − 84th percentiles of RPAH.
+Although we observe a decreasing trend in these two
+galaxies towards the center, it is as of yet uncertain how
+the presence of an AGN may be related to this decrease.
+Additional work will be required to clearly understand
+the scales on which AGN impact the global fraction of
+PAHs.
+We also show the profile of RPAH with 12 + log(O/H)
+metallicity, in the top right panel of Figure 2. Global
+trends of the PAH fraction with metallicity have been
+studied on integrated scales in several works (Draine
+et al. 2007; R´emy-Ruyer et al. 2015; Chastenet et al.
+2019; Galliano et al. 2021).
+Here, we see that this
+trend is similar on small resolved scales (∼ tens of par-
+secs). The turning point in the metallicity value, where
+the abundance ratio starts to decrease with increasing
+metallicity, is similar to the down-turn at small r/r25 as
+these are primarily radial metallicity gradients.
+3.2. Variation of PAH fraction with ISM environment
+In Figure 2, middle and bottom rows, we present
+the variation of RPAH as a function of the ISM en-
+vironment.
+We combine the Hi and H2 maps to es-
+timate the total gas surface density, and use the ra-
+tio of Hα intensity to total gas to follow the varia-
+tions of the ISM conditions in each target (in units of
+erg s−1 kpc−2 (M⊙ pc−2)−1). As we expect PAH emis-
+sion to arise from a range of ISM phases, the ratio of Hα
+intensity to total gas (IHα/ΣHI+H2) is used as a proxy for
+environments dominated by the ionized phase and can
+provide a clear indication of what ISM environments the
+PAH emission is coming from. For example, high val-
+ues of IHα/ΣHI+H2 are indicative of regions dominated
+by ionized gas, while low values of IHα/ΣHI+H2 suggest
+the dominance of neutral gas. This comparison works
+well because neutral and ionized gas are decorrelated on
+small spatial scales and the clearing of neutral gas due
+to rapid ionizing feedback (< 5 Myr, Kruijssen et al.
+2019; Chevance et al. 2020, 2022; Kim et al. 2022). By
+comparing RPAH to IHα/ΣHI+H2 we can better under-
+stand what ISM conditions, i.e.
+what fraction of the
+line of sight is ionized gas free of PAHs, could lead to
+an observed dearth of PAH emission.
+This is further
+investigated within H II regions in Egorov et al. (2022).
+The middle left panel of Figure 2 shows the varia-
+tions of RPAH as a function of IHα/ΣHI+H2. The abun-
+dance of PAHs appears to stay rather flat until a thresh-
+old in IHα/ΣHI+H2, where it decreases steeply. This is
+expected from the destruction of PAHs in harsh en-
+vironments traced by high-intensity Hα (e.g., Groves
+et al. 2008; Micelotta et al. 2010; Bocchio et al. 2012;
+Egorov et al. 2022), and has been observed in Galac-
+tic PDRs (Pety et al. 2005; Compi`egne et al. 2008).
+Interestingly, it appears that all galaxies share similar
+thresholds in IHα/ΣHI+H2 at which the PAH fraction
+starts to decrease. These inflection points seem to be
+around ∼ 37.5 erg s−1 kpc−2 (M⊙ pc−2)−1 for all tar-
+gets. However, it should be pointed out that there are
+(currently) no similar resolution Hi data for IC 5332 and
+NGC 1365, and therefore a universal threshold across
+all environments is left for future studies. Considering
+IHα/ΣHI+H2 traces the fraction of ionized gas per unit
+of total gas, this common threshold could be a limit at
+which the amount of radiation producing the intense Hα
+emission is able to destroy the PAHs through sputtering,
+overcoming shielding from molecular gas and reducing
+the observed PAH fraction.
+We can also see that there is an offset in PAH frac-
+tion, on average, between each galaxy (see also Ta-
+ble 1). Overall, this offset nicely tracks the global galaxy
+metallicity gradients (Kreckel et al. 2019; Santoro et al.
+2022; Groves et al. subm.), in the lower Hα intensity
+regions. As we move towards higher IHα/ΣHI+H2 val-
+ues, the offset gets minimized. This relates to results
+seen by Egorov et al. (2022), where no metallicity trend
+is found with RPAH within H II regions. This suggests
+that the offset in PAH fraction between the four galaxies
+may be driven by a difference in the PAH population in
+the diffuse or neutral ISM set by the average metallic-
+ity of the galaxy. This general offset also follows trends
+observed in previous works that found that the PAH
+fraction correlates positively with metallicity, in nearby
+galaxies (Draine et al. 2007; R´emy-Ruyer et al. 2015;
+Chastenet et al. 2019; Galliano et al. 2021), although it
+
+6
+Chastenet, Sutter, Sandstrom et al.
+IC5332
+5 kpc
+NGC0628
+5 kpc
+NGC1365
+5 kpc
+NGC7496
+5 kpc
+2
+3
+4
+5
+6
+7
+RPAH = (F770W + F1130W)/F2100W
+Figure 1. Maps of RPAH in IC 5332 (top left), NGC 628 (top right), NGC 1365 (bottom left), and NGC 7496 (bottom right).
+We mask the pixels with a S/N < 3 in all bands (gray uniform background). We also mask the central pixels in NGC 1365
+and NGC 7496 which are saturated, using instrument PSFs, and perform a by-hand additional masking to remove conspicuous
+saturation artifacts (shown in gray scale, not included in the analysis). We plot contours for a few of the brightest H II regions.
+They are clearly visible as depressions (darker colors) in RPAH, especially in NGC 628 and NGC 7496.
+
+PAH Fraction and ISM in PHANGS-JWST
+7
+0.0
+0.2
+0.4
+0.6
+0.8
+r/r25
+1.0
+2.0
+3.0
+4.0
+5.0
+RPAH
+IC5332
+NGC0628
+NGC1365
+NGC7496
+8.35
+8.4
+8.45
+8.5
+8.55
+8.6
+12 + log(O/H)
+36.5
+37.0
+37.5
+38.0
+log10(IH /
+H I + H2 [erg s
+1 kpc
+2 (M
+pc
+2)
+1])
+1.0
+2.0
+3.0
+4.0
+5.0
+RPAH
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+H2/
+H I + H2
+Center
+Bar
+Interarm
+Spiral arms
+Disk
+w/o spirals
+1.0
+2.0
+3.0
+4.0
+5.0
+RPAH
+Figure 2. Running medians of RPAH, as a function of (top left:) r/r25; (top right:) 12 + log(O/H) using metallicity maps
+from Williams et al. (2022); (middle left:) IHα/ΣHI+H2 in units of erg s−1 kpc−2 (M⊙ pc−2)−1; (middle right:) the fraction of
+molecular gas; (bottom left:) the environmental masks from Querejeta et al. (2021), with a black star symbol showing the median
+for all pixels within each category. The error-bars show 3× standard error of the mean in each bin (except for the bottom left
+panel, only 1 SEM). Note that the middle panels involve the flat Hi distribution assumption in IC 5332 and NGC 1365, which
+may shift the curves horizontally.
+
+8
+Chastenet, Sutter, Sandstrom et al.
+should be noted that the sample included in this work
+covers a small range of metallicity.
+Figure 3 shows the same variations, with metallicity
+information. We show the 2D-histograms of RPAH as
+a function of IHα/ΣHI+H2, color-coded by the median
+12 + log(O/H) metallicity in each bin. There is a visible
+color difference between each galaxy, but no clear gra-
+dient. This implies that while the average metallicity
+of each galaxy seems to influence the PAH fraction, the
+moderate local metallicity variations are not having a
+large effect on RPAH.
+In the middle right panel of Figure 2, we show the vari-
+ations of RPAH as a function of the fraction of molecular
+gas to total cold gas fraction, as traced by CO. Behav-
+iors vary between each galaxy, showing either a similar
+profile to that of RPAH with IHα/ΣHI+H2 (NGC 1365,
+NGC 628, reflecting the similarity in spatial distribu-
+tion of CO and Hα emission at these scales Schinnerer
+et al. 2019), a rather flat trend (IC 5332, which has little
+CO), or a ∼ 20% increase to a maximum value, followed
+a decrease in RPAH (NGC 7496).
+This could suggest
+that the abundance of PAHs is not particularly sensi-
+tive to the molecular fraction, compared to the ionized
+gas content. Chastenet et al. (2022) found that the av-
+erage grain size of the global PAH population (as traced
+by the 3.3/7.7 µm ratio; e.g., Maragkoudakis et al. 2020;
+Draine et al. 2021; Rigopoulou et al. 2021) is more sen-
+sitive to the fraction of molecular gas.
+In the bottom left panel of Figure 2, we plot the me-
+dian values of RPAH in different galactic environments,
+identified by Querejeta et al. (2021). We use their spa-
+tial mask to separate pixels in 5 different categories (see
+their Table 1). Figure 4 shows the masks projected to
+the RPAH maps, for a visual representation of the differ-
+ent environments. In this panel, it appears that there
+is no striking differences between environments, again
+showing minimal variations of a few tens of percent, for
+individual galaxies. We also show the median and asso-
+ciated standard error of the mean (SEM) for all pixels
+falling into each category, with black symbols.
+Here,
+we can see that RPAH is the highest in the bar, and
+interarm regions, with lower values in the spiral arms,
+followed by the center and finally in the disk. Although
+this approach is limited by the number of targets at this
+stage, it shows promising results. The higher fraction
+of PAHs in the interarms tracks with a less harsh envi-
+ronment due to star formation, and a possibly more Hi
+dominated ISM. Adding more targets to this approach
+will provide a generic view of the variation of the PAH
+fraction in the different phases of nearby galaxies.
+Future work will improve on this work by more finely
+binning the data. For example, it will be interesting to
+investigate the sensitivity of each parameter to the S/N
+measured in the MIRI data. It will also be possible to
+test a Vorono¨ı binning, to check whether the trends seen
+in Figure 2 would be significantly pulled down by taking
+into account more low-S/N pixels.
+4. CONCLUSIONS
+With the advent of the JWST, we can now probe indi-
+vidual mid-IR emission features on spatial scales never
+achieved before outside the Local Group. In this letter,
+we have used a combination of the JWST/MIRI F770W,
+F1130W, and F2100W filters to trace the abundance of
+PAHs relative to small dust grains in four nearby galax-
+ies, IC 5332, NGC 628, NGC 1365, and NGC 7496,
+as part of the Treasury GO program PHANGS-JWST
+#2107.
+We present maps of RPAH ≡ (F770W + F1130W)/F2100W
+in these first four targets.
+This ratio traces the rela-
+tive fraction of PAHs (the F770W and F1130W bands)
+to small dust grains (from the F2100W band).
+The
+ratio RPAH decreases in H II regions, showing that
+the PAH fraction drops there, which is further dis-
+cussed in Egorov et al. (2022).
+We track the vari-
+ations of the abundance ratio as a function of the
+ISM content as traced by CO, Hα, Hi, and metallic-
+ity measurements.
+We find that RPAH as a function
+of ionized gas fraction (traced by IHα/ΣHI+H2, Fig-
+ure 2, bottom left panel) shows a similar trend in all
+the targets:
+a rather flat distribution up to a value
+of IHα/ΣHI+H2
+∼
+1037.5 erg s−1 kpc−2 (M⊙ pc−2)−1
+for all galaxies, at which the abundance ratio system-
+atically decreases. The variations with the fraction of
+molecular gas (Figure 2, bottom right panel) are rather
+small. This work sets the stage for future research to
+refine how the local environment influences the relative
+PAH fraction. As JWST data reduction methods are
+improved and the sample of galaxies with this cover-
+age expands, the conditions in which PAHs are found
+will be better established.
+This early study provides
+insights into how global metallicity and ISM environ-
+ment can effect the relative PAH population, and shows
+the improvements that JWST observations bring to
+determining the answers to these questions.
+ACKNOWLEDGMENTS
+We thank the anonymous referee for their careful
+reading and comments that helped improve the clar-
+ity of the paper.
+This work was carried out as part
+of the PHANGS collaboration, associated with JWST
+program 2107.
+This work is based on observations
+made with the NASA/ESA/CSA JWST. Some/all of
+the data presented in this paper were obtained from
+
+PAH Fraction and ISM in PHANGS-JWST
+9
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+6.0
+7.0
+RPAH
+IC5332
+NGC0628
+36.0
+36.5
+37.0
+37.5
+38.0
+38.5
+39.0
+log10(IH /
+H I + H2 [erg s
+1 kpc
+2 (M
+pc
+2)
+1])
+0.0
+1.0
+2.0
+3.0
+4.0
+5.0
+6.0
+7.0
+RPAH
+NGC1365
+36.0
+36.5
+37.0
+37.5
+38.0
+38.5
+39.0
+log10(IH /
+H I + H2 [erg s
+1 kpc
+2 (M
+pc
+2)
+1])
+NGC7496
+8.350
+8.375
+8.400
+8.425
+8.450
+8.475
+8.500
+8.525
+8.550
+12 + log OH
+Figure 3. 2D histograms of the IHα/ΣHI+H2 and the RPAH, color-coded by metallicity, using the gradient from PHANGS-
+MUSE. The more transparent colors indicate bins with at least 10 hits, while the solid colors with at least 100 hits per bin.
+
+10
+Chastenet, Sutter, Sandstrom et al.
+IC5332
+NGC0628
+NGC1365
+NGC7496
+2
+3
+4
+5
+6
+7
+RPAH = (F770W + F1130W)/F2100W
+Center
+Bar
+Interarm
+Spiral arms
+Disk
+w/o spirals
+Figure 4. RPAH maps colored by the environmental masks from Querejeta et al. (2021): center in red, bar in green, interarms
+in gray, spiral arms in blue and disk in orange. We use this separation to measure the median RPAH in each phase individually,
+and collectively, in Figure 2.
+
+PAH Fraction and ISM in PHANGS-JWST
+11
+the Mikulski Archive for Space Telescopes (MAST) at
+the Space Telescope Science Institute, which is oper-
+ated by the Association of Universities for Research in
+Astronomy, Inc., under NASA contract NAS 5-03127.
+The specific observations analyzed can be accessed via
+10.17909/9bdf-jn24. Based on observations collected at
+the European Southern Observatory under ESO pro-
+grammes 094.C-0623 (PI: Kreckel), 095.C-0473, 098.C-
+0484 (PI: Blanc), 1100.B-0651 (PHANGS-MUSE; PI:
+Schinnerer), as well as 094.B-0321 (MAGNUM; PI:
+Marconi), 099.B-0242, 0100.B-0116, 098.B-0551 (MAD;
+PI: Carollo) and 097.B-0640 (TIMER; PI: Gadotti).
+This paper makes use of the following ALMA data:
+ADS/JAO.ALMA#2012.1.00650.S,
+ADS/JAO.ALMA#2013.1.01161.S,
+ADS/JAO.ALMA#2015.1.00925.S,
+ADS/JAO.ALMA#2015.1.00956.S,
+ADS/JAO.ALMA#2017.1.00392.S,
+ADS/JAO.ALMA#2017.1.00766.S,
+ADS/JAO.ALMA#2017.1.00886.L,
+ADS/JAO.ALMA#2018.1.01651.S.
+ADS/JAO.ALMA#2018.A.00062.S.
+ALMA is a partnership of ESO (representing its mem-
+ber states), NSF (USA) and NINS (Japan), together
+with NRC (Canada), MOST and ASIAA (Taiwan), and
+KASI (Republic of Korea), in cooperation with the
+Republic of Chile.
+The Joint ALMA Observatory is
+operated by ESO, AUI/NRAO and NAOJ.
+JC acknowledges support from ERC starting grant
+#851622 DustOrigin. EJW acknowledges funding from
+the Deutsche Forschungsgemeinschaft (DFG, German
+Research Foundation) – Project-ID 138713538 – SFB
+881 (“The Milky Way System”, subproject P1).
+MC
+gratefully acknowledges funding from the DFG through
+an Emmy Noether Research Group (grant number
+CH2137/1-1). COOL Research DAO is a Decentralized
+Autonomous Organization supporting research in astro-
+physics aimed at uncovering our cosmic origins. JMDK
+gratefully acknowledges funding from the European Re-
+search Council (ERC) under the European Union’s Hori-
+zon 2020 research and innovation programme via the
+ERC Starting Grant MUSTANG (grant agreement num-
+ber 714907). TGW and ES acknowledge funding from
+the European Research Council (ERC) under the Euro-
+pean Union’s Horizon 2020 research and innovation pro-
+gramme (grant agreement No. 694343). MB acknowl-
+edges support from FONDECYT regular grant 1211000
+and by the ANID BASAL project FB210003. IC thanks
+the National Science and Technology Council for sup-
+port through grants 108-2112-M-001-007-MY3 and 111-
+2112-M-001-038-MY3, and the Academia Sinica for In-
+vestigator Award AS-IA-109-M02. KK, OE gratefully
+acknowledge funding from the Deutsche Forschungsge-
+meinschaft (DFG, German Research Foundation) in the
+form of an Emmy Noether Research Group (grant num-
+ber KR4598/2-1, PI Kreckel).
+FB would like to ac-
+knowledge funding from the European Research Coun-
+cil (ERC) under the European Union’s Horizon 2020
+research and innovation programme (grant agreement
+No.726384/Empire). ER and HH acknowledge the sup-
+port of the Natural Sciences and Engineering Research
+Council of Canada (NSERC), funding reference number
+RGPIN-2022-03499. KG is supported by the Australian
+Research Council through the Discovery Early Career
+Researcher Award (DECRA) Fellowship DE220100766
+funded by the Australian Government. KG is supported
+by the Australian Research Council Centre of Excellence
+for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
+through project number CE170100013. AS is supported
+by an NSF Astronomy and Astrophysics Postdoctoral
+Fellowship under award AST-1903834. AKL gratefully
+acknowledges support by grants 1653300 and 2205628
+from the National Science Foundation, by award JWST-
+GO-02107.009-A, and by a Humboldt Research Award
+from the Alexander von Humboldt Foundation.
+Facilities: JWST (MIRI), MUSE, ALMA
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+arXiv:2301.11643v1  [math.NT]  27 Jan 2023
+Primes, knots and periodic orbits
+Christopher Deninger∗
+1
+Introduction
+We begin by sketching some analogies between number theory and knot theory originally
+pointed out by Manin, Mazur [Maz] and Mumford. Those were later developed by Kapra-
+nov, Morishita, Reznikov, Sikora and many other reseachers.
+We then explain a more
+structured analogy where the knots arise from the closed orbits of an R-dynamical system.
+Finally, we explain our recent construction of dynamical systems for arithmetic schemes
+which realize some aspects - but not all - of these analogies. At least in the beginning we
+assume more knowledge of analysis than of algebra on the part of the reader.
+I am very grateful to Umberto Zannier for the invitation to Pisa and to the organizers of
+the Colloquium de Giorgi for giving me the occasion to contribute to this series.
+2
+Primes and Knots
+We first explain how a prime number p can be viewed as a 1-dimensional object and the
+ring of integers Z as 3-dimensional. For this we need to introduce schemes and the étale
+(Grothendieck) topology.
+For a commutative unital ring R, the spectrum spec R consists of the prime ideals p of R.
+It is equipped with the Zariski topology whose closed subsets are of the form
+V (S) = {p ∈ spec R | p ⊃ S}
+with S any subset of R. The Zariski topology is almost never Hausdorff and the topological
+space spec R is only a weak invariant of R. Grothendieck equipped X = spec R with a
+sheaf of rings OX. An element f ∈ R gives a global section of OX and we may view f as a
+function on spec R by sending p to the residue class f mod p in κ(p) = Quot(R/p). Thus
+∗Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Ger-
+many’s Excellence Strategy EXC 2044–390685587, Mathematics Münster: Dynamics–Geometry–Structure
+and the CRC 1442 Geometry: Deformations and Rigidity
+1
+
+we do not have only one field where our functions can take values but many. For example
+f ∈ Z takes values f((p)) ∈ Fp for the prime numbers p and f((0)) ∈ Q. In this sense
+the important idea that “numbers are functions” has been realized. However, working with
+spec Z in isolation, this elementary geometric point of view does not give new arithmetic
+insights. Using functoriality the situation can be improved. Every homomorphism of rings
+ϕ : R1 → R2 induces a continuous map
+spec R2 → spec R1
+via p �→ ϕ−1(p) .
+One even obtains a map of locally ringed spaces between the “affine schemes” spec Ri. Affine
+schemes can be glued to give new “spaces”, the schemes of Grothendieck. Moreover, us-
+ing the entire web of schemes which map to spec R one can refine the Zariski topologies
+to Grothendieck-topologies from which very interesting homotopical and cohomological in-
+variants of the original ring R can be obtained. Let us motivate the basic idea behind
+the étale topology: The usual topology underlying our geometric intuition on a complex
+manifold has the following property: Any analytic map whose Jacobian is everywhere in-
+vertible is itself locally invertible. In the context of algebraic geometry this statement is
+no longer true for the Zariski topology. The map x �→ x2 has the square root as a local
+inverse around x = 1 but the square root is not a polynomial map. One can translate the
+invertibility of the Jacobian into commutative algebra terms and one arrives at the class
+of “etale” maps of schemes. Without giving the formal definition, let us just note that the
+map spec L → spec K corresponding to an inclusion of fields K ⊂ L is étale if and only if
+the extension L/K is finite and separable. For a finite extension of number fields L/K with
+rings of integers oL and oK the map spec oL → spec oK is étale if and only if the extension
+L/K is unramified at all primes. In general it is étale away from the ramified primes.
+The above map x �→ x2 gives a ring homomorphism C[x] → C[x] and the induced map
+spec C[x] ← spec C[x] is étale around the point p = (x − 1) of spec C[x]. For a topology
+closer to our intuition than the Zariski topology, all étale maps and in particular the ones
+in the example should have local inverses. This can only be achieved by generalizing the
+notion of a topology on a set X. In the usual context, the open sets U are subsets of X.
+Instead Grothendieck considered the class of étale maps f : U → X as the “open sets”
+of the étale “topology” on X. Finite intersections are defined by iterated fibre products
+over X, coverings are also easy to define, and many notions of topology can be extended
+to this more general setting, including cohomology and fundamental groups. Now we have
+achieved our goal: In the étale topology, étale maps f are locally invertible! The reason is
+trivial: Consider the diagram:
+U
+id
+⑦⑦⑦⑦⑦⑦⑦⑦
+⑦⑦⑦⑦⑦⑦⑦⑦
+f
+�
+U
+f
+� X
+Here the vertical map f defines an “open set” on which the horizontal étale map f which we
+want to invert has an inverse, namely the identity on U! Historically, in [Ser58] Serre had
+introduced the condition of being “locally isotrivial” for fibre bundles on algebraic varieties.
+This is a weaker condition than local triviality for the Zariski topology. Inspired by this
+work, Grothendieck invented the étale topology and used it to define étale cohomology and
+the étale fundamental group.
+2
+
+Let us now look at the inclusion
+spec Fp ֒→ spec Z , (0) �→ (p)
+coming from the projection Z → Z/p = Fp. The étale fundamental group ˆπ1(spec Fp) is the
+automorphism group of the universal (pro-)étale covering of spec Fp i.e. of spec Fp. Hence
+we have
+ˆπ1(spec Fp) = Aut(Fp) .
+This is a (pro-)cyclic group generated by the Frobenius automorphism x �→ xp. We may
+therefore view
+(1)
+ˆπ1(spec Fp) = ˆZ := lim
+←−
+N
+Z/N
+as an analogue of
+(2)
+π1(S1) = Z
+where S1 is the cicle.
+By the way, the ultimate reason why étale fundamental groups are always pro-finite like
+ˆZ = lim
+←−N Z/N is this: polynomials in one variable have only finite many zeroes. Note that
+the group Z in (2) occurs as the automorphism group of the covering R → S1, t �→ exp 2πit,
+and the deck transformations t �→ t + n correspond to the infinitely many zeroes n ∈ Z of
+the function 1 − exp(2πit).
+It turns out that the higher étale homotopy groups of spec Fp vanish so that for the étale
+topology, spec Fp is a K(ˆZ, 1)-space just as S1 is a K(Z, 1), space for the ordinary topology.
+Let us now turn to spec Z. Its Zariski dimension which intuitively corresponds to the com-
+plex dimension is one. So its étale cohomological dimension which intuitively corresponds to
+the usual real dimension should be 2. However the residue fields Fp of the closed points (p)
+are not separably closed and the étale topological dimension 1 of the spec Fp’s adds to the
+étale topological dimension of spec Z. These heuristics suggest that the étale cohomogical
+dimension of spec Z should be 2 · 1 + 1 = 3. A proof using class field theory can be found in
+[Maz73]. From an arithmetic point of view spec Z is not compact, because it only contains
+the (equivalence classes of the) non-archimedean absolute values of Z, which correspond to
+the prime ideals (p), but not the archimedean absolute value. The latter is usually denoted
+by the symbol ∞ and we may view spec Z = spec Z ∪ {∞} as a compactification of spec Z.
+Artin and Verdier extended the étale topology in a natural way to spec Z. The resulting
+cohomology groups differ from the ones of spec Z only by 2-torsion, which for the purposes
+of this section is harmless.
+In the next section, the role of ∞ will be more important.
+Intuitively we may view spec Z as a compact 3-manifold M3 (without boundary). By a
+theorem of Minkowski, there are no non-trivial everywhere unramified extensions of the
+number field Q. In terms of the étale topology this result can be restated as ˆπ1(spec Z) = 1
+which implies ˆπ1(spec Z) = 1. In this regard spec Z resembles a closed simply connected 3-
+manifold, i.e. S3 by the Poincaré conjecture. Let us write Kp for the embedding spec Fp ֒→
+spec Z. Since the image is (p), we may view Kp as (p) embedded into spec Z. As explained
+above this is reminiscent of an embedding S1 ֒→ S3 i.e. of a knot K. Apparently, this
+observation was first made by Mumford [Maz, Introduction].
+3
+
+Here are some analogies between the topological and the arithmetical situation where for
+simplicity we deal with spec Z instead of spec Z. The abelianized fundamental group of
+the knot complement is cyclic, πab
+1 (S3 \ K) ∼= Z. On the other hand, the abelianized étale
+fundamental group of spec Z \ Kp is the Galois group of the maximal abelian extension of
+Q which is unramified away from the prime p. By a theorem of Kronecker the latter is the
+field Q(µp∞) obtained by adjoining all pn-th roots of unity to Q for n ≥ 1. Hence we have
+ˆπab
+1 (spec Z \ Kp) = Gal(Q(µp∞)/Q) = ˆZ×
+p .
+Here ˆZp is the ring of p-adic numbers. The group ˆZ×
+p is not quite (pro p) cyclic, but it
+almost is. The Alexander polynomial of the knot K is the characteristic polynomial of a
+generator of πab
+1 (S3 \ K) ∼= Z acting on πab
+1 ( �
+S3 \ K) ⊗Z Q. Here �
+S3 \ K is the Z-covering of
+S3 \ K corresponding to the quotient
+π1(S3 \ K) −→ πab
+1 (S3 \ K) ∼= Z .
+Correspondingly, the quotient
+ˆπ1(spec Z \ Kp) −→ ˆπab
+1 (spec Z \ Kp) ∼= ˆZ×
+p
+corresponds to a pro-étale ˆZ×
+p -covering
+�
+spec Z \ Kp of spec Z\Kp. The action of ˆπab
+1 (spec Z\
+Kp) ∼= Z×
+p on ˆπab
+1 (
+�
+spec Z \ Kp) which is the Galois group of the maximal abelian outside of
+p unramified extension of Q(µp∞), gives rise to the Iwasawa zeta function. Iwasawa thought
+in terms of Galois groups, the analogy with knot theory via the étale topology was observed
+only later by Mazur [Maz]. The work of Iwasawa gave rise to a most fruitful direction in
+algebraic number theory. Morishita and his co-authors have also applied techniques from
+knot theory to the study of non-abelian Iwasawa theory [MT17].
+Here is another well known analogy. The mod 2 linking number l(K, L) of two knots K, L
+may be defined by counting mod 2 the number of times that L intersects a Seifert surface
+with boundary K. With this definition it is a nontrivial fact that we have
+(3)
+l(K, L) = l(L, K)
+in Z/2 .
+For odd prime numbers p ̸= l the quadratic residue symbol
+�p
+l
+�
+is 1 if p is a square mod l
+and −1 if it is not. If p or l is congruent to 1 mod 4 the famous Gauss reciprocity law
+asserts that
+(4)
+�p
+l
+�
+=
+� l
+p
+�
+.
+Viewing primes as knots, there are good reasons to view
+� p
+l
+�
+as an analogue of (−1)l(K,L).
+In fact there is a proof of (3) which can be translated to the arithmetic context via our
+dictionary and which then yields a proof of (4).
+The limitations of viewing spec Z as
+analogous to S3 become apparent though. If both p and l are not congruent to 1 mod 4,
+quadratic reciprocity assert that
+�p
+l
+�
+= −
+� l
+p
+�
+.
+4
+
+According to Kapranov and Smirnov [KS, § 3], the reason for this assymmetry is the follow-
+ing: the primes p which are not congruent to 1 mod 4 should correspond to knots which are
+not homologous to zero in the 3-manifold M3 corresponding to spec Z in our analogy. Tak-
+ing S3 for M3 is an oversimplification in this regard because in S3 all knots are homologous
+to zero. In [KS] it is explained how to restore the analogy between quadratic reciprocity
+and mod 2 linking of knots if more general 3-manifolds M3 are allowed.
+For three knots in S3 which are pairwise unlinked their union can still be a non-trivial link,
+as for the Borromean rings. This can be shown by the non-triviality of certain triple Massey
+products for example. In 1938 Redei published a symbol (p, l, q) defined for three prime
+numbers p, l, q congruent to 1 mod 4 whose quadratic residue symbols satisfy ( p
+l ) = 1 and
+( p
+q) = 1. Morishita observed that non-triviality of the Redei symbol could be interpreted
+as the three primes being non-trivially linked while being pairwise unlinked. For example,
+we may view the primes 5, 41, 61 as an analogue of the Borromean link. He also gave an
+entirely parallel construction of “higher Milnor invariants” both for n-tuples of knots and of
+primes, which reduce to the Redei symbol for n = 3, [Mor02]. The relation of Redei symbols
+with Massey products in Galois cohomology is explained in [Mor04]. Recently [AC22] gives
+an interesting application of Massey products to infinite class field towers. There are very
+many further analogies between knot theory and prime numbers and more generally between
+three dimensional topology and algebraic number theory. We refer to the book [Mor12] for
+a great overview of this branch of mathematics, called arithmetic topology.
+3
+Primes and periodic orbits
+In this section we explain certain analogies between number theory and the theory of R-
+dynamical systems on 3-manifolds M3 with a 1-codimensional foliation F. Up to isotopy
+the periodic orbits give embedded circles i.e. knots in M3. The analogies in the present
+section enhance the ones of section 2. They are mostly of an analytic nature and relate
+to analytic number theory contrary to the topological analogies of the previous section.
+I arrived at these analogies by a long detour via cohomological considerations [Den00].
+However they can be much more easily motived by the following argument, [Kop06]. For a
+rational number f ∈ Q× the p-adic absolute values of f are
+|f|p = p−ordpf
+if f = pordpf a
+b
+with 0 ̸= b, a ∈ Z prime to p .
+We also have the ordinary archimedean absolute value |f|∞ := |f|. The product formula
+which immediately follows from these definitions asserts that
+�
+p≤∞
+|f|p = 1 .
+Taking the log, we get
+(5)
+�
+p̸=∞
+ordpf · log p − log |f|∞ = 0 .
+5
+
+This reminds of, but is more complicated than the formula
+(6)
+�
+x∈X
+ordxf = 0
+for a meromorphic function f on a Riemann surface X. In (6) only integers are added,
+whereas in (5) the integers are multiplied with the Q-linearly independent transcendental
+numbers log p, and there is also the term log |f|∞ which seems of a different nature. A
+formula like (5) but without a term corresponding to − log |f|∞ was observed by Ghys
+in the context of Riemann surface laminations.
+Kopei later showed how to include an
+analogue of − log |f|∞: As in [Den08] consider triples (X, F, φ) where X is a closed smooth
+3-manifold with a 1-codimensional foliation F by Riemann surfaces, and φ is a flow, such
+that each φt maps leaves of F to (possibly other) leaves. The flow should be non-degenerate
+which implies that there are only finitely many fixed points x and for any C > 0 there are
+only finitely many closed orbits γ with length l(γ) ≤ C. The fixed points of φ should lie in
+finitely many compact leaves. All other leaves should be non-compact and the flow should
+be transversal to them. Let X be the open complement in X of the union over the finitely
+many compact leaves of F. Since the compact leaves contain fixed points, they are invariant
+under the flow and become sub-dynamical systems of codimension one. The open manifold
+X is invariant under the flow and foliated by the non-compact leaves of F.
+The triple (X, F, φt) can be described more explicitely as follows.
+Pick a leaf F of F
+(they are all diffeomorphic), let Yφ be the vector field generated by the flow and let ωφ be
+the 1-form on X which is zero on the tangent bundle TF to the foliation and such that
+⟨ωφ, Yφ⟩ = 1. For the period group
+Λ =
+� �
+γ
+ωφ | γ ∈ πab
+1 (X)
+�
+⊂ R
+we have Λ = {t ∈ R | φt(F) = F}. For example, if γ is a periodic orbit then l(γ) ∈ Λ and
+φl(γ) is the Poincaré return map on F. The group Λ acts diagonally on F ×R. Moreover the
+action is fixed point free and properly discontinuous. The map F × R → X, (x, t) �→ φt(x)
+induces a diffeomorphism of F ×Λ R with X. Here the leaves of F correspond to the images
+of F × {s} in F ×Λ R and φt becomes translation by t via the second factor. In [ALKL22]
+and [KMNT21] the possible shapes of the triples (X, F, φt) are determined. For example
+the Reeb foliation F on X = S3 with suitable flows are possibilities. Recall that the Reeb
+foliation of S3 is obtained by glueing two solid tori S1 × D foliated by concentric infinitely
+extended paraboloids, along their common boundary. Thus there is one compact leaf, the
+2-torus S1 × S1 and all the other leaves are diffeomorphic to the plane.
+Any choice of a smooth metric on TF over X defines conformal structures on the leaves
+which vary smoothly in the transversal direction. Thus for a smooth 3-manifold with a
+smooth 1-codimensional foliation one obtains the structure of a Riemann surface lamination
+[Ghy99]. Let f be a smooth P1(C)-valued function whose restrictions to all leaves F of F
+are holomorphic and let dFf be the exterior derivative of f along the leaves. Applying
+Stoke’s theorem to the differential form
+1
+2πif −1dFf ∧ωφ on the complement in X of disjoint
+open tubular neighborhoods of the periodic orbits and of ε-neighborhoods Uε(K) of the
+compact leaves K, Kopei obtained the following formula for small ε > 0 in [Kop06]
+(7)
+�
+γ
+ordγf l(γ) −
+�
+x
+1
+2πi
+�
+∂Uε(Kx)
+f −1dFf ∧ ωφ = 0 .
+6
+
+Here γ runs over the periodic orbits and ordγf ∈ Z is defined to be the order of the
+meromorphic function f |F in a point z ∈ F ∩γ ̸= ∅. Since ordz(f |F) ∈ Z varies continuously
+with z ∈ γ, this is independent of the leaf F and the chosen point z ∈ F ∩ γ. Moreover
+x runs over the fixed points and Kx is the compact leaf containing x. Comparing (5) and
+(7), we see that in the analogy prime numbers p correspond to periodic orbits γ where log p
+corresponds to l(γ). Incidentally, note that if we had a bijection p ↔ γ with log p = l(γ),
+then the Riemann zeta function would be a Ruelle zeta function
+ζ(s) =
+�
+γ
+(1 − e−sl(γ))−1
+for Re s > 1 .
+The question whether a natural dynamical system with this property exists is quite old and
+we will discuss our progress in this direction in the last section. Back to our analogy. From
+(5) and (7) we also see that the archimedean absolute value | |∞ of spec Z should correspond
+to a fixed point x. More generally, there is an analogue of (5) for number fields K. It shows
+that the prime ideals 0 ̸= p ∈ spec oK correspond to periodic orbits γ where log Np �= l(γ)
+and the finitely many archimedean absolute values correspond to the finitely many fixed
+points. Note that every periodic orbit γ comes with embeddings
+R/l(γ)Z ֒→ X t mod l(γ) �→ φt(x)
+for the choices of points x ∈ γ. The embeddings for different choices of x are isotopic (via
+φt for a suitable t) and therefore define the same knot in X. Thus the dynamical systems
+analogy between primes and periodic orbits refines the previous analogy between primes
+and knots.
+There are several further analogies between spec Z and systems of the form (X, F, φt). For
+example, Hilbert reciprocity has a dynamical analogue as shown in [KMNT21]. Lichten-
+baum’s conjecture on the order of vanishing and the leading coefficient of the Hasse-Weil
+zeta function of a regular algebraic scheme over spec Z in terms of Weil-étale cohomology
+can be translated almost verbally to the dynamical context, where it can be proved under
+certain conditions, c.f. [Den08], also [Den06]. The proof uses work of Álvarez-Lopez and
+Kordyukov and the Cheeger-Müller theorem.
+Here is another analogy which was found earlier [Den01, 3.5 Corollary].
+The “explicit
+formulas of analytic number theory” are an equality of two distributions on the real line,
+c.f.
+[Wei52].
+Restricted to R>0 they have a particularly simple form: In the space of
+distributions D′(R>0) we have
+(8)
+1 −
+�
+ρ
+etρ + et =
+�
+p
+log p
+�
+k≥1
+δk log p + (1 − e−2t)−1 .
+Here ρ runs over the zeroes of ζ(s) in 0 < Re s < 1, for α ∈ C the locally integrable
+function etα is viewed as a distribution, and δx is the Dirac delta distribution supported in
+x. Evaluating on a test function ϕ ∈ C∞
+c (R>0) and setting Φ(α) =
+�
+R etαϕ(t) dt formula (8)
+takes the more familiar form
+Φ(0) −
+�
+ρ
+Φ(p) + Φ(1) =
+�
+p
+log p
+�
+k≥1
+ϕ(k log p) +
+� ∞
+0
+ϕ(t)
+1 − e−2t dt .
+7
+
+For the foliation analogue, assume that there exists a metric gF of TF such that φt acts
+with conformal factor eαt for some α ∈ R. Then the spectrum of the infinitesimal generator
+θ of the group of operators (φt)∗ for t ∈ R on the space of global leafwise harmonic L2-forms
+ker ∆1
+F,(2) consists of eigenvalues ρ. If dim X = 3 and there are not compact leaves, so that
+X = X and the flow is everywhere transverse to the leaves, then the following formula holds
+in D′(R>0) for certain signs ± which can easily be made explicit
+(9)
+1 −
+�
+ρ
+etρ + eαt =
+�
+γ
+l(γ)
+�
+k≥1
+±δkl(γ) .
+Moreover we have Re ρ = α
+2 for all the eigenvalues ρ. Equation (9) is a transversal index
+theorem for the R-action φt and the complex (Λ• T ∗F, dF) which is elliptic in the leaf
+direction. For fixed t the operator φt∗ on ker ∆i
+F,(2) is not trace class if ker ∆i
+F,(2) is infinite
+dimensional. However for ϕ ∈ C∞
+c (R) the trace of the mollified operator
+�
+R φt∗ϕ(t) dt on
+ker ∆i
+F,(2) exists and by our assumptions we have
+⟨Tr(φ∗ | ker ∆i
+F,(2), ϕ⟩ := Tr
+�
+R
+φt∗ϕ(t) dt
+=
+�
+ρ
+�
+R
+etθϕ(t) dt
+= ⟨
+�
+ρ
+etρ, ϕ⟩ .
+The only eigenvalues ρ of θ on ker ∆i
+F,(2) for i = 0, 2 are 0 and α by our conditions. Thus
+the left hand side of (9) may be viewed as the transversal index defined by Atiyah for
+compact Lie group actions and by Hörmander in general.
+It is given by the following
+Euler-characteristic
+�
+i
+(−1)iTr(φ∗ | ker ∆i
+F,(2)) ∈ D′(R) .
+Incidentally, ker ∆i
+F,(2) may also be interpreted as the maximal Hausdorff quotient of the
+leafwise L2-cohomology
+Hi
+F,(2)(X) = ker di
+F,(2)/im di−1
+F,(2) .
+Formula (9) does not contain a term corresponding to (1−e−2t)−1 in (8) because we assumed
+that φt had no fixed points. If we allow fixed points, then the distributional trace defined
+above may no longer exist. An extension of transverse index theory to such a more general
+situation has only recently been accomplished by Álvarez-Lopez, Leichtnam and Kordyukov,
+c.f. [ALKL23], [ALKL21]. It is much more involved then the transversal case and leads
+to formulas which are quite similar to the explicit formulas of number theory, even as
+distributions on all of R.
+One can show that the conformal factor eαt for a metric gF as above necessarily has to
+be 1, i.e. α = 0 which implies that the eigenvalues ρ of θ on ker ∆1
+F,(2) have real part
+Re ρ = 0. By comparison with the Riemann hypothesis, we would obviously want α = 1
+to be a possibility in the geometric analogue (X, F, φt). This is only one instance which
+shows that the class of smooth compact manifolds is too restrictive to actually rewrite the
+explicit formulas of analytic number theory as a transversal index theorem. One can show
+8
+
+that α = 1 can be achieved if for X we allow the local structure (totally disconnected) ×
+(3-dimensional ball), c.f. [Lei07]. In [Ghy99] and [MS06] it is explained how to do analyis
+of PDE on such spaces.
+Analogies of Arakelov theory with foliated dynamical systems
+were obtained in [Kop11]. Recently we found an argument why a possible complex valued
+Weil-type cohomology theory for arithmetic curves spec oK cannot have a functorial real
+structure, [Den22b].
+Since leafwise cohomology groups or the spaces of global leafwise
+harmonic forms always have natural real structures, this shows that our analogies are not
+perfect in this regard. We are missing a fundamental “twist” excluding real structures on
+cohomology somewhere.
+4
+Dynamical systems for arithmetic schemes
+In this section we recall the construction of “foliated” topological dynamical systems for
+arithmetic schemes given in [Den22a] and explain their basic properties. We first found an
+extrinsic construction of the typical leaf with its Poincaré return maps guided by the idea
+to use Frobenius elements in Galois groups to generate periodic orbits. The following more
+general and conceptual intrinsic definition using C-valued points of rational Witt spaces
+came later after understanding the work of Kucharczyk and Scholze [KS18]. The previous
+extrinsic definition then became a theorem, namely Theorem 4.1 below. Our dynamical
+systems have some but by no means all of the properties that we expect from the analogies
+in section 3. We view them as a first but important step in our quest to apply analysis
+on dynamical systems to number theory. Since arithmetic over p-adic fields is much better
+understood than over number fields we also studied a p-adic version of our construction. In
+that situation there is a very natural modification of the dynamical system and it turned
+out to be closely related to the Fargues-Fontaine curve, one of the fundamental objects in
+p-adic Hodge theory.
+Before we can define rational Witt spaces, we have to recall the rational Witt vectors Wrat(R)
+of a commutative unital ring R. As a set, Wrat(R) consists of the rational functions f among
+the power series f ∈ R[[t]], with f(0) = 1, i.e. f = P/Q with P, Q ∈ R[t], P(0) = 1 = Q(0).
+Addition in Wrat(R) is defined to be multiplication of rational functions. Following Almkvist
+[Alm74] we write f ∈ Wrat(R) as a quotient
+f = det(1 − tϕ | V )
+det(1 − tψ | W) ,
+where V and W are projective R-modules of finite rank equipped with endomorphisms ϕ
+and ψ. If ˜f has a corresponding expression then the sum f + ˜f in Wrat(R) will be represented
+by (ϕ ⊕ ˜ϕ, V ⊕ ˜V ) and (ψ ⊕ ˜ψ, W ⊕ ˜W). By definition the product of f and ˜f in Wrat(R) is
+represented by (ϕ⊗ ˜ϕ, V ⊗ ˜V ) and (ψ ⊗ ˜ψ, W ⊗ ˜W). One can check that this is independent
+of all choices, and in fact the product in Wrat(R) coincides with the product in the big Witt
+ring W(R) = 1 + tR[[t]]. This leads to the K-theoretical description of Wrat(R) in [Alm74].
+Now let X be a scheme, for example X = spec R and let Wrat(OX) be the sheafification of
+the Zariski presheaf U �→ Wrat(OX(U)). For all x ∈ X we have Wrat(OX)x = Wrat(OX,x).
+We call the ringed space
+Wrat(X) = (Xtop, Wrat(OX)) ,
+9
+
+the rational Witt space of X, where Xtop is the underlying topological space of X. If S is
+another scheme we define a morphism P : S → Wrat(X) as a morphism of ringed spaces
+(P, P ♯) such that for all s ∈ S there is a (uniquely determined) ring homomorphism ˜P ♯
+s
+making the following diagram commutative
+(10)
+Wrat(OX,f(s))
+P ♯
+s
+�
+��
+OS,s
+��
+Wrat(κ(f(s)))
+˜P ♯
+s
+� κ(s) .
+Here κ denotes the residue field. The stalks Wrat(OX)s are not local rings in general and
+Wrat(X) is not a locally ringed space. Condition (10) replaces the usual locality condition
+in scheme theory. The set of S-valued points of Wrat(X) is
+Wrat(X)(S) = Mor(S, Wrat(X)) .
+If S = spec A is affine we set Wrat(X)(A) = Wrat(X)(S). The canonical ring homomor-
+phisms
+Wrat(R) −→ R , f �−→ −f ′(0)/f(0)
+induce a canonical morphism
+X −→ Wrat(X) .
+By composition we get an injection
+X(S) ֒→ Wrat(X)(S) .
+Hence Wrat(X)(S) contains the classical S-valued points of X.
+Morphisms Wrat(Y ) → Wrat(X) between two rational Witt spaces are defined similarly as
+above. Every morphism α : Y → X of schemes induces a morphism Wrat(α) : Wrat(Y ) →
+Wrat(X) and one obtains a faithful but not fully faithful functor from schemes to rational
+Witt spaces.
+For example, the commuting Frobenius endomorphisms Fν for ν ≥ 1 of
+Witt vector theory induce commuting Frobenius endomorphisms Fν on Wrat(X). We have
+Fpn = Wrat(F n) if X is an Fp-scheme and F : X → X is the absolute Frobenius, but in
+general the Fν are not induced by morphisms of schemes. The multiplicative monoid N of
+positive integers therefore acts on Wrat(X) and hence also on Wrat(X)(S) for all S. We now
+describe how global sections f of X give rise to A-valued functions on Wrat(X)(A) using
+the multiplicative (Teichmüller-)map
+[ ] : R −→ Wrat(R)
+with [r] = 1 − rt .
+Namely, for f ∈ Γ(X, OX) consider its images [f] in Wrat(OX(X)) and hence in Wrat(OX)(X).
+A point P ∈ Wrat(X)(C) gives a morphism of sheaves P ♯ : Wrat(OX) → P∗Ospec C. We de-
+fine the value of f in P to be the image of [f] ∈ Wrat(OX)(X) in (P∗Ospec C)(X) = C under
+this map. In this way we obtain a multiplicative map from Γ(X, OX) into the C-algebra of
+C-valued function on Wrat(X)(C).
+For normal schemes X with countable function field we defined a natural topology on
+Wrat(X)(C) in [Den22a, section 7] and the previous construction gives a multiplicative map
+(11)
+[ ] : Γ(X, O) −→ C0(Wrat(X)(C), C) .
+10
+
+For X = spec Z this map interprets numbers i.e. elements of Γ(spec Z, O) = Z as highly
+non-trivial C-valued continuous functions on the infinite dimensional connected Hausdorff
+space Wrat(spec Z)(C). The following result which follows from [KS18, Lemma 4.9] makes
+the points of Wrat(X)(C) somewhat more explicit but even for X = spec Z we do not have
+a satisfactory understanding of that space. We change the notation somewhat.
+Theorem 4.1. Let X0 be a normal scheme with function field K0 and let X be the normal-
+ization of X0 in an algebraic closure K of K0. Let G = Aut(K/K0) be the absolute Galois
+group of K0 and set
+•
+X(C) = {(x, P
+×) | x ∈ X, P
+× ∈ Hom(κ(x)×, C×)} .
+Then there is a natural N -equivariant bijection
+Wrat(X0)(C)
+∼−→
+•
+X0(C) :=
+•
+X(C)/G .
+Here σ ∈ G acts on
+•
+X(C) via (x, P
+×)σ = (xσ, P
+×
+◦σ) and ν ∈ N acts G-equivariantly via
+Fν(x, P
+×) = (x, P
+×
+◦( )ν).
+The space Wrat(X0)(C) =
+•
+X0(C) is almost the typical leaf of the “foliated” dynamical system
+that we will construct. The monoid N acts on
+•
+X0(C) by injective maps and we need to
+invert them to get a group action by (Q>0, ·) via homeomorphisms. This is done by forming
+the colimit space over N viewed as a poset ordered by divisibility
+ˇX0(C) = colimN
+•
+X0(C) .
+Note that on the level of functions we have a limit over Frobenii, which reminds of the
+tilting process in p-adic geometry. Set
+X0 = (ˇX0(C) × R>0)/Q>0
+where Q>0 acts diagonally. Let t ∈ R act on X0 by setting φt[P, u] = [P, etu]. The 1-
+codimensional “foliation” F has leaves the images of ˇX0(C) × {u} in X0 for u ∈ R>0. It
+is everywhere transversal to the flow and each φt maps leaves to leaves. In general, the
+dynamical system (X0, φt) has too many periodic orbits, since the N -space Wrat(X0)(C)
+does not know enough about the addition in OX0. In the local p-adic situation below, we
+know the right modification to make. However in the global case presently we can only
+impose an “admissible” condition E on the characters P
+× : κ(x)× → C× in the description
+of Theorem 4.1. What we know for certain is that the restrictions of P
+× to µ(κ(x)) must
+have finite kernels (condition Etors) and that for char κ(x) > 0 the image of P
+× must not
+be torsion unless κ(x)× is itself torsion. For example E can be the conditions that ker P
+×
+is always finite resp.
+finitely generated.
+We refer to [Den22a, section 4] for a detailed
+discussion. With the obvious modifictions we get a G×N -space
+•
+X(C)E, an N -space
+•
+X0(C)E
+and a “foliated” dynamical system (X0E, F, φt).
+Theorem 4.2. Let X0 be normal of finite type over spec Z, e.g.
+X0 = spec R0 for an
+integrally closed finitely generated ring R0. Then we have
+{x0 ∈ X0E | φt(x0) = x0
+for some t > 0} =
+�
+x0
+Γx0 .
+11
+
+Here x0 runs over the closed points of X0, i.e. the maximal ideals m0 of R0 if X0 = spec R0.
+The compact subsets Γx0 ⊂ X0 consist of periodic orbits of length log Nx0 where Nx0 =
+|κ(x0)| (= |R0/m0|) and they are pairwise disjoint. In fact Γx0 is a fibre space over the
+compact group Aut(F
+×
+p )/Aut(Fp) where p = char κ(x) with fibres the compact orbits in Γx0.
+The theorem asserts that the closed points of X0 e.g. the prime numbers p if X0 = spec Z
+correspond not to individual periodic orbits γ as in the analogies of section 3 but to compact
+packets of periodic orbits all of which have length log Nx0 i.e. log p if X0 = spec Z. This is
+reminiscient of the invariant tori of Hamiltonian dynamics.
+It follows from the existence of Frobenius elements in the Galois group G that closed points
+give periodic orbits in X0E. It is more difficult to show that any periodic orbit comes from
+a closed point i.e. lies in Γx0 for some x0, c.f. [Den22a, Theorems 5.2 and 6.1].
+The proof of the following result requires de Jong’s theory of alterations if dim X0 ≥ 2 and
+some approximation theorems from number theory.
+Theorem 4.3. Let X0 be an integral normal scheme which is flat of finite type over spec Z.
+Then the spaces X0 and X0E are connected. In fact they are almost pathwise connected: for
+any two points x0 and x′
+0 in X0E and any two neighborhoods U0 ∋ x0 and U′
+0 ∋ x′
+0 there are
+points y0 ∈ U0 and y′
+0 ∈ U′
+0 which can be connected by a continuous path.
+Another result, which is stronger than connectedness asserts that the leafwise cohomology
+group H0
+F(X0E) is one dimensional. It is essentially the group of global sections of the sheaf
+of continuous R-valued functions on X0E which are locally constant along the leaves of F,
+c.f. [Den22a, section 10].
+The space X0E is infinite dimensional if dim X0 ≥ 1 and one could hope that the sub-
+dynamical system obtained as the closure of the union of all its compact orbits might be
+significantly smaller. However, this is not the case as follows from [Den22a, Theorem 8.2].
+The closure is the subsystem obtained by replacing
+•
+X(C)E in the previous constructions with
+the subspace of pairs (x, P
+×) with P
+× : κ(x)× → S1 a unitary character. For dim X0 ≥ 2
+this is conditional on a result in Diophantine approximation which should be provable and
+which is known for dim X0 = 1 and X0 flat over spec Z.
+The following result makes the structure of the dynamical system X0 clearer. Consider the
+natural projection
+•
+X(C) → X mapping (x, P
+×) to x. Let
+•
+X(C)p resp.
+•
+X(C)Q be the fibres of
+the composition
+•
+X(C) → X → spec Z over (p) resp. (0). Consider the G-invariant subspace
+•
+X(C)in = {(x, P
+×) ∈
+•
+X(C) | P
+× |µ(κ(x)) is injective }
+and set
+•
+X(C)p,in =
+•
+X(C)in ∩
+•
+X(C)p
+and
+•
+X(C)Q,in =
+•
+X(C)in ∩
+•
+X(C)Q .
+These are subspaces of
+•
+X(C) and the quotient topologies on
+•
+X0(C)p,in =
+•
+X(C)p,in/G
+and
+•
+X0(C)Q,in =
+•
+X(C)Q,in/G
+12
+
+agree with the subspace topologies within
+•
+X0(C). For any p, the Frobenius endomorphism
+Fp of
+•
+X0(C) restricts to a homeomorphism of
+•
+X0(C)p,in. Hence pZ ⊂ Q>0 acts on
+•
+X0(C)p,in
+via p ↔ Fp.
+Theorem 4.4. The following canonical φt-equivariant map is a continuous bijection (and
+similarly after imposing an admissible condition E)
+•
+X0(C)Q,in × R>0 ∐
+�
+p
+•
+X0(C)p,in ×pZ R>0
+∼−→ X0 = ˇX0(C)Etors ×Q>0 R>0 .
+In view of Theorem 4.3 the map in the theorem is not a homeomorphism since X0 is
+connected, whereas the left hand side is disconnected.
+We now discuss the relation of our construction with the work of Kucharczyk and Scholze
+[KS18].
+Consider a field of characteristic zero containing all roots of unity, and fix an
+injective homomorphism ι : µ(F) = µ(F) ֒→ C×.
+The multiplicative Teichmüller map
+[ ] : F × ֒→ Wrat(F) and the multiplicative map ι give ring homomorphisms
+Z[µ(F)] −→ Wrat(F)
+and
+Z[µ(F)] −→ C
+on the group ring of µ(F). Set
+XF = spec (Wrat(F) ⊗Z[µ(F )] C) .
+One can show that the connected components of Wrat(spec F)(C) are parametrized by the
+embeddings µ(F) ֒→ C× and that XF(C) is the component corresponding to ι. For arbitrary
+connected pointed topological spaces (Z, z) Kucharczyk and Scholze defined a pro-finite
+fundamental group π´et
+1 (Z, z) which classifies the finite coverings of Z. One of their main
+results is an isomorphism of π´et
+1 (XF(C), x) with the absolute Galois group of F. They also
+calculate the sheaf cohomology of XF(C) with coefficients in Z/n and Q. For Z/n they prove
+that it is isomorphic to the Galois cohomology of F with Z/n-coefficients using the work
+of Rost and Voevodsky on the Bloch-Kato conjecture. For Q-coefficients the cohomology
+of XF(C) is the Galois fixed part of Λ• (F
+× ⊗ Q).
+A more natural group would be the
+Milnor K-group of F tensored with Q. The absence of the Steinberg relations in rational
+cohomology is an indication that the space XF(C) and hence also our space Wrat(X)(C) do
+not encode enough information about the additive structure of F resp. OX.
+In [DW23] we introduced and studied a pro-algebraic fundamental group πE(Z, z) over any
+field E for pointed connected topological spaces (Z, z). Its maximal pro-étale quotient is
+π´et
+1 (Z, z) viewed as a group scheme over E. For the “right” version of XF(C) discussed in
+the introduction of [KS18], we expect the pro-algebraic fundamental group to be deeply
+related to the motivic Galois group for motives over F with coefficients in E. In [DW23,
+section 5] we calculated πE(XF(C), x) for algebraically closed fields E of characteristic zero.
+Its connected component is commutative and its unipotent part is related to extensions of
+Tate motives. For the reductive part we could see no relation with motives, i.e. to the Serre
+group. In our opinion this just shows that XF(C) and our spaces Wrat(X0)(C) are only first
+steps (but in the right direction).
+For inspiration, we looked at the much better understood situation where X0 is a normal
+scheme of finite type over spec Zp instead of spec Z. In order to compare with p-adic Hodge
+13
+
+theory it is reasonable to study Wrat(X0)(o) where o is the valuation ring in the completion
+Cp of Qp. We proved that
+Wrat(X0)(o) = Wrat(X)(o)/G .
+Here, as before X is the normalization of X0 in K0, with K0 the function field of X0 and
+G its absolute Galois group. The points of Wrat(X)(o) are certain diagrams determined by
+points x,y ∈ X with y ∈ {x} and multiplicative maps Py : O{x},y → o sending 1 to 1 and 0
+to 0. Here we know how to modify our constructions in order to get the right Fp-dynamical
+system: We have to impose the condition that the mod p reduction of the multiplicative
+map Py i.e. the composition
+(12)
+O{x},y
+Py
+−→ o −→ o/p
+is additive. The technical details are more involved but this is the heart of the matter. It
+now turns out that the elements of Fontaine’s Ainf-rings become ordinary o-valued functions
+on the resulting sub-dynamical systems. Judith Lutz has proved that this representation
+of Ainf by functions is faithful. Also, e.g. for X0 = spec Zp we obtained a natural bijection
+of the subsystem with the points of the Fontaine-Fargue curve.
+This uses the work of
+Fontaine-Wintenberger in the reformulation by Scholze and Kedlaya. One can reformulate
+additivity of (12) in terms of absolute values but it is still unclear what the right analogue
+is to impose on Wrat(X)(C) for X0/Z.
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+16
+

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+Learn to Rapidly and Robustly Optimize
+Hybrid Precoding
+Ortal Lavi and Nir Shlezinger
+Abstract
+Hybrid precoding plays a key role in realizing massive multiple-input multiple-output (MIMO)
+transmitters with controllable cost. MIMO precoders are required to frequently adapt based on the
+variations in the channel conditions. In hybrid MIMO, where precoding is comprised of digital and
+analog beamforming, such an adaptation involves lengthy optimization and depends on accurate channel
+state information (CSI). This affects the spectral efficiency when the channel varies rapidly and when
+operating with noisy CSI. In this work we employ deep learning techniques to learn how to rapidly and
+robustly optimize hybrid precoders, while being fully interpretable. We leverage data to learn iteration-
+dependent hyperparameter settings of projected gradient sum-rate optimization with a predefined number
+of iterations. The algorithm maps channel realizations into hybrid precoding settings while preserving
+the interpretable flow of the optimizer and improving its convergence speed. To cope with noisy CSI,
+we learn to optimize the minimal achievable sum-rate among all tolerable errors, proposing a robust
+hybrid precoding based on the projected conceptual mirror prox minimax optimizer. Numerical results
+demonstrate that our approach allows using over ten times less iterations compared to that required
+by conventional optimization with shared hyperparameters, while achieving similar and even improved
+sum-rate performance.
+I. INTRODUCTION
+Wireless communication networks are subject to constantly growing requirements in terms of
+connectivity, throughput, and reliability. One of the emerging technologies which is expected to
+play a key role in meeting these demands is based on equipping wireless base stations (BSs) with
+large-scale antenna arrays, resulting in massive multiple-input multiple-output (MIMO) networks
+[2]. While the theoretical gains of massive MIMO are well-established [3]–[5], implementing
+such large scale arrays in a power and cost efficient manner is associated with several core
+Parts of this work were presented at the 2022 IEEE Workshop on Signal Processing Advances in Wireless Communications
+(SPAWC) as the paper [1]. This work was supported in part by the Israeli Innovation Authority through the 5G-WIN consortium.
+The authors are with the School of ECE, Ben-Gurion University of the Negev (e-mail: [email protected]; [email protected]).
+1
+arXiv:2301.00369v1  [eess.SP]  1 Jan 2023
+
+challenges. Among these challenges is the conventional need to feed each antenna element with
+a dedicated RF chain, which tend to be costly and consume notable power [6].
+A leading approach to tackle the cost and power challenges of massive MIMO is to utilize
+hybrid analog/digital MIMO transceivers [7], [8]. Hybrid MIMO transceivers carry out part of
+the processing of the transmitted and received signals in the analog domain, enabling operation
+with less RF chains than antennas [9]–[11]. As a result, hybrid MIMO transmitters implement
+precoding partially in digital and partially in analog. Analog processing is dictated by the
+circuitry, often implemented using vector modulators [12] or phase shifters [9]. Consequently,
+analog precoding is typically more constrained compared with digital processing, where, e.g.,
+one can typically apply different precoders in each frequency [13].
+The constrained form of hybrid MIMO makes the setting of the precoding pattern for a given
+channel realization notably more challenging compared with costly fully-digital architectures.
+Various methods have been proposed for designing hybrid precoding systems, optimizing their
+analog and digital processing to meet the communication demands [14]. The common approach
+formulates the objective of the precoders, e.g., sum-rate maximization or minimizing the distance
+from the fully-digital precoder [15], [16], as an optimization problem. The resulting optimization
+is then tackled using iterative solvers which vary based on the objective and the specific con-
+straints induced by the analog circuitry and the antenna architecture. Iterative algorithms were
+proposed for tunning hybrid MIMO systems with controllable gains [11], phase-shifting structure
+[10], [13], partial-connectivity [9], [15], discretized vector modulators [12], and Lorentzian-
+constrained metasurface antennas [17], [18]. While iterative optimizers are interpretable, being
+derived as the solution to the formulated problem, they are often slow in terms of convergence.
+This can be a major limitation as this setting is based on the instantaneous channel state
+information (CSI), and thus must be done in real-time to cope with the frequent variations
+of wireless channels, and its performance depends on the accuracy of the CSI.
+An alternative emerging approach to tuning hybrid precoders is based on deep learning. This
+approach builds upon the ability of deep neural networks (DNNs) to learn complex mapping
+while inferring at controllable speed dictated by the number of layers, which is often much
+faster compared with conventional iterative optimizers [19]. The usage of deep learning to tackle
+optimization problems is referred to as learn-to-optimize [20]. DNN-aided optimization of hybrid
+precoders was considered in [21]–[27], which employed multi-layered perceptrons [21], [22]
+and convolutional neural networks [23]–[27] as the optimizer, while [28], [29] employed deep
+2
+
+reinforcement learning techniques. While such deep models are able to map channel estimates
+into precoder structure, they lack of transparency or interpretability in how input data are
+transformed into the precoder setting. Furthermore, the resulting precoding scheme is geared
+towards the number of users and the distribution of the channels used for its training. When
+the channel distribution or the number of users changes, the DNN needs to be trained anew,
+which is a lengthy procedure. Moreover, highly-parameterized DNNs may be too computationally
+complex to deploy on hardware-limited wireless communication devices.
+Interpretable deep models with a small number of parameters can be obtained from iterative
+optimization algorithms via the deep unfolding methodology [30]. Deep unfolding leverages
+data-driven deep learning techniques to improve an iterative optimizer, rather than replace its
+operation with DNNs [31], [32]. Deep unfolded optimizers were utilized for configuring one-bit
+hybrid precoders in [33]; for fully-digital (non-hybrid) narrowband precoders in [34]; for narrow-
+band phase shifter based hybrid precoders in [35], [36]; and for narrowband hybrid predcoders
+with limited feedback in [37]. The latter unrolled an optimization algorithm using generative
+networks for handling the limited feedback and overparametrized ResNets for optimizing the
+precoders, resulting in a highly-parameterized DNN, which may not be suitable for application on
+hardware limited MIMO devices. The work [38] used dedicated DNNs integrated into unfolded
+optimization of analog precoders in hybrid MIMO transmitters with full CSI, at the cost of
+additional complexity during inference. This motivates designing interpretable light-weight learn-
+to-optimize algorithms for rapidly translating CSI into multi-band hybrid precoders, while being
+robust to inaccurate CSI.
+In this work, we propose a learn-to-optimize algorithm for tuning multi-band downlink hybrid
+MIMO precoders in a manner that is rapid, robust, and interpretable. We consider different analog
+architectures, including analog precoders with controllable gains, as well as architectures based on
+fully-connected phase shifters. Our approach leverages data in the form of past channel realization
+to accelerate the convergence of conventional iterative optimization of the achievable sum-rate.
+In order to design an unfolded algorithm for dealing with the challenging and practical setting
+where the CSI may be noisy, we commence by treating the setting where full CSI is available,
+for which we formulate an unfolded optimizer which naturally extends to cope with noisy CSI.
+Our proposed data-aided algorithms fully preserve the interpretable and flexible operation of
+conventional optimizers from which they originate, while being light-weight and operating with
+fixed run-time and complexity.
+3
+
+In particular, we adopt the projected gradient ascent (PGA) algorithm as the hybrid precoding
+designing optimizer for maximizing the achievable sum-rate with full CSI. Our motivation here
+stems from the ability of this approach to directly optimize the sum-rate, rather than adopting
+some simpler surrogate objective as done in, e.g., [10], [15], which can thus be extended to
+robust optimization of this measure. We unfold the PGA algorithm, by fixing a small number of
+iterations and train only its iteration-dependent hyperparameters which can be tuned from data
+without deviating from its overall flow and operation.
+Our selection of PGA optimization with the sum-rate objective as our basis for hybrid pre-
+coding design with full CSI facilitates extending our approach to noisy CSI settings: we identify
+the projected conceptual mirror prox (PCMP) algorithm for maximin optimization [39] as a
+suitable optimizer for hybrid precoding with noisy CSI, whose algorithmic steps involve com-
+putations of gradients of projection of the sum-rate objective, as we used by PGA with full
+CSI. Consequently, we propose to cope with noisy CSI by formulating our design objective as
+maximizing the minimal achievable sum-rate within a given tolerable CSI error margin. We then
+unfold PCMP, resulting in a trainable architecture involving similar steps as those used when
+unfolding PGA, where again data is leveraged to tune the hyperparameters of the optimizer for
+each iteration. Our experimental results evaluate our proposed unfolded algorithms compared
+with the conventional optimizers from which they originate for hybrid precoding in multi-band
+synthetic Rayleigh channels as well as channels generated via the Quasi Deterministic Radio
+channel Generator (QuaDRiGa) channel simulator [40] in various signal-to-noise ratios (SNRs).
+There, we systematically show that the proposed unfolded algorithms achieve similar and even
+improved sum-rates compared with their iterative counterparts, while operating with over ten
+times less iterations and computational complexity.
+The rest of this work is organized as follows: Section II describes the system model; the
+proposed learn-to-optimize algorithm for tuning hybrid precoders with full CSI is derived in
+Section III, while Section IV derives the learned optimizer for noisy CSI. Our proposed algo-
+rithms are numerically evaluated in Section V, and Section VI provides concluding remarks.
+Throughout the paper, we use lowercase boldface letters for vectors, e.g., x; [x]i denotes the
+ith element of x. Uppercase boldface letters are used for matrices, e.g., X, with [X]i,j being its
+(i, j)th entry and In denoting the n×n identity matrix. Calligraphic letters, such as X, are used
+for sets, and C is the set of complex numbers. The transpose, Hermitian transpose, Frobenius
+norm, and stochastic expectation are denoted by (·)T, (·)H, ∥ · ∥F, and E{·}, respectively.
+4
+
+Fig. 1: Architecture of MIMO system using hybrid precoding
+II. SYSTEM MODEL
+In this section we present the system model. We start by reviewing the model for wideband
+downlink MIMO communications with hybrid beamforming in Subsection II-A, after which we
+discuss the considered analog precoder architectures in Subsection II-B. Then, we formulate the
+considered problem of rapid and robust hybrid precoder setting in Subsection II-C.
+A. Downlink MIMO Communications with Hybrid Beamforming
+1) Channel Model: We consider a single-cell downlink hybrid MIMO system with N single-
+antenna users. The BS is equipped with M transmitting antennas, and utilizes B frequency bands
+for communications, where the spectrum is shared among all users in a non-orthogonal fashion.
+The overall system is illustrated in Fig. 1.
+The BS employs multi-carrier signaling, and we use sb ∈ CN to denote the multiuser trans-
+mitted signal vector that is being transmitted in the bth frequency bin, b ∈ {1, 2, . . . , B} ≜ B.
+The transmitted symbols are i.i.d. and of equal power, such that E[sbsH
+b ] = 1
+N IN for each b ∈ B.
+At each frequency bin b ∈ B, for a channel input xb ∈ CM, the channel output is given by
+yb = Hb · xb + nb ∈ CN,
+(1)
+where Hb ∈ CN×M is the bth frequency band sub-channel, and nb ∈ CN is additive white
+Gaussian noise (AWGN) with i.i.d. entries of variance σ2.
+2) Hybrid Beamforming: The BS has L < M RF chains, and thus employs hybrid precoding.
+Here, the multiuser transmitted signal vectors {sb}b∈B are precoded in two stages. First, a digital
+5
+
+Channel
+Receivers
+Digital Precoder
+Analog Precoder
+→Y1,1
+S1
+RF Chain
+S2
++YB,1
+Wa
+.
+Wa,B
+HB
+.Y1,N
+SB
+RF Chain
+→yB,N
+N
+B
+L
+M
+B
+N
+Users
+FreguencyBands
+RF Chains
+Tx Antennas
+Frequency Bands
+Receiversprecoder Wd,b ∈ CL×N is applied to sb in each frequency bin b ∈ B, i.e., Wd,b is the digital
+precoding matrix of the bth bin. Next, the digital symbols pass through L RF chains, and are
+combined into the channel input xb using an analog precoder. Unlike digital processing, analog
+precoding is carried out using dedicated hardware assumed to be static in frequency. Hence,
+analog precoding is modeled using the matrix Wa ∈ A ⊆ CM×L, where A represents the set of
+feasible analog precoder settings, discussed in the sequel.
+The output of the transmitter in the bth bin is given by
+xb = WaWd,b · sb.
+(2)
+We require the precoders to satisfy
+1
+B
+B
+�
+j=1
+∥WaWd,j∥2
+F ≤ N,
+(3)
+as the transmitter’s total power constraint. Substituting (2) into (1), the channel output at the N
+users at frequency bin b can be written as
+yb = HbWaWd,b · sb + nb ∈ CN.
+(4)
+B. Analog Precoders
+The feasible mappings that can be realized by the analog precoder Wa, encapsulated in the set
+A, depend on the hardware architecture. We consider two different architectures for the analog
+precoder: unconstrained precodersZ and fully-connected phase shifter networks.
+1) Unconstrained Precoder: Here, the analog precoding matrix, Wa, has no hardware con-
+straints, and can realize any complex matrix (while satisfying the overall power constraint (3)).
+For unconstrained precoders, we use A = CM×L. Such analog processing with configurable
+attenuation and phase shifting can be implemented using, e.g., vector modulators, as in [11],
+[12]. While this formulation does not assume any constraints, other than the power constraint, it is
+emphasized that unconstrained solutions can be useful for constrained analog hardware designs,
+which often involve a preliminary step of obtaining an unconstrained combiner followed by its
+projection to the constrained set, as done in, e.g., [10].
+2) Phase Shifter Networks: A common implementation of analog precoders utilizes phase
+shifters [9], [10]. In fully-connected phase shifter networks, every antenna element is connected
+6
+
+to each RF chain via a dedicated controllable phase shifter. Such precoders are modelled as
+matrices whose entries have a unit magnitude, namely,
+A =
+�
+A ∈ CM×L���
+��[A]m,l
+�� = 1,
+∀(m, l)
+�
+.
+(5)
+C. Problem Formulation
+We aim at designing the hybrid precoding operation given a channel realization to maximize
+the achievable sum-rate of multiuser downlink MIMO communications. Defining the sum-rate as
+the precoding objective is a common approach [13], [18], [41], being a communication measure
+of the overall combined effect of the channel and the hybrid precoder. We particularly focus on
+two scenarios – sum-rate maximization (given accurate CSI) and robust sum-rate maximization
+(given noisy CSI).
+1) Sum-Rate Maximization: Since [sbsH
+b ] =
+1
+N IN, it holds that the following sum-rate is
+achievable [42]
+R (Wa, {Wd,b}b∈B, {Hb}b∈B) = 1
+B
+B
+�
+b=1
+log
+����IN +
+1
+Nσ2HbWaWd,bWH
+d,bWH
+a HH
+b
+���� .
+(6)
+Consequently, for a given channel realization {Hb}b∈B, we aim to find W(o)
+a
+and {W(o)
+d,b}b∈B, that
+are the solution to the following optimization problem
+�
+W(o)
+a , {W(o)
+d,b}b∈B
+�
+=
+argmax
+Wa,{Wd,b}b∈B
+R (Wa, {Wd,b}, {Hb})
+(7)
+s.t.
+�
+�
+�
+�
+�
+1
+B
+�B
+b=1 ∥WaWd,b∥2
+F ≤ N
+Wa ∈ A
+.
+In (7), A is the set of matrices corresponding to the analog precoding models described above, i.e.,
+we seek to find a suitable precoders under the different constraints detailed in Subsection II-B.
+Since the optimization problem (7) must be tackled each time the channel realization changes,
+our proposed solution should not only tackle (7), but also to do it rapidly, i.e., with a predefined
+amount of computations. To achieve this aim, we assume access to a set D of previously
+encountered or simulated channel realizations {Hr}|D|
+r=1, which can be exploited to facilitate
+optimization within a fixed number of operations (dictated by, e.g., the coherence duration of
+the channel).
+7
+
+2) Robust Sum-Rate Maximization: The optimization problem in (7) relies on knowledge of
+the channel in each frequency bin, i.e., on full CSI. Using (6) as our objective is expected to
+affect the performance of the hybrid precoders when the channel matrices provided as inputs
+are (possibly inaccurate) estimates of the true channel. Consequently, we seek to optimize the
+hybrid precoders in a manner that is robust to a predefined level of inaccuracies in the CSI.
+To formulate this, we consider a noisy sub-channel estimation denoted {Hb + Eb}b∈B, where
+Hb is the true channel realization, and Eb is the estimation error. We wish to design the hybrid
+precoders to be robust to estimation errors within a predefined level ε, i.e., when ∥Eb∥F < ε for
+each b ∈ B. Consequently, we convert (7) into a maximin problem
+�
+W(o)
+a , {W(o)
+d,b}b∈B
+�
+=
+argmax
+Wa,{Wd,b}b∈B
+�
+min
+∥Eb∥F <ε R (Wa, {Wd,b}, {Hb + Eb})
+�
+(8)
+s.t.
+�
+�
+�
+�
+�
+1
+B
+�B
+b=1 ∥WaWd,b∥2
+F ≤ N
+Wa ∈ A
+.
+In (8), we seek to maximize the minimal rate resulting from a tolerable estimation error of the
+channel, i.e., an estimation error within the predefined level ε. Again, we wish to carry out robust
+optimization rapidly, and can leverage the set of past channel realizations D to that aim.
+III. RAPID HYBRID PRECODER LEARNED OPTIMIZATION WITH FULL CSI
+We first consider the rapid tuning of the hybrid precoder with full CSI. For each realization
+of the wireless channel {Hb}b∈B, the configuration of the hybrid precoder can be formulated
+as a constrained optimization problem (7). Consequently, one can design a hybrid precoder
+using suitable iterative optimization methods. A candidate method for tackling the optimization
+problem in (7) is PGA. PGA can directly tackle the multi-carrier sum-rate objective in (7), as
+opposed to alternative iterative optimizers for hybrid beamforming which use a relaxed objective
+aiming to approach the fully-digital beamformer as in [10], [15]. An additional motivation for
+aiming at directly optimizing the sum-rate rather than an alternative surrogate objective (which
+may be desired to optimize) stems from the fact that the resulting derivation can be extended to
+robust optimization, i.e., to cope with noisy CSI, as detailed in Section IV. Setting the hybrid
+precoder setup is a channel-dependent task. Consequently, one would have to carry out the
+optimization procedure each time the channel changes, i.e., on each coherence duration, while
+principled iterative optimizers such as PGA tend to be slow and lengthy. To cope with this
+challenge, in this section we present the method of learn-to-optimize hybrid precoding with full
+8
+
+CSI, which is specifically designed to rapidly configure hybrid precoders. Our scheme is based
+on the application of PGA to (7), as detailed in Subsection III-A, whose convergence speed we
+optimize without compromising on interpretability and suitability using deep unfolding [31], as
+we derive in Subsection III-B and discuss in Subsection III-C.
+A. Projected Gradient Ascent
+Problem (7) represents constrained maximization with B + 1 optimization matrix variables
+Wa, {Wd,b}b∈B. Such problems can be tackled using the PGA algorithm combined with alter-
+nating optimization. In this iterative method, each iteration first optimizes Wa while keeping
+{Wd,b}b∈B fixed, then repeats this process for every Wd,b. The optimized matrices are projected
+to guarantee the constraints are not violated.
+To formulate this operation mathematically, we define ˜Hb ≜
+�
+1
+Nσ2Hb. Each iteration of index
+k + 1 is comprised of two alternating stages: analog PGA and digital PGA.
+Analog PGA: The update of Wa is carried out according to
+W(k+1)
+a
+= ΠA
+�
+W(k)
+a
++ µ(k)
+a
+∂
+∂Wa
+R(W(k)
+a , {W(k)
+d,b}, {Hb})
+�
+,
+(9)
+where µ(k)
+a
+is the step size of the gradient step, and ΠA{·} is the projection operator onto the
+set A. The gradient of R with respect to Wa is given by (see Appendix A)
+∂
+∂Wa
+R(Wa, {Wd,b}, {Hb}) = 1
+B
+B
+�
+b=1
+˜HT
+b Gb(Wa, Wd,b, Hb)−T ˜H∗
+bW∗
+aW∗
+d,bWT
+d,b,
+(10)
+where Gb(Wa, Wd,b, Hb) ≜ (IN + ˜HbWaWd,bWH
+d,bWH
+a ˜HH
+b ).
+The projection operator ΠA in (9) depends on the feasible analog mappings. For unconstrained
+architectures, clearly ΠA(A) = A. For fully connected phase shifters based hardware, the
+projection is given by
+ΠA{A} = ˜A,
+[˜A]m,l = [A]m,l
+|[A]m,l|, ∀(m, l).
+(11)
+Digital PGA: The digital precoder is updated after the analog precoder, where the update is
+carried out for all frequencies in parallel. Namely, the gradient step for each b ∈ B is computed
+as
+ˆW(k+1)
+d,b
+= W(k)
+d,b + µ(k)
+d,b
+∂
+∂Wd,b
+R
+�
+W(k+1)
+a
+, {W(k)
+d,b}, {Hb}
+�
+,
+(12)
+9
+
+where µ(k)
+d,b is the step size. The gradient of R with respect to Wd,b for each b ∈ B is computed
+as (see Appendix A)
+∂
+∂Wd,b
+R(Wa, {Wd,b}, {Hb}) = 1
+B WT
+a ˜HT
+b Gb(Wa, Wd,b, Hb)−T ˜H∗
+bW∗
+aW∗
+d,b.
+(13)
+After the gradient step is taken for all frequency bins, the solution is projected to meet the
+power constraint via
+W(k+1)
+d,b
+=
+�
+NB
+�B
+b=1 ∥W(k+1)
+a
+ˆW(k+1)
+d,b
+∥2
+F
+· ˆW(k+1)
+d,b
+.
+(14)
+The overall procedure is summarized as Algorithm 1. While the initial settings of {W(0)
+d,b}b∈B
+are taken to be random, the initial analog combiner W(0)
+a
+is set to be the first L right-singular
+vectors of
+1
+B
+�B
+b=1 ˜Hb (the frequency bands sub-channels average). This setting corresponds
+to analog beamforming towards the eigenmodes of the (frequency-average) channel, being the
+part of the capacity achieving precoding method for frequency flat MIMO channels [43, Ch.
+10]. This principled initialization is numerically shown to notably improve the performance of
+hybrid precoders tuned to directly optimize the rate in (7).
+Algorithm 1: Projected Gradient Ascent for Hybrid Precoding
+Init: Randomize {W(0)
+d,b}b∈B
+W(0)
+a
+← first L right-singular vectors of
+1
+B
+�
+b ˜Hb
+Set step sizes {µ(k)
+d,b}b∈B, µ(k)
+a
+Input: Channel matrices {˜Hb}b∈B
+1 for k = 0, 1, . . . until convergence do
+2
+Update W(k+1)
+a
+via (9)
+3
+for b = 1, . . . , B do
+4
+Calculate ˆW(k+1)
+d,b
+by (12)
+5
+end
+6
+for b = 1, . . . , B do
+7
+Update W(k+1)
+d,b
+via (14)
+8
+end
+9 end
+10 return {W(k)
+d,b}b∈B and W(k)
+a
+The convergence speed of gradient-based optimizers largely depends on the step sizes, i.e.,
+���
+µ(k)
+d,b
+�
+b∈B, µ(k)
+a
+��
+in Algorithm 1. However, conventional step size optimization methods based
+on, e.g., line search and backtracking [44, Ch. 9], typically involve additional per-iteration
+10
+
+processing which increases the overall complexity and run-time. Hence, a common practice
+is to use pre-defined hand-tuned constant step sizes, which may result in lengthy convergence.
+B. Learn-to-Optimize Hybrid Precoding
+Algorithm 1 optimizes hybrid precoders for a given channel realization. However, its conver-
+gence speed largely depends on its hyperparameters, i.e., the step sizes, which tend to be difficult
+to set. Here, we propose to leverage automated data-based optimizers used in deep learning to
+tune iteration-dependent step sizes, i.e., to learn-to-optimize hybrid precoders in a small and
+predefined number of iterations.
+Our design follows the deep unfolding methodology [30], [31], which designs DNNs as
+iterative optimizers with a fixed number of iterations. In particular, we use as our optimizer
+the PGA method in Algorithm 1 with exactly K iterations. By doing so, we guarantee an exact
+pre-known, and typically small, run-time and complexity in real-time. While the accuracy of
+first-order optimizers such as PGA is typically invariant of the setting of the hyperparameters
+when allowed to run until convergence (under mild conditions, e.g., that the step sizes are
+sufficiently small), their performance is largely affected by these values when the number of
+iterations is fixed. Consequently, our design treats the hyperparameters of an iterative optimizer
+with K iterations as the parameters of a DNN with K layers, and tunes them via end-to-end
+training, based on the available data set D, thus converting PGA into a trainable discriminative
+model [45].
+To formulate this, the step sizes vector of the kth iteration is defined as µk ≜ (µ(k)
+a , . . . , µ(k)
+d,B),
+and the step sizes matrix defined as µ ≜ (µ0, . . . , µK−1)T for K iterations of Algorithm 1. The
+entries of this K ×(B +1) matrix are trainable parameters, that are learned from data. Note that
+end-to-end training is feasible despite the fact that D does not hold the ground truth precoders.
+This follows since the performance of hybrid precoders can be evaluated using the differentiable
+measure in (6), that is used to define a loss function with which the hyperparameters µ are
+trained in an unsupervised manner.
+The loss function, for a given normalized channel ˜H = {˜Hb}b∈B and step sizes µ, is computed
+as a weighted average of the negative resulting achievable sum-rates of this channel (6) in
+each PGAK(˜H, µ) iteration. This implies that the kth iteration sum-rate is computed when the
+precoders are
+�
+W(k)
+a , {W(k)
+d,b}b∈B
+�
+≜ PGAk(˜H, µ), i.e., the precoders obtained via Algorithm 1
+after k < K iterations and step sizes µ. Since each iteration is required to provide a setting of
+11
+
+the hybrid beamformer which gradually improves along the iterative procedure, we adopt the
+following loss, inspired by [46]
+L(˜H, µ) = 1
+K
+K
+�
+k=1
+log(1 + k) ·
+�−1
+B
+B
+�
+b=1
+log
+���IN + ˜HbW(k)
+a W(k)
+d,b
+�˜HbW(k)
+a W(k)
+d,b
+�H���
+�
+.
+(15)
+The data set D includes channel realizations. Since ˜Hb ≜
+�
+1
+Nσ2Hb with known N and σ2, we
+henceforth write the entries of the data set D as {˜Hr}|D|
+r=1.
+The learn-to-optimize method seeks to tune the hyperparameters vector µ to best fit the data
+set D in the sense of the loss measure (15). Namely, we aim at setting
+µ(o) = arg min
+µ
+1
+|D|
+|D|
+�
+r=1
+L(˜Hr, µ).
+(16)
+We tackle (16) using deep learning optimization techniques based on, e.g., mini-batch stochastic
+gradient descent, to tune µ based on the data set D. The resulting procedure is summarized
+as Algorithm 2. We initialize µ before the training process, with fixed step sizes with which
+PGA converges. After training, which is based on past channel realization and can thus be done
+offline, the learned µ is used as hyperparameters for rapidly converting a channel realization
+into a hybrid precoding setting via K of Algorithm 1. The resulting unfolded PGA algorithm is
+illustrated in Fig. 2.
+Algorithm 2: Learn-to-Optimize Hybrid Precoding with Full CSI
+Init: Set µ as fixed step sizes.
+Fix learning rate η
+Input: Training set D = {˜Hr}|D|
+r=1
+1 for epoch = 0, 1, . . . , epochmax − 1 do
+2
+Randomly divide D into Q batches {Dq}Q
+q=1
+3
+for q = 1, . . . , Q do
+4
+Compute precoders via PGAK(Dq, µ)
+5
+Compute the average loss of the batch: L(µ) =
+1
+|Dq|
+�
+˜H∈Dq L(˜H, µ)
+6
+Update µ ← µ − η∇µL(µ)
+7
+end
+8 end
+9 return µ
+12
+
+Fig. 2: Unfolded PGA block illustration. The learned parameters are marked in red fonts.
+C. Discussion
+The proposed learn-to-optimize method leverages data to improve the performance and con-
+vergence speed of iterative PGA-based optimization. The resulting precoder design preserves the
+interpretability and simplicity of classic PGA optimization, while inferring at a fixed and low
+delay as done by DNNs applied for such tasks. We thus benefit from the best of both worlds of
+model-based optimization and data-driven deep learning.
+The fact that the number of iterations is fixed and limited is reflected upon the computational
+complexity associated with setting a hybrid precoder for a given channel realization. To quantify
+the complexity, we examine one iteration of Algorithm 1; its complexity is dominated by the
+gradient computation (whose complexity order is O
+�
+BM 2(N + L)
+�
+) in Step 2 (which is of
+the same complexity order as the B gradient computations in Step 12), and by the projection
+required in Step 7 (whose complexity order is O
+�
+BNML
+�
+). For K iterations of Algorithm 1
+and when signaling over B frequency bands, it follows that the overall complexity of the PGA
+algorithm with pre-defined K iterations, as is done by the proposed learned optimizer, is of the
+order of
+CPGA = O
+�
+K · BM
+�
+NL + M(N + L)
+��
+.
+(17)
+The fact that we set our objective to be the rate R(·) results in its gradient computation yielding a
+higher complexity per iteration compared with that used when taking the gradients of a surrogate
+objective, e.g., [15] (where the quadratic dependence is on the number of RF chains rather than
+the number of antennas), while being of a similar order of that used in [16]. However, recall that
+13
+
+H
+Unfolded PGA
+Iteration#1
+Iteration#K
+aR
+(0)
+(K-1)
+aR
+I1.A(-)
+(K-1)
+aw.
+Ha
+aw.
+11.^()
++W(K)
++ 
++
+H
+++
+aR
+,(0)
+(K-1)
+aR
+Ha,1
+.(K-1)
+aWa,1
+Pa,i
+d.1
+d,1
+aWa.
+W(R)
+d,1
++ +
++ +
+...
+Digital
+Digital
+Projection
+Projection
+H
+aR
+(0)
+aR
+d.BT
+awd.
+Pa,B
+awa.
+d,B
+d,B
+W(R)
+d,B
++ +
++our derivation of Algorithm 2 serves as the first step towards rapid and robust optimization of
+the rate, for which these gradient computations are useful, as shown in the sequel. Furthermore,
+the proposed learn-to-optimize framework facilitates implementing the optimizer with a fixed
+and small number of iterations, allowing to limit the overall complexity.
+Our approach follows the deep unfolding methodology [30]–[32]. As opposed to other forms
+of deep unfolded networks, which designed DNNs to imitate the operation of a model-based
+optimizer while modifying its operation, as in, e.g., [46], our design is geared to preserve the
+operation of model-based iterative optimization. We use automated training capabilities of deep
+learning tools to tune the hyperparameters of the optimizer. By doing so, we improve upon
+the conventional usage of fixed hyperparameters, as demonstrated in Section V, and avoid the
+excessive delay of implementing hyperparameter search in each iteration.
+The design objective used in our derivation is the achievable sum-rate (6), which is approached
+using dedicated coding over asymptotically large blocks. While one is likely to deviate from (6)
+in practice, it serves as a fundamental characteristic of the overall channel which encompasses
+hybrid precoding, transmission, and reception, making it a relevant figure-of-merit for optimizing
+hybrid MIMO systems. Furthermore, as the objective in (6) is explicit and differentiable, it
+enables our method to learn-to-optimize in an unsupervised manner, i.e., one does not need
+access to ground-truth precoders as in [19], and can train using solely channel realizations as
+data. Yet, using (6) as our objective is expected to affect the performance of the hybrid precoders
+when the channel matrices provided as inputs are inaccurate estimates of the true channel, in
+the same manner that such CSI errors affect the performance of model-based optimizers. We
+discuss this extension of our method to handling CSI errors in the following section.
+IV. RAPID AND ROBUST HYBRID PRECODER LEARNED OPTIMIZATION WITH NOISY CSI
+In Section III, we designed the hybrid precoder assuming full CSI. The resulting Algorithm 2
+is compatible for a specific channel, therefore, when the estimation of the channel is noisy,
+a performance degradation is expected. When dealing with mismatched CSI, the optimization
+problem can be formulated as (8), where the minimal rate over all bounded errors is maximized.
+In this section we present a method which builds upon our derivation of Algorithm 2 for
+tackling (8) via rapid optimization with a fixed run-time. Our approach is based on the PCMP
+algorithm [39], which is an iterative optimizer suitable for maximin objectives as in (8). As
+detailed in Subsection IV-A, PCMP relies on projected gradient steps, and thus its derivation
+can utilize steps obtained for non-robust optimization via PGA. Thus, following the approach
+14
+
+used in Section III, we employ deep unfolding, leveraging data to optimize its performance within
+a fixed number of iterations, as described in Subsection IV-B. Then, we provide a discussion in
+Subsection IV-C.
+A. Projected Conceptual Mirror Prox Robust Optimization
+The formulation in (8) represents a constrained maximin optimization problem with 2B + 1
+optimization (and auxiliary) matrix variables Wa, {Wd,b}b∈B, {Eb}b∈B. Such problems can be
+tackled by combining the conceptual mirror prox (CMP) algorithm [39] with alternating opti-
+mization and projections, resulting in the PCMP method.
+PCMP is an iterative method. Each iteration is comprised of two stages: CMP and projection.
+We next formulate these stages in the context of (8).
+CMP: The objective in (8) is maximized according to the analog and digital precoding
+matrices, and minimized according to the error matrices. CMP aims at iteratively refining the
+optimization variables via gradient steps, which at iteration index k + 1 take the form
+�
+W(k+1)
+a
+, {W(k+1)
+d,b
+}b∈B, {E(k+1)
+b
+}b∈B
+�
+=
+�
+W(k)
+a , {W(k)
+d,b}b∈B, {E(k)
+b }b∈B
+�
++ µ(k)
+�
+∂
+∂Wa
+,
+�
+∂
+∂Wd,b
+�
+, −
+� ∂
+∂Eb
+��
+R
+�
+W(k+1)
+a
+, {W(k+1)
+d,b
+}b∈B, {Hb + E(k+1)
+b
+}b∈B
+�
+.
+(18)
+The computation in (18) cannot be directly implemented in general since the gradients are
+taken with respect to the updated optimization variables (i.e., the iteration index k + 1 appears
+on both sides of the update equation). However, it can be approached via additional iterative
+updates with index i of the form [39, Eq. (6)]
+ˆWa,(i) = W(k)
+a
++ µ(k)
+a,(i) ·
+∂
+∂Wa
+R
+�
+ˆWa,(i−1), { ˆWd,b,(i−1)}, {Hb + ˆEb,(i−1)}
+�
+,
+(19a)
+ˆWd,b,(i) = W(k)
+d,b + µ(k)
+d,b(i) ·
+∂
+∂Wd,b
+R
+�
+ˆWa,(i−1), { ˆWd,b,(i−1)}, {Hb + ˆEb,(i−1)}
+�
+, ∀b ∈ B,
+(19b)
+ˆEb,(i) = E(k)
+b
+− µ(k)
+e,b(i) ·
+∂
+∂Eb
+R
+�
+ˆWa,(i−1), { ˆWd,b,(i���1)}, {Hb + ˆEb,(i−1)}
+�
+, ∀b ∈ B.
+(19c)
+In (19), µ(k)
+a(i), µ(k)
+d,b(i), µ(k)
+e,b(i) are the step sizes. The optimization variables in (19) are initialized to
+�
+ˆWa,(0), { ˆWd,b,(0)}b∈B, {ˆEb,(0)}b∈B
+�
+=
+�
+W(k)
+a , {W(k)
+d,b}b∈B, {E(k)
+b }b∈B
+�
+.
+(20)
+15
+
+The computation of the gradients in (19) with respect to the analog and digital precoders can
+use the gradient formulations derived for PGA in (10) and (13), respectively. The gradient of R
+with respect to Eb for each b ∈ B is computed as (see Appendix B)
+∂
+∂Eb
+R(Wa, {Wd,b}, {Hb+Eb}) = 1
+B Gb(Wa, Wd,b, Hb+Eb)−T(˜H∗
+b+E∗
+b)W∗
+aW∗
+d,bWT
+d,bWT
+a .
+(21)
+While the above CMP procedure introduces an additional iterative procedure which has to be
+carried out at each iteration of index k, it is typically sufficient to only carry out two iterations
+of (19) [47], i.e., repeat (19) for i = 1, . . . , imax with imax = 2.
+Projection: After the CMP is conducted for two iterations, the resulting matrices, denoted
+�
+ˆWa,(imax), { ˆWd,b,(imax)}b∈B, {ˆEb,(imax)}b∈B
+�
+, are projected to meet the constraints. Thus, the
+update rule at iteration k + 1 is
+W(k+1)
+a
+= ΠA
+�
+ˆWa,(imax)
+�
+,
+(22a)
+W(k+1)
+d,b
+=
+�
+NB
+�B
+b=1 ∥W(k+1)
+a
+ˆWd,b,(imax)∥2
+F
+· ˆWd,b,(imax),
+∀b ∈ B,
+(22b)
+E(k+1)
+b
+= min
+�
+ε · NM
+∥ˆEb,(imax)∥F
+, 1
+�
+· ˆEb,(imax),
+∀b ∈ B,
+(22c)
+where ε is the error bound on the entries of the sub-channel matrix, {Hb}b∈B ∈ CN×M.
+The overall procedure is summarized as Algorithm 3. The matrices W(0)
+a , {W(0)
+d,b}b∈B are
+initialized in the same manner as in Algorithm 1, while {E(0)
+b }b∈B is generated randomly while
+normalizing to guarantee that ∥Eb∥F < ϵ for each b ∈ B. Notice that Algorithm 3 describes
+a gradient-based optimizer, similar to Algorithm 1. Therefore, its convergence speed depends
+as well on the step sizes µ(k)
+a(i), {µ(k)
+d,b(i)}b∈B, {µ(k)
+e,b(i)}b∈B, which are hard to select manually. In
+Subsection IV-B we will describe the learn-to-rapidly-optimize method, extending the rationale
+used with full CSI in Algorithm 2, for accelerating Algorithm 3 convergence via data-aided
+hyperparameters tuning.
+B. Robust Learn-to-Optimize Hybrid Precoding
+Algorithm 3 optimizes hybrid precoders for a given noisy channel realization. Nevertheless, it
+needs to be executed in real-time, whenever there is a change in the channel, and thus the need
+to obtain reliable hybrid precoders rapidly, i.e., within a fixed and small number of iterations,
+still applies. To accomplish this, we use the deep unfolding methodology following the approach
+16
+
+Algorithm 3: Projected Conceptual Mirror Prox for Robust Hybrid Precoding
+Init: Randomize {W(0)
+d,b}b∈B, {E(0)
+b }b∈B
+W(0)
+a
+← first L right-singular vectors of
+1
+B
+�
+b ˜Hb
+Set step sizes µ(k)
+a(i), {µ(k)
+d,b(i)}b∈B, {µ(k)
+e,b(i)}b∈B
+Input: Channel matrices {˜Hb}b∈B
+1 for k = 0, 1, . . . until convergence do
+2
+Set (20)
+3
+for i = 1, . . . , imax do
+4
+Calculate
+�
+ˆWa,(i), { ˆWd,b,(i)}b∈B, {ˆEb,(i)}b∈B
+�
+by (19)
+5
+end
+6
+Update
+�
+W(k+1)
+a
+, {W(k+1)
+d,b
+}b∈B, {E(k+1)
+b
+}b∈B
+�
+via (22)
+7 end
+8 return {W(k)
+d,b}b∈B and W(k)
+a
+as in Subsection III-B, leveraging its ability to tune hyperparameters in optimization involving
+projected gradients of the rate function, where the main differences are in the number of learned
+parameters and in the algorithm structure.
+We use the PCMP method as the optimizer with exactly K iterations and imax = 2 in-
+ternal iterations. Let us define the step sizes vector for the analog update of the kth itera-
+tion as µk
+a ≜ (µ(k)
+a(1), µ(k)
+a(2))T; step sizes vectors for the digital updates of the kth iteration as
+{µk
+d,b}b∈B ≜ {(µ(k)
+d,b(1), µ(k)
+d,b(2))T}b∈B; and the step sizes vector for the error updates as {µk
+e,b}b∈B ≜
+{(µ(k)
+e,b(1), µ(k)
+e,b(2))T}b∈B. Accordingly, we define the kth iteration 2 × 2B + 1 step sizes matrix
+as µk ≜ (µk
+a, µk
+d,1, . . . , µk
+d,B, µk
+e,1, . . . , µk
+e,B), obtaining the K × 2 × 2B + 1 step sizes tensor
+µ ≜ (µ0, . . . , µK−1) for K iterations of Algorithm 3.
+The unfolded architecture converts the PCMP method with K iteration into a discriminative
+algorithm whose trainable parameters are the entries of µ. Similarly to the approach used in
+Subsection III-B, we train the unfolded PCMP in an unsupervised manner, i.e., the data set only
+includes channel realizations, while the level of tolerable error ε is given. The loss function
+for a given normalized channel ˜H = {˜Hb}b∈B and step sizes µ is computed as the maximal
+negative resulting achievable sum-rate of this channel, (6) when the maximum is taken over all
+∥Eb∥F < ε, and the precoders are
+�
+W(K)
+a
+, {W(K)
+d,b }b∈B
+�
+≜ PCMPK(˜H, µ), i.e., the precoders
+obtained via Algorithm 3 with K iterations and step sizes µ. The resulting loss is
+L(˜H, µ)= max
+∥Eb∥F <ε
+�
+−1
+B
+B
+�
+b=1
+log
+����IN +
+1
+Nσ2(Hb+Eb)W(K)
+a
+W(K)
+d,b
+�
+(Hb+Eb)W(K)
+a
+W(K)
+d,b
+�H
+����
+�
+. (23)
+The maximization term in (23) notably complicates its usage as a training objective for learning
+17
+
+the PCMP hyperparameters. To overcome this, we generate a finite set of ne error patterns
+(satisfying ∥Eb∥F < ε) via random generation, to which we add the zero error, i.e., E ≜
+�
+{Et
+b}b∈B
+�ne
+t=1∪{0}, and seek the setting which minimizes the maximal loss among these error
+patterns. Namely, to maintain feasible training, the objective used for training the unfolded
+algorithm evaluates the loss based on its output after K iterations and is given by
+˜L(˜H, µ)=max
+Eb∈E
+�
+−1
+B
+B
+�
+b=1
+log
+����IN +
+1
+Nσ2(Hb+Eb)W(K)
+a
+W(K)
+d,b
+�
+(Hb+Eb)W(K)
+a
+W(K)
+d,b
+�H
+����
+�
+.
+(24)
+The robust learn-to-optimize method, summarized as Algorithm 4, uses data to tune µ. We
+initialize µ before the training process, with fixed step sizes with which PCMP converges (though
+not necessarily rapidly). After training, the learned matrix µ is used as hyperparameters for
+rapidly converting a noisy channel realization into a hybrid precoding setting via K iterations
+of Algorithm 3.
+Algorithm 4: Robust Learn-to-Optimize Hybrid Precoding
+Init: Set µ as fixed step sizes. Fix learning rate η
+Input : Training set D = {˜Hr}|D|
+r=1
+1 for epoch = 0, 1, . . . , epochmax − 1 do
+2
+Randomly divide D into Q batches {Dq}Q
+q=1
+3
+for q = 1, . . . , Q do
+4
+Compute precoders via PCMPK(Dq, µ)
+5
+Compute the average loss of the batch: L(µ) =
+1
+|Dq|
+�
+˜H∈Dq ˜L(˜H, µ)
+6
+Update µ ← µ − η∇µL(µ)
+7
+end
+8 end
+9 return µ
+C. Discussion
+The proposed robust learn-to-optimize method in Algorithm 4 extends the method in Algo-
+rithm 2 to cope with noisy CSI. In particular, for the special case of ε = 0 and imax = 1,
+it holds that the PCMP algorithm (Algorithm 3) reduces to the PGA method (Algorithm 1).
+Accordingly, the data-aided optimized Algorithm 4 specializes Algorithm 2. Consequently, our
+gradual design, which started with designing learned PGA optimization applied to the rate
+function in Section III, allowed us to extend its derivation to realize robust learned PCMP
+algorithm for maximin rate optimization. In addition, this implementation shows that the general
+18
+
+learn-to-optimize method can be adapted for speeding the convergence of a broad range iterative
+model-based hyperparameters-depending algorithms, and particularly those utilizing gradient and
+projection steps.
+In terms of complexity, the unfolded PCMP with K iterations and imax = 2 shares the same
+complexity order as that the PGA method with full CSI. In particular, Step 4 of Algorithm 3
+repeats the gradient computations of Algorithm 1 imax times per iteration and includes additional
+gradients with respect to Eb (which is of the same complexity as taking the gradients with respect
+to Wd,b). Thus, the complexity order of the unfolded PCMP algorithm is
+CPCMP = O
+�
+K · BM
+�
+NL + M(N + L)imax
+��
+,
+(25)
+which, for imax = 2, yields a similar complexity as that of the unfolded PGA in (17).
+The ability to rapidly optimize in a robust manner via the unfolded PCMP results in hybrid
+precoders that are less sensitive to channel estimation errors. The PCMP algorithm is coping
+with mismatched channels by considering a wide range of channel errors when setting the hybrid
+precoders. While the PGA algorithm is carried out each time the channel changes, the PCMP
+algorithm can be carried out less frequently since it is more robust and stable. Accordingly, the
+PCMP can be also used to reduce the rate in which the optimization procedure is carried out,
+and not only for coping with mismatched channels.
+The error bound value, ε, plays a key role in the robust algorithm. It affects the robustness,
+reliability, and stability of the solution, and therefore, it needs to be chosen based on some prior
+knowledge, or on application needs. Another approach is to use channel estimation methods prior
+to the precoders design we presented in this work, and by this overcome the mismatched channel
+problem. An additional important parameter is the setting of the expected error patterns evaluated
+during training, i.e., E. While in our evaluation we generated this set in a random fashion and
+included in it the zero-error pattern to guarantee suitability for error-free case, one can consider
+alternative methods for preparing this set such that the resulting optimized hyperparameters
+would be most suitable for the desired robust optimization metric. Since this setting is carried
+out offline and thus complexity is not a key issue here, one can possibly explore the usage of
+sequential sampling and Bayesian optimization techniques [48] for setting E. We leave the study
+of extensions of our method to future work.
+19
+
+V. NUMERICAL EVALUATIONS
+We next numerically evaluate the proposed unfolded optimization framework1. Our main
+purpose is to numerically demonstrate the ability of the proposed learn-to-optimize framework
+to notably reduce the number of iterations compared with conventional optimization with fixed
+hyperparameters.
+Our numerical evaluations commence with settings with full CSI in Subsection V-A. As the
+full CSI setting (with analog combiners constrained to represent phase shifter network) is the
+common setup considered for hybrid precoding, this numerical evaluation allows us to compare
+the first step of our design, i.e., the unfolded PGA method, to existing optimization methods.
+Then, we consider robust optimization based on the unfolded PCMP in Subsection V-B. For
+both CSI settings, we simulate two different models for the downlink hybrid MIMO system as
+described in Subsection II-A: Rayleigh fading, where the channel matrix in each frequency bin is
+randomized from an i.i.d. Gaussian distribution; and the QuaDRiGa model [40], an open source
+geometry-based stochastic channel model, relevant for simulating realistic MIMO systems.
+A. Hybrid Precoding with Full CSI
+For full CSI, the optimizer for designing the hybrid precoder is PGA, and its data-aided
+learn-to-optimize version is detailed Subsection III-B. The implementation of this method can
+be carried out for both unconstrained analog combiners (for which the only constraint is the
+power constraint), as well as hardware-limited designs, where a common constraint is that of
+phase shifter networks. We thus divide our evaluation into learned PGA with unconstrained
+analog combiners and PGA with phase shifters.
+1) PGA using unconstrained analog combiners: In this case we use the analog projection
+operator as ΠA(A) = A, i.e., we do not impose any constraints on the analog architecture. Such
+unconstrained combiners can be approximated using architectures proposed in, e.g. [11], [12].
+We consider three different settings for the number frequency bands B, the number of users
+N, the number of RF chains L, the number of transmit antennas M, and for the data source:
+1) A Rayleigh channel with 8 frequency bins, configured with B = 8, N = 6, L = 10, M = 12;
+2) A QuaDRiGa channel with B = 16 frequency bins, where we use N = 4, L = 6, M = 12;
+3) A wideband QuaDRiGa channel with B = 128, N = 4, L = 8, M = 16.
+For each configuration, we applied Algorithm 2 to learn to set hybrid precoders with merely
+1The complete source is available at https://github.com/ortalagiv/Learn-to-Rapidly-Optimize-Hybrid-Precoding.
+20
+
+K = 5, 10 iterations based on |D| = 1000 channel realization, where we used 50 − 70 epochs
+with batch size |Dq| = 100 and used Adam for the update in Step 6 of Algorithm 2.
+The PGA hybrid precoding design with K optimized iterations is compared with applying
+PGA with fixed hyperparameters, where we used a constant step size chosen based on empirical
+trials. Both hybrid precoders are evaluated over 100 unseen test channels. The simulation results
+for the settings of (B = 8, N = 6, L = 10, M = 12), (B = 16, N = 4, L = 6, M = 12), and
+(B = 128, N = 4, L = 8, M = 16) are depicted in Fig. 3, Fig. 4, and Fig. 5, respectively,
+where each figure compares both the convergence of the algorithms averaged over all channel
+realizations and for the randomly chose realizations, as well as the resulting sum-rates versus
+SNR.
+The resulting sum-rates (for SNR= 0dB) versus the number of PGA iterations, of both
+the unfolded PGA and the classical PGA (with manually chosen constant step sizes), when
+averaged over all unseen channels, and for two random channel realizations, are depicted in
+Fig. 3a, Fig. 4a, and Fig. 5a. We observe in these figures that the proposed learn-to-optimize
+consistently facilitates simple PGA optimization of the hybrid precoders, achieving similar
+and even surpassing the sum-rates achieved by conventional PGA with fixed step sizes, while
+requiring much fewer iterations. The reduction in the number of iterations due to learn-to-
+optimize is by factors of 20 (5 iterations vs. 100 iterations) and 10 (10 iterations vs. 100
+iterations) in speed, compared to conventional fixed-size optimization. These gains are consistent
+over the different channel models and configurations considered. Fig. 3b, Fig. 4b, and Fig. 5b
+illustrate the sum-rate for different values of SNR, of the proposed unfolded PGA compared
+with that of the classical PGA and the sum-rate resulting from fully digital baseband precoding,
+serving as an upper bound on the achievable sum-rate. The figures show the gain of converting
+the classical PGA algorithm in a discriminative trainable model. In particular, the unfolded
+algorithm implements PGA with merely few iterations while systematically improving upon
+conventional PGA with fixed step size and 100 iterations. These results demonstrate the benefits
+of the proposed approach in leveraging data to improve both performance and convergence speed
+while preserving the interpretability and suitability of conventional iterative optimizers.
+2) PGA using phase shifters networks: The numerical evaluation above showed that the
+considered deep unfolding methodology indeed facilitates rapid optimization of hybrid precoders.
+To demonstrate that these gains are not unique to unconstrained analog combiners, as well as to
+compare with alternative iterative optimizers, we next consider phase shifters networks for the
+21
+
+(a) Sum-rate per PGA iteration.
+(b) Sum-rate vs. SNR.
+Fig. 3: Rayleigh channel; unconstrained analog precoder; B = 8, N = 6, L = 10, M = 12.
+(a) Sum-rate per PGA iteration.
+(b) Sum-rate vs. SNR.
+Fig. 4: QuaDRiGa channel; unconstrained analog precoder; B = 16, N = 4, L = 6, M = 12.
+(a) Sum-rate per PGA iteration.
+(b) Sum-rate vs. SNR.
+Fig. 5: QuaDRiGa channel; unconstrained analog precoder; B = 128, N = 4, L = 8, M = 16.
+22
+
+19.5
+19
+19.0
+18
+18.5
+Achievable Rate
+17
+16
+18.0
+15
+17.5
+14
+: Average on all realizations - Classical PGA
+Average on all realizations - Unfolded PGA
+?Realization#1-UnfoldedPGA
+17.0
+13
+Realization#1-ClassicalPGA
+Realization #2 - Unfolded PGA
+12
+Realization #2 -Classical PGA
+345
+1
+20
+40
+60
+80
+100
+NumberofIterationClassical PGA - 1oo iterations
+Classical PGA - 5 iterations
+Unfolded PGA - 5 iterations
+25
+Fully digital
+Rate
+20
+Achievable
+15
+10
+5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+SNR[dB]14
+14
+13
+13
+Achievable Rate
+12
+12
+11
+11
+Average on all realizations - Classical PGA
+Average on all realizations - Unfolded PGA
+←Realization#1-UnfoldedPGA
+10 +
+Realization #1 - Classical PGA
+10
+Realization #2 - Unfolded PGA
+Realization #2 - Classical PGA
+1
+5
+10
+1
+20
+40
+60
+80
+100
+NumberofIterationClassical PGA - 1oo iterations
+20
+ClassicalPGA - 10 iterations
+Unfolded PGA - 1o iterations
+18
+Fully digital
+13
+Rate
+16
+Achievable
+12
+14
+11
+12
+.1
+0
+10
+8
+-5
+-4
+3
+-2
+-1
+0
+1
+2
+m
+4
+5
+SNR [dB]16
+16
+15
+15
+14
+Rate
+14
+Achievable I
+13
+13
+12
+Average on all realizations - Classical PGA
+12
+Average on all realizations - Unfolded PGA
+11
+Realization#1-UnfoldedPGA
+Realization #1 - Classical PGA
+11
+Realization #2 - Unfolded PGA
+10
+Realization #2 - Classical PGA
+1
+5
+10
+L
+20
+40
+60
+80
+100
+NumberofIterationClassical PGA - 1oo iterations
+22
+Classical PGA - 10 iterations
+Unfolded PGA - 1o iterations
+20
++Fully digital
+18
+16
+Rate
+Achievable
+16
+15
+14
+0
+12
+10
+8
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+SNR [dB]implementation of the analog architecture, as often considered in the literature. This means that
+the projection in (11) is used in Step 2 of Algorithm 1. We used as a benchmark the MO-AltMin
+algorithm of [15], which is designed to iteratively optimize such constrained hybrid precoders.
+We consider the QuaDRiGa channel model for two different settings of the number frequency
+bands B, users N, RF chains L, and transmit antennasM: 1) B = 16, N = 4, L = 2, M = 12;
+and 2) B = 128, N = 4, L = 2, M = 16. Similarly to the unconstrained case, for each setting,
+we apply Algorithm 2 to learn the hyperparameters for which the PGA converges with K = 5
+iterations. In the training procedure we use |D| = 1000 channels for each setting, where we
+used 50 epochs with batch size |Dq| = 100 and used Adam for the update in Step 6. The
+PGA with K = 5 learned iterations is compared with the PGA with fixed hyperparameters,
+chosen empirically. The hybrid precoders optimizers are evaluated over 100 unseen test channels.
+The performance evaluation results are shown in Fig. 6, and Fig. 7, for the settings of B =
+16, N = 4, L = 2, M = 12, and B = 128, N = 4, L = 2, M = 16, respectively, where each
+figure examines both the convergence curves as well as the sum-rate achieved at the end of the
+optimization procedure versus SNR.
+The resulting sum-rate (for SNR= 0dB) versus the number of PGA iterations, when averaged
+over all unseen channels, and of two random channel realizations, is shown in Fig. 6a, and
+Fig. 7a. We observe in these figures that the learn-to-optimize method is able to accelerate and
+outperform the PGA optimization, when compared to the standard PGA, with constant, manually
+chosen, step sizes. Observe that the number of iterations is reduced by 20 (5 iterations vs. 100),
+compared to standard optimization. The gain in performance systematically observed here follows
+from the ability of data-driven optimization to facilitate coping with the non-convex nature of
+the resulting optimization problem. Fig. 6b, and Fig. 7b demonstrate the comparison between
+the proposed unfolded PGA, the classical PGA, the MO-Altmin algorithm of [15], and fully
+digital baseband precoding. It is shown that the gap from the fully digital baseband precoding is
+significant, due to the use of a relatively small number of RF chains in the hybrid architecture.
+When comparing the iterative algorithms we see a small difference in terms of sum-rate, but
+it is important to consider the fact that the proposed unfolded PGA operates with the least
+number of iterations, therefor it achieves improvement in terms of speed when comparing to the
+benchmark and to the standard PGA. These results show the ability of the proposed learn-to-
+optimize technique to tackle and incorporate different constraints into the PGA algorithm while
+enabling rapid performance and fully preserving interpretability.
+23
+
+(a) Sum-rate per PGA iteration.
+(b) Sum-rate vs. SNR.
+Fig. 6: QuaDRiGa channel; constrained analog precoder; B = 16, N = 4, L = 2, M = 12.
+(a) Sum-rate per PGA iteration.
+(b) Sum-rate vs. SNR.
+Fig. 7: QuaDRiGa channel; constrained analog precoder; B = 128, N = 4, L = 2, M = 16.
+B. Hybrid Precoding with Noisy CSI
+The numerical studies reported in Subsection V-A empirically validate the ability of the pro-
+posed methodology to facilitate rapid optimization under different settings. We next demonstrate
+its ability to simultaneously enable rapid and robust hybrid precoding. Hence, in the following,
+we consider the unfolding of the PCMP algorithm for precoding under noisy channel estimation.
+Having demonstrated the ability of our approach to deal with constrained precoders, we focus
+here on unconstrained analog combiners. According, we implement of the learn-to-optimize
+methodology, i.e. Algorithm 4, for Algorithm 3, when the projection operator applied on the
+analog precoder in Step 6 is set to be ΠA(A) = A. The set of considered error patterns E is
+comprised of ne = 20 patterns randomly generated with Frobenius norm values in the range
+(0, ϵ), and we evaluate the maximin rate over E.
+We examine two different settings of the number frequency bands B, users N, RF chains
+24
+
+Classical PGA - 1oo iterations
+20
+Classical PGA - 5 iteriterations
+MO AltMin - 3o iterations (average)
+18
+Unfolded PGA - 5 iterations
+Fully digital
+16
+10
+Rate
+Achievable
+14
+12
+10
+8
+6
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+SNR [dB]10.4
+10.4
+-: Average on all realizations - Classical PGA
+Average on all realizations - Unfolded PGA
+Realization #1 - Unfolded PGA
+10.2
+Realization #1 - Classical PGA
+10.2
+Realization #2 -Unfolded PGA
+Realization #2 - Classical PGA
+10.0
+AchievableRate
+10.0
+9.8 -
+9.8
+F 9'6
+9.6
+9.4 -
+9.4 
+12345
+1
+20
+40
+60
+80
+100
+Numberof Iteration22
+Classical PGA - 1oo iterations
+Classical PGA - 5 iteriterations
+MO AltMin - 74 iterations (average)
+20
+Unfolded PGA - 5 iterations
+Fully digital
+18
+Rate
+10
+16
+Achievable
+14
+9
+12
+10
+8
+6
+5
+-2
+-1
+0
+1
+2
+m
+4
+5
+SNR [dB]9.8
+9.8
+9.6
+9.6
+Achievable Rate
+9.4
+9.4 
+9.2
+9.2
+Average on all realizations - Classical PGA
+9.0
+Average on all realizations - Unfolded PGA
+Realization #1-Unfolded PGA
+Realization #1 - Classical PGA
+9.0
+Realization #2 - Unfolded PGA
+8.8
+Realization #2 - Classical PGA
+12345
+1
+20
+40
+60
+80
+100
+NumberofIterationL, transmit antennas M, and the data source: B = 8, N = 6, L = 10, M = 12, with Rayleigh
+channels; and B = 16, N = 4, L = 6, M = 12 with QuaDRiGa channel model, the simulations
+results for each setting are demonstrated in Fig. 8, and Fig. 9, respectively. Again, we evaluate
+both the convergence rate and the minimal rate (within the tolerable error regime) at the end of
+the optimization procedure versus SNR, and compare our results with the conventional PCMP
+optimizer. In the simulations, three error bound values are considered, ε = 0.005, 0.05, 0.5. For
+each setting, we applied Algorithm 4 to learn to set hybrid precoders with K = 5 iterations based
+on |D| = 1000 channels, where we used 40 − 50 epochs with batch size |Dq| = 100, and used
+Adam for the update in Step 6. We compared the standard PCMP, with constant hyperparameters,
+to the optimized PCMP with exactly K = 5 iterations.
+Fig. 8a and Fig. 9a depict the minimal sum-rates, averaged on 100 unseen test channels
+results, for the three error bound values, vs. the number of PCMP iterations. In these figures,
+we observe notable gains in convergence speed of a factor of 20 (5 iterations vs. 100 iterations),
+and it is shown that the performances of the learned PCMP algorithm consistently surpasses
+the performances of the conventional PCMP with constant step sizes. In Fig. 8b, and Fig. 9b,
+the sum-rate for different values of SNR is shown. These figures include the fully digital
+baseband precoding error-free sum-rates, which are calculated with full CSI, and reflect on the
+best performance one can achieve given full CSI and without RF chain reduction. We observe
+that the proposed robust optimization method, the unfolded PCMP, achieves relatively close
+performances to the full-CSI fully digital baseband precoding, systematically outperforming the
+model-based PCMP operating with much more iterations. These results demonstrates the gains
+and advantages of our proposed learn-to-optimize method in enabling optimization of hybrid
+precoders which is both rapid and robust.
+VI. CONCLUSIONS
+In this work we proposed a method to leverage data to enable rapid, robust, and interpretable
+tuning of hybrid precoders. Our approach unfolds a suitable optimizer for maximizing the
+minimal sum-rate within a given tolerable CSI error into a fixed and small number of iterations.
+Then, we use data to tune the hyperparameters of each iteration. Our method is shown to
+notably improve convergence speed while setting hybrid precoders which achieve similar and
+even improved sum-rates compared to those tuned via lengthy non-learned optimization.
+25
+
+(a) Minimal sum-rate per PCMP iteration.
+(b) Minimal sum-rate vs. SNR.
+Fig. 8: Rayleigh channel; unconstrained analog precoder; B = 8, N = 6, L = 10, M = 12.
+(a) Minimal sum-rate per PCMP iteration.
+(b) Minimal sum-rate vs. SNR.
+Fig. 9: QuaDRiGa channel; unconstrained analog precoder; B = 16, N = 4, L = 6, M = 12.
+VII. ACKNOWLEDGEMENTS
+The authors would like to thank Tomer Yeblonka from CEVA for his valuable inputs and
+meaningful discussions.
+APPENDIX A
+COMPLEX GRADIENTS OF R(·) WITH RESPECT TO Wa, {Wd,b}b∈B
+We first derive (10) following the computation of complex-valued gradients of a general scalar
+function detailed in [49]. The complex differential of R (Wa, {Wd,b}, {Hb}) with respect to Wa
+26
+
+17.5
+17
+17.0
+IAchievableRate
+16
+16.5
+15
+16.0
+Minimal
+15.5
+Classical PCMP - =0.5
+13
+Unfolded PCMP - E=0.5
+15.0
+←UnfoldedPCMP-=0.05
+Classical PCMP - =0.05
+12
+Unfolded PCMP-E=0.005
+14.5
+Classical PCMP - =0.005
+2345
+20
+40
+60
+80
+100
+Numberof IterationClassical PCMP - 100 iter, E=0.05
+Classical PCMP -5 iter. =0.05
+Unfolded PCMP - 5 iter, =0.05
+25
+Classical PCMP - 100 iter, =0.005
+Rate
+Classical PCMP - 5 iter, =0.005
+Unfolded PCMP - 5 iter, =0.005
+ble
+Classical PCMP - 100 iter, E=0.5
+20
+IAchievab
+Classical PCMP - 5 iter, =0.5
+Unfolded PCMP - 5 iter, =0.5
+Error Free Fully digital
+Minimal
+15
+18
+10
+16
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+SNRdB]12.0
+11.75
+11.5
+11.50
+11.0
+Minimal AchievableRate
+11.25
+10.5
+11.00
+10.0
+10.75
+9.5
+10.50
+Classical PCMP - =0.5
+9.0
+Unfolded PCMP - E=0.5
+10.25
+Unfolded PCMP - E=0.05
+8.5
+Classical PCMP - E=0.05
+Unfolded PCMP - E=0.005
+10.00
+8.0
+Classical PCMP - =0.005
+12345
+1
+20
+40
+60
+80
+100
+Numberof IterationClassical PCMP - 100 iter, E=0.05
+20
+Classical PCMP - 5 iter, =0.05
+Unfolded PCMP - 5 iter, =0.05
+18
+Classical PCMP - 100 iter, =0.005
+Classical PCMP - 5 iter, =0.005
+Rate
++
+16
+Unfolded PCMP - 5 iter, =0.005
+Minimal Achievable 
+Classical PCMP - 100 iter, E=0.5
+14
+Classical PCMP - 5 iter, =0.5
+Unfolded PCMP - 5 iter, =0.5
+12
+Error Free Fully digital
+10
+8
+12
+11
+6
+4
+0
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+SNR[dB]is
+dWaR (Wa, {Wd,b}, {Hb})
+= dWa
+�
+1
+B
+B
+�
+b=1
+log
+��IN + ˜HbWaWd,bWH
+d,bWH
+a ˜HH
+b
+��
+�
+= Tr
+� 1
+B
+B
+�
+b=1
+Wd,bWH
+d,bWH
+a ˜HH
+b Gb(Wa, Wd,b, Hb)−1 ˜Hb(dWa)
+�
++ Tr
+� 1
+B
+B
+�
+b=1
+W∗
+d,bWT
+d,bWT
+a ˜HT
+b Gb(Wa, Wd,b, Hb)−1 ˜H∗
+b(dW∗
+a)
+�
+.
+Using [49, Table 3.2], this yields (10).
+Similarly, the differential of R (Wa, {Wd,b}, {Hb}) with respect to Wd,b is
+dWd,bR (Wa, {Wd,b}, {Hb})
+= dWd,b
+�
+1
+B
+B
+�
+n=1
+log
+��IN + ˜HnWaWd,nWH
+d,nWH
+a ˜HH
+n
+��
+�
+= Tr
+� 1
+B WH
+d,bWH
+a ˜HH
+b ˜HbWa(dWd,b)
+�
++ Tr
+� 1
+B WT
+d,bWT
+a ˜HT
+b Gb(Wa, Wd,b, Hb)−T ˜H∗
+bW∗
+a(dW∗
+d,b)
+�
+.
+Using [49, Table 3.2], this yields (13).
+APPENDIX B
+COMPLEX GRADIENT OF R(·) WITH RESPECT TO {Eb}b∈B
+The complex differential of R(Wa, {Wd,b}, {Hb + Eb}) with respect to Eb is given by
+dEbR(Wa, {Wd,b}, {Hb + Eb})
+= dEb
+�
+1
+B
+B
+�
+n=1
+log
+���IN +
+�˜Hn + En
+�
+WaWd,nWH
+d,nWH
+a
+�˜Hn + En
+�H���
+�
+= Tr
+� 1
+B
+�
+WaWd,bWH
+d,bWH
+a ˜HH
+b Gb(Wa, Wd,b, Hb + Eb)−1
++ WaWd,bWH
+d,bWH
+a EH
+b Gb(Wa, Wd,b, Hb + Eb)−1�
+(dEb)
+�
++ Tr
+� 1
+B
+�
+W∗
+aW∗
+d,bWT
+d,bWT
+a ˜HT
+b Gb(Wa, Wd,b, Hb + Eb)−T
++ W∗
+aW∗
+d,bWT
+d,bWT
+a ET
+b Gb(Wa, Wd,b, Hb + Eb)−T�
+(dE∗
+b)
+�
+.
+Using [49, Table 3.2], this yields (21).
+27
+
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+

TdE0T4oBgHgl3EQflQEG/content/tmp_files/2301.02482v1.pdf.txt

@@ -0,0 +1,2395 @@
+Astronomy & Astrophysics manuscript no. main
+©ESO 2023
+January 9, 2023
+Evolution of the reservoirs of volatiles in the protosolar nebula
+Antoine Schneeberger1, Olivier Mousis12, Artyom Aguichine1, and Jonathan I. Lunine3
+1 Aix- Marseille Université, CNRS, CNES, Institut Origines, LAM, Marseille, France
+e-mail: [email protected]
+2 Institut Universitaire de France (IUF), France
+3 Cornell University, Department of Astronomy, Ithaca NY, USA
+Received August 02, 2022; accepted November 30, 2022
+ABSTRACT
+The supersolar abundances of volatiles observed in giant planets suggest that a compositional gradient was present at the time of
+their formation in the protosolar nebula. To explain this gradient, several studies have investigated the radial transport of trace species
+and the effect of icelines on the abundance profiles of solids and vapors formed in the disk. However, these models only consider
+the presence of solids in the forms of pure condensates or amorphous ice during the evolution of the protosolar nebula. They usually
+neglect the possible crystallization and destabilization of clathrates, along with the resulting interplay between the abundance of water
+and those of these crystalline forms. This study is aimed at pushing this kind of investigation further by considering all possible solid
+phases together in the protosolar nebula: pure condensates, amorphous ice, and clathrates. To this end, we used a one-dimensional
+(1D) protoplanetary disk model coupled with modules describing the evolution of trace species in the vapor phase, as well as the
+dynamics of dust and pebbles. Eleven key species are considered here, including H2O, CO, CO2, CH4, H2S, N2, NH3, Ar, Kr, Xe, and
+PH3. Two sets of initial conditions are explored for the protosolar nebula. In a first scenario, the disk is initially filled with icy grains
+in the forms of pure condensates. In this case, we show that clathrates can crystallize and form enrichment peaks up to about ten times
+the initial abundances at their crystallization lines. In a second scenario, the volatiles were delivered to the protosolar nebula in the
+forms of amorphous grains. In this case, the presence of clathrates is not possible because there is no available crystalline water ice
+in their formation region. Enrichment peaks of pure condensates also form beyond the snowline up to about seven times the initial
+abundances. Our model can then be used to compare the compositions of its different volatile reservoirs with those of comet C/2016
+R2 PanSTARRS, Jupiter, Uranus, and Neptune. We find that the two investigated scenarios provide compositions of solids and vapors
+consistent with those observed in the bodies considered.
+Key words. Solar system formation; Planetary system formation; Protoplanetary disks; Comets origins; Solar system planets
+1. Introduction
+It is commonly assumed that Solar System bodies have bulk
+compositions that are representative of the material present in the
+protosolar nebula (PSN) from which they formed. If the PSN was
+homogeneous in composition, gas and ice giants would be ex-
+pected to reflect this homogeneity. However, observations show
+that giant planets present a range of supersolar metallicities. In
+Jupiter’s atmosphere, the abundances of volatile elements were
+found to be ∼1.5–6.1 times higher than their protosolar values
+(Atreya et al. 2003; Mousis et al. 2018; Li et al. 2020), with a
+few exceptions attributed to interior processes. In Uranus and
+Neptune, volatile abundances can reach up to about 100 times
+their protosolar values (Lindal et al. 1987, 1990; Baines et al.
+1995; Karkoschka & Tomasko 2009; Sromovsky et al. 2014).
+Numerical models show that the dynamics of icy pebbles and
+their vapors around icelines is an efficient mechanism to produce
+local changes in the composition of the PSN (Booth et al. 2017;
+Desch et al. 2017). This process efficiently concentrates species
+around their respective icelines, creating the compositional gra-
+dient that may be responsible for the volatile enrichment in giant
+planets of our Solar System. The radial transport of trace species
+and the effect of icelines have been investigated using accretion
+disk models to assess the composition of the PSN. The resulting
+compositional profiles would then be used to constrain the for-
+mation conditions of gas giants (Mousis et al. 2019; Schneider
+& Bitsch 2021; Aguichine et al. 2022) or ice giants (Owen et al.
+1999; Monga & Desch 2015; Mousis et al. 2020). This approach
+has also been used to explain the diversity among the cometary
+compositions and to determine their source or the general origins
+of their building blocks (Mandt et al. 2020; Mousis et al. 2021a).
+The form taken by volatiles that have fallen from the inter-
+stellar medium (ISM) onto the PSN is still an open question. Icy
+pebbles that are present at the earliest stages of the PSN may
+have formed in the very cold environment of the ISM, at tem-
+peratures of 10K or below (Gibb et al. 2004). At such low tem-
+peratures, H2O condenses in an amorphous structure that can
+efficiently trap other volatile species (Mayer & Pletzer 1986;
+Jenniskens et al. 1995). At ∼135 K, the amorphous ice transi-
+tions to crystalline ice and releases trapped volatiles (Bar-Nun et
+al. 2007). This temperature is much higher than the usual subli-
+mation temperature of pure condensates. In the PSN, the helio-
+centric distance at which this release occurs is called the Amor-
+phous to Crystalline Transition Zone (ACTZ), and is located at
+approximately 5 au (Mousis et al. 2019). However, it is not clear
+whether this amorphous ice survives the fall onto the PSN. For
+instance, it has been proposed that amorphous dust was heated
+up to crystallization when entering the PSN (Visser et al. 2013),
+as a consequence of the presence of shockwaves in the accreting
+Article number, page 1 of 19
+arXiv:2301.02482v1  [astro-ph.EP]  6 Jan 2023
+
+A&A proofs: manuscript no. main
+PSN (Miura et al. 2017; Burkhardt et al. 2019). As a result, many
+circumstellar disk models treat volatile species as pure conden-
+sates, with sublimation temperatures computed from thermody-
+namic tables (Dodson-Robinson et al. 2009; Ciesla et al. 2015;
+Öberg & Wordsworth 2019). Alternatively, volatile species may
+also be trapped in clathrate form in the PSN, when sufficient
+amounts of H2O are available (Lunine & Stevenson 1985; Gau-
+tier & Hersant 2005; Mousis et al. 2021b). A species, i, is usually
+trapped in clathrate at a higher temperature than the one needed
+to sublimate its pure condensate form, except for the cases of
+CO2 and NH3. Because CO2 clathrate and NH3 monohydrate
+form at lower temperatures than their respective condensates at
+nebular pressures, they are not considered in our model. In the
+following, we refer to the heliocentric distance at which volatiles
+are entrapped in or released from clathrates as the clathration
+line. For this reason, volatile species can remain adsorbed on
+amorphous ice or trapped in clathrates, then they are released
+several au closer to the Sun – than they otherwise would be if
+they were in the form of pure condensates.
+In this study, we aim to quantify the influence of the presence
+of various icelines in the PSN, including the ACTZ and multiple
+clathration lines, on the nature of the main volatile reservoirs
+that were at play during the formation of the first icy grains in
+the disk. To do so, we use an existing PSN model that already
+describes the condensation and sublimation of pure ices, as well
+as the transport of species in solid and vapor forms (Aguichine
+et al. 2020, 2022), along with prescriptions of amorphous ice
+destabilization, as well as clathrate crystallization and dissocia-
+tion added. Eleven key species are considered in our approach,
+namely: H2O, CO, CO2, CH4, H2S, N2, NH3, Ar, Kr, Xe, and
+PH3. Each of these species can exist in the forms of vapor, crys-
+talline ice, clathrate (monohydrate in the case of NH3), or amor-
+phous ice in the PSN. Two scenarios are investigated, each of
+them corresponding to a different initial state of the system. In
+scenario 1: the PSN is filled with volatile species in pure con-
+densates or vapor form, depending on their location in the PSN.
+In scenario 2: the PSN is filled with volatile species adsorbed
+on amorphous ice beyond the ACTZ, and in vapor form in re-
+gions closer to the Sun. Because the ACTZ is located beyond the
+snowline, H2O is found in crystalline form between the snow-
+line and the ACTZ. Figure 1 represents the different forms of
+volatiles in the PSN in these two cases. In scenario 1, volatile
+species are in the vapor phase between the snowline and their
+respective icelines. If enough H2O is available, these vapors can
+form clathrates. In scenario 2, icy grains of amorphous ice that
+drifted inward of the ACTZ release the adsorbed volatile species
+as vapors. The outward diffusion of these vapors can lead to the
+formation of clathrates or pure condensates (or both).
+Section 2 describes the PSN model, as well as the transport
+modules used in our calculations. It also depicts the different for-
+malisms that have been added to the disk model to mimic the for-
+mation and destabilization of clathrates, as well as the desorption
+of volatiles from the amorphous ice particles crossing the ACTZ.
+In Section 3, the radial profiles of the abundances of the different
+species are represented in various forms as a function of time in
+the PSN and in the individual cases described by the two sce-
+narios. Section 4 is devoted to a discussion of the sensitivity of
+our results in light of the variation of the disk parameters. Our
+model is then used to compare the compositions of its resulting
+volatile reservoirs with those of the H2O–poor comet C/2016 R2
+PanSTARRS (R2), Jupiter, Uranus, and Neptune. This allows us
+to discuss the formation conditions of those bodies in the context
+of the two scenarios. Section 5 presents our conclusions.
+2. Volatile transport and evolution model
+In this section, we describe the protosolar nebula model em-
+ployed in our simulations, along with the modules calculating
+the transport of dust particles and vapors within the disk. Source
+and sink terms related to sublimation and condensation of pure
+ices as well as to clathrate destabilization and formation within
+the disk are also depicted.
+2.1. Protoplanetary disk model
+The disk model used here is the one described in Aguichine et
+al. (2020) and Mousis et al. (2020). The evolution of the PSN is
+governed by the following differential equation (Lynden-Bell &
+Pringle 1974):
+∂Σg
+∂t = 3
+r
+∂
+∂r
+�
+r1/2 ∂
+∂r
+�
+r1/2Σgν
+��
+,
+(1)
+which describes the time evolution of a viscous accretion disk
+of surface density, Σg, and viscosity, ν, assuming invariance in
+the orbital direction and hydrostatic equilibrium in the azimuthal
+direction. This equation can be rewritten as a set of two first-
+order differential equations coupling the gas surface density, Σg,
+field and mass accretion rate, ˙M:
+�������������
+∂Σg
+∂t =
+1
+2πr
+∂ ˙M
+∂r
+˙M = 3πΣgν
+�
+1 + 2∂ ln νΣg
+∂ ln r
+�
+.
+(2)
+The first equation is a mass conservation law and the second
+one is a diffusion equation. The mass accretion rate is expressed
+as a function of the gas velocity field, vg, and the radius, r, as
+˙M = −2πΣgvgr.
+The dynamical viscosity ν is calculated using the prescrip-
+tion of Shakura & Sunyaev (1973):
+ν = α c2
+s
+ΩK
+,
+(3)
+where α is the viscosity coefficient, cs is the sound speed in the
+PSN, and ΩK is the Keplerian frequency; also, α is estimated to
+be in the 10−4–10−2 range, based on models calibrated on disk
+observations (Hartmann et al. 1998; Hersant et al. 2004; Gautier
+& Hersant 2005; Birnstiel et al. 2012; Dr˛a˙zkowska & Alibert
+2017; Armitage 2019). The sound speed, cs, is expressed as fol-
+lows:
+cs =
+�
+RT
+µg
+,
+(4)
+where µg is the mean molecular mass of the gas in the PSN, as-
+sumed here to be equal to 2.31 g·mol−1, T is the midplane tem-
+perature, and R is the ideal gas constant.
+Two energy sources are considered in our model, namely vis-
+cous heating, and the constant irradiation by the local environ-
+ment of ambient temperature, Tamb = 10 K. Irradiation from the
+young Sun is neglected because the presence of shadowing is as-
+sumed in the outer part of the disk (Ohno & Ueda 2021). This
+allows the disk temperature to decrease to the condensation tem-
+perature of Ar (∼20 K), allowing this species to be trapped in
+Article number, page 2 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+Scenario 1
+Scenario 2
+Pure condensates 
+pebbles
+Clathrate pebbles
+Condensation line
+Clathration line
+Vapor
+Radial drift (inward)
+.
+. :
+'
+\
+- r
+-
+.
+.
+I
+'
+- ,
+\
+.
+' /
+-
+y
+'
+,
+a
+/
+\
+'
+s
+-
+-
+/
+-
+\ ,
+"
+"
+°
+. .
+-
+I
+/
+✓
+Pure condensates 
+pebbles
+Condensation line
+Clathration line
+Vapor
+Radial drift (inward)
+Amorphous ice 
+pebbles
+ACTZ
+.
+.
+.
+'
+'
+I
+- r
+- -
+/
+-
+-
+-
+I
+/
+/
+I
+•
+.
+..
+'.
+"
+I
+-
+y
+'
+.
+.
+-
+/
+<
+\
+I
+-
+1
+-
+-
+-
+/
+- ,
+I
+✓
+-
+/
+-
+-
+-
+/
+-
+Iceline
+Pure condensates 
+pebbles
+Clathrate pebbles
+Iceline
+Clathration line
+Vapor
+Radial drift (inward)
+.
+. :
+'
+\
+- r
+-
+.
+.
+I
+'
+- ,
+\
+.
+' /
+-
+y
+'
+,
+a
+/
+\
+'
+s
+-
+-
+/
+-
+\ ,
+"
+"
+°
+. .
+-
+I
+/
+✓
+Pure condensates 
+pebbles
+Iceline 
+Clathration line
+Vapor
+Radial drift (inward)
+Amorphous ice 
+pebbles
+ACTZ
+.
+.
+.
+'
+'
+I
+- r
+- -
+/
+-
+-
+-
+I
+/
+/
+I
+•
+.
+..
+'.
+"
+I
+-
+y
+'
+.
+.
+-
+/
+<
+\
+I
+-
+1
+-
+-
+-
+/
+- ,
+I
+✓
+-
+/
+-
+-
+-
+/
+-
+Fig. 1. Two outcome scenarios for volatile species explored in this paper. Top panel represents the case where volatiles are initially delivered
+in pure condensate form to the PSN (scenario 1). Bottom panel represents the case where volatiles are released in vapor form in the PSN when
+amorphous grains cross the ACTZ region (scenario 2). Pure condensates, clathrate, and amorphous ice pebbles are represented as blue, brown and
+red circles, respectively. Vapor is represented as purple dots. The iceline, clathration line, and ACTZ are represented as blue, brown, and red solid
+lines, respectively. Once delivered to the disk, the phase (solid or gaseous) of each species is determined by the positions of the corresponding
+condensation, hydration, or clathration lines. Except for the case of CO2, which vaporises at a higher temperature than its clathrate form, hydration,
+or clathration lines of the volatiles considered are closer to the Sun than their respective icelines. Gaseous volatiles condense or become entrapped
+(depending on the availability of water ice) when diffusing outward of the locations of their condensation, hydration, or clathration lines. Con-
+versely, volatiles condensed or entrapped in grains or pebbles are released in vapor form when drifting inward of their lines. Peaks of abundances
+form close to each phase-transition line (see text). Those enrichments are represented by higher solid and vapor concentrations in the panels.
+Jupiter’s building blocks in a way consistent with the supersolar
+abundance observed in its envelope (Mousis et al. 2009, 2012,
+2021b). The temperature profile is computed by summing the
+energy production rates of both energy sources (Hueso & Guil-
+lot 2005):
+T 4 =
+1
+2σSB
+�3
+8τR + 1
+2τP
+�
+ΣgνΩ2
+K + T 4
+amb,
+(5)
+where σsb is the Stefan-Boltzmann constant, while τR and τP are
+the Rosseland and Planck optical depth, respectively. Here, we
+assume τP = 2.4τR, a case corresponding to the opacity gener-
+ated by dust grains smaller than 10 µm (Nakamoto & Nakagawa
+1994); τR is derived from the Rosseland mean opacity, κR, via
+the following expression (Hueso & Guillot 2005):
+τR = ΣgκR
+2
+.
+(6)
+Here, κR is computed as a sequence of power laws of the form
+κR = κ0ρaT b, where ρ denotes the gas density at the midplane,
+and κ0, a, and b are constants that are obtained by fits on obser-
+vational data in different opacity regimes (Bell & Lin 1994).
+The initial state of the model is computed from the self-
+similar solution derived by Lynden-Bell & Pringle (1974):
+Σgν ∝ exp
+�������−
+� r
+rc
+�2−p�������.
+(7)
+By combining Eqs. 7 and 2, and assuming p =
+3
+2, which cor-
+responds to the case for an early disk (Lynden-Bell & Pringle
+1974), the initial profiles of the dust surface density and mass
+accretion rate are given by:
+�����������������
+Σg,0 =
+˙Macc,0
+3πν exp
+�������−
+� r
+rc
+�0.5�������
+˙M0 = ˙Macc,0
+�������1 −
+� r
+rc
+�0.5������� exp
+�������−
+� r
+rc
+�0.5�������
+,
+(8)
+where rc is the centrifugal radius and ˙Macc,0 is the initial mass
+accretion rate onto the central star, set to 10−7.6 M⊙·yr−1 (Hart-
+Article number, page 3 of 19
+
+A&A proofs: manuscript no. main
+mann et al. 1998). The disk mass is related to the surface density
+profile via the following expression:
+Mdisk = 2π
+� Rmax
+Rmin
+Σrdr,
+(9)
+where Rmin and Rmax are the inner and outer bounds of our
+model. The total disk mass is set to 0.1M⊙ and most of it (99%)
+is encapsulated within ∼200 au. The centrifugal radius, rc, is de-
+termined by solving Eqs. 8 with 9 for the chosen values of mass
+accretion rate and disk mass. Figure 2 represents the thermody-
+namic profiles of our PSN model assuming α = 10−3, and at t =
+104, 105, and 106 yr of the disk evolution.
+100
+101
+102
+101
+102
+103
+Temperature [K]
+10.0 kyr
+100.0 kyr
+1000.0 kyr
+100
+101
+102
+10
+10
+10
+7
+10
+4
+10
+1
+102
+Midplane Pressure [Pa]
+100
+101
+102
+Distance to the star [AU]
+10
+3
+10
+1
+101
+103
+105
+Surface density [Kg. m
+2]
+Fig. 2. Profiles of the disk midplane temperature, pressure and surface
+density calculated at t = 104, 105, and 106 yr as a function of heliocentric
+distance, assuming α = 10−3, shown from top to bottom.
+2.2. Dust dynamics
+To determine the size of the dust pebbles, we rely heavily on the
+two-population algorithm developed by Birnstiel et al. (2012).
+This algorithm relies on the key idea that the dynamics of dust
+pebbles of many different sizes can be well approximated by the
+dynamics of only two populations of particles, in which all par-
+ticles have the same representative sizes. The first group cor-
+responds to the small population, where grains are of constant
+size: a0 = 0.1 µm. The second group represents a large popula-
+tion, where pebbles have a representative size a1, which depends
+on the characteristics of the flow.
+In the disk, pebbles grow by sticking collisions via the fol-
+lowing law:
+a1(t) = a0 exp
+�
+t
+τgrowth
+�
+,
+(10)
+where τgrowth is the growth timescale,
+τgrow =
+4Σg
+√
+3ϵgΣbΩK
+,
+(11)
+where Σb is the total surface density of solids, and ϵg is the dust
+growth efficiency through mutual sticking set to 0.5 (Lambrechts
+& Johansen 2014). Then, we compute the Stokes number of peb-
+bles as a function of their sizes (Johansen et al. 2014):
+St =
+���������������
+√
+2πa1ρb
+Σg
+If a1 ≤ 9
+4λ
+8
+9
+a2
+1ρbcs
+Σgν
+If a1 ≥ 9
+4λ
+.
+(12)
+The top and bottom lines of Eq. 12 correspond to the Epstein and
+Stokes regimes, respectively. The limit between both regimes is
+fixed by the gas mean free path λ = √π/2 · ν/cs, computed by
+equating the two terms in Eq.(12). The term ρb is the pebbles
+mean bulk density:
+ρb =
+�
+i Σb,iρb,i
+�
+i Σb,i
+,
+(13)
+computed as the average of each species’ bulk density, ρb,i,
+weighted by their solid surface density, Σb,i.
+Observations indicate that disks are rich in small dust and
+suggest that fragmentation is a dominant process (Williams &
+Cieza 2011). Based on this observation, our approach consid-
+ers fragmentation and radial drift as the growth-limiting mech-
+anisms. These mechanisms set an upper limit on the highest
+Stokes number that particles can achieve. The first limitation re-
+sults from the fragmentation occurring when the relative speed
+between two pebbles due to turbulent motion exceeds the frag-
+mentation velocity, uf. This upper limit is given by (Birnstiel et
+al. 2012):
+Stfrag = ff
+1
+3α
+u2
+f
+c2s
+,
+(14)
+where uf is set to 10 m·s−1 and the factor ff = 0.37 accounts
+for the fact that the representative size of the large population
+is smaller than the biggest size particles can achieve before they
+fragment.
+A second limitation for dust growth is determined by the drift
+velocities of the different pebbles. When pebbles drift faster than
+they grow, this sets another upper limit for the Stokes number
+(Birnstiel et al. 2012):
+Stdrift = fd
+Σbv2
+K
+Σgc2s
+�����
+d ln P
+d ln r
+�����
+−1
+,
+(15)
+where P is the disk midplane pressure, vK the keplerian velocity,
+and fd = 0.55 has the same origin as ff.
+Article number, page 4 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+When dust grains drift at a high velocity and collide with
+other particles on their path, they can fragment. This induces a
+third upper limit for the Stokes number (Birnstiel et al. 2012),
+given by:
+Stdf =
+1
+1 − N
+ufvK
+cs
+�dP
+dr
+�−1
+,
+(16)
+where the factor N = 0.5 accounts for the fact that only larger
+grains fragment when colliding.
+In the algorithm, all limiting Stokes numbers are computed
+and compared with the Stokes number derived from Eq. 12. At
+each time step, the smallest Stokes number found in this compar-
+ison becomes the reference Stokes number which, in turn, sets
+the value for the representative size, a1, of the large population.
+The representative size of the small population is always a0, and
+their Stokes number is always computed in the Epstein regime
+(top line of Eq. 12).
+Finally, the two-population algorithm of Birnstiel et al.
+(2012) introduces fm the fraction of the mass contained in
+the large population. Among the three size-limiting mecha-
+nisms, if particle drift is the most limiting one (Stdrift
+=
+min
+�
+Stfrag, Stdrift, Stdf
+�
+), then the fraction of the mass contained
+in the large population is fm = 0.97. Otherwise, fm is set to 0.75
+(Birnstiel et al. 2012). The mean grain size ¯a is then given by:
+¯a = fma1 + (1 − fm)a0.
+(17)
+2.3. Trace species evolution model
+Trace species are considered in four distinct forms: vapors, pure
+condensates, entrapped in clathrates, or forming a monohydrate
+(case for NH3 only), and adsorbed in amorphous ice. In our
+model, a distinct surface density is attributed to each of these
+forms, with Σv,i, Σp,i, Σc,i, and Σa,i corresponding to species i in
+vapor, pure condensate, clathrate or hydrate, or amorphous ice
+phases, respectively. Their time and radial evolution is governed
+by the advection-diffusion equation (Birnstiel et al. 2012; Desch
+et al. 2017):
+∂Σi
+∂t + 1
+r
+∂
+∂r
+�
+r
+�
+Σivi − DiΣg
+∂
+∂r
+� Σi
+Σg
+���
+− ˙Qi = 0,
+(18)
+where Di is the diffusion coefficient and is vi is the radial speed.
+˙Qi is a source or sink term that accounts for phase changes,
+counted positive/negative when some matter is created or lost.
+For surface densities of vapors, we assume Di = Dg and vi
+= vg because vapors are well coupled to the PSN gas and evolve
+similarly. The gas diffusivity, Dg, is assumed to be equal to the
+viscosity, ν, and the gas velocity is (Shakura & Sunyaev 1973):
+vg = −
+˙Macc
+2πrΣg
+.
+(19)
+At each time and location, we assume that dust particles
+are formed from a mixture of all available solids. As a conse-
+quence, surface densities of solid phases, namely clathrates (and
+NH3 monohydrate), amorphous ices, and pure condensates, are
+evolved with the same diffusion coefficient, Ds, and radial veloc-
+ity, vs. For particles of a given size, a, and Stokes number, St, the
+diffusion coefficient is given by (Birnstiel et al. 2012):
+Ds =
+Dg
+1 + St2 .
+(20)
+The dust radial velocity is expressed as the sum of gas drag and
+drift velocities (Birnstiel et al. 2012):
+vs =
+1
+1 + St2 vg +
+2St
+1 + St2 vdrift,
+(21)
+where the drift velocity is (Weidenschilling 1997):
+vdrift = c2
+s
+vK
+d ln P
+d ln r .
+(22)
+The diffusion coefficient and radial velocity of solids are com-
+puted for the small and the large populations, that is, for particles
+of sizes a0 and a1. The diffusion coefficient, Ds, and radial veloc-
+ity vs used to evolve surface densities of solids are then given by
+mass-averaged diffusivities and velocities of the small and large
+population (Birnstiel et al. 2012):
+�vs = fmvd,a1 + (1 − fm)vd,a0.
+Ds = fmDd,a1 + (1 − fm)Dd,a0.
+(23)
+2.4. Sources and sinks of trace species
+We follow the approach of Aguichine et al. (2020) to depict the
+sources and sinks for both the solid and vapor phases of the dif-
+ferent species. A pure condensate of species i undergoes sub-
+limation if its partial pressure is lower than the corresponding
+equilibrium pressure. Sublimation results in a sink term for pure
+condensates during the time step, ∆t (Dr˛a˙zkowska & Alibert
+2017):
+˙Qp,i = − min
+�������
+�
+8πµi
+RT
+3
+π¯a¯ρPeq,iΣp,i; Σp,i
+∆t
+������� ,
+(24)
+where µi is the molar mass of species i, ¯ρ is the mean bulk den-
+sity of grains, Peq,i is the equilibrium pressure, and ¯a is the mean
+size of grains. The second part of the minimum function ensures
+that no more than the available quantity of pure condensate sub-
+limates. Equilibrium curves of pure condensates are given in Ap-
+pendix A.
+Conversely, a gas of species i forms a pure condensate if
+its partial pressure is larger than the corresponding equilibrium
+pressure. The condensation rates results in a source term for pure
+condensates (Dr˛a˙zkowska & Alibert 2017):
+˙Qp,i = min
+��
+Pi − Peq,i
+� 2Hµi
+RT∆t; Σv,i
+∆t
+�
+.
+(25)
+We also added the possibility of NH3 monohydrate and
+clathrate crystallization in the PSN. Assuming that enough crys-
+talline water is available, these solids form first during the cool-
+ing of the disk because their crystallization temperatures are
+higher than those of the corresponding pure condensates (see
+Fig. 3). The only exceptions to that rule are CO2 and NH3,
+namely the only species that condense at a higher temperature
+than their hydrates at nebular conditions (see Fig. 3). To compute
+the source and sink terms of trace species in clathrates, we used
+the same prescription as that used for pure condensates (Eqs.
+24 and 25). The equilibrium pressures of pure condensates are
+replaced by those of clathrates and NH3 monohydrate (see Ap-
+pendix B). The formation of clathrates and NH3 monohydrate
+Article number, page 5 of 19
+
+A&A proofs: manuscript no. main
+also requires the presence of a minimum amount of crystalline
+water, resulting in a limit for their source term ˙Qc,i:
+˙Qc,i∆t ≤
+µi
+S iµH2O
+�������Σp,H2O −
+�
+k
+Σc,k
+S kµH2O
+µk
+������� .
+(26)
+This expression takes the amount of available crystalline wa-
+ter Σp,H2O and subtracts the amount of water that is already used
+to trap currently existing clathrates. In this expression, S k is the
+stoichiometric ratio between the species k and water. This ra-
+tio is set to 5.75, 5.66, and 1 in the cases of type I clathrate,
+type II clathrate, and NH3 monohydrate, respectively. This con-
+dition sets an upper limit on the clathrate source term, that can be
+equal to 0 if all the crystalline water is already holding clathrates.
+If all conditions are met for a trace species to be able to form
+pure condensates, clathrates, and monohydrates, we prioritize
+the solid phase that has the greatest Pi − Peq,i value. Prioritizing
+the largest Pi −Peq,i value is equivalent to prioritizing the highest
+solidification rate. Such considerations are not taken into account
+when performing a prioritization test among clathrate formation
+and condensation of pure condensates, since clathrates only form
+when pure condensates are not stable.
+10
+9
+10
+8
+10
+7
+10
+6
+Midplane Pressure [bar]
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Temperature [K]
+Condensation lines
+Fig. 3. Equilibrium curves of pure condensates (solid lines), clathrates,
+and NH3 monohydrate (dashed lines) in a pressure-temperature domain
+relevant to PSN conditions (see the appendix for the relevant data). Two
+clathrate structures are considered in our model, namely, type I and type
+II with stoichiometric factors of 5.75 and 5.66, respectively. Partial pres-
+sures are calculated by considering the species abundances given in Ta-
+ble 1.
+When grains containing amorphous ice cross the ACTZ,
+water crystallizes and releases all the adsorbed volatiles irre-
+versibly. The corresponding sink term is then:
+˙Qa = −Σa
+∆t,
+(27)
+with the only condition that must be satisfied is: T > TACTZ.
+From the various rates of condensation, crystallization and
+sublimation, we can finally derive the overall sink and source
+term for the vapor:
+˙Qv,i = − ˙Qp,i − ˙Qc,i − ˙Qa,i.
+(28)
+3. Results
+Simulations have been performed in the case of two distinct sce-
+narios. In scenario 1, particles initially released in the PSN are
+only made from pure condensates. During their inward drift, they
+can sublimate, condense again and/or form various clathrates
+and a NH3 monohydrate, depending on the local temperature and
+pressure conditions of the disk. In scenario 2, particles initially
+released in the PSN are only made from amorphous ice. During
+their inward drift, water contained in these particles transitions
+to a crystalline structure when crossing the ACTZ and leads to
+the release of adsorbed volatiles in the gas phase. The released
+vapors can in turn condense into pure ices and/or form various
+clathrates and NH3 monohydrate, following the prescription de-
+scribed in the previous section. Both scenarios are explored with
+the assumption of 0.1M⊙ for the mass of the PSN and a viscosity
+parameter of α = 10−3.
+The initial surface density of a species, i, is:
+Σ0,i = x0,iΣg,
+(29)
+where x0,i is the initial mass fraction of the species, i. At t = 0,
+the partial pressures of the different species are computed at each
+point of the grid. In scenario 1, if the partial pressure of a given
+species is below its equilibrium pressure, then the correspond-
+ing location is filled with vapor. Otherwise, this location filled
+with pure ice. In scenario 2, species are in the vapor phase where
+T ≥ TACTZ and in an amorphous ice phrase otherwise. In both
+scenarios, thermodynamic equilibrium is established after a few
+time steps (∼ 1 yr).
+The initial PSN composition is derived from the protosolar
+elemental abundances tabulated by Lodders et al. (2009). We as-
+sume that all C is distributed between CO, CO2, or CH4, with
+the remaining O forming H2O. We have set CO:CO2:CH4 =
+10:4:1 in the PSN gas phase. The CO:CO2 ratio is derived from
+ROSINA measurements of comet 67P/C-G between 2014 Au-
+gust and 2016 September (Mousis et al. 2014). The CO:CH4 ra-
+tio is consistent with the production rates measured in the south-
+ern hemisphere of the 67P/C-G in October 2014 by the ROSINA
+instrument (Le Roy et al. 2015). Sulfur is assumed to be half
+in H2S form and half in refractory sulfide components (Pasek
+et al. 2005). We also assumed N2:NH3 = 1:1, a value predicted
+by thermochemical models that take into account catalytic ef-
+fects of Fe grains on the kinetics of N2 to NH3 conversion of
+the PSN (Fegley 2000; Mousis et al. 2009). The molar abun-
+dances of the different species are derived from the gas phase
+abundances given in Table 1.
+Figure 4 represents the radial evolution of the water mass
+abundance in the PSN at different epochs of its evolution in sce-
+nario 1 and scenario 2 (top row and bottom row, respectively).
+Article number, page 6 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+Table 1. Initial molar abundances of the considered trace species.
+Trace species
+(X/H2)⊙
+Trace species
+(X/H2)⊙
+H2O
+5.479 × 10−4
+NH3
+5.456 × 10−5
+CO
+3.698 × 10−4
+PH3
+6.368 × 10−7
+CO2
+1.479 × 10−4
+Ar
+7.150 × 10−6
+CH4
+3.698 × 10−5
+Kr
+4.310 × 10−9
+H2S
+1.633 × 10−5
+Xe
+4.210 × 10−10
+N2
+5.456 × 10−5
+Both cases produce very similar water abundances and show a
+water enrichment peak that is about 10 times its initial abun-
+dance at the location of the snowline (∼2.8 au). However, the
+availability of crystalline ice is much more limited in scenario 2
+due to H2O being mostly in an amorphous state. Indeed, in this
+case, crystalline ice is present only in the ∼2.8–4 au region.
+3.1. Scenario 1: Initial delivery of pure condensates
+Figure 5 represents the time and radial evolution of the abun-
+dance ratios (with respect to initial abundances) of the differ-
+ent species existing in various phases in the case of scenario 1,
+respectively. After 10 kyr of PSN evolution, the species con-
+sidered are present under all their possible forms – except for
+CO2 and NH3, which are never enclathrated. By successive order
+with progressing heliocentric distance (and decreasing tempera-
+ture and pressure conditions), we first find the different vapors,
+then narrow regions corresponding to the presence of clathrates,
+and, finally, an outer region that is only populated with pure con-
+densates. Regions where clathrates exist expand from 7 to 12 au,
+depending on the species considered. At this early stage of the
+PSN evolution, there is no significant enrichment that can be ob-
+served for any species.
+After 0.1 Myr of disk evolution, Figure 5 also shows that
+beyond 6 au, clathrates coexist with pure condensates and their
+abundances decrease steeply – except for CO2 and NH3, which
+that exist only as pure ices. A few au beyond that, pure con-
+densates become the only solid structures existing in the outer
+PSN. Depending on the PSN ther modynamic conditions and
+the availability of crystalline water, when the different species
+become fully enclathrated, their abundances form unique enrich-
+ment peaks located at their clathration lines, reaching up to ∼15
+times their initial values. On the other hand, if the budget of crys-
+talline water is not high enough, then the species are only partly
+enclathrated. They then form two narrow enrichment peaks at
+their condensation and clathration lines, reaching ∼5 and ∼15
+times their initial abundances, respectively. Table 2 displays the
+heliocentric distance and the value of the enrichment peak (rel-
+ative to the initial abundance) for each species under consider-
+ation at t = 0.1 Myr of the PSN evolution. Depending on the
+species considered, the enrichment peaks range between 2 and
+18 times the protosolar values and are located in the 2–11 au re-
+gion. The closest and furthest peaks from the Sun are those of the
+water snow line at 2.8 au and N2 iceline at 10.8 au, respectively.
+Two enrichment peaks, located at the clathration and icelines,
+are found in the cases of CH4 and Kr.
+In Figure 5, all species, except CO, N2, and Ar, exhibit a
+dip in the surface densities of the pebbles of pure condensates
+around 10.5 au. In this region, the total surface density of icy
+pebbles is increased due to the combined actions of CO, N2, and
+Ar icelines located at 10.4, 10.9, and 10.7 au, respectively. The
+condensations rates of CO, N2, and Ar locally increase the sur-
+face density of solids, which are in excess compared with their
+loss rates via inward drift. On the other hand, because the ice-
+lines of the other species are located closer to the Sun, their sur-
+face densities progressively decrease in this region, as a result of
+the inward drift of particles.
+After 1 Myr of PSN evolution, all peaks are smoothed out
+in the gas phase. Because of the inward drift of pebbles, most of
+the species are in vapor form when the solids start to deplete. The
+total mass of solids has decreased by a factor of four compared
+to the beginning of the simulation. This effect is more significant
+in the case of clathrates forming at very low temperatures (less
+than 50 K), since the fraction of crystalline water used to form
+higher temperature clathrates increases with time.
+Table 2. Heliocentric distance and value of the enrichment peak (rela-
+tive to the initial abundance) for each species under consideration at t =
+0.1 Myr in the case of scenario 1 (middle column of Figure 5).
+Element
+Peak location (au)
+Peak value
+H2O
+2.7
+11.0
+CO
+10.4
+8.1
+CO2
+6.0
+16.0
+CH4
+7.1
+4.7
+9.5
+9.4
+H2S
+5.9
+14.0
+N2
+10.9
+5.6
+NH3
+5.6
+17.2
+Ar
+10.7
+7.3
+Kr
+8.2
+9.12
+9.8
+23.5
+Xe
+7.0
+14.3
+PH3
+6.4
+15.4
+Both vapor and solid forms are considered.
+3.2. Scenario 2: initial delivery of amorphous ices
+Figure 6 represents the time and radial evolutions of the abun-
+dance ratios (with respect to the initial abundances) of the dif-
+ferent species existing in various phases in the case of scenario
+2. After 10 kyr of PSN evolution, with progressing heliocentric
+distance, first the different vapors are found, then an outer region
+that is only populated with amorphous ice. The budget of solid
+phase is dominated by species trapped in amorphous ice. Pure
+condensates only form when the vapors desorb from amorphous
+ice at the ACTZ (shown at ∼5 au), diffuse outward and cross an
+iceline. Such a process is efficient only if the condensation line
+is close to the ACTZ, leading to very narrow regions with the
+presence of pure condensates only in the case of NH3.
+After 0.1 Myr of disk evolution, it is notable that only pure
+condensates of NH3, H2S, CO2, and PH3 form. Table 3 displays
+the heliocentric distance and the value of the enrichment peak
+(relative to the initial abundance) for each species under consid-
+eration at this epoch of the PSN evolution. All peaks are located
+in a much narrower region, centered at the location of the ACTZ,
+compared with the scenario 1, with values ranging from 7 to 10
+times the initial abundances. The peak locations are influenced
+by the presence of narrow (less than 1 au) regions filled with pure
+condensates. In those regions, corresponding to the icelines, the
+abundance of pure condensates exceed that of amorphous ice,
+thus influencing the location of the enrichment peaks.
+After 1 Myr of PSN evolution, as a result of pebble drift, all
+peaks have been smoothed and the volatiles trapped in amor-
+phous water ice phase are strongly depleted by factors reach-
+Article number, page 7 of 19
+
+A&A proofs: manuscript no. main
+100
+101
+Heliocentric distance [au]
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+Water mass abundance
+Scenario I
+Scenario II
+t = 1.00e+04 yr
+100
+101
+Heliocentric distance [au]
+t = 1.00e+05 yr
+100
+101
+Heliocentric distance [au]
+t = 1.00e+06 yr
+100
+101
+Heliocentric distance [au]
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+Water mass abundance
+t = 1.00e+04 yr
+100
+101
+Heliocentric distance [au]
+t = 1.00e+05 yr
+100
+101
+Heliocentric distance [au]
+t = 1.00e+06 yr
+Fig. 4. Time evolution of the mass abundance of water, defined as the radial profile of ΣH2O/Σg, for both scenarios at t = 104, 105, and 106 yr. Solid
+lines represent H2O in the gaseous phase (orange line), crystalline phase (blue line), and in amorphous phase (red line).
+ing more than 100, compared with their gaseous abundances in
+the inner disk. The amount of pure condensates exceeds that
+of amorphous ice in some narrow regions, except in the cases
+of H2S and PH3. For reasons identical to those invoked at the
+same epoch of PSN evolution in scenario 1, the surface density
+of solids is strongly decreased in the region centered at ∼7 au.
+Table 3. Heliocentric distance and value of the enrichment peak (rela-
+tive to the initial abundance) for each species under consideration at t =
+0.1 Myr in the case of scenario 2 (middle column of Figure 6).
+Element
+Peak location (au)
+Peak value
+H2O
+2.7
+8.5
+CO
+4.8
+7.0
+CO2
+4.7
+7.5
+CH4
+4.7
+7.0
+H2S
+4.7
+7.1
+N2
+4.7
+7.0
+NH3
+5.6
+10.1
+Ar
+4.7
+7.0
+Kr
+4.7
+7.0
+Xe
+4.7
+7.0
+PH3
+4.7
+7.1
+Both vapor and solid forms are considered.
+4. Discussion
+In this section, we discuss the implications of our model for var-
+ious bodies of the solar system. We first investigate the sensitiv-
+ity of our model to the variation of its input parameters and then
+provide fits of the volatile composition of comet R2 and those of
+Uranus and Neptune.
+4.1. Sensitivity to parameters
+The stability of our results has been tested against the variations
+of the pebble density, disk’s mass, and the viscosity parameter in
+the 0.1–1 g cm−2, 10−2–10−1 M⊙, and 10−4–10−2 ranges, respec-
+tively. Variation in pebble density leads to results similar to those
+presented in Sec. 3.1 and 3.2. Lower density pebbles drift over
+shorter timescales at given size, but are also smaller because of
+a lower fragmentation limit (see Sect. 2.2). On the other hand,
+smaller pebbles drift over longer timescales, implying that both
+effects are (roughly) mutually counterbalanced and produce only
+minor variations in the abundance profiles. Both the mass and α
+viscosity parameter of the disk affect its viscous evolution and,
+thus, the locations of the various icelines. Less massive disks
+are cooler because of their reduced viscous dissipation (see Eq.
+5) and display their icelines closer to the host star. For exam-
+ple, we find that the condensation and clathration lines of the
+species investigated in our study are ∼2 au closer to the Sun in a
+Article number, page 8 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+10−2 M⊙ disk, compared with a 10−1 M⊙ disk. As another exam-
+ple, the enrichment peak associated with the CO iceline ranges
+between 7 and 11 au from the Sun when the α–value is varied
+between 10−4 and 10−2. The magnitude of the abundance peaks
+is also affected by the variation of α. The abundance of the CO
+peak ranges between about 10 and 5 times, respectively, its ini-
+tial PSN abundance when the α value is varied between 10−4 and
+10−2. In our disk model, the contribution from the irradiation by
+the Sun to the disk’s midplane temperature is not considered be-
+cause it is assumed that the outer PSN is shadowed by its inner
+region (Ohno & Ueda 2021). This allows the disk to reach tem-
+peratures low enough to enable the condensation of ultravolatiles
+(CO, N2, Ar, etc.) at the current locations of the giant planets,
+assuming the absence of migration during formation. When this
+contribution is included by considering the formalism depicted
+in Adams et al. (1988) and Ruden & Pollack (1991), the temper-
+ature profile becomes slightly warmer, implying that the icelines
+and clathration lines are moved outward by 1–2 au. Despite these
+changes, the general trend of our results is not impacted.
+4.2. Implications for comet R2
+R2 is a long-period comet displaying an unusually high N2/CO
+ratio of 0.006–0.008 (Biver et al. 2018; Opitom et al. 2019). An-
+other peculiar characteristic of this comet is its heavy depletion
+in terms of H2O, with a CO/H2O ratio of about 312 (McKay et
+al. 2019).
+Figures 7 and 8 represent the radial profiles of the CO/H2O
+and N2/CO ratios calculated in the PSN pebbles with our nomi-
+nal model as a function of time in the cases of scenario 1 and
+scenario 2, respectively. Both ratios are compared with those
+measured in R2’s coma (blue horizontal bar). Assuming that R2
+formed from a unique set of building blocks, we require both
+ratios to be reproduced by our model at the same heliocentric
+distance and epoch. Simulations performed in the case of sce-
+nario 1 reproduce both ratios at a heliocentric distance of 7 au,
+after 1 Myr of PSN evolution. Our result is then consistent with
+those derived from the study of Mousis et al. (2021b), based on
+a simple approach that does not consider the interplay between
+the clathrate and water ice reservoirs.
+Simulations performed in the case of scenario 2 reproduce
+the N2/CO ratio at 8 au after 1 Myr of PSN evolution. However,
+the CO/H2O ratio is not matched by our model, even if a peak
+is seen at 8 au and 1 Myr. We should note that the positions and
+magnitudes of those peaks can change when the α–parameter
+and mass of our disk model are varied. This implies that, even
+if scenario 1 provides a better match of R2’s composition with
+our nominal model, scenario 2 cannot be excluded. Interestingly,
+our calculated peaks are within a zone of dynamic instability in
+the early Solar System, which is more likely to result in ejection
+of planetesimals than capture by the Oort Cloud. This is a pos-
+sible explanation for the lack of R2-like comets observed today
+(Anderson et al. 2022).
+4.3. Implications for the composition of Jupiter
+One-σ error bar measurements made at Jupiter by the Galileo
+probe and the Juno spacecraft indicate C, N, O, S, P, Ar, Kr, and
+Xe abundances that are ∼1.5 to 6 times higher than the proto-
+solar values (Atreya et al. 2003; Wong et al. 2004; Mousis et
+al. 2018; Li et al. 2020). To explain those features, it has been
+proposed that Jupiter’s atmosphere could reflect the composition
+of icy planetesimals either made of amorphous ice (Owen et al.
+1999) or from pure condensates and/or clathrates (Gautier et al.
+2001; Gautier & Hersant 2005; Mousis et al. 2018, 2021b). Al-
+ternatively, it has been proposed that this supersolar metallicity
+could result from the accretion of already pre-enriched PSN gas
+(Mousis et al. 2019; Aguichine et al. 2022).
+Figure 9 represents the time evolution of the sum of the el-
+emental enrichments calculated in vapor and solid phases at the
+heliocetric distance of 4 au, compared with their protosolar val-
+ues, and in the cases of our two scenarios. Following the ap-
+proach of Aguichine et al. (2022), we focused on the composi-
+tion of the PSN at 4 au, chosen as the location of Jupiter’s for-
+mation. This distance is, in the model, beyond the water iceline
+but inward of the icelines of all other trace species. The dust-to-
+gas ratio can easily become greater than 2 to 3 times the pro-
+tosolar composition in this region and could ease the formation
+of a proto-Jupiter core via the streaming instability (Yang et al.
+2017).
+Figure 9 shows that the measured elemental enrichments are
+all matched by our model after 0.8–1 Myr and 50–100 kyr of the
+PSN evolution in scenario 1 and scenario 2, respectively. Our
+models suggest that the consideration of clathrate formation in
+addition to the crystallization of pure condensates in the PSN
+(scenario 1) still allow for the formation of a Jupiter-like planet
+from supersolar gases originating from the disk, compared with
+models considering the crystallization of pure condensates only,
+such as that developed by Aguichine et al. (2022). Our model
+also suggests that the presence of multiple condensation and
+clathration lines does not alter the formation of supersolar va-
+pors subsequent to their release from amorphous ice (scenario
+2). Previous works exploring this possibility did not consider the
+formation of icelines in their models (Monga & Desch 2015;
+Mousis et al. 2019). In particular, Saturn’s tropospheric abun-
+dances of C, N, S, and P have been measured to be between 3
+and 13 times their protosolar abundances (Atreya et al. 2018),
+suggesting that its metallicity is higher than that of Jupiter. As-
+suming that Saturn formed in the vicinity of the CO2 iceline to
+account for its high carbon enrichment, which corresponds to a
+heliocentric distance of ∼6 au at early epochs in our PSN model,
+our calculations show that this range of enrichments is repro-
+duced within 0.2–0.3 Myr in scenario 1 and 0.2 Myr in scenario
+2. This implies that Saturn could have formed earlier than Jupiter
+in scenario 1, whereas in scenario 2, it could have formed later.
+An earlier formation of Saturn could have reduced the inward
+flux of pebble and vapors, lowering the value of elemental en-
+richment peaks at Jupiter’s location. However, the height of en-
+richment peaks is highly sensitive to the disk parameters (e.g.,
+the α-value; see Aguichine et al. (2022)). Therefore, if the ac-
+cretion of pebbles by Saturn reduces the height of the enrich-
+ment peaks at Jupiter’s location, it is still possible to achieve a
+metallicity similar to what is measured in its atmosphere.
+4.4. Implications for the composition of Ice Giants
+The composition of the deep atmospheres of Uranus and Nep-
+tune is shrouded in mystery since most of the heavy constituents
+condense at pressures deeper than may readily be probed re-
+motely (Mousis et al. 2020). The only determinations that can
+be used so far in our model are the C/N and C/S ratios, which
+have been found equal to or higher than ∼175 and ∼35, respec-
+tively (Asplund et al. 2009; Karkoschka & Tomasko 2009, 2011;
+Irwin et al. 2018, 2019a,b). Figures 10 and 11 represent the ra-
+dial profiles of the C/N and C/S elemental ratios in pebbles as a
+function of time in the PSN, compared with the minimum val-
+ues measured in the tropospheres of the two ice giants. In both
+Article number, page 9 of 19
+
+A&A proofs: manuscript no. main
+cases, these ratios can be reproduced in the ∼7–8 au region after
+1 Myr of PSN evolution. The formation distance of solids in the
+PSN is model-dependent, but our simulations suggest that the
+giant planets did form in a more compact configuration than cur-
+rent the one – which is also in agreement with several dynamical
+models (Tsiganis et al. 2005; Lykawka et al. 2010; Guilera et
+al. 2011). Given their high metallicities, the formation of Uranus
+and Neptune requires a higher surface density of solids than the
+values derived here for the outer PSN. This assumption is at odds
+with the fact that the planetesimal density and collision proba-
+bility are both low in the outer disk. One way to overcome this
+difficulty would be to assume the formation of the two giants
+via the accretion of pebbles directly onto the planetary embryo,
+which can still work efficiently far from the host star (Helled et
+al. 2014; Bitsch et al. 2015, 2018; Armitage 2019).
+4.5. Limitations of the model
+Over the recent years, substructures have been largely observed
+in protoplanetary disks. The ALMA/DSHARP survey showed
+that protoplanetary disks are not smooth and that substructures
+are ubiquitous (Andrews et al. 2020; Jennings et al. 2022). Sub-
+structures can be produces by planet-disk interactions, in the
+form of spiral density waves (Muto et al. 2012; Zhang et al.
+2021) and gaps (Bae et al. 2017). Radiative hydrodynamic mod-
+els show that these substructures can be also formed by insta-
+bilities in the disk (Lovelace et al. 1999; Lovelace & Romanova
+2014; Blanco et al. 2021). Such features translate into local den-
+sity and pressure variations that act as dust traps which locally
+enhance the dust surface density. Although our model does not
+take into account such disk substructures, we show that the ice-
+lines, clathration lines, and the ACTZ correspond to vapor and
+pebble enrichment peaks, which can lead to instabilities.
+In this work, giant planets are assumed to be formed in situ.
+In scenario 1, Jupiter is formed within 0.8–1 Myr. This timescale
+is compatible with a formation via pebble accretion (Bitsch et
+al. 2015, 2018; Alibert et al. 2018; Armitage 2019; Venturini &
+Helled 2020) and gravitational instability (Boss 1997; Zhu et al.
+2012; Kratter & Lodato 2016). On the other hand, in scenario
+2, Jupiter forms in less than 0.1 Myr. This timescale is consis-
+tent with a gravitational instability which can be triggered as
+early as 0.1 Myr (Zhu et al. 2012). An early formation of Jupiter
+allows us to account for the observed carbonaceous chondrites
+dichotomy (Kleine et al. 2020). Although, the disk instability
+model is consistent with formation timescales found in both sce-
+nario 1 and scenario 2, it is important to note that observations
+and models suggest that, at minimum, amorphous ice should be
+heated up to its crystallization temperature when falling from the
+presolar cloud onto the PSN (Visser et al. 2013). Those findings
+suggest that scenario 1 is the most likely scenario, implying that
+our results remain consistent with the core accretion model.
+One limitation of our model is the fact that planet migration
+is not considered. During or after formation, planets migrate in-
+ward or outward (Masset & Papaloizou 2003; Bitsch et al. 2015;
+Schneider & Bitsch 2021). During migration, planets accrete
+material from different parts of the disk with various compo-
+sitions. Although planetary migration contradicts an in situ for-
+mation hypothesis, 3D hydrodynamical simulations indicate that
+the rates of type I and type II migrations could be several times
+slower than the prescription usually employed in the literature of
+planet formation (Chrenko & Nesvorný 2020; Lega et al. 2021;
+Chametla & Chrenko 2022). Assuming an in situ formation of
+Jupiter in our model to reproduce its observed metallicity is ex-
+pected to remain valid in light of these revised migration rates
+because our derived timescales are still quite short, compared to
+the PSN evolution.
+5. Conclusion
+In this work, we investigate the impact of clathrate formation
+on the radial distribution of volatiles in the PSN, considering
+two scenarios, each of them corresponding to a distinct initial
+condition. To do so, we used a 1D protoplanetary disk model
+coupled with modules describing the evolution of trace species
+in the vapor phase, as well as the dynamics of dust and peb-
+bles. This model also considers the different sources and sinks
+for the volatile phases considered (vapors, pure condensates, or
+clathrates).
+In scenario 1, we assume that the volatiles were delivered to
+the PSN in the form of pure condensate grains. In this case, we
+show that clathrates can crystallize and form enrichment peaks
+at about 10 times the value of the initial abundances at their
+clathration lines, which are closer to the Sun than their corre-
+sponding icelines. The amount of clathrates formed in the PSN
+depends on the local abundance of crystalline water, which in
+many cases, acts as a limiting factor in our model. In scenario
+2, we assumed that the volatiles were delivered to the PSN in
+the form of amorphous grains. Under those conditions, volatiles
+are only released from amorphous ice when the icy grains are
+heated up to ∼135K, namely, at the ACTZ location. An enrich-
+ment peak up to about seven times the initial abundances then
+forms at the ACTZ location. In this case, clathrate formation is
+not possible because there is no crystalline water ice available
+beyond the ACTZ in the PSN. All the enrichment peaks of pure
+condensates are also located close to the ACTZ. Our investiga-
+tion shows that both scenario 1 and scenario 2 can reproduce the
+known compositions of comet R2, Jupiter, Uranus, and Neptune.
+This implies that our model does not allow us to formally rule
+out the presence of amorphous ice during the early phases of the
+PSN, assuming that those planetary bodies accreted from peb-
+bles or pebble-made planetesimals. More planetary composition
+data, such as the in situ measurements of Saturn’s atmosphere,
+are needed to understand the initial formation conditions of the
+PSN.
+6. Acknowledgments
+The project leading to this publication has received fund-
+ing from the Excellence Initiative of Aix-Marseille Université
+– A*Midex, a French ”Investissements d’Avenir programme”
+AMX-21-IET-018. This research holds as part of the project FA-
+COM (ANR-22-CE49-0005-01_ACT) and has benefited from a
+funding provided by l’Agence Nationale de la Recherche (ANR)
+under the Generic Call for Proposals 2022. OM acknowledges
+support from CNES. JIL was supported by the Juno project. We
+acknowledge Sarah E. Anderson for helpful discussions about
+her dynamical simulations regarding the origin of comet R2. We
+thank the anonymous referee for their useful comments that im-
+proved the quality of this paper.
+Article number, page 10 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+10
+2
+10
+1
+100
+101
+f
+NH3
+t = 1.00e+04 yr
+t = 1.00e+05 yr
+t = 1.00e+06 yr
+10
+2
+10
+1
+100
+101
+f
+H2S
+10
+2
+10
+1
+100
+101
+f
+CO2
+10
+2
+10
+1
+100
+101
+f
+PH3
+10
+2
+10
+1
+100
+101
+f
+CH4
+10
+2
+10
+1
+100
+101
+f
+N2
+10
+2
+10
+1
+100
+101
+f
+CO
+10
+2
+10
+1
+100
+101
+f
+Ar
+10
+2
+10
+1
+100
+101
+f
+Kr
+100
+101
+Heliocentric distance [au]
+10
+2
+10
+1
+100
+101
+f
+Xe
+100
+101
+Heliocentric distance [au]
+100
+101
+Heliocentric distance [au]
+Fig. 5. Time and radial evolution of species’ mass abundances normalized to their initial values in gaseous phase (orange line), pure condensate
+form (blue line), and clathrate (green line), at t = 104, 105, and 106 yr in scenario 1.
+Article number, page 11 of 19
+
+A&A proofs: manuscript no. main
+10
+2
+10
+1
+100
+101
+f
+NH3
+t = 1.00e+04 yr
+t = 1.00e+05 yr
+t = 1.00e+06 yr
+10
+2
+10
+1
+100
+101
+f
+H2S
+10
+2
+10
+1
+100
+101
+f
+CO2
+10
+2
+10
+1
+100
+101
+f
+PH3
+10
+2
+10
+1
+100
+101
+f
+CH4
+10
+2
+10
+1
+100
+101
+f
+N2
+10
+2
+10
+1
+100
+101
+f
+CO
+10
+2
+10
+1
+100
+101
+f
+Ar
+10
+2
+10
+1
+100
+101
+f
+Kr
+100
+101
+Heliocentric distance [au]
+10
+2
+10
+1
+100
+101
+f
+Xe
+100
+101
+Heliocentric distance [au]
+100
+101
+Heliocentric distance [au]
+Fig. 6. Time and radial evolution of species’ mass abundances normalized to their initial values in gaseous phase (orange line), pure condensate
+form (blue line), and amorphous form (red line), compared with their initial mass abundances, at t = 104, 105, and 106 yr in scenario 2.
+Article number, page 12 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+100
+101
+Heliocentric distance [au]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+103
+[N2]/[CO]
+R2
+0.1 Myr
+0.5 Myr
+1.0 Myr
+100
+101
+Heliocentric distance [au]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+103
+[CO]/[H2O]
+R2
+0.1 Myr
+0.5 Myr
+1.0 Myr
+Fig. 7. N2/CO (top panel) and CO/H2O (bottom panel) abundance ratios
+in pebbles represented as a function of heliocentric distance in the case
+of scenario 1, and compared with those measured in R2’s coma (blue
+bar) at 0.1, 0.5, and 1 Myr of the PSN evolution. Both ratios measured
+in R2 are simultaneously matched by our model at a distance of 7 au
+and 1 Myr.
+100
+101
+Heliocentric distance [au]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+103
+[N2]/[CO]
+R2
+0.1 Myr
+0.5 Myr
+1.0 Myr
+100
+101
+Heliocentric distance [au]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+103
+[CO]/[H2O]
+R2
+0.1 Myr
+0.5 Myr
+1.0 Myr
+Fig. 8. N2/CO (top panel) and CO/H2O (bottom panel) abundance ratios
+in pebbles represented as a function of heliocentric distance in the case
+of scenario 2, and compared with those measured in R2’s coma (blue
+bar) at 0.1, 0.5 and 1 Myr of the PSN evolution. Only the N2/CO ratio
+is reproduced in R2 at 8 au and 1 Myr. The CO/H2O ratio is however
+approached by our model at the same location and epoch of the PSN
+evolution.
+Article number, page 13 of 19
+
+A&A proofs: manuscript no. main
+104
+105
+time (yr)
+100
+101
+fvapor + solid
+Scenario I, r = 4.0 au
+C
+S
+N
+P
+O
+Ar
+Kr
+Xe
+104
+105
+time (yr)
+100
+101
+fvapor + solid
+Scenario II, r = 4.0 au
+C
+S
+N
+P
+O
+Ar
+Kr
+Xe
+Fig. 9. Time evolution of the elemental abundances (relatives to the pro-
+tosolar values) at a heliocentric distance of 4 au in the cases of scenario
+1 (top panel) and scenario 2 (bottom panel). The blue area corresponds
+to the range covered by the elemental abundances derived from space-
+craft measurements (see text).
+100
+101
+Heliocentric distance [au]
+10
+1
+100
+101
+102
+103
+104
+C/N
+0.1 Myr
+0.5 Myr
+1.0 Myr
+100
+101
+Heliocentric distance [au]
+10
+1
+100
+101
+102
+103
+104
+C/S
+0.1 Myr
+0.5 Myr
+1.0 Myr
+Fig. 10. Radial profiles of the C/N (top panel) and C/S (bottom panel)
+ratios calculated in pebbles at different epochs of the PSN evolution in
+the case of scenario 1. The horizontal dashed line represents the min-
+imum ratio measured in the tropospheres of Uranus and Neptune. The
+orange area encompasses the current locations of the ice giants in the
+solar system.
+Article number, page 14 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+100
+101
+Heliocentric distance [au]
+10
+1
+100
+101
+102
+103
+104
+C/N
+0.1 Myr
+0.5 Myr
+1.0 Myr
+100
+101
+Heliocentric distance [au]
+10
+1
+100
+101
+102
+103
+104
+C/S
+0.1 Myr
+0.5 Myr
+1.0 Myr
+Fig. 11. Radial profiles of the C/N (top panel) and C/S (bottom panel)
+ratios calculated in pebbles at different epochs of the PSN evolution in
+the case of scenario 2. The horizontal dashed line represents the min-
+imum ratio measured in the tropospheres of Uranus and Neptune. The
+orange area encompasses the current locations of the ice giants in the
+Solar System.
+Article number, page 15 of 19
+
+A&A proofs: manuscript no. main
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+A&A proofs: manuscript no. main
+Appendix A: Vapor pressures of pure condensates
+The vapor pressures of the pure condensates considered in this
+work, except H2O and NH3, follow the law:
+ln Peq =
+�
+k
+ak
+� 1
+T
+�k
+,
+(A.1)
+where the ak factors have been determined experimentally (Fray
+& Schmitt 2009) and are summarized in Table A.1.
+The vapor pressures of H2O is given by (Wagner et al. 2011):
+Peq =Ptp exp
+�Ttp
+T
+� �������−21.2144006
+�Ttp
+T
+�1/300
++ 27.3203819
+�Ttp
+T
+�2.10666667
+−6.10598130
+�Ttp
+T
+�1.70333333������� ,
+(A.2)
+where Ptp and Ttp are the pressure and the temperature of the
+triple point of water, respectively.
+The phosphine equilibrium pressure is given by Stull (1947):
+log10 Peq = 4.02591 −
+702.651
+T − 11.065.
+(A.3)
+Appendix B: Dissociation pressures of NH3
+monohydrate and clathrates
+The dissociation pressures of NH3 monohydrate and clathrates
+follow the Antoine law:
+ln Peq = A
+T + B,
+(B.1)
+with A and B parameters determined from experiments and sum-
+marized in Table B.1.
+Article number, page 18 of 19
+
+A. Schneeberger et al.: Evolution of the reservoirs of volatiles in the protosolar nebula
+Table A.1. Polynomial factors for the vapor pressure equations of pure condensates
+Element
+Temperature range (K)
+a0
+a1
+a2
+a3
+a4
+a5
+a6
+CO
+T ≤ 61.55
+10.43
+-721.3
+-10740
+2.341 × 105
+−2.392 × 106
+9.478 × 106
+∅
+T > 61.55
+10.25
+-748.2
+-5843
+3.939 × 104
+∅
+∅
+∅
+CO2
+T ≤ 40
+10−40
+∅
+∅
+∅
+∅
+∅
+∅
+40.0 < T ≤ 194.7
+14.76
+-2571
+−7.781 × 104
+4.325 × 106
+−1.207 × 108
+1.350 × 109
+∅
+T > 194.7
+18.61
+-4154
+1.041 × 105
+∅
+∅
+∅
+∅
+CH4
+all
+10.51
+-1110
+-4341
+1.035 × 105
+−7.910 × 105
+∅
+∅
+H2S
+T ≤ 127.0
+12.98
+-2707
+∅
+∅
+∅
+∅
+∅
+T > 127.0
+8.933
+-726.0
+−3.504 × 105
+2.724 × 107
+−8.582 × 108
+∅
+∅
+N2
+T ≤ 35.61
+12.40
+-80.74
+-3926
+6.297 × 104
+−4.633 × 105
+1.325 × 105
+∅
+T > 35.61
+8.514
+-456.4
+−1.987 × 104
+4.800 × 105
+−4.524 × 106
+∅
+∅
+NH3
+all
+15.96
+-3537
+−3.310 × 104
+1.742 × 106
+−2.995 × 107
+∅
+∅
+Ar
+all
+10.69
+-893.2
+-3567
+6.574 × 104
+−4.280 × 105
+∅
+∅
+Kr
+all
+10.77
+-1223
+-8903
+2.635 × 105
+−4.260 × 106
+3.575 × 107
+−1.210 × 108
+Xe
+all
+10.698
+-1737
+−1.332 × 104
+4.349 × 105
+−7.027 × 106
+4.447 × 107
+∅
+Table B.1. Parameters for the dissociation pressure equations of NH3
+monohydrate and clathrates
+Element
+A
+B
+Reference
+CO
+-1685.54
+10.9946
+Hersant et al. (2004)
+CO2
+-2544.395
+11.411518
+Longhi (2005)
+CH4
+-2161.81
+11.1249
+Hersant et al. (2004)
+H2S
+-3111.02
+11.3801
+Hersant et al. (2004)
+N2
+-1677.62
+11.1919
+Hersant et al. (2004)
+NH3
+-2878.28
+8.00205
+Hersant et al. (2004)
+Ar
+-1481.78
+9.95523
+Hersant et al. (2004)
+Kr
+-1987.5
+9.99046
+Hersant et al. (2004)
+Xe
+-2899.19
+11.0354
+Hersant et al. (2004)
+PH3
+-3011.28
+11.95
+Lunine & Stevenson (1985)
+Article number, page 19 of 19
+

UNFKT4oBgHgl3EQflS6S/content/tmp_files/2301.11853v1.pdf.txt

@@ -0,0 +1,1599 @@
+Competition between the Spin and Pseudospin Subsystems 
+in a Model Cuprate
+Yu. D. Panova, *, V. A. Ulitkoa, K. S. Budrina, D. N. Yasinskayaa, and A. A. Chikova
+a Ural Federal University Named after the First President of the Russian Federation B.N. Yeltsin,
+Yekaterinburg, 620002 Russia
+*e-mail: [email protected]
+Abstract—The competition between the magnetic and charge orderings in a model cuprate is considered in
+terms of a simplified static 2D spin–pseudospin model. This model is equivalent to the 2D dilute antiferro-
+magnetic (AFM) Ising model with charged impurities. The mean-field approximation results are presented
+for the system under study and briefly compared to the classical Monte Carlo (MC) calculations. The numer-
+ical simulation shows that the cases of the strong exchange and the strong charge correlation differ qualita-
+tively. In the case of a strong exchange, the AMF phase is instable with respect to the phase separation (PS)
+into the pseudospin (charge) and magnetic (spin) subsystems that behave as immiscible quantum liquids. The
+analytical expression has been obtained for the PS temperature.
+1. INTRODUCTION
+A topical problem of the physics of superconduct-
+ing cuprates is the coexistence and the competition of
+the spin, superconducting, and charge orderings. The
+correlation between the magnetism and the supercon-
+ductivity in cuprates has been studied for a long time
+[1, 2]. For the last fifteen years, many experimental
+results indicating the existence of the charge ordering
+[3–8] and the mutual influence of the spin and charge
+orderings in cuprates [9–14] were obtained. The
+model that relates the unique properties of cuprates to
+their instability to the charge transfer of the CuO4 cen-
+ter states in the CuO2 planes was proposed in [15]. This
+model makes it possible to consider, for CuO4 centers
+in the CuO2 plane, three many-electron valence states
+CuO
+ (that formally correspond to the states of
+Cu1+, 2+, 3+ cooper ions) as components of the pseudo-
+spin triplet S = 1 with MS = –1, 0, +1, respectively,
+and enables us to use the pseudospin formalism for
+pseudospin S = 1 [15, 16]. To consider the competi-
+tion between the spin and charge orderings in cupra-
+tes, the simplified static 2D spin–pseudospin model
+that is a limiting case of the general pseudospin model
+was proposed in [17–19].
+Hamiltonian of a static spin–pseudospin model is
+(1)
+where Szi is the z component of pseudospin S = 1 on a
+site, and σzi = P0iszi/s is the normalized z component of
+spin s = 1/2 multiplied by the projection operator P0i =
+1 – 
+. Parameters Δ = U/2 and V > 0 determine the
+on-site and the inter-site density-density interactions,
+respectively; J = 
+ > 0 is the Ising exchange inter-
+action between Cu2+ ions, h = 
+ is external magnetic
+field, μ is the chemical potential necessary for the
+inclusion of the condition of the doping charge con-
+stancy, and nN = 
+ = const, where n is the dop-
+ing charge density. The summation is carried out over
+the 2D square lattice and ij implies the nearest
+neighbors. This spin–pseudospin model generalizes
+the 2D dilute antiferromagnetic (AFM) Ising model
+with charged impurities. In the limit Δ → –∞, it
+reduces to the Ising model for spin S = 1/2 with a fixed
+magnetization. At Δ > 0, the results can be compared
+to the Blume–Capel [20, 21] or to the Blume–
+Emery–Griffiths [22] model. The Ising model with
+mobile charged impurities was also considered in [23].
+It is apparent that the most important restriction of
+our model for its comparison with real cuprates is the
+absence of charge transfer in the Hamiltonian.
+The phase diagrams of the ground state were con-
+sidered in the mean-field approximation in [17, 18]. It
+was shown that in all five phases of the ground state are
+realized in two limits. In the weak exchange limit at
+< V, all the ground state phases (COI, COII, COIII,
+FIM) correspond to the charge ordering (CO) of a
+− − −
+7 ,6 ,5
+4
+= Δ
++
++
+σ σ
+−
+σ
+− μ
+
+
+
+
+
+�
+�
+*
+2
+,
+zi
+zi
+zj
+zi
+zj
+i
+ij
+ij
+zi
+zi
+i
+i
+S
+V
+S S
+J
+h
+S
+2
+zi
+S
+�
+2
+/
+J s
+�/
+h s
+
+zi
+S
+�J
+1
+
+ 
+ 
+ 
+ 
+ 
+ 
+checker-board type at average charge density n.
+Whereas there are no spin centers (Cu2+) in phase
+COI, phases COII and COIII are diluted with nonin-
+teracting spin centers distributed only in one sublat-
+tice. Such a ferrimagnetic spin ordering is a result of
+the mean-field approximation; because of this, the
+calculations by the classical Monte Carlo (MC)
+method in these cases show a paramagnetic response
+at low temperatures. The FIM phase is also formally
+ferrimagnetic. In this case, the spin AFM ordering is
+diluted with noninteracting charge centers (Cu1+, 3+)
+distributed only in one sublattice. In the limit of strong
+exchange, at  > V, we observe only COI and AFM
+phases in which charge centers are homogeneously
+distributed in both sublattices.
+This report is organized as follows. We present the
+results of the calculation of the thermodynamic prop-
+erties of the system under study in the mean-field
+approximation and concisely compare them with the
+calculations by the classical MC method in Section 2.
+The MC calculations show that, in the limit of the
+strong exchange, the AFM phase is instable with
+respect to the phase separation (PS) into the subsys-
+tems of charge and spin centers. In Section 3, we ana-
+lyze the thermodynamic properties of the PS state in a
+framework of coexistence of two homogeneous
+phases. Section 4 presents the conclusions.
+2. MEAN-FIELD APPROXIMATION
+In this section, we briefly present the results of the
+calculations of the thermodynamic properties in a
+mean-field (MF) approximation. We use the Bogo-
+lyubov inequality for the grand potential Ω(
+):
+Ω(
+) ≤ Ω = Ω(
+) + 
+ – 
+. In the standard way,
+we introduce two sublattices A and B on a square lat-
+tice and choose
+(2)
+where β = 1/T, δ = βΔ, βα and γα are the molecular
+fields, α = A, B. We obtain the expression for estima-
+tion of ω = Ω/N
+(3)
+where ξ = βμ, ν = βV, j = 
+, η = 
+, z = 4 is the
+number of the nearest neighbors, and the average
+(pseudo)magnetizations for sublattices 
+ = Sα and
+ = σα have the form
+(4)
+�J
+*
+*
+* 0
+*
+* 0
+α
+α
+α
+α
+α
+α
+α
+α
+β
+= δ
+−
+β
+−
+γ σ
+
+
+
+*
+2
+0
+,
+,
+,
+zi
+zi
+zi
+i
+i
+i
+S
+S
+α
+α
+α
+α
+α
+−δ
+α
+α
+βω =
+β − ξ
++ γ − η σ
+−
+β +
+γ
++ ν
++
+σ σ
+
+2
+[(
+)
+(
+)
+ln 2(
+cosh
+cosh
+)]
+,
+A
+B
+A
+B
+S
+e
+z S S
+zi
+β �J
+β �h
+α
+zi
+S
+α
+σz
+α
+α
+δ
+α
+α
+α
+α
+−δ
+α
+α
+β
+=
+β +
+γ
+γ
+σ
+=
+β +
+γ
+sinh
+,
+cosh
+cosh
+sinh
+.
+cosh
+cosh
+S
+e
+e
+Minimizing ω with respect to βα and γα, we obtain the
+system of the MF equations
+(5)
+where 
+ = B and 
+ = A.
+Equations (5) must be complemented by charge
+restriction SA + SB = 2n. To explicitly include this con-
+dition, we can introduce the charge order parameter
+a = (SA – SB)/2 and to write the free energy f = ω + μn
+as a function of n, a, and σα using the inverse relation-
+ships for Eqs. (4):
+(6)
+where
+For the nonordered (NO) high-temperature solu-
+tion at h = 0, we have a = 0 and σα = 0, and the free
+energy per one site takes the form
+(7)
+where g0 = ((1 – n2)e–2δ + n2)1/2. This enables us to cal-
+culate all the thermodynamic functions of the NO
+phase. The entropy, the internal energy, and the spe-
+cific heat per one site are
+(8)
+(9)
+(10)
+Equations (7)–(10) correspond to the thermodynamic
+characteristics of the ideal system of noninteracting
+pseudospin (charge) and spin doublets separated in
+energy by the value Δ up to the temperature indepen-
+dent term 
+. At Δ = 0, the entropy and the internal
+α
+α
+α
+α
+β − ξ = − ν
+γ − η = −
+σ
+,
+,
+z S
+z j
+A
+B
+α
+α
+δ
+− δ
+β
+α
+α
+α
+α
+α
+−δ
+δ
+γ
+α
+α
+α
+α
+α
++
+− σ
+=
+−
+− σ
+σ
++
+−
+=
+− σ
+−
+2
+2
+2
+2
+2
+2
+2
+2 2
+2
+2
+2
+(
+)
+,
+(1
+)
+(
+)
+,
+(1
+)
+S e
+G
+e
+e
+S
+e
+G
+S e
+e
+S
+δ
+− δ
+α
+α
+α
+α
+α
+=
+−
+− σ +
++ σ
+2
+2
+2 2
+2
+2
+1/2
+(1
+)
+.
+G
+S
+S e
+e
++
+
+
+=
++ Δ
+−
+
+
+β
+
+
+−
+
+
++
++
+
+
+β
+−
+
+
+2
+0
+NO
+2
+0
+1
+1 ln 2
+2
+1
+ln
+,
+1
+g
+z
+f
+Vn
+n
+n
+n
+n
+g
+n
+−
+−
++
+
+
+= δ
++
+
+
++
+
+
+−
+
+
++
+−
+
+
+−
+
+
+0
+0
+NO
+2
+0
+0
+(1
+)(
+)
+1
+ln 2
+1
+1
+ln
+,
+1
+n
+g
+n
+g
+s
+g
+n
+n
+g
+n
+n
++
+=
++ Δ
++
+2
+2
+0
+NO
+0
+,
+2
+1
+n
+g
+z
+e
+Vn
+g
+− δ
+−
+= δ
++
+2 2
+2
+2
+NO
+2
+0
+0
+(1
+)
+.
+(1
+)
+n
+e
+c
+g
+g
+2
+2
+zVn
+2
+
+energy become constant; thus, the specific heat is a
+zero. If Δ ≠ 0, the specific heat has a maximum at
+T ∝ |Δ|. In particular, if n = 0, then
+(11)
+and the maximum is in point T = |Δ|/(2x), where x is
+the root of equation x = cothx.
+We can also write the explicit form of the magnetic
+susceptibility at h = 0 in the NO phase. Assuming that
+SA = SB = n and σA = σB = σ at h ≠ 0, we eliminate ξ
+from system (5) and obtain equation
+(12)
+where we introduced the following notations:
+(13)
+(14)
+After the standard calculations, we obtain
+(15)
+where χ0(n) is the normalized susceptibility in a zero
+external field for the ideal system of noninteracting
+pseudospin and spin doublets. System of equations (5)
+has the ferrimagnetic solutions with σA + σB ≠ 0 at
+h = 0 [17] that are a result of the MF approximation
+and do not appear in MC calculations. Due to the
+short-range character of the exchange interaction in
+our model, these solutions can appear upon a numer-
+ical simulation as a mixture of the antiferromagnetic
+and paramagnetic phases. The underestimating of the
+paramagnetic response is the systematic error of the
+MF method in these cases. In what follows, we will
+consider only the AFM types of solutions with σA = –
+σB = σ at h = 0. In this case, according to Eqs. (5),
+relationship γA = –γB is fulfilled. The magnetizations
+of sublattices σα are monotonic functions of molecular
+fields γα, according to Eqs. (4), therefore, only the case
+βA = ±βB is possible for σ ≠ 0. This implies that, at
+n ≠ 0, there are only AFM solutions with a = 0, σ ≠ 0
+and CO solutions with a ≠ 0 and σ = 0. The case n = 0
+should be considered individually, since it gives a pos-
+sibility for frustrated states as the CO- and AFM-types
+solutions become degenerate.
+The thermodynamic properties of the AFM and
+CO phases suggest knowledge of the roots of Eqs. (5)
+and can be calculated numerically. In addition, the
+( )
+−
+δ
+δ
+=
+2
+2
+NO
+cosh
+,
+2
+2
+c
+σ = ψ η −
+σ
+(
+, ),
+xj
+n
+−
+ψ
+=
++
+2
+(1
+)sinh
+( , )
+,
+cosh
+( , )
+n
+x
+x n
+x
+g x n
+− δ
+=
+−
++
+=
+2
+2
+2
+2
+1/2
+0
+( , )
+((1
+)
+cosh
+)
+,
+(0, )
+.
+g x n
+n e
+n
+x
+g
+n
+g
+=
+η=
+χ
+∂σ
+χ
+= β
+=
+∂η
++
+χ
+−
+χ
+= β +
+�
+2
+2
+0
+NO
+0
+0
+0
+2
+0
+0
+( )
+,
+1
+( )
+1
+( )
+,
+1
+h
+n
+s
+s
+zJ
+n
+n
+n
+g
+equations for the second-order phase transition tem-
+peratures and the critical points can be found analyti-
+cally.
+For the AFM phase, we use condition 
+ = 0
+at σ = 0, which gives
+(16)
+With accounting for Eqs. (6), we obtain the equation
+for the NO–AFM transition temperature
+(17)
+In particular, we obtain for Δ → +∞ that
+(18)
+which coincides with the results of [22]. Substituting
+Eq. (17) into Eq. (15), we find the susceptibility in the
+transition point χNO = s2/(
+).
+To find the critical point that separates the first and
+second order transitions, we use equation 
+ =
+0 in the coexistence curve. After some manipulations,
+we obtain
+(19)
+Taking into account Eq. (17), we obtain the critical
+point position
+(20)
+In particular, at n = 0, Tc1 = 
+, Δc1/Tc1 = –ln2,
+which agrees with the results of [22].
+The susceptibility in a zero field in the AFM phase
+has the form
+(21)
+where
+(22)
+The order parameter of the AFM phase σ at h = 0 can
+be found from equation
+(23)
+∂
+∂σ
+2
+2
+/
+f
+α
+α
+α σ =
+∂γ
+=
+∂σ
+0
+.
+zj
+−
+=
++
+2
+0
+(1
+)
+1
+.
+n zj
+g
+=
+−
+�
+AFM
+(1
+)
+,
+T
+n zJ
+�
+2zJ
+∂
+∂σ
+4
+4
+/
+f
+−
+−
+=
+2
+2
+0
+0
+2
+3
+0.
+g
+g
+n
+−
+=
++
++
+Δ
+−
+=
++
++
++
+�
+2
+1
+2
+2
+1
+2
+2
+1
+1
+,
+2
+1
+3
+1
+1
+ln
+.
+2
+2(1
+1
+3
+)
+c
+c
+c
+n
+T
+zJ
+n
+n
+T
+n
+n
+�/3
+zJ
+=
+βψ
+σ
+χ
+=
++
+ψ
+σ
+2
+AFM
+0
+'(
+, )
+,
+1
+'(
+, )
+h
+z j
+n
+s
+z j
+z j
+n
+−
++
+ψ
+=
++
+2
+2
+0
+2
+(1
+)( ( , )
+cosh )
+'( , )
+.
+( , )(cosh
+( , ))
+n
+g x n
+g
+x
+x n
+g x n
+x
+g x n
+σ = ψ
+σ
+(
+, ).
+zj
+n
+3
+
+ 
+ 
+ 
+ 
+ 
+ 
+By analogy, for the CO phase, condition 
+ =
+0 at a = 0 gives
+(24)
+and we obtain equation for the NO–CO transition
+temperature
+(25)
+In particular, for Δ → –∞, we obtain
+(26)
+In the CO phase, the equation for the critical point is
+more complex
+(27)
+but, at n = 0, this gives Tc2 = zV/3, Δc2/Tc2 = ln2.
+In a zero field the susceptibility in the CO phase is
+(28)
+where the CO-phase order parameter obeys equation
+(29)
+∂
+∂
+2
+2
+/
+f
+a
+α
+σ =
+∂γ
+∂γ
+
+
+−
+= ν
+
+
+
+
+∂
+∂
+0
+1
+,
+2
+A
+B
+z
+a
+a
+−
+−
+ν =
++
+2
+1
+0
+(1
+)
+1
+.
+n z
+g
+=
+−
+2
+CO
+(1
+)
+.
+T
+n zV
++
+−
+−
++
+=
+2
+3
+2
+2
+2
+0
+0
+0
+2(1
+3
+)
+6
+3
+0,
+n g
+g
+n g
+n
+=
+χ
++
++ χ
+−
+χ
+=
++
+χ
++
++ χ
+−
+�
+0
+0
+2
+CO
+0
+0
+0
+1(
+(
+)
+(
+))
+2
+,
+1
+1
+(
+(
+)
+(
+))
+2
+h
+n
+a
+n
+a
+s
+zJ
+m
+a
+n
+a
+
+
++
++
++
+−
++
+=
+
+
+ν
+−
++
+−
+−
+−
+
+
+(
+(0,
+))(1
+)
+1 ln
+.
+2
+(
+(0,
+))(1
+)
+n
+a
+g
+n
+a
+n
+a
+a
+z
+n
+a
+g
+n
+a
+n
+a
+The general formula for the susceptibility in a zero
+field combining the cases of the NO, AFM, and CO
+phases is given by relationship
+(30)
+where σ is the AFM-phase order parameter, and a is
+the CO-phase order parameter.
+The numerical simulation was carried out using a
+high-efficient algorithm of parallel calculations and
+the classical MC method. Figure 1 shows the results of
+the MC calculations. The peak position in the tem-
+perature dependence of the specific heat approxi-
+mately (because of a finite size of the system) corre-
+sponds to the disorder–order transition temperature.
+These values are indicated by points. The transition
+temperatures obtained in the MF approximation are
+shown by solid lines. Figure 1 clearly demonstrates the
+typical (by a factor of slightly lower than two times)
+overestimation of the critical temperature value in the
+MF approximation. Figure 3 shows the dependence of
+the AFM-transition critical temperature on n in the
+MF approximation. Its shape qualitatively agrees with
+the results obtained in the Bethe approximation
+in [23].
+Figure 2 presents the comparison of the results for
+the susceptibility and the specific heat obtained by the
+MF approximation and by MC method. The analyti-
+cal MF-dependences show a qualitative agreement to
+the results of the numerical simulation and, in some
+cases, even a quantitative coincidence of the results for
+the high-temperature region. The main discrepancies
+are due to the difference in the critical temperature
+β ψ
+σ
++
++ ψ
+σ
+−
+χ =
++
+ψ
+σ
++
++ψ
+σ
+−
+2
+1 (
+'(
+,
+)
+'(
+,
+))
+2
+,
+1
+1
+(
+'(
+,
+)(
+'(
+,
+))
+2
+z j
+n
+a
+z j
+n
+a
+s
+z j
+z j
+n
+a
+z j
+n
+a
+Fig. 1. (a) Left panel: the case of weak exchange: n = 0.1,  = 0.25, and V = 1; (b) the case of the strong exchange: n = 0.1,  =
+0.25, and V = 0.1. Points correspond to the MC critical temperatures. The values of the MF critical temperature given by
+Eqs. (17), (25), and (34) are indicated in lines 1–3.
+2
+1
+NO
+CO
+AFM
+�1
+0
+1
+2
+�/J
+�/J
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+T/J
+T/J
+�2
+�1
+0
+1
+2
+CO
+2
+1
+NO
+AFM
+3
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+AFM + charge droplets
+(a)
+(b)
+�J
+�J
+4
+
+and systematic inaccuracies in the MF approximation
+in the case of description of the critical fluctuations
+and the paramagnetic response at low temperatures.
+3. THE PHASE SEPARATION 
+CRITICAL TEMPERATURE
+According to the results of the numerical simula-
+tion by the MC method, the temperature dependences
+of the specific heat demonstrate in the limit of strong
+exchange at positive Δ two successive phase transi-
+tions. The immediate study of the system state shows
+that the first transition is the AFM ordering. With low-
+ering temperature in the spin subsystem that is an
+AFM matrix diluted by randomly distributed charged
+impurities, the impurities are condensed into charge
+droplets. It means that in the limit of the strong
+exchange, the diluted AFM-phase is instable with
+respect to the macroscopic (PS) into the pseudospin
+(charge) and magnetic (spin) subsystems. At this
+stage, the AFM matrix forces out charged nonmag-
+netic impurities to minimize the surface energy. Note
+that, in the limit of weak exchange, the charged impu-
+rities are randomly distributed over the AFM matrix
+up to T = 0 and also the charged impurities are ran-
+domly distributed in the CO phase, since the energies
+of all possible distributions of additional charges over
+the CO matrix are the same in the approximation of
+interaction of only the nearest neighbors. The results
+of our numerical simulation are similar to the results
+obtained for binary alloys in [24, 25].
+To describe the thermodynamic properties of the
+heterogeneous state, we use the model developed in
+[26–28] for a macroscopic PS in electron systems.
+This model is based on the Maxwell construction.
+Assuming that there are two coexisting macroscopic
+homogeneous phases 1 and 2, we write the free energy
+of the PS state per one site as
+(31)
+where m is the system fraction with density n1, 1 – m is
+the system fraction with density n2, so that mn1 + (1 –
+m)n2 = n. In our case, one phase consists of charged
+centers (C), and another phase is the spin AFM phase
+without impurities; therefore, n1 = sgnn, n2 = 0, and
+m = |n|. The transition point is determined by equation
+(32)
+=
++
+−
+PS
+1
+1
+2
+2
+( )
+(1
+) (
+),
+f
+mf n
+m f n
++
+−
+=
+AFM
+AFM
+(1)
+(1
+)
+(0)
+( ).
+C
+n f
+n f
+f
+n
+Fig. 2. Susceptibility and the specific heat obtained (solid lines) in the MF approximation and (points) by MC method: (a) the
+case of weak exchange: n = 0.1,  = 0.25, and V = 1 and (b) the case of the strong exchange: n = 0.1,  = 0.25, and V = 0.1.
+�/J = 0
+�/J = 0
+�/J = 0
+�/J = 1
+�/J = 1
+�/J = 1
+�/J = �1
+�/J = �1
+�/J = 2
+�/J = 0
+�/J = 1
+�/J = 2
+ 
+0
+1
+2
+0.2
+0.1
+0
+�, arb.units
+c, arb.units
+�, arb.units
+c, arb.units
+0
+1
+2
+3
+T/J
+T/J
+0.10
+0.05
+0
+0
+1
+2
+3
+(a)
+(b)
+0
+2
+1
+0
+1
+2
+2
+2
+2
+0
+1
+0
+1
+0
+1
+�J
+�J
+Fig. 3. The bright circles correspond to the susceptibility
+maxima upon the AFM ordering; dark cycles correspond
+to the specific heat maxima upon PS obtained by the MC
+method. The model parameters are Δ = 1,  = 0.25, and
+V = 0.14. Lines 1 and 3 show the critical temperatures
+given by Eqs. (17) and (34).
+NO
+AFM
+1
+3
+n
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+AFM droplets
+AFM + charge droplets
+0
+0.2
+0.4
+0.6
+0.8
+T/J
+�J
+5
+
+ 
+ 
+ 
+ 
+ 
+ 
+The free energy of charge centers fC(1) = 2V + Δ. The
+free energy of the AFM phase can be written as
+(33)
+We assume that Δ > 0 and consider the case of low
+temperatures, thus, δ 
+ 1 and j 
+ 1. In this approxi-
+mation, with the inclusion of Eq. (23), we obtain |σ| =
+1 – |n|. As a result, Eq. (32) gives the following rela-
+tionship for PS temperature
+(34)
+This expression does not depend on Δ, which agrees
+with the MC results for TPS in Fig. 1.
+Figure 3 shows the concentration dependences of
+the critical temperatures for the AFM transition and
+the PS. The circles denote the MC results for the sus-
+ceptibility maxima upon the AFM ordering, and the
+points show the specific heat maxima upon the PS.
+Solid line 3 shows the PS temperature determined by
+Eq. (34). This temperature agrees unexpectedly well
+with the MC results, while the dependence in the MF
+approximation (line 2) for the AFM-ordering tem-
+perature determined by Eq. (17) becomes qualitatively
+wrong at |n| > 0.5. Note as well that Eq. (34) for the PS
+temperature based on the Maxwell construction gives
+lower values as compared to the values found in [22].
+4. CONCLUSIONS
+We considered the static 2D spin–pseudospin
+model on a square lattice that generalizes the dilute
+antiferromagnetic Ising model. We compared similar
+results in the MF approximation with the results of the
+numerical simulation by the classical MC method.
+The analysis of the specific heat and the susceptibility
+obtained by the MC method showed that the MF crit-
+ical temperature of CO and AFM ordering qualita-
+tively reproduce the numerical results, but they sys-
+tematically give higher values. The MC calculations
+show that the cases of the strong and weak exchange
+are qualitatively different. In the case of the weak
+exchange, a frustration appears in the charge-ordered
+ground state of the system. In the limit of the strong
+exchange, the homogeneous AFM phase is instable
+with respect to the PS of the pseudospin and spin sub-
+systems. We obtained the analytical expression for the
+PS temperature and revealed that it agrees well with
+the numerical simulation by the classical MC method.
+=
++
+σ
++
+Δ
+σ +
+σ
+
+
+−
+
+
+β
+−
+
+
+σ +
+σ
+
+
++
+
+
+β
+−
+
+
+�
+2
+2
+AFM
+2
+( )
+(
+)
+2
+cosh(
+)
+(
+, )
+1 ln 2
+1
+cosh(
+)
+(
+, )
+ln
+.
+1
+z
+f
+n
+Vn
+J
+n
+z j
+g z j
+n
+n
+n
+z j
+g z j
+n
+n
+n
+≫
+≫
+−
+−
+=
++
+−
+−
+�
+PS
+(1
+)
+(
+).
+ln
+(1
+)ln(1
+)
+2
+n
+n
+z V
+J
+T
+n
+n
+n
+n
+FUNDING
+This work was supported by Program 211 of the
+Government of the Russian Federation (Agreement
+02.A03.21.0006), the Ministry of Education and Sci-
+ence of the Russian federation (projects nos. 2277 and
+5719), and the Russian Foundation for Basic Research
+(project no. 18-32-00837\18).
+REFERENCES
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+5, 3390 (2014).
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+7
+

UNFKT4oBgHgl3EQflS6S/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf,len=542
+page_content='Competition between the Spin and Pseudospin Subsystems  in a Model Cuprate  Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Panova, *, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Ulitkoa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Budrina, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yasinskayaa, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Chikova  a Ural Federal University Named after the First President of the Russian Federation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yeltsin,  Yekaterinburg, 620002 Russia  e-mail: yuri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='panov@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='ru  Abstract—The competition between the magnetic and charge orderings in a model cuprate is considered in  terms of a simplified static 2D spin–pseudospin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' This model is equivalent to the 2D dilute antiferro-  magnetic (AFM) Ising model with charged impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The mean-field approximation results are presented  for the system under study and briefly compared to the classical Monte Carlo (MC) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The numer-  ical simulation shows that the cases of the strong exchange and the strong charge correlation differ qualita-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In the case of a strong exchange, the AMF phase is instable with respect to the phase separation (PS)  into the pseudospin (charge) and magnetic (spin) subsystems that behave as immiscible quantum liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The  analytical expression has been obtained for the PS temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' INTRODUCTION  A topical problem of the physics of superconduct-  ing cuprates is the coexistence and the competition of  the spin, superconducting, and charge orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The  correlation between the magnetism and the supercon-  ductivity in cuprates has been studied for a long time  [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' For the last fifteen years, many experimental  results indicating the existence of the charge ordering  [3–8] and the mutual influence of the spin and charge  orderings in cuprates [9–14] were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The  model that relates the unique properties of cuprates to  their instability to the charge transfer of the CuO4 cen-  ter states in the CuO2 planes was proposed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' This  model makes it possible to consider, for CuO4 centers  in the CuO2 plane, three many-electron valence states  CuO  (that formally correspond to the states of  Cu1+, 2+, 3+ cooper ions) as components of the pseudo-  spin triplet S = 1 with MS = –1, 0, +1, respectively,  and enables us to use the pseudospin formalism for  pseudospin S = 1 [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' To consider the competi-  tion between the spin and charge orderings in cupra-  tes, the simplified static 2D spin–pseudospin model  that is a limiting case of the general pseudospin model  was proposed in [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Hamiltonian of a static spin–pseudospin model is  (1)  where Szi is the z component of pseudospin S = 1 on a  site, and σzi = P0iszi/s is the normalized z component of  spin s = 1/2 multiplied by the projection operator P0i =  1 –  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Parameters Δ = U/2 and V > 0 determine the  on-site and the inter-site density-density interactions,  respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' J =  > 0 is the Ising exchange inter-  action between Cu2+ ions, h =  is external magnetic  field, μ is the chemical potential necessary for the  inclusion of the condition of the doping charge con-  stancy, and nN =  = const, where n is the dop-  ing charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The summation is carried out over  the 2D square lattice and \uf0e1ij\uf0f1 implies the nearest  neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' This spin–pseudospin model generalizes  the 2D dilute antiferromagnetic (AFM) Ising model  with charged impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In the limit Δ → –∞, it  reduces to the Ising model for spin S = 1/2 with a fixed  magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' At Δ > 0, the results can be compared  to the Blume–Capel [20, 21] or to the Blume–  Emery–Griffiths [22] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The Ising model with  mobile charged impurities was also considered in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  It is apparent that the most important restriction of  our model for its comparison with real cuprates is the  absence of charge transfer in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  The phase diagrams of the ground state were con-  sidered in the mean-field approximation in [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' It  was shown that in all five phases of the ground state are  realized in two limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In the weak exchange limit at  < V, all the ground state phases (COI, COII, COIII,  FIM) correspond to the charge ordering (CO) of a  − − −  7 ,6 ,5  4  = Δ  +  +  σ σ  −  σ  − μ  \uf0e5  \uf0e5  \uf0e5  \uf0e5  \uf0e5  �  � 2  ,  zi  zi  zj  zi  zj  i  ij  ij  zi  zi  i  i  S  V  S S  J  h  S  2  zi  S  �  2  /  J s  �/  h s  \uf0e5  zi  S  �J  1  checker-board type at average charge density n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Whereas there are no spin centers (Cu2+) in phase  COI, phases COII and COIII are diluted with nonin-  teracting spin centers distributed only in one sublat-  tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Such a ferrimagnetic spin ordering is a result of  the mean-field approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' because of this, the  calculations by the classical Monte Carlo (MC)  method in these cases show a paramagnetic response  at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The FIM phase is also formally  ferrimagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In this case, the spin AFM ordering is  diluted with noninteracting charge centers (Cu1+, 3+)  distributed only in one sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In the limit of strong  exchange, at  > V, we observe only COI and AFM  phases in which charge centers are homogeneously  distributed in both sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  This report is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' We present the  results of the calculation of the thermodynamic prop-  erties of the system under study in the mean-field  approximation and concisely compare them with the  calculations by the classical MC method in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  The MC calculations show that, in the limit of the  strong exchange, the AFM phase is instable with  respect to the phase separation (PS) into the subsys-  tems of charge and spin centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In Section 3, we ana-  lyze the thermodynamic properties of the PS state in a  framework of coexistence of two homogeneous  phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Section 4 presents the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' MEAN-FIELD APPROXIMATION  In this section, we briefly present the results of the  calculations of the thermodynamic properties in a  mean-field (MF) approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' We use the Bogo-  lyubov inequality for the grand potential Ω(  ):  Ω(  ) ≤ Ω = Ω(  ) + \uf0e1  –  \uf0f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In the standard way,  we introduce two sublattices A and B on a square lat-  tice and choose  (2)  where β = 1/T, δ = βΔ, βα and γα are the molecular  fields, α = A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' We obtain the expression for estima-  tion of ω = Ω/N  (3)  where ξ = βμ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' ν = βV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' j =  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' η =  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' z = 4 is the  number of the nearest neighbors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' and the average  (pseudo)magnetizations for sublattices  = Sα and  = σα have the form  (4)  �J 0 0  α  α  α  α  α  α  α  α  β  = δ  −  β  −  γ σ  \uf0e5  \uf0e5  \uf0e5 2  0  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  zi  zi  zi  i  i  i  S  S  α  α  α  α  α  −δ  α  α  βω =  β − ξ  + γ − η σ  −  β +  γ  + ν  +  σ σ  \uf0e5  2  [(  )  (  )  ln 2(  cosh  cosh  )]  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  A  B  A  B  S  e  z S S  zi  β �J  β �h  α  zi  S  α  σz  α  α  δ  α  α  α  α  −δ  α  α  β  =  β +  γ  γ  σ  =  β +  γ  sinh  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  cosh  cosh  sinh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  cosh  cosh  S  e  e  Minimizing ω with respect to βα and γα, we obtain the  system of the MF equations  (5)  where  = B and  = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Equations (5) must be complemented by charge  restriction SA + SB = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' To explicitly include this con-  dition, we can introduce the charge order parameter  a = (SA – SB)/2 and to write the free energy f = ω + μn  as a function of n, a, and σα using the inverse relation-  ships for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (4):  (6)  where  For the nonordered (NO) high-temperature solu-  tion at h = 0, we have a = 0 and σα = 0, and the free  energy per one site takes the form  (7)  where g0 = ((1 – n2)e–2δ + n2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' This enables us to cal-  culate all the thermodynamic functions of the NO  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The entropy, the internal energy, and the spe-  cific heat per one site are  (8)  (9)  (10)  Equations (7)–(10) correspond to the thermodynamic  characteristics of the ideal system of noninteracting  pseudospin (charge) and spin doublets separated in  energy by the value Δ up to the temperature indepen-  dent term  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' At Δ = 0, the entropy and the internal  α  α  α  α  β − ξ = − ν  γ − η = −  σ  ,  ,  z S  z j  A  B  α  α  δ  − δ  β  α  α  α  α  α  −δ  δ  γ  α  α  α  α  α  +  − σ  =  −  − σ  σ  +  −  =  − σ  −  2  2  2  2  2  2  2  2 2  2  2  2  (  )  ,  (1  )  (  )  ,  (1  )  S e  G  e  e  S  e  G  S e  e  S  δ  − δ  α  α  α  α  α  =  −  − σ +  + σ  2  2  2 2  2  2  1/2  (1  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  G  S  S e  e  +  \uf0e6  \uf0f6  =  + Δ  −  \uf0e7  \uf0f7  β  \uf0e8  \uf0f8  −  \uf0e6  \uf0f6  +  +  \uf0e7  \uf0f7  β  −  \uf0e8  \uf0f8  2  0  NO  2  0  1  1 ln 2  2  1  ln  ,  1  g  z  f  Vn  n  n  n  n  g  n  −  −  +  \uf0e6  \uf0f6  = δ  +  \uf0e7  \uf0f7  +  \uf0e8  \uf0f8  −  \uf0e6  \uf0f6  +  −  \uf0e7  \uf0f7  −  \uf0e8  \uf0f8  0  0  NO  2  0  0  (1  )(  )  1  ln 2  1  1  ln  ,  1  n  g  n  g  s  g  n  n  g  n  n  +  =  + Δ  +  2  2  0  NO  0  ,  2  1  n  g  z  e  Vn  g  − δ  −  = δ  +  2 2  2  2  NO  2  0  0  (1  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  (1  )  n  e  c  g  g  2  2  zVn  2  energy become constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' thus, the specific heat is a  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' If Δ ≠ 0, the specific heat has a maximum at  T ∝ |Δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In particular, if n = 0, then  (11)  and the maximum is in point T = |Δ|/(2x), where x is  the root of equation x = cothx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  We can also write the explicit form of the magnetic  susceptibility at h = 0 in the NO phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Assuming that  SA = SB = n and σA = σB = σ at h ≠ 0, we eliminate ξ  from system (5) and obtain equation  (12)  where we introduced the following notations:  (13)  (14)  After the standard calculations, we obtain  (15)  where χ0(n) is the normalized susceptibility in a zero  external field for the ideal system of noninteracting  pseudospin and spin doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' System of equations (5)  has the ferrimagnetic solutions with σA + σB ≠ 0 at  h = 0 [17] that are a result of the MF approximation  and do not appear in MC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Due to the  short-range character of the exchange interaction in  our model, these solutions can appear upon a numer-  ical simulation as a mixture of the antiferromagnetic  and paramagnetic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The underestimating of the  paramagnetic response is the systematic error of the  MF method in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In what follows, we will  consider only the AFM types of solutions with σA = –  σB = σ at h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In this case, according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (5),  relationship γA = –γB is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The magnetizations  of sublattices σα are monotonic functions of molecular  fields γα, according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (4), therefore, only the case  βA = ±βB is possible for σ ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' This implies that, at  n ≠ 0, there are only AFM solutions with a = 0, σ ≠ 0  and CO solutions with a ≠ 0 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The case n = 0  should be considered individually, since it gives a pos-  sibility for frustrated states as the CO- and AFM-types  solutions become degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  The thermodynamic properties of the AFM and  CO phases suggest knowledge of the roots of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (5)  and can be calculated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In addition, the  ( )  −  δ  δ  =  2  2  NO  cosh  ,  2  2  c  σ = ψ η −  σ  (  , ),  xj  n  −  ψ  =  +  2  (1  )sinh  ( , )  ,  cosh  ( , )  n  x  x n  x  g x n  − δ  =  −  +  =  2  2  2  2  1/2  0  ( , )  ((1  )  cosh  )  ,  (0, )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  g x n  n e  n  x  g  n  g  =  η=  χ  ∂σ  χ  = β  =  ∂η  +  χ  −  χ  = β +  �  2  2  0  NO  0  0  0  2  0  0  ( )  ,  1  ( )  1  ( )  ,  1  h  n  s  s  zJ  n  n  n  g  equations for the second-order phase transition tem-  peratures and the critical points can be found analyti-  cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  For the AFM phase, we use condition  = 0  at σ = 0, which gives  (16)  With accounting for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (6), we obtain the equation  for the NO–AFM transition temperature  (17)  In particular, we obtain for Δ → +∞ that  (18)  which coincides with the results of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Substituting  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (17) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (15), we find the susceptibility in the  transition point χNO = s2/(  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  To find the critical point that separates the first and  second order transitions, we use equation  =  0 in the coexistence curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' After some manipulations,  we obtain  (19)  Taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (17), we obtain the critical  point position  (20)  In particular, at n = 0, Tc1 =  , Δc1/Tc1 = –ln2,  which agrees with the results of [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  The susceptibility in a zero field in the AFM phase  has the form  (21)  where  (22)  The order parameter of the AFM phase σ at h = 0 can  be found from equation  (23)  ∂  ∂σ  2  2  /  f  α  α  α σ =  ∂γ  =  ∂σ  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  zj  −  =  +  2  0  (1  )  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  n zj  g  =  −  �  AFM  (1  )  ,  T  n zJ  �  2zJ  ∂  ∂σ  4  4  /  f  −  −  =  2  2  0  0  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  g  g  n  −  =  +  +  Δ  −  =  +  +  +  �  2  1  2  2  1  2  2  1  1  ,  2  1  3  1  1  ln  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content="  2  2(1  1  3  )  c  c  c  n  T  zJ  n  n  T  n  n  �/3  zJ  =  βψ  σ  χ  =  +  ψ  σ  2  AFM  0  '(  , )  ,  1  '(  , )  h  z j  n  s  z j  z j  n  −  +  ψ  =  +  2  2  0  2  (1  )( ( , )  cosh )  '( , )  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  ( , )(cosh  ( , ))  n  g x n  g  x  x n  g x n  x  g x n  σ = ψ  σ  (  , ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  zj  n  3  By analogy, for the CO phase, condition  =  0 at a = 0 gives  (24)  and we obtain equation for the NO–CO transition  temperature  (25)  In particular, for Δ → –∞, we obtain  (26)  In the CO phase, the equation for the critical point is  more complex  (27)  but, at n = 0, this gives Tc2 = zV/3, Δc2/Tc2 = ln2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  In a zero field the susceptibility in the CO phase is  (28)  where the CO-phase order parameter obeys equation  (29)  ∂  ∂  2  2  /  f  a  α  σ =  ∂γ  ∂γ  \uf0e6  \uf0f6  −  = ν  \uf0e7  \uf0f7  \uf0e8  \uf0f8  ∂  ∂  0  1  ,  2  A  B  z  a  a  −  −  ν =  +  2  1  0  (1  )  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  n z  g  =  −  2  CO  (1  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  T  n zV  +  −  −  +  =  2  3  2  2  2  0  0  0  2(1  3  )  6  3  0,  n g  g  n g  n  =  χ  +  + χ  −  χ  =  +  χ  +  + χ  −  �  0  0  2  CO  0  0  0  1(  (  )  (  ))  2  ,  1  1  (  (  )  (  ))  2  h  n  a  n  a  s  zJ  m  a  n  a  \uf0e6  \uf0f6  +  +  +  −  +  =  \uf0e7  \uf0f7  ν  −  +  −  −  −  \uf0e8  \uf0f8  (  (0,  ))(1  )  1 ln  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  2  (  (0,  ))(1  )  n  a  g  n  a  n  a  a  z  n  a  g  n  a  n  a  The general formula for the susceptibility in a zero  field combining the cases of the NO, AFM, and CO  phases is given by relationship  (30)  where σ is the AFM-phase order parameter, and a is  the CO-phase order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  The numerical simulation was carried out using a  high-efficient algorithm of parallel calculations and  the classical MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Figure 1 shows the results of  the MC calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The peak position in the tem-  perature dependence of the specific heat approxi-  mately (because of a finite size of the system) corre-  sponds to the disorder–order transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  These values are indicated by points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The transition  temperatures obtained in the MF approximation are  shown by solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Figure 1 clearly demonstrates the  typical (by a factor of slightly lower than two times)  overestimation of the critical temperature value in the  MF approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Figure 3 shows the dependence of  the AFM-transition critical temperature on n in the  MF approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Its shape qualitatively agrees with  the results obtained in the Bethe approximation  in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Figure 2 presents the comparison of the results for  the susceptibility and the specific heat obtained by the  MF approximation and by MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The analyti-  cal MF-dependences show a qualitative agreement to  the results of the numerical simulation and, in some  cases, even a quantitative coincidence of the results for  the high-temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=" The main discrepancies  are due to the difference in the critical temperature  β ψ  σ  +  + ψ  σ  −  χ =  +  ψ  σ  +  +ψ  σ  −  2  1 (  '(  ,  )  '(  ,  ))  2  ,  1  1  (  '(  ,  )(  '(  ,  ))  2  z j  n  a  z j  n  a  s  z j  z j  n  a  z j  n  a  Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (a) Left panel: the case of weak exchange: n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='1,  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='25, and V = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (b) the case of the strong exchange: n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='1,  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='25, and V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Points correspond to the MC critical temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The values of the MF critical temperature given by  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (17), (25), and (34) are indicated in lines 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  2  1  NO  CO  AFM  �1  0  1  2  �/J  �/J  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='5  T/J  T/J  �2  �1  0  1  2  CO  2  1  NO  AFM  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='0  AFM + charge droplets  (a)  (b)  �J  �J  4  and systematic inaccuracies in the MF approximation  in the case of description of the critical fluctuations  and the paramagnetic response at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' THE PHASE SEPARATION  CRITICAL TEMPERATURE  According to the results of the numerical simula-  tion by the MC method, the temperature dependences  of the specific heat demonstrate in the limit of strong  exchange at positive Δ two successive phase transi-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The immediate study of the system state shows  that the first transition is the AFM ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' With low-  ering temperature in the spin subsystem that is an  AFM matrix diluted by randomly distributed charged  impurities, the impurities are condensed into charge  droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' It means that in the limit of the strong  exchange, the diluted AFM-phase is instable with  respect to the macroscopic (PS) into the pseudospin  (charge) and magnetic (spin) subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' At this  stage, the AFM matrix forces out charged nonmag-  netic impurities to minimize the surface energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Note  that, in the limit of weak exchange, the charged impu-  rities are randomly distributed over the AFM matrix  up to T = 0 and also the charged impurities are ran-  domly distributed in the CO phase, since the energies  of all possible distributions of additional charges over  the CO matrix are the same in the approximation of  interaction of only the nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The results  of our numerical simulation are similar to the results  obtained for binary alloys in [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  To describe the thermodynamic properties of the  heterogeneous state, we use the model developed in  [26–28] for a macroscopic PS in electron systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  This model is based on the Maxwell construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Assuming that there are two coexisting macroscopic  homogeneous phases 1 and 2, we write the free energy  of the PS state per one site as  (31)  where m is the system fraction with density n1, 1 – m is  the system fraction with density n2, so that mn1 + (1 –  m)n2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In our case, one phase consists of charged  centers (C), and another phase is the spin AFM phase  without impurities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' therefore, n1 = sgnn, n2 = 0, and  m = |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The transition point is determined by equation  (32)  =  +  −  PS  1  1  2  2  ( )  (1  ) (  ),  f  mf n  m f n  +  −  =  AFM  AFM  (1)  (1  )  (0)  ( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  C  n f  n f  f  n  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Susceptibility and the specific heat obtained (solid lines) in the MF approximation and (points) by MC method: (a) the  case of weak exchange: n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='1,  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='25, and V = 1 and (b) the case of the strong exchange: n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='1,  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='25, and V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  �/J = 0  �/J = 0  �/J = 0  �/J = 1  �/J = 1  �/J = 1  �/J = �1  �/J = �1  �/J = 2  �/J = 0  �/J = 1  �/J = 2  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='1  0  �, arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='units  c, arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='units  �, arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='units  c, arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='units  0  1  2  3  T/J  T/J  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='05  0  0  1  2  3  (a)  (b)  0  2  1  0  1  2  2  2  2  0  1  0  1  0  1  �J  �J  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The bright circles correspond to the susceptibility  maxima upon the AFM ordering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' dark cycles correspond  to the specific heat maxima upon PS obtained by the MC  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The model parameters are Δ = 1,  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='25, and  V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Lines 1 and 3 show the critical temperatures  given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (17) and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  NO  AFM  1  3  n  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='0  AFM droplets  AFM + charge droplets  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='8  T/J  �J  5  The free energy of charge centers fC(1) = 2V + Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The  free energy of the AFM phase can be written as  (33)  We assume that Δ > 0 and consider the case of low  temperatures, thus, δ  1 and j  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In this approxi-  mation, with the inclusion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (23), we obtain |σ| =  1 – |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' As a result, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (32) gives the following rela-  tionship for PS temperature  (34)  This expression does not depend on Δ, which agrees  with the MC results for TPS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Figure 3 shows the concentration dependences of  the critical temperatures for the AFM transition and  the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The circles denote the MC results for the sus-  ceptibility maxima upon the AFM ordering, and the  points show the specific heat maxima upon the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Solid line 3 shows the PS temperature determined by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' This temperature agrees unexpectedly well  with the MC results, while the dependence in the MF  approximation (line 2) for the AFM-ordering tem-  perature determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (17) becomes qualitatively  wrong at |n| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Note as well that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' (34) for the PS  temperature based on the Maxwell construction gives  lower values as compared to the values found in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' CONCLUSIONS  We considered the static 2D spin–pseudospin  model on a square lattice that generalizes the dilute  antiferromagnetic Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' We compared similar  results in the MF approximation with the results of the  numerical simulation by the classical MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  The analysis of the specific heat and the susceptibility  obtained by the MC method showed that the MF crit-  ical temperature of CO and AFM ordering qualita-  tively reproduce the numerical results, but they sys-  tematically give higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' The MC calculations  show that the cases of the strong and weak exchange  are qualitatively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In the case of the weak  exchange, a frustration appears in the charge-ordered  ground state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' In the limit of the strong  exchange, the homogeneous AFM phase is instable  with respect to the PS of the pseudospin and spin sub-  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' We obtained the analytical expression for the  PS temperature and revealed that it agrees well with  the numerical simulation by the classical MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  =  +  σ  +  Δ  σ +  σ  \uf0e6  \uf0f6  −  \uf0e7  \uf0f7  β  −  \uf0e8  \uf0f8  σ +  σ  \uf0e6  \uf0f6  +  \uf0e7  \uf0f7  β  −  \uf0e8  \uf0f8  �  2  2  AFM  2  ( )  (  )  2  cosh(  )  (  , )  1 ln 2  1  cosh(  )  (  , )  ln  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  1  z  f  n  Vn  J  n  z j  g z j  n  n  n  z j  g z j  n  n  n  ≫  ≫  −  −  =  +  −  −  �  PS  (1  )  (  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  ln  (1  )ln(1  )  2  n  n  z V  J  T  n  n  n  n  FUNDING  This work was supported by Program 211 of the  Government of the Russian Federation (Agreement  02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='A03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='0006), the Ministry of Education and Sci-  ence of the Russian federation (projects nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 2277 and  5719), and the Russian Foundation for Basic Research  (project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 18-32-00837\\18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content=' Liang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Bonn, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Julien, Nature (London, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Ghiringhelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' le Tacon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Minola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Blanco-  Canosa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Mazzoli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Brookes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' de Luca,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Frano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Hawthorn, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' He, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Loew, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Mo-  retti Sala, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content=' Salluzzo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Schierle, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content=' Blackburn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Holmes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Chris-  tensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content=' Liang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Bonn,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Hardy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Watenphul, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
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+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' da Silva Neto, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Aynajian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Frano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Comin,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Schierle, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Weschke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Gyenis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Wen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Schnee-  loch, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Xu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Ono, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Gu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' le Tacon, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yazdani, Science (Washington, DC, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=') 343, 393  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Comin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Damascelli, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 7, 369 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Laliberté, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Frachet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Benhabib, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Borgnic,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Loew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Porras, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Le Tacon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Keimer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Wied-  mann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Proust, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' LeBoeuf, npj Quantum  Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 3, 11 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Abbamonte, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Rusydi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Smadici, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Gu,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Sawatzky, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Feng, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 1, 155  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Berg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Fradkin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Kivelson, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Tran-  quada, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 11, 115004 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Fujita, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Hiraka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Matsuda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Matsuura,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Tranquada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Wakimoto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Xu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yama-  da, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 81, 011007 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Fradkin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Kivelson, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 8, 864 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Drachuck, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Razzoli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Bazalitski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Kanigel,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Niedermayer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Shi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Keren, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  5, 3390 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Cyr-Choinière, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Grissonnanche, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Badoux,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Day, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Bonn, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Hardy, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Liang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Doi-  ron-Leyraud, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Taillefer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' B 92, 224502  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Moskvin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' B 84, 075116 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Moskvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Matter 25, 085601  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Panov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Chikov, and  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Avvakumov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Supercond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 29, 1077  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  6  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Panov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Chikov, and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Budrin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 187, 646 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Panov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Budrin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Chikov, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Moskvin, JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 106, 440 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Blume, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 141, 517 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Capel, Physica 32, 966 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Blume, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Emery, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Griffiths, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  A 4, 1071 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Semkin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Smagin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Solid State 57,  943 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yaldram, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Khalil, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Sadiq, Solid State  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' 87, 1045 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Khalil, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Yaldram, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Sadiq, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' C 08, 139 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Kapcia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Robaszkiewicz, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Micnas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' :  Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Matter 24, 215601 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Kapcia and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Robaszkiewicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  Matter 25, 065603 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Kapcia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Murawski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Klobus, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' Robasz-  kiewicz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=' A (Amsterdam, Neth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content=') 437, 218 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}
+page_content='  7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFKT4oBgHgl3EQflS6S/content/2301.11853v1.pdf'}

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf,len=423
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='04541v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='CT]  11 Jan 2023  ALMOST MATHEMATICS OF POINTED SYMMETRIC MONOIDAL MODEL  CATEGORIES BY SMITH IDEAL THEORY  YUKI KATO  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' This article is a generalization of a result in Quillen’s note [Qui96] “Module theory over non-  unital rings” giving a one-to-one correspondence between bilocalization of abelian categories of modules  and idempotent ideals of the base ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Faltings [Fal88];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Gabber and Ramero [GR03] established almost  mathematics, the same as Quillen’s bilocalization of a category of modules by nil modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  In this paper, by using the theory of Smith ideals mentioned in Hovey [Hov14], we consider almost  mathematics of symmetric monoidal pointed model categories and prove a weak analogue of the one-to-  one correspondence in Quillen [Qui96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Introduction  Almost mathematics is firstly introduced by Faltings [Fal88], proving almost purity in his article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Gab-  ber and Ramero [GR03] provided a detailed foundation of almost mathematics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' they called it almost ring  theory: almost modules, almost algebras, and almost homotopy algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let V be a unital commutative  ring with an idempotent ideal I of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A V-module M is said to be almost I-zero (or simply, almost zero)  if M is killed by I, and almost mathematics is a theory of algebra working by localizing the category of  V-modules by the Serre subcategory of almost zero modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Independently, Quillen considered linear algebra over non-unital rings, the same as almost mathe-  matics in his unpublished note [Qui96]: Almost mathematics in Quillen [Qui96] is characterized as  bilocalization (both localization and colocalization) of an abelian category of modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' ([Qui96] Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='5) Let V be a commutative ring with a multiplicative unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' There  is a one-to-one correspondence between Serre subcategories S of ModV which the localization F :  ModV → ModV/S is also a colocalization, and idempotent ideals of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  This article aims to prove a symmetric monoidal model categorical analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' To intro-  duce a theory of idempotent ideals for model categories, we use the theory of Smith ideals Hovey [Hov14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  In the theory of algebra, a two-sided ideal of a ring can be defined to be a kernel of some ring homo-  morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' We can interpret that Smith ideals theory is based on the correspondence between two-sided  ideals of rings and kernels of ring homomorphisms (See Hovey [Hov14, Section 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  A category is pointed if it has an initial object ∅ and a final object ∗ such that the unique morphism  ∅ → ∗ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' We call the initial object the zero object of a pointed category, and 0 denotes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Throughout this paper, we consider pointed (model) categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let C be a pointed symmetric monoidal  Date: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Symmetric monoidal model categories, Smith ideals, almost mathematics, Bousfield localization  and colocalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  1  category with a monoidal unit object V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let ∆1 denote the category with two objects 0 and 1, and only  one non-trivial morphism 0 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The functor category Fun(∆1, C) from ∆1 to C is called the arrow  category of C, and Ar(C) denotes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' By Hovey [Hov14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2], the category Ar(C) has two  distinct symmetric monoidal structures derived from C’s: For any two morphisms in C, f : X0 → X1 and  g : Y0 → Y1, one has a commutative square:  X0 ⊗ Y0  f⊗id �  id⊗g  �  X1 ⊗ Y0  id⊗g  �  X0 ⊗ Y1  f⊗id  �X1 ⊗ Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  One of them is the diagonal (or tensor) product monoidal structure is defined by f ⊗g as the composition  f ⊗ g = ( f ⊗ id) ◦ (id ⊗ g) = (id ⊗ g) ◦ ( f ⊗ id) : X0 ⊗ Y0 → X1 ⊗ Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  The other is the push-out product monoidal structure is defined by the induced morphism:  f□g : (X0 ⊗ Y1) ∐X0⊗Y0 (X1 ⊗ Y0) → X1 ⊗ Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  We can define Smith ideals by the push-out product monoidal structure: A Smith ideal j : I → R is  defined to be a commutative monoid object of Ar(C) with respect to the push-out product monoidal  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In the case of a pointed symmetric monoidal category C, the cokernel functor  cok : Ar(C) ∋ ( f : X → Y) �→ (Y → Coker( f)) ∈ Ar(C)  is symmetric monoidal from the pushout product monoidal structure to the diagonal product monoidal  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Further, the right adjoint of cok is the kernel functor  ker : Ar(C) ∋ ( f : X → Y) �→ (Ker( f) → X) ∈ Ar(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (See Hovey [Hov14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In the special case, a Smith ideal j : I → R with an isomorphism  j → (ker ◦ cok)( j) is isomorphic to the kernel of the monoid morphism cok( j) : R → Coker( j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  For any class T of morphisms in a category, �T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' T�) denotes the class of morphisms which have  right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' left) lifting property with respect to all morphisms in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then �(T�) is said to be the weakly  saturated class of morphisms generated by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In this paper, we always assume that all model categories  are cofibrantly generated: there exist small subsets I and J of the set of morphisms of the model category  such that the collection of all cofibrations (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' trivial cofibrations) is the weakly saturated class of  morphisms generated by I (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  In this paper, we define a homotopical analogue of almost mathematics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' idempotent ideals, almost  zero-modules, and almost isomorphisms by using the Smith ideal theory of model categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In Sec-  tion 2, we recall the definition of Smith ideals of model categories and some properties of the homotopy  cokernel and kernel functors to work in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In Section 3, we introduce the theory of almost  mathematics of pointed symmetric monoidal model categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' homotopically idempotent Smith ideals,  homotopically almost zero objects, and homotopically almost weak equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  As the main results, we prove a weak version of a model categorical analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1: Given  a pointed symmetric monoidal model category with a homotopically idempotent Smith ideal of the  monoidal unit object, we obtain that the Bousfield localization on the model category by homotopically  2  almost weak equivalences is also a Bousfield colocalization (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Conversely, in the case that  the monoidal unit generates the stable symmetric monoidal model category in the sense of MacLane [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=',  Chapter V, Section 7], we can construct a homotopically idempotent ideal (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' I would like to thank a colleague for giving me an innovative idea to use Hovey’s  Smith ideal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Smith ideal theory of symmetric monoidal model categories  Following Hovey [Hov14], we explain concise of the theory of Smith ideals to generalize almost  mathematics theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The arrow categories of pointed model categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A symmetric monoidal model category M is  a model category with a symmetric monoidal structure − ⊗ − : M × M → M such that, for any object M  of M, those functors (−) ⊗ M and M ⊗ (−) are left Quillen functors on themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' This paper assumes  that symmetric monoidal model categories are always closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' That is, Madmits internal hom-objects  MapM(M, N) for any couple (M, N) of objects of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  The arrow category Ar(M) has the evaluation functors Evi( f : X0 → X1) = Xi (i = 0, 1), which has a  left adjoint and a right adjoint:  Evi : Ar(M)  �M : Li, Ui (i = 0, 1),  ��  where Li denotes the left adjoint of Evi and Ui the right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  The category Ar(M) has two canonical model structures the injective model structure and the projec-  tive model structure induced by M’s:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The arrow category Ar(M) has the following two model  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (Injective model structure) A morphism α : ( f : X0 → X1) → (g : Y0 → Y1) is a cofibrations  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' weak equivalence) in Ar(M) if and only if so is each Evi(α) for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Fibrations are  morphisms with the right lifting property for all trivial cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (Projective model structure) A morphism α : ( f : X0 → X1) → (g : Y0 → Y1) is a fibrations  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' weak equivalence) in Ar(M) if and only if so is each Evi(α) for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Cofibrations are  morphisms with the right lifting property for all trivial fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  By the definition of the injective and the projective model structures, for each i = 0, 1, Evi is both a  left Quillen functor with respect to the injective model structure and a right Quillen functor with respect  to the projective model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore one has the following:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (1) The injective model structure of Ar(M) admits a Quillen adjunction Li : M ⇄ Ar(M) : Evi for each  i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (2) The projective model structure of Ar(M) admits a Quillen adjunction Evi : Ar(M) ⇄ M : Ui for  each i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  3  In the allow category Ar(M), fibrations of the injective model structure and cofibrations of the projec-  tive model structure are characterized as follows:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3 ([Hov14] Thereom 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1(2) and Therem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1 (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (1) On the injective model structure of Ar(M), α : f → g is a (trivial) fibration if and only if so are  Ev1(α) and the induced morphism ( f, Ev1(α)) : Ev1( f) → Ev0(g) ×Ev1(g) Ev1( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In particular, if α  is a (trivial) fibration, so is Evi(α) for each i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (2) On the projective model structure of Ar(M), α : f → g is a (trivial) cofibration if and only if so are  Ev0(α) and the induced morphism (Ev0(α), g) : Ev1( f) ∐Ev0(f) Ev0(g) → Ev1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In particular, if α  is a (trivial) cofibration, so is Evi(α) for each i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  In a pointed model category M, we consider a homotopically commutative diagram:  X  f  �  �  Y  g  �  0  �Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  If the diagram is a homotopy Cartesian square, then X is said to be a homotopy kernel of g, and if it  is a homotopy coCartesian sequence, then Z is a homotopy cokernel of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' We can consider homotopy  image objects and homotopy coimage objects as additive categories: Let f : X → Y be a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The  homotopy image object of f is the homotopy kernel of Y → Coker f, and Im( f) denotes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Dually, the  homotopy coimage object is the homotopy cokernel of Ker f → X, and Coim( f) denotes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The two monoidal structure of the arrow category of symmetric monoidal pointed model  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A symmetric monoidal category can be regarded as a symmetric multicategory (See Le-  inster [Lei04, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let 1 denote the terminal multicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A commutative  monoid object R of M is a covariant functor from 1 to M of symmetric multi-categories [Lei04, Example  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let CAlg(M) denote the category of commutative monoid object of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Directly explaining, the  commutative monoid R is equipped with a unit morphism η : V → R and a multiplication µ : R ⊗ R → R  satisfying commutativity, associativity, and unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  On the arrow category Ar(M), those functors cok : Ar(M) → Ar(M) and ker : Ar(M) → Ar(M)  are defined as follows: For a morphism f : X → Y in M, the arrow cok( f) is Y → Coker f and ker( f)  Ker( f) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the pair  cok : Ar(M) ⇄ Ar(M) : ker  is a Quillen adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The diagonal monoidal structure is compatible with the injective model structure,  and the push-out monoidal structure is compatible with the projective model structure:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='4 ([Hov14] Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1 (4), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1 (5), and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a cofi-  brantly generated symmetric monoidal model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the category Ar(M) has the following two  symmetric monoidal model structures:  (1) The injective model structure of Ar(M) is symmetric monoidal with respect to the diagonal  monoidal model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  4  (2) The projective model structure of Ar(M) is symmetric monoidal with respect to the push-out  monoidal model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (3) Further, if M is pointed, then the Quillen adjunction  cok : Ar(M) ⇄ Ar(M) : ker  is compatible with those monoidal model structures the push-out and the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The assertions (1) and (2) are Hovey [Hov14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1 (4)] and [Hov14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1 (5)],  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The third is obtained by the homotopical analogues of the proof of Hovey [Hov14, Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='4]: Let V be a unit object of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then U0(V) = (0 → V) and L0(V) = idV : V → V are unit  objects of the push-out monoidal structure and the diagonal monoidal structure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The functor  cok : Ar(M) → Ar(M) sends U0(V) to (V → Coker(0 → V)) ≃ (idV : V → V) = L0(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore,  the functor cok is homotopically unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' For two morphisms f : X0 → X1 and g : Y0 → Y1, we show  that those objects cok( f)□cok(g) and cok( f ⊗ g) are isomorphic in the homotopy category of Ar(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Consider the following commutative diagram:  X1 ⊗ Y1  X0 ⊗ Y1  �  �0  Coker( f) ⊗ Y1  X1 ⊗ Y0  �  �  X0 ⊗ Y0  �  �  �  �  0  �  �  Coker( f) ⊗ Y0  id⊗g  �  �  X1 ⊗ Y0  X1 ⊗ Y0  �  �0  0  X1 ⊗ Y1  (X1 ⊗ Y1) ∐X0⊗Y0 (X1 ⊗ Y0)  �  f□g  �  0,  where the bottom horizontal diagram of the out of squares is induced by the homotopy push-outs of  the vertical arrows in the squares, and the right vertical diagram is induced by the homotopy-push outs  of horizontal arrows in the squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since the homotopy colimit of the whole squares are uniquely  determined up to homotopy equivalence, those push-outs of the diagrams out of the squares are weakly  equivalent: one has a zig-zag of weak equivalences  Coker( f□g) ← Z → Coker( f) ⊗ Coker(g),  where Z denotes the homotopy colimit of whole the squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a pointed symmetric monoidal model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A Smith ideal in M is a monoid  object j : I → R in the symmetric monoidal model category Ar(M) with respect to the push-out product  monoidal model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  A stable model category is a model category whose homotopy category is triangulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In the case  that M is further stable, the cokernel functor is a left Quillen equivalence:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='6 ([Hov14] Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a cofibrantly generated stable symmetric monoidal  model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  The cokernel functor cok : Ar(M) → Ar(M) is a left Quillen equivalence form the projective  model structure to the injective model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  5  If M is a cofibrantly generated symmetric monoidal stable model category, the cokernel functor  is a Quillen equivalence from the model category of Smith ideals to the model category of monoid  morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Almost mathematics of pointed symmetric monoidal model categories  On the model category setting, we will consider almost mathematics theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Fix a pointed closed  symmetric monoidal model category M with a zero object 0, a unital commutative monoid object V, and  a Smith ideal j : I → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A Smith ideal j : I → V is homotopically idempotent if it satisfies the following proper-  ties:  The homotopically coCartesian square  I  j  �  �  V  cok(j)  �  0  �Coker( j)  is also homotopy Cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  The push-out product µ j : j□ j → j is a weak equivalence in the model category Ar(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Equiv-  alently, the homotopy cokernel I ∐I⊗I 0 of the product µ : I ⊗ I → I is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  The tensor product ˜I := I ⊗ I is homotopically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' That is, the functor ˜I ⊗ (−) : M → M  preserves all finite homotopy limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let V be a unital commutative ring and m an idempotent ideal of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In almost mathematics  theory, we usually consider the case ˜m = m ⊗ m is a flat V-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore, in this paper, we include  the homotopically flatness of ˜I in the definition of homotopically idempotentness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let j : I → V be a homotopically idempotent Smith ideal of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' For any M ∈ M, We say  that M is homotopically almost zero if the homotopy image of the composition µM : I⊗M → V⊗M → M  is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Let V/I denote the homotopy cokernel of j : I → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' By the homotopically idempotentness of I, one  has I ⊗ V/I ≃ I ⊗ (V ∐I 0) ≃ I ∐I⊗I 0 ≃ 0, implying that V/I is homotopically almost zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Furthermore,  since I ⊗ I is homotopically flat and j : I → V is the homotopy kernel of cok( j) : V → Coker( j), the  composition ε′  ˜I ◦ (µ ⊗ j) : ˜I ⊗ I → ˜I ⊗ V → ˜I is a weak equivalence, where ε′  ˜I is the inverse of the unit  isomorphism ε˜I : ˜I → ˜I ⊗ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A morphism f : M → N in M is a homotopically almost weak equivalence, if the  homotopy kernel and cokernel are both homotopically almost zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' An object M of M is homotopically almost local if, for any homotopically almost weak  equivalence f : N1 → N2, the induced map f∗ : HomHoM(M, N1) → HomHoM(M, N2) is an isomor-  phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Dually, M is homotopically almost colocal if the induced morphism f ∗ : HomHoM(M, N2) →  HomHoM(M, N1) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  6  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a pointed symmetric monoidal model category with a monoidal unit V and a  homotopically idempotent Smith ideal j : I → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the following conditions are equivalent:  (1) An object M is homotopically almost local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (2) For any homotopically almost zero object N, the set HomM(M, N) has only one point in the  homotopy category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Dually, the following conditions are equivalent:  (1)’ An object M is homotopically almost colocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (2)’ For any homotopically almost zero object N, the set HomM(N, M) has only one point in the  homotopy category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let N be a homotopically almost zero object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the trivial morphisms 0 → N and N → 0 are  almost weak equivalences by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore, the implication (1) to (2) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' (1)’ to (2)’) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  We assume that the homotopy kernel and cokernel of f : X → Y are homotopically almost zero, and we  have zig-zags of weak equivalences Y ≃ X∐X Y ≃ (Y ∐X 0)∐X ← X and X ≃ X×Y Y ≃ (X×Y 0)×Y → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Therefore, under the condition (2) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' (2)’), the induced map by f  f∗ : HomM(M, X) → HomM(M, Y)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' f ∗ : HomM(Y, M) → HomM(X, M))  is an isomorphism in the homotopy category of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' An object M of M is homotopically firm if the product morphism µM : I ⊗ M → M is a  weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Dually, M is homotopically closed if the induced morphism µ∗  M : M → MapM(I, M)  is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let j : I → V be a homotopically idempotent Smith ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then an object M is homotopi-  cally almost zero if and only if ˜I ⊗ M is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since j : I → V is homotopically idempotent, the morphism µM : I ⊗ M → M induces an weak  equivalence id˜I ⊗ µM : ˜I ⊗ I ⊗ M → ˜I ⊗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' If µM is null-homotopic, the ˜I ⊗ M is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Conversely, if the bar construction ˜I ⊗ M is contractible, the morphism µM ◦ (µ ⊗ idM) : I ⊗ I ⊗ M →  I ⊗ M → M is null-homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since the homotopy cokernel of µ ⊗ idM is contractible, the induced  morphism Coker(µM◦(µ⊗idM)) → Coker(µM) is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore the canonical morphism  cok(µM) : M → Coker(µM) is also a weak equivalence, entailing µM : I ⊗ M → M is null-homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' □  We recall the definition of lax Quillen adjunctions:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='9 ([SS03a] Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let L : M ⇄ N : R be a Quillen adjunction of monoidal model  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then we say that (L, R) is lax Quillen monoidal if it satisfies the following properties:  (1) The right Quillen functor R is lax monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (2) For any cofibrant objects M and N in M, the induced morphism  ∇M, N : L(M ⊗ N) → L(M) ⊗ L(N)  by the canonical morphism ∆M, N : M ⊗ N → R(L(M)) ⊗ R(L(N)) → R(L(M) ⊗ L(N)) is a weak  equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  7  (3) For any cofibrant replacement c : C(1M) → 1M of the monoidal unit 1M of M, the composition  L(C(1M)) → L(1M) → 1M  is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Assume that ˜I is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be an object of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the following conditions are  equivalent:  (1) The object M is almost colocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (2) The object M is homotopically firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (3) The canonical morphism ˜I ⊗ M → M is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since ˜I ⊗ I → ˜I is a weak equivalence, I ⊗ ˜I ⊗ M → ˜I ⊗ M is also a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore  the conditions (2) and (3) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  For any homotopically almost zero object N, the identity morphism on N factors through V/I ⊗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' A  homotopical section s : V/I ⊗ N → N corresponds to the morphism s∗ : N → MapM(V/I, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' If M is  firm, since V/I ⊗ M is contractible, any morphism M → MapM(V/I, N) is null-homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Hence, any  morphism from M to N is also null-homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' This means that M is almost colocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Finally, we assume that M is almost colocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then µM : I ⊗ M → M has a homotopical section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Equivalently, any morphism from coker(µM) is null-homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore coker(µM) is contractible and  µM : I ⊗ M → M is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Dually, we have the following:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be an object of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the following conditions are equivalent:  (1) The object M is almost local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (2) The object M is homotopically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  (3) The canonical morphism M → MapM(˜I, M) is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let j : I → V be a homotopically idempotent Smith ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the functor ˜I ⊗ (−) :  M → M is a left lax monoidal Quillen functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Indeed, the product µ˜I : (˜I ⊗ M) ⊗ (˜I ⊗ N) → ˜I ⊗ M ⊗ N is a weak equivalence, letting the functor  ˜I ⊗ (−) : M → M be homotopically monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a pointed symmetric monoidal model category and V a unit object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Assume that  V has an idempotent Smith ideal j : I → V and ˜I is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the Bousfield localization La : M → Ma  by almost weak equivalence admits homotopically fully faithful left Quillen adjoint and right Quillen  adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Furthermore, let Mfirm and Mclosed denote the full subcategory spanned by all firm objects and  closed objects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the left Quillen functor ˜I ⊗ (−) : M → Mfirm and the right Quillen  functor MapM(˜I, −) : M → Mclosed are a left adjoint and a right adjoint of La:  Mclosed La  �Ma  MapM(˜I, −)  �  ˜I⊗(−)�Mfirm  La  �  8  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' We only prove that the right Quillen adjunction ˜I ⊗ (−) : Ma ⇄: Mfirm is a Quillen equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='8 and the homotopically idempotentness of the functor ˜I ⊗ (−), the natural transformation  ˜I ⊗ (−) → IdM is a homotopically almost equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore the unit IdMa → La ◦ ˜I ⊗ (−) is a weak  equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' By the equivalence of conditions (2) and (3) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='10, the counit ˜I ⊗ (−) ◦ La →  IdMfirm is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let V be a commutative unital monoid object of M and j : I → V a homotopically  idempotent Smith ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then ˜I ⊗ M is contractible if and only if so is MapM(˜I, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Bousfield bilocalization of stable symmetric monoidal model categories generated by the  monoidal unit  Following the previous section, we consider a stable symmetric monoidal model category M with a  unit object V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Given a Bousfield bilocalization functor F : M → M, we show that F determines an  idempotent Smith ideal j : I → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let F∗ : M → M denote the homotopically fully faithful left adjoint  of F and F!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' the homotopically fully faithful right adjoint:  M  F  �M  F!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  �  F∗ �M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  F  �  Further, we assume that, in this section, the left adjoint F∗ : M → M preserves all finite homotopy limits  and  F∗ : M ⇄ M : F  is a lax Quillen monoidal adjunction and the counit c : F∗ ◦ F → IdM induces a Smith ideal cV :  F∗(F(V)) → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let j : I → V denote the homotopy image of cV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since M is stable, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='6, the  unit morphism cV → j is a projective weak equivalence in the arrow category of Ar(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' We will prove  that I is homotopically idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' By the assumption of F∗, one has a weak equivalence:  ∇ : F∗(F(V) ⊗ F(V)) → F∗(F(V)) ⊗ F∗(F(V)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  The product µ : F∗(F(V)) ⊗ F∗(F(V)) → F∗(F(V)) is homotopic to the push-out of ∇ along the isomor-  phism F∗(F(µ)) : F∗(F(V) ⊗ F(V)) → F∗(F(V)), letting µ : I ⊗ I → I be a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Following MacLane [Mac88], we recall the definition of generators of categories: Let C be a category  and G a class of objects of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The set G is a generator of C if for any parallel of morphisms h, h′ : X → Y  in C, h � h′ implies that there exists S ∈ G and a morphism f : S → X such that h ◦ f � h′ ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a model category with a class G of objects of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' We say that G generates M if  the equivalence class of G generates the homotopy category of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a pointed closed symmetric monoidal model category with a class G of objects  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Assume that G generates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then an object M is contractible if and only if so is the internal  hom-object MapM(S, M) for any S ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' The if-direction is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be an object of M such that the internal-hom object MapM(S, M)  is contractible for any S ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since G generates M, for any object N and morphism f : N → M,  9  the condition that f is not null-homotopic implies that there exists an object S ∈ G and a morphism  i : S → N and f ◦ i : S → M is not null-homotopic, contradicting the generating condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a pointed closed symmetric monoidal model category generated by the monoidal  unit V and F : M → M a symmetric monoidal Bousfield bilocalization, and F∗ : M → M denote the  homotopically fully faithful left adjoint and F!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' : M → M the homotopically fully faithful right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Then, for any object M, F(M) is contractible if and only if so is F∗(F(V)) ⊗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since the induced morphism F(cV) is to homotopic the identity on F(V), the if-direction is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Assume that F(M) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Since F is monoidal, the counit cV : F∗(F(V)) → V induces a  weak equivalence F(M) → F(F∗(F(V)) ⊗ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Therefore, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2, the internal-hom object  MapM(V, F!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' (F(F∗(F(V) ⊗ M)))) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Note that, for any object N, one has a zig-zag of iso-  morphisms:  HomHo(M)(V, MapM(F∗(F(V)), N)) ≃ HomHo(M)(F∗(F(V)), N)  ≃ HomHo(M)(V, F!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='F(N)) ≃ HomHo(M)(V, MapM(V, F!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' (F(N))))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Applying the case N = F∗(F(V)) ⊗ M to the isomorphisms, we obtain that the internal-hom object  MapM(F∗(F(V)), F∗(F(V) ⊗ M))) is contractible, implying that, by the canonical isomorphism  HomHo(M)(F∗(F(V)) ⊗ M, F∗(F(V)) ⊗ M) ≃ HomHo(M)(M, MapM(F∗(F(V)), F∗(F(V)) ⊗ M)))),  F∗(F(V)) ⊗ M is also contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a stable closed symmetric monoidal model category generated by the monoidal  unit V and F : M → M a symmetric monoidal Bousfield bilocalization, and F∗ : M → M denote the  homotopically fully faithful left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then, for any object M of M, the counit cM : F∗(F(M)) → M of  M induces a zig-zag of weak equivalences:  F∗(F(M)) ≃ F∗(F(V) ⊗ F(M)) ≃ F∗(F(V)) ⊗ F∗(F(M)) → F∗(F(V)) ⊗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let C denote the homotopy cokernel of the counit cM : F∗(F(M)) → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then F(C) is con-  tractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3, the homotopy cokernel C ⊗ F(F∗(V)) of the induced morphism idF∗(F(V)) ⊗cM :  F∗(F(V)) ⊗ F∗(F(M)) → F∗(F(V)) ⊗ M is also contractible, implying that idF∗(F(V)) ⊗ cM : F∗(F(V)) ⊗  F∗(F(M)) → F∗(F(V)) ⊗ M is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  In the case that M is stable and F∗ : M → M preserves all finite homotopy limits, the composition  F∗ ◦ F also preserves all finite homotopy limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Immediately, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='4, the counit F∗(F(V)) is  a flat object of M if the monoidal unit V generates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Thus, we obtain the following theorem:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Let M be a stable closed symmetric monoidal model category with a monoidal unit object  V and F : M → M a symmetric monoidal Bousfield bilocalization, and F∗ : M → M denote the  homotopically fully faithful left adjoint of F and c : F∗ ◦ F → Id the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Assume that the left adjoint  F∗ : M → M preserves all finite homotopy limits and the monoidal unit V generates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Then the counit  morphism cV : F∗(F(V)) → V of the Quillen adjunction (F∗, F) determines a homotopically idempotent  Smith ideal j : I → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  □  10  References  [Fal88]  Faltings, Gerd: p-adic Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 1 (1988), Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 255–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISSN 0894–0347  [GR03]  Gabber, Ofer ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Ramero, Lorenzo: Lecture Notes in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 1800: Almost ring theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Springer-Verlag,  Berlin, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – vi+307 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISBN 3–540–40594–1  [Hir03]  Hirschhorn, Philip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=': Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Monogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='. Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 99: Model categories and their localizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Providence, RI:  American Mathematical Society (AMS), 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISBN 0–8218–3279–4  [Hov99] Hovey, Mark: Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 63: Model categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' American Mathematical Society,  Providence, RI, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – xii+209 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISBN 0–8218–1359–5  [Hov14] Hovey, Mark: Smith ideals of structured ring spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Available at:https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='org/abs/1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2850, 2014  [Lei04]  Leinster, Tom: London Mathematical Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 298: Higher operads, higher categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Cambridge University Press, Cambridge, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – xiv+433 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1017/CBO9780511525896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1017/CBO9780511525896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISBN 0–521–53215–9  [Lur09]  Lurie, Jacob: Annals of Mathematics Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 170: Higher topos theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Princeton, NJ : Princeton University  Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – xviii+925 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISBN 978–0–691–14049–0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 0–691–14049–9  [Lur17]  Lurie, Jacob: Higher Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' available at:https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='ias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='edu/˜lurie/papers/HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='pdf, 2017  [Mac88] MacLane, Saunders: Grad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Texts Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='. Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 5: Categories for the working mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 4th corrected printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  New York etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' : Springer-Verlag, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISBN 3–540–90035–7  [Qui96] Quillen, Daniel: Module theory over nonunital rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Available at:  https://ncatlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='org/nlab/files/QuillenModulesOverRngs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='pdf, 1996  [SS03a] Schwede, Stefan ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Shipley, Brooke: Equivalences of monoidal model categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In: Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 3 (2003), S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  287–334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2140/agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2140/agt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISSN 1472–2747  [SS03b] Schwede, Stefan ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' Shipley, Brooke: Stable model categories are categories of modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' In: Topology 42 (2003), Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 1,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' 103–153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1016/S0040-9383(02)00006-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='1016/S0040–9383(02)00006–  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content=' – ISSN 0040–9383  National institute of technology, Ube college, 2-14-1, Tokiwadai, Ube, Yamaguchi, JAPAN 755-8555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='  Email address: ykato@ube-k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}
+page_content='jp  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf'}

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+First search for ultralight dark matter with a space-based gravitational-wave antenna:
+LISA Pathfinder
+Andrew L. Miller
+1, 2, 3, ∗ and Luis Mendes4, †
+1Universit´e catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
+2Nikhef – National Institute for Subatomic Physics,
+Science Park 105, 1098 XG Amsterdam, The Netherlands
+3Institute for Gravitational and Subatomic Physics (GRASP),
+Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands
+4RHEA Group for ESA, Camino bajo del Castillo s/n,
+Urb. Villafranca del Castillo, Villanueva de la Ca˜nada, 28692 Madrid, Spain
+We present here results from the first-ever search for dark photon dark matter that could have
+coupled to baryons in LISA Pathfinder, the technology demonstrator for a space-based gravitational-
+wave antenna. After analyzing approximately three months of data taken by LISA Pathfinder in
+the frequency range [2 × 10−5, 5] Hz, corresponding to dark photon masses of [8 × 10−20, 2 × 10−14]
+eV/c2, we find no evidence of a dark-matter signal, and set upper limits on the strength of the
+dark photon/baryon coupling. To perform this search, we leveraged methods that search for quasi-
+monochromatic gravitational-wave signals in ground-based interferometers, and are robust against
+non-Gaussianities and gaps in the data. Our work therefore represents a proof-of-concept test of
+search methods in LISA to find persistent, quasi-monochromatic signals, and shows our ability to
+handle non-Guassian artifacts and gaps while maintaining good sensitivity compared to the optimal
+matched filter. The results also indicate that these methods can be powerful tools in LISA to not
+only find dark matter, but also look for other persistent signals from e.g. intermediate-mass black
+hole inspirals and galactic white dwarf binaries.
+I.
+INTRODUCTION
+LISA Pathfinder [1–3] was a demonstrator of some of
+the technologies to be used in LISA, the space-based
+gravitational-wave antenna to be launched in the sec-
+ond half of the next decade [4, 5].
+Its main goal, the
+demonstration that the noise in the separation of two
+test masses ≈ 40cm apart could be kept at a suitably
+low level to allow the detection of gravitational waves
+by LISA, was successfully achieved by LISA Pathfinder.
+In fact, LISA Pathfinder was a remarkable success and
+produced a noise power spectral density that surpassed
+the LPF mission requirement by more than an order of
+magnitude for some frequency regions and by a factor of
+a few compared to the LISA requirement [6, 7].
+Though designed for the purpose described above, the
+data from LISA Pathfinder have been used in different
+ways to probe other areas of physics, such as quan-
+tifying possible noise correlations in future stochastic
+gravitational-wave background searches [8], measuring
+the value of Newton’s gravitational constant [9], testing
+the strong equivalence principle when spacecrafts orbit
+Lagrange points [10], and bounding collapse models [11].
+Such studies show that high-precision measurements of
+acceleration or displacement can be employed to tackle
+interesting physics problems for which the experiment
+was not primarily designed, and motivate the need for
+further analyses of LISA Pathfinder data [12] to see what
+∗ [email protected]
+† [email protected]
+other kinds of questions that this data, and the future
+LISA mission, can answer.
+LISA Pathfinder data can be used to probe the exis-
+tence of ultralight dark matter directly via its interac-
+tions with the test masses. Essentially, the instruments
+exists in a field or “wind” of dark matter, and this dark
+matter would couple to standard model particles in the
+test masses, causing sinusoidal oscillations of their posi-
+tions over time at a frequency fixed by the dark matter
+mass. Scalar, dilaton dark matter would cause oscilla-
+tions in the electron mass or other fundamental constants
+[13–15], resulting in a change of the size of an object [16];
+axions [17] would alter the phase velocity of light [18, 19],
+affecting the roundtrip time of laser light in LISA; dark
+photons could couple to baryons or baryon-lepton num-
+ber in the test masses, leading to a sinusoidal force ex-
+erted on them [20]. While the physics of each model is
+different, the observable is the same: time-varying posi-
+tions or accelerations of test masses relative to one an-
+other.
+The mass range to which space-based gravitational-
+wave detectors are sensitive is a function of the frequency
+range: O(10−5 − 1) Hz corresponds to O(10−19 − 10−14)
+eV/c2.
+At such low masses – compared to those to
+which ground-based gravitational-wave detectors such as
+LIGO [21], Virgo [22], and KAGRA [23] could probe
+– the expected signal is monochromatic, since the fre-
+quency shift canonically introduced by the motion of the
+earth through the dark matter “wind” is very small com-
+pared to the frequency resolution of a search. Therefore,
+we cannot differentiate between a monochromatic and a
+“quasi”-monochromatic ultralight dark-matter signal at
+most frequencies to which LISA Pathfinder and LISA are
+arXiv:2301.08736v1  [gr-qc]  20 Jan 2023
+
+2
+sensitive1.
+There is a growing interest in using gravitational-wave
+detectors to search for ultralight dark matter. GEO600
+data [24] was recently analyzed using a Logarithmic fre-
+quency axis Power Spectral Density (LPSD) method
+[25, 26], yielding strong constraints on scalar, dilaton
+dark matter coupling to the beam splitter [27]. Further-
+more, constraints on vector dark matter, i.e. dark pho-
+tons, were placed using data from the first [28] and third
+[29] observing runs of advanced LIGO/Virgo/KAGRA
+that surpassed upper limits from the E¨ot-Wash [30]
+and MICROSCOPE [31] experiments by a few orders
+of magnitude at frequencies between ∼ 100 − 1000 Hz
+(4×10−13 −4×10−12 eV/c2). Finally, methods to search
+for axions and dilatons in different interferometer chan-
+nels [18, 32], for vector bosons with KAGRA [33], and
+tensor bosons [34] in LIGO/Virgo/KAGRA and pulsar
+timing arrays [35] have been developed for such kinds of
+searches.
+What has not yet been done in this field of direct dark-
+matter searches with gravitational-wave detectors is to
+apply these methods to space-based instruments. While
+projected sensitivities have been estimated for a variety
+of dark-matter models [20, 36, 37], the development of al-
+gorithms tuned towards monochromatic signals in LISA,
+DECIGO [38], and TianQin [39] has not yet followed. In
+this work, we take a first step in this direction by per-
+forming a search of LISA Pathfinder data, a precursor to
+the kind of data that we expect in LISA, for ultralight
+dark matter, which requires that we develop methods
+to handle specific problems that LISA Pathfinder faced
+and LISA will face, such as non-Gaussianities, gaps, and
+sparse sampling at low frequencies [40–42] Specifically,
+we adapt methods developed in the context of quasi-
+monochromatic, persistent signals emitted by isolated
+neutron stars [43], planetary- and asteroid-mass primor-
+dial black hole binaries [44–46], depleting boson clouds
+around black holes [47, 48], and dark matter that cou-
+ples to ground-based gravitational-wave interferometers
+[20, 49], to look for dark matter in space with LISA
+Pathfinder.
+The methods presented here are generically good at
+finding quasi-monochromatic signals in any dataset, re-
+gardless of the underlying physics [43, 50].
+They are
+also (1) robust against noise disturbances such as pow-
+erful lines, (2) efficiently deal with gaps in data collec-
+tion, and (3) are computationally cheap compared to
+matched filtering, the optimal method to find weak sig-
+nals buried in noise.
+In space-based detectors, gravi-
+tational waves from a variety of astrophysical sources,
+such as galactic white dwarf binaries or black hole inspi-
+rals [51, 52], will also be quasi-monochromatic and last
+for durations longer than the observing time [49, 53],
+1 This depends on the duration of data analyzed, as will be ex-
+plained in section II, but is generally true for the expected life-
+time of LISA.
+or at least for greater lengths of time than in ground-
+based detectors, of O(hours-days). Currently, proposals
+for parameter estimation and matched-filtering searches
+for gravitational waves with LISA struggle with each of
+the three aforementioned points [54–56]; therefore, the
+work presented here has much farther reaching implica-
+tions than “just” a search for dark matter; it represents
+a comprehensive analysis scheme that can be applied to
+any quasi-monochromatic signal embedded in imperfect,
+non-Gaussian, gapped LISA data.
+This paper is organized as follows:
+in section II,
+we describe the dark-matter signal expected at LISA
+Pathfinder when it flew; in section III, we explain which
+data segments from the ∼ 1.5 year run we analyze. Sec-
+tion IV focuses on the methods that we use to search
+for ultralight dark matter, one that breaks the data into
+smaller, Gaussian chunks, and another that match filters
+the data with a signal model coherently. We present our
+results in section V, which include rejecting strong out-
+liers in the data and upper limits on the strength of the
+coupling of dark matter to baryons.
+Finally, we draw
+some conclusions in section VI and discuss opportunities
+for future work.
+II.
+DARK MATTER SIGNAL
+Dark matter could be composed of spin-1 particles,
+which we denote as dark photons. The relic abundance
+of dark matter can be explained entirely by dark photons,
+which could arise from the misalignment mechanism [57–
+59], parametric resonance or the tachyonic instability of a
+scalar field [60–63], or from cosmic string network decays
+[64]. Dark photons could couple directly to baryon or
+baryon-lepton number in the two LISA Pathfinder test
+masses, and exert a “dark” force on the them, causing
+quasi-sinusoidal oscillations [20, 49].
+Similarly to the ordinary photon, the vector potential
+for a single dark photon particle can be written as:
+⃗A(t, ⃗x) =
+�ℏ√2ρDM
+mc2
+1
+√ϵ0
+�
+sin
+�
+ωt − ⃗k · ⃗x + φ
+�
+,
+(1)
+where ω = (mc2)/ℏ is the angular Compton frequency,
+⃗k = (m⃗vobs)/ℏ is the wave vector, m is the mass of the
+vector field, ϵ0 is the permittivity of free space and φ is
+a random phase.
+The Lagrangian L that characterizes the dark photon
+coupling to a number current density Jµ of baryons or
+baryons minus leptons is:
+L = − 1
+4µ0
+F µνFµν +
+1
+2µ0
+�mc
+ℏ
+�2
+AµAµ − ϵeJµAµ, (2)
+where Fµν indicates the dark electromagnetic field ten-
+sor, µ0 is the magnetic permeability in vacuum, Aµ is the
+four-vector potential of the dark photon, e is the electric
+
+3
+charge, and ϵ is the strength of the particle/dark pho-
+ton coupling normalized by the electromagnetic coupling
+constant.
+Dark photons would cause small motions in each of
+the test masses, and lead to an observable effect be-
+cause the test masses are separated from each other and
+hence experience slightly different dark photon dark mat-
+ter phases. Such a phase difference leads to a change of
+the arm length over time. What is needed, therefore, is
+very precise measurements of the positions of the two test
+masses, something which LISA Pathfinder provides.
+Each test mass experiences an almost identical accel-
+eration:
+⃗a(t, ⃗x) ≃ ϵe q
+M ω ⃗A cos(ωt − ⃗k · ⃗x + φ),
+(3)
+where q/M is the charge-to-mass ratio of the test masses.
+This effect is in fact a residual one: if the test masses were
+made of different materials, then the signal induced from
+dark photons coupling to baryon-lepton number would be
+enhanced [33]. A simple relation between dark photon
+parameters and the effective strain hD experienced by
+LISA Pathfinder can be written as [20, 29, 37]:
+�
+⟨h2
+D⟩ = CLPF
+q
+M
+v0
+2πc2
+�
+2ρDM
+ϵ0
+eϵ
+fA
+,
+(4)
+where fA = ω/(2π) is the frequency of the dark-matter
+particle, and CLPF = 1/3 is a geometrical factor obtained
+by averaging the acceleration over all possible dark pho-
+ton propagation and polarization directions for a single
+arm, using the appendix in [20] to aid with this calcula-
+tion. We note CLPF is a factor of
+√
+2 smaller than the
+geometrical factor in LIGO CLIGO =
+√
+2/3, which indi-
+cates that an L-shaped interferometer would observe a
+signal
+√
+2 ∼ 1.4 times stronger than a single arm would.
+As we can see from equation 3, the test masses will
+experience a sinusoidal oscillation of their positions over
+time, at a frequency set by the mass of the dark-matter
+particle. However, if we observe for longer than the dark-
+matter signal coherence time Tcoh, then we will resolve
+stochastic frequency fluctuations ∆f about the dark-
+matter mass, induced by the motion of the spacecraft
+relative to the dark-matter field. In other words, we wish
+to contain these fluctuations to one frequency bin δf, so
+we require:
+∆f = 1
+2
+�v0
+c
+�2
+fA < δf =
+1
+TFFT
+(5)
+which leads to:
+TFFT < 107 s
+�0.1 Hz
+fA
+�
+∼ Tcoh
+(6)
+where TFFT is the fast Fourier transform length, and v0 ≃
+220 km/s is the velocity at which dark matter orbits the
+number
+date
+duration (days) TFFT (s) Temp (C)
+1
+2016-03-21
+5.08
+16384
+22
+2
+2016-04-04
+9.33
+16384
+22
+3
+2016-07-24
+5.29
+4096
+22
+4
+2016-11-16
+9.82
+8192
+22
+5
+2016-12-26
+17.86
+8192
+22
+6
+2017-02-14
+13.33
+8192
+11
+7
+2017-05-29
+7.04
+131072
+22
+8
+2017-06-08
+8.4
+262144
+22
+TABLE I. Data segments considered in this search obtained
+from [12]. Segment # 6 is treated as a separate dataset here:
+we perform coincidences of the other seven datasets with seg-
+ment # 6 in order to reduce the number of outliers.
+center of our galaxy, i.e. the virial velocity [65]. Here,
+we observe for a duration much less than Tcoh, which
+implies that the signal will be fully contained within one
+frequency bin, i.e.
+it is purely monochromatic for our
+purposes.
+III.
+DATA
+We analyze eight periods of differential acceleration
+time-series LISA Pathfinder data, whose details are given
+in table I, each of which lasts from ∼ 5 days to ∼ 2.5
+weeks [12]. One segment, #6, that began on 14 February
+2017, was obtained at a much lower temperature than the
+others, which resulted in a different noise level [7]. We
+therefore treat it as a separate dataset from the others,
+as will be detailed in Sec. IV, and present all constraints
+based on this dataset.
+In table I, we also report the TFFT such that the data
+within that segment remain Gaussian at the 5% signifi-
+cance level, based on the Kolmogorov-Smirnov test [66].
+We see here that some segments remain Gaussian for
+longer compared to others, but in general, the duration
+of Gaussian data is short compared to the segment dura-
+tion. Therefore, for the semi-coherent search, we consider
+TFFT = 8192 s for all segments, as explained in the next
+section.
+IV.
+METHOD
+.
+A.
+Semi-coherent method: projection
+Due to the non-Gaussian nature of the noise when ob-
+serving for longer than ∼ 8192 s, we break the dataset
+into smaller chunks, of duration TFFT = 8192 s, analyze
+these chunks coherently, and then combine their power
+incoherently, that is, without the phase information. To
+perform this analysis, we leverage existing methods used
+
+4
+in continuous-wave data analysis for the search of deplet-
+ing boson clouds around black holes [47], and for dark
+matter that could couple to ground-based gravitational-
+wave detectors [49].
+1.
+Creation of time/frequency peakmaps
+This semi-coherent method [49] operates on data in
+the time/frequency plane, so we take fast Fourier Trans-
+forms with TFFT = 8192 s, divide the square modulus of
+the fast Fourier transform by a running-median estima-
+tion of the power spectral density to obtain “equalized
+power” (whose mean value in noise should be 1), set a
+threshold θthr = 2.5 to remove spurious noise peaks, and
+select local maxima in the power spectra. Each of these
+choices is motivated in [67], and is meant to be robust
+against the presence of non-stationarities in the noise.
+This threshold in fact comes from empirical studies in
+[67] showing that while the ideal threshold in Gaussian
+noise is 2, a threshold of 2.5 allows for a higher value
+of the detection statistic, and a reduction in the num-
+ber of outliers in the presence of disturbances in real
+LIGO/Virgo data. The “local maxima” criteria is ap-
+plied because spectral disturbances in real data may not
+be well localized to one frequency bin (i.e.
+they have
+finite coherence times that do not match the TFFT cho-
+sen), or they may turn on and off over the course of the
+run (or even within one TFFT. Therefore, selecting local
+maxima helps to minimize contamination of nearby bins
+from noise lines. The thresholded time/frequency maps
+are called “peakmaps”, in which only powerful “peaks”,
+i.e. points in the time/frequency plane, remain after the
+thresholding and local maxima selection.
+This procedure results in the left-hand panel of figure
+1, for segment #6. We employ the semi-coherent method
+at frequencies above ∼ 1 mHz, since below this value, it
+becomes difficult to estimate the power spectral density
+using a simple running median, due to the lack of data
+points at such low frequencies.
+2.
+Projection and selection of candidates
+After we have created the time/frequency peakmaps,
+we then attempt to recover some signal power that has
+been lost due to dividing the data into chunks. To do
+this, we integrate the peakmap over time, i.e. we project
+it onto the frequency axis, summing only whether or not
+a peak appears at a given time and frequency, not the
+equalized power that is given in the color bar of the left-
+hand panel of figure 1. This is a choice that is, again,
+motivated by the presence of noise artifacts:
+in pure
+Gaussian noise, summing signal power provides the best
+chances of detection. However, there are powerful noise
+lines, e.g. the one at 1 Hz, that we do not want to blind
+us to possible signals. In this way, we reduce the signal-
+to-noise ratio of the instrumental lines, while preserv-
+ing sensitivity towards weak, monochromatic signals. We
+sum each peakmap from datasets 1-5; 7-8 together in this
+method, allowing us to recover some signal-to-noise ratio
+that is lost by breaking the data into chunks of length
+TFFT.
+At this point, we obtain a “histogram” of the number
+of peaks at a given frequency, in the right-hand panel of
+figure 1. It is on this projected peakmap that we select
+possible significant candidates, i.e.
+particular frequen-
+cies, whose number counts are high relative to other fre-
+quencies. Since the frequency range spans three orders of
+magnitude, we select candidates uniformly in the log of
+frequency. We decide on a certain number of candidates
+Ncand to select such that we would expect a certain num-
+ber of coincidences Ncoin between the combined dataset
+(segments #1-5; 7-8) and segment #6 if the data were
+purely Gaussian in the frequency band of width B:
+Ncand ≈
+�
+NcoinTFFTB.
+(7)
+Here, we set Ncoin = 1, B = 5 Hz, and TFFT = 8192 s, so
+we select Ncand ≈ 200 candidates uniformly in logarith-
+mic frequency. We select the strongest candidate in each
+sub-band.
+Once we have these candidates, we compute our detec-
+tion statistic, the critical ratio (CR):
+CR = n − µ
+σ
+,
+(8)
+where n is the number of peaks at a given frequency, and
+µ and σ are the average number and standard deviation
+of peaks across the sub-band, respectively. The CR is a
+random variable with mean 0 and standard devitation of
+1; therefore, we set a threshold CRthr = 5, corresponding
+to 5 standard deviations from the mean, that indicates
+that a particular candidate is “significant” and needs to
+be studied further.
+B.
+Fully-coherent method: matched-filter
+For the frequencies considered in this analysis, and for
+the durations analyzed (see table I), equation 6 indicates
+that the signal will be purely monochromatic; hence, we
+also perform a fully coherent search of each segment, by
+simply taking a fast Fourier Transform and computing
+the so-called matched filter signal-to-noise ratio (SNR)
+ρ:
+ρ2 = 4 Re
+�� ∞
+0
+df
+˜h(f) ˜d(f)
+Sn(f)
+�
+(9)
+= 4 Re
+� ˜d(f)2
+Sn(f)
+�
+(10)
+where the tilde denotes the Fourier Transform of the cor-
+responding quantity, ˜d(f) is the Fourier transform of the
+
+5
+FIG. 1. Left: example time/frequency peakmap used in the semi-coherent search. Right: projection of the peakmap onto the
+frequency axis, corresponding to an integration over time.
+time-series ∆g acceleration data d(t), ˜h(f) is the Fourier
+Transform of the waveform for the dark-photon signal
+h(t), and Sn is the power spectral density of the noise. To
+pass from the first to the second line in the above equa-
+tion, we note that our desired signal is purely monochro-
+matic. Therefore, the best filter, at each frequency, is
+simply a monochromatic signal at that frequency, and the
+number of templates used is simply equal to the number
+of frequencies analyzed, Nf ∼ 3.4 × 106. We impose a
+threshold ρthr = 8 to indicate “significant” events, which
+corresponds to a false alarm probability, in pure Gaussian
+noise accounting for the trials factor, of 1 per 109.
+1.
+PSD estimation at low frequencies
+The sparseness of LISA Pathfinder data at low fre-
+quencies (defined here as less than 1 mHz) is problem-
+atic for a running-median estimation of the PSD. Hence,
+we employ a method developed in [68] in order to ob-
+tain a robust low-frequency estimation of the PSD. The
+concept is to average Black-Harris windowed, 50% in-
+terlaced power spectra, obtained with a fixed TFFT at
+a given frequency, and then subsequently decrease TFFT
+at higher frequencies, resulting in more spectra to aver-
+age. The frequencies fj at which the PSD estimation are
+performed are given by a recursive formula:
+fj =
+2
+(3/5)j−1 f0
+(11)
+f0 =
+M
+Nmax∆T
+(12)
+where M is the spacing between frequency bins needed to
+avoid spectral leakage from the averaging window among
+neighboring frequencies, Nmax is the maximum number
+of samples to fast Fourier transform (for the starting
+TFFT = T0 = 2×105 s), and ∆T = 1/10 s is the sampling
+time. Here, Nmax =
+T0
+∆T = 2 × 106 samples, and M is
+set to 4 bins at f0, and 8 bins for all others, obtained by
+ensuring a lack of correlation between neighboring bins
+[69]. The factor 3/5 is obtained by avoiding correlations
+between two neighboring frequency bins:
+M − α
+M + α = 3
+5,
+(13)
+where α = 2 ensures that no correlations in the PSD exist
+between frequencies spaced by with a spacing δf =
+2α
+TFFT .
+TFFT scales with the same relation as:
+TFFT,j =
+�3
+5
+�j−1
+Nmax∆T
+(14)
+By windowing, Fourier transforming and averaging the
+data in chunks of length TFFT,j, we obtain the val-
+ues of the PSD at frequencies between 20µHz and ∼
+1mHz, shown in figure 2 as black dots. For the rest of
+the parameter space, a running median estimation over
+20 bins is employed to estimate the PSD, as in many
+LIGO/Virgo/KAGRA searches.
+C.
+Coincidences
+We have indicated in table I eight segments that have
+been analyzed in this work. However, we only have data
+from one detector. Typically, in gravitational-wave data
+analysis, we look for similar signals in a collection of de-
+tectors to rule out the possibility of false alarms. In this
+case, though, we can look for coincidences between two
+datasets if their noise distributions are sufficiently differ-
+ent, such that they “function” as independent probes of
+dark matter. Furthermore, the dark-matter signal should
+always be present in the data; therefore, if a candidate
+is seen in one segment but not in another, this would
+indicate that it is due to something artificial, not astro-
+physical.
+
+100
+80
+equalized power spectrum
+frequency (Hz)
+60
+40
+20
+10-3
+0
+2
+4
+6
+8
+10
+12
+days since 14 Feb. 2017250
+number of peaks
+200
+150
+100
+50
+0
+10-3
+10-2
+10-1
+100
+frequency (Hz)6
+FIG. 2. Estimation of the amplitude spectral density at low
+and high frequencies, using the method described in [68] and
+a running median, respectively.
+In the semi-coherent case, we sum the peakmaps from
+segments #1−5; 7−8 and require that a candidate in one
+detector appear within 2 frequency bins of a candidate
+in another. This choice of 2 bins is in fact very generous:
+the dark-matter signal should be exactly at the same fre-
+quency in each segment.
+We then impose a threshold
+on the critical ratio, requiring that a candidate’s CR is
+greater than CRthr = 5.
+In the fully-coherent searches, we do coincidences be-
+tween each segment and segment #6, and again require
+that candidates be within 2 frequency bins of each other.
+In this case, however, the size of the frequency bins of
+each segment will be slightly different, since their dura-
+tions are not equal. We therefore use the larger of the
+two frequency bins to determine whether a coincidence
+has occurred. Then, we impose that the SNR must be
+greater than ρthr = 8. As a last stringent check, we will
+also require that a candidate be present in each data seg-
+ment for it to be considered as an “outlier” worthy of
+further study.
+V.
+RESULTS
+We present here the results of our search for dark mat-
+ter interacting with the test masses in LISA Pathfinder.
+In the semi-coherent search in the high-frequency
+regime, the only coincident outliers above CRthr were
+at ∼ 70 mHz, 1 Hz and 3 Hz.
+These frequencies are
+contaminated with very strong noise lines and can thus
+be discarded. At 70 mHz, there is a known noise distur-
+bacne due to the thrusters [70]; at 1 Hz and 3 Hz, these
+are harmonics arising from electrical cross-couplings from
+the pulse-per-second timing signal present on the space-
+craft [71].
+In the fully-coherent search, across all frequencies, we
+only obtained three coincident outliers in at least two of
+the segments analyzed that were not due to a particular
+frequency (Hz) SNR
+0.474766
+8.57
+1.213422
+8.50
+2.584182
+8.26
+TABLE II.
+Remaining candidates from the fully coher-
+ent, high-frequency search that appeared in at least two the
+datasets analyzed. We note that these candidates did not ap-
+pear in all segments analyzed, as we would expect for a true
+dark photon signal; thus, we veto them.
+FIG. 3. Distribution of the semi-coherent detection statistic
+CR, with a Gaussian curve overlayed for data segment #6
+starting on 14 Feb. 2017.
+known noise disturbance. They are given in table II, and
+are barely above the SNR threshold set. We note, how-
+ever, that these outliers did not appear in each segment
+that we separately analyzed, indicating that they are due
+to noise disturbances.
+For the critical ratio and the matched-filter SNR, we
+provide the statistical distributions obtained from the
+search in figures 3 and 4, respectively. We can see here
+that these distributions are not exactly Gaussian, due to
+the presence of non-stationarities in the data.
+After vetoing all candidates, we then set upper lim-
+its on the coupling of dark matter to baryons in the
+test masses. We provide these limits in figure 5 in the
+low- and high-frequency regime. Specifically, in the high-
+frequency regime, we calculate these limits with both the
+results of the projection method and the matched filter
+for comparison. To obtain these limits, we employ the
+Feldman-Cousins approach [72] to map values of the crit-
+ical ratio and SNR to “inferred” ones to ensure complete
+coverage at the chosen confidence level (95% in this case)
+using table 10 of [72]. This procedure inherently assumes
+that the critical ratio and SNR follow Gaussian distribu-
+tions, but is robust against non-Gaussianities: it provides
+conservative results with respect to simulations of dark-
+photon signals injected in real data (see figure 12 in [49]).
+From these “inferred” values of our detection statistics,
+we compute the constraint on the coupling strength us-
+ing equation 30 in [49], and equation 4 here. Equation 30
+
+7/T-
+amplitude spectral density (Hz-
+low-freguency estimation
+10-12
+median estimation
+10-13
+10-14
+10-15
+10-16
+10-17
+10-4
+10-3
+10-2
+10-1
+100
+frequency (Hz)104
+103
+count
+102
+101
+100
+-2
+0
+2
+-8
+-6
+-4
+4
+6
+8
+critical ratio (CR)7
+FIG. 4.
+Distribution of the matched-filter detection statis-
+tic ρ across the whole frequency range, for data segment #6
+starting on 14 Feb 2017.
+in [49] arises from theoretically calculating the minimum
+detectable amplitude of a monochromatic signal that our
+search could see in Gaussian noise.
+We note that the upper limits for the semi-coherent
+method, which combine the peaks in each segment, are
+shown for the “limiting” segment #6, since the duration
+and noise spectral density of this segment impede the sen-
+sitivity of the semi-coherent search relative to the other
+segments. If we were, instead, to use the critical ratios
+arising from integrating over each of the other segments,
+the limits would be lower by ∼ 2, representing the ra-
+tio between the total observation time of segments #1-5;
+7-8, and segment #6.
+There is an additional effect that must be accounted
+for when calculating these upper limits. Due to the fact
+that Tobs << Tcoh for the parameter space considered
+here, the stochastic nature of the signal amplitude af-
+fects the strength of the dark-matter coupling that we
+could observe. In other words, it could be possible that
+we would observe the dark-matter field amplitude at a
+near-zero value, something that does not happen in the
+regime Tobs >> Tcoh, since we break the data into chunks
+of length TFFT ∼ Tcoh, i.e. we observe for a full coher-
+ence time. These effects have been calculated for both
+scalar, axion dark matter [73] and vector dark matter
+(dark photons) [74]. This stochastic effect amounts to an
+O(1) loosening of the upper limits in amplitude, and in
+our case, our limits on ϵ2 must be increased by a factor
+of 9. This correction has been applied in figure 5.
+VI.
+CONCLUSIONS
+We have performed the first search for dark photon
+dark matter with LISA Pathfinder data, and have set
+upper limits on the coupling of dark photons to baryons
+in the test masses used in this mission.
+While these
+limits are not stringent compared to those from tor-
+FIG. 5.
+Upper limits in both the fully-coherent and semi-
+coherent regimes, with the red indicating the regime in which
+the low-frequency PSD estimation was used, for the period
+starting 14 Feb. 2017. The mass range shown on the x-axis
+corresponds to the frequency range [2 × 10−5, 5] Hz.
+sion balance experiments [30, 31], the work here pro-
+vides a proof-of-concept pipeline to analyze future LISA
+data and perform dark-matter searches.
+Interestingly,
+we had to consider problems such as PSD estimation
+at very low frequencies, performing coincidences using
+one instrument, dealing with non-Gaussian artifacts aris-
+ing from a gravitational-wave detector in space, and
+breaking the search into semi-coherent and fully-coherent
+regimes. Even though there were not many visible (by
+eye) non-Gaussian disturbances, the Gaussianity of this
+data broke down after O(hours − days) in each seg-
+ment analyzed.
+Our work therefore motivates the de-
+velopment of more advanced PSD estimation algorithms
+that can handle future data in LISA, the need for semi-
+coherent, time/frequency-based analyses of the data, and
+the power that search techniques designed for quasi-
+monochromatic signals in ground-based gravitational-
+wave detectors have when applied to a LISA-like mission.
+Furthermore, in LISA, many astrophysical signals will
+be almost monochromatic – light binary black hole in-
+spirals, inspiraling galactic white dwarf binaries, etc.–;
+therefore, our work shows how effective time/frequency
+analyses, as opposed to pure, computationally expensive
+matched filters, can be to look for different kinds of sig-
+nals in future LISA data.
+While there is of course a
+sacrifice in sensitivity, evidenced by comparing the pur-
+ple and magenta curves in the right-hand panel of figure
+5, the difference is, at most, an order of magnitude in ϵ2
+at ∼ 1 mHz, meaning a factor of a few in strain, which is
+consistent with the comparison of semi-coherent methods
+to matched filtering given in [75].
+For dark-matter searches, LISA will provide access to
+masses that ground-based gravitational-wave detectors
+simply cannot probe, due to seismic activity and Newto-
+
+105
+104
+103
+102
+101
+100
+0
+-40
+-20
+20
+40
+signal-to-noise ratio (SNR)semi-coherent, high-frequency
+10-14
+fully-coherent, high-frequency
+fully-coherent, low-frequency
+10-18
+95%
+29
+3
+10-22
+coupling strength
+10-26
+10-30
+10-34
+10-38
+10-19
+10-18
+10-17
+10-16
+10-15
+10-14
+dark photon mass (eV/c2)8
+nian noise on earth. The sensitivity achievable in LISA
+is expected to surpass by several orders of magnitude
+the existing constraints on ϵ2 from torsion-balance ex-
+periments [20]. It is therefore worth investing in data-
+analysis pipelines in LISA, such as the one employed
+here, to perform searches to potentially directly detect
+dark matter, and also other astrophysical sources that
+would emit quasi-monochromatic signals.
+Future work will include implementing a more robust
+estimation of the background of our detection statistics
+as described in [76], extending our analysis to the whole
+LISA Pathfinder dataset, performing simulations of dark-
+matter particles interacting with the detector, and apply-
+ing this analysis to other types of dark-matter that could
+be detected, e.g. the dark photon arising from kinetic
+mixing that couples to the ordinary photon [77]. All of
+these avenues of research will be relevant for when LISA
+eventually flies.
+ACKNOWLEDGMENTS
+LIGO Laboratory which is a major facility fully funded
+by the National Science Foundation.
+Computational resources have been provided by the
+supercomputing facilities of the Universit´e catholique
+de Louvain (CISM/UCL) and the Consortium des
+´Equipements de Calcul Intensif en F´ed´eration Wallonie
+Bruxelles (C´ECI) funded by the Fond de la Recherche
+Scientifique de Belgique (F.R.S.-FNRS) under conven-
+tion 2.5020.11 and by the Walloon Region.
+We would like to thank Eleonora Castelli for useful
+discussions on the noise disturbances in Lisa Pathfinder
+data, and Yue Zhao for a Mathematica code to perform
+the geometric factor calculation.
+This work has made use of the LTPDA MATLAB tool-
+box.
+All plots were made with the Python tools Matplotlib
+[78], Numpy [79], and Pandas [80, 81].
+A.L.M. is a beneficiary of a FSR Incoming Post-
+doctoral Fellowship. We acknowledge support from the
+ESA Archival Research Visitor Programme, that allowed
+us to carry out this work.
+We would like to thank all of the essential workers who
+put their health at risk during the COVID-19 pandemic,
+without whom we would not have been able to complete
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+

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+TRINET: STABILIZING SELF-SUPERVISED LEARNING FROM COMPLETE OR SLOW
+COLLAPSE
+Lixin Cao 1†
+Jun Wang 1†
+Ben Yang 1,2‡
+Dan Su1
+Dong Yu3
+1Tencent AI Lab, China
+2 Peking University
+3Tencent AI Lab, USA
+ABSTRACT
+Self-supervised learning (SSL) models confront challenges of
+abrupt informational collapse or slow dimensional collapse.
+We propose TriNet, which introduces a novel triple-branch
+architecture for preventing collapse and stabilizing the pre-
+training. Our experimental results show that the proposed
+method notably stabilizes and accelerates pre-training and
+achieves a relative word error rate reduction (WERR) of
+5.32% compared to the state-of-the-art (SOTA) Data2vec for
+a downstream benchmark ASR task. We will release our code
+at https://github.com/tencent-ailab/.
+Index Terms— Self-supervised learning, collapse, pseudo
+label, self-learning, bootstrapping
+1. INTRODUCTION
+Self-supervised learning (SSL) models leverage unlabeled
+data, which makes significant advances [1] and reaches per-
+formances almost on par with supervised baselines on many
+downstream tasks such as speech processing [2, 3, 4]. Among
+these models, state-of-the-art contrastive learning methods
+[3, 5, 1, 6] learn to reduce the distance between positive pairs
+of a sample and its distorted version, while increasing the dis-
+tance between negative pairs of different samples. They yield
+good performance with large amounts of contrastive pairs[1],
+which are difficult to mine and computationally intensive for
+training.
+These challenges motivate alternative methods.
+Boot-
+strapping approaches [7, 2, 8] emerge to avoid using negative
+examples. Two networks are used to predict the same repre-
+sentation from augmented pairs. One is the teacher network
+with a stop-gradient (SG) operation (otherwise, a complete
+informational collapse may happen where the learned rep-
+resentations would rapidly collapse towards a single vector
+regardless of the inputs), and the other is the student network
+updating online. Among these approaches, SimSiam [8] sim-
+ply copied the student network’s weights over to the teacher
+network; BYOL [7] updated the teacher network by track-
+ing the exponential moving average (EMA) of the student
+network’s weights. Data2vec[2] also took EMA to update
+† Equal Contribution.
+‡ Contribution made during internship in Tencent
+the teacher network, but it used a masking prediction task
+similar to Wav2vec2[3] by feeding the student network with
+the masked data and the teacher network with the original
+data. Its objective is to predict the averaged embedding of
+several top layers of the teacher network, which is different
+from using only the top layer in BYOL.
+As reported in Data2vec [2], a collapse issue is more pro-
+nounced for speech tasks than computer vision or natural lan-
+guage processing tasks, due to the very correlated adjacent
+targets of the speech modality. It may come from two dif-
+ferent natures [9]: 1) the complete collapse; 2) a slow col-
+lapse like the observation made in [10] that the architectural
+tricks such as BYOL, Data2vec, and SiaSiam are not perfectly
+maintaining the variance of the representations, i.e., very slow
+collapse is still happening with these methods. In this pa-
+per, we propose a novel network structure TriNet, an analogy
+with a three-legged stabilizing stand “Trivet”, to address the
+main challenge for the above joint embedding architectures
+by preventing the complete or slow collapse. After introduc-
+ing related work in Section 2, we summarize our contribu-
+tions. Then we describe the method about its detailed network
+architecture in Section 3.1, its pre-training process in 3.2 and
+regularization objectives in 3.3. Experimental setups and re-
+sults are presented in 4, where we demonstrate our proposed
+method remarkably stabilizes the pre-training and speeds up
+the convergence, and meanwhile achieves a WERR of 5.32%
+compared to the state-of-the-art (SOTA) Data2vec model in a
+benchmark ASR task.
+2. RELATED WORK
+Aside from contrastive methods for preventing informational
+collapse, other main trends are regularization methods for
+maximizing the information content of the embedding to pre-
+vent collapse.
+Recently, various regularization approaches
+are proposed to prevent the collapse in which the embed-
+ding variables contain highly redundant information. Among
+them, W-MSE[11], Barlow-Twinss[12], and VICReg[10] at-
+tempt to produce embedding variables that are decorrelated
+from each other, whereas CorInfoMax[13] does not constrain
+the variables to be uncorrelated but instead avoids covariance
+matrix degeneracy by using log-determinant as a regular-
+izer loss function. However, recent investigations show that
+arXiv:2301.00656v1  [eess.AS]  12 Dec 2022
+
+these regularization terms worked effectively only if given
+specific SSL structural settings [10] and strong data aug-
+mentation [14]. Note that all these regularization methods
+[10, 13, 11, 12] adopt an SSL-no-SG structure, where “no-
+SG” means the branch networks are both learnable with no
+stop-gradient. Instead, optimization of some regularization
+terms together with SSL-SG structures ([7, 8]) was found
+hard[10]. We also empirically observed that adding covari-
+ance regularization terms was not as effective in an SSL-SG
+structure. Data2vec[2] employs the SSL-SG structural tricks
+akin to BYOL[7] and Simsiam [8] that rely on a mechanism
+of normalizing the target to prevent collapse. This strategy
+seems effective but difficult to interpret and may lead to
+instabilities during the training[10, 2].
+Given the above challenges, we are motivated to study
+novel regularization methods that are effective and prac-
+tical for SSL models that are susceptible to complete or
+slow collapse.
+Our idea is also related to a different re-
+search area on pseudo-labeling. BEST-RQ [15] employs a
+random-projection quantizer to generate discrete pseudo la-
+bels. Hubert[4] uses an offline K-means clustering step to
+provide discrete pseudo labels for the masked regions, and
+takes an iterative re-clustering and re-training process. These
+pseudo-labeling methods simplify the SSL targets to the level
+of clusters but essentially require the downstream tasks to
+be at the appropriate clustering level for the model to learn
+well. Another related idea is a combination of SSL and self-
+training [16, 17, 18, 19]. A fine-tuned SSL model [20, 21] or
+a supervised teacher model [22] is used as the initial teacher
+model for pseudo-labeling the unlabeled set. Then a student
+model is trained on the combined labeled and pseudo-labeled
+data. Different from these approaches, our proposed TriNet
+has contributions mainly as follows:
+• In contrast to most other pseudo-labeling approaches,
+TriNet does not require techniques such as K-means clus-
+tering, frame-level alignment, etc.
+For example, unlike
+Hubert [4], which builds a fixed set of discrete target units
+by clustering, TriNet learns the SSL latent embedding space
+and incorporates it to a higher level space for constructing
+target vectors with no limitation on the number of target
+units.
+• Not requiring to distract from any negative samples like
+Wav2vec2 does, nor requiring any statistical assumption
+as the other advanced regularization approaches do (such
+as decorrelation [10] or maximizing log determinant [13]
+which may not always be tenable for the sequences and
+tasks at hand), TriNet instead employs a third branch to
+generate stable and stale target vectors from the sequences
+themselves in the high-level space to construct regulariza-
+tion loss, which acts effectively as barriers against embed-
+ding space degeneracy.
+• We show that our regularization method stabilizes and ac-
+Fig. 1. Illustration of TriNet with its three-legged networks:
+the left and right teacher networks perform in different modes
+to produce representations based on the original input, which
+are then predicted by the same middle network in student
+mode based on a perturbed version of the input.
+Bottom
+is t-SNE visualization of latent embedding of Data2vec and
+TriNet.
+celerates 1 the pre-training and leads to significant perfor-
+mance improvements, with no requirement for more data
+augmentation or larger model capacities.
+To the best of our knowledge, TriNet is the first work that of-
+fers successful regularization and stabilization for speech pro-
+cessing in an SSL-SG framework. Meanwhile, we would like
+to point out that TriNet achieves the above advances provided
+a frozen teacher model, although TriNet will notably surpass
+the frozen teacher, as we will demonstrate in the paper.
+3. METHOD
+3.1. Network Architecture
+As illustrated on top of Fig. 1, the proposed TriNet consists of
+three supporting networks. The middle “leg” represents a stu-
+dent network that simultaneously regresses and predicts tar-
+gets from the left and right teacher networks. The left teacher
+tracks the student parameters and generates the regression tar-
+get, while the right teacher is a frozen fine-tuned model for
+automatic speech recognition (ASR) to generate high-level
+target vectors for stabilizing the whole training. Due to the
+different nature of the targets, we project both the student and
+the right teacher’s embedding to a pseudo-class space.
+1Comparing the pretraining time of SSL model with the frozen teacher to
+that without, while not counting the training of the frozen teacher model.
+
+original input x
+Perturbation
+x
+x
+Encoder
+Encoder
+Encoder
+inEMAteachermode
+instudentmode
+infrozenteachetmode
+trackingstu&entparameters
+Projector
+Projector
+predicting averaged latent
+predicting pseudo
+embeddingof originalinput
+class of original input
+Zstruc.
+z'
+Yreg
+Date2vecemb.
+TriNet emb. z'
+TriNet emb.z"!We mask spans of the input sequence x to generate the
+perturbed sequence x′ and feed it to a standard Conformer en-
+coder [23] of the student. The target zstruc. is constructed by
+encoding the intact input x with the same network but param-
+eterized as an EMA teacher, as shown as the left “leg” in Fig.
+1, and summarizing the teacher’s top-K layer outputs [24, 7].
+This “leg” adopts the same SSL structure as in Data2vec[2]
+and BYOL[7] for a straightforward comparison in this paper,
+whereas alternative SSL structures should be equally applica-
+ble. Meanwhile, TriNet stabilizes the training by introducing
+the third “leg”, as shown as the right branch in Fig. 1, which
+takes the fine-tuned teacher to encode the intact input x and
+generates pseudo target yregul. of the original input data. The
+design prevents the rest joint embedding architectures from
+abrupt or very slow collapse, in which output vectors pro-
+duced by the branches are identical and constant, or end up
+spanning a low-dimension subspace.
+3.2. Pre-training
+In the proposed TriNet, we pretrain the student encoder to
+simultaneously learn contextualized representations of differ-
+ent levels and structural natures. The EMA teacher relies on
+the structural tricks of averaging (including both the moving
+average of model weights and the averaging of top-K layer
+outputs) to keep the prediction targets relatively stable while
+allowing the student to evolve freely and hopefully learn mid-
+level contextual representations. This freedom is a double-
+edged sword, though: the downside is that once the student
+starts to collapse, the EMA teacher will end up collapsing
+albeit very slowly, which will be demonstrated in our experi-
+ment Section 4.2.1.
+To address either abrupt or slow collapse, the third branch
+plays the important role of “anchor” by regularizing and
+avoiding cases in which the student and the EMA teacher
+degenerate together. TriNet employs the frozen teacher to
+provide high-level targets for regularization in a pseudo-class
+space, which is different from the mid-level embedding space
+between the student and the EMA teacher that maintains the
+SSL property. At the bottom of Fig. 1, we use t-SNE [25]
+to visualize the latent embedding generated by Data2vec and
+TriNet. Each point denotes a sample in a random batch and
+its color denotes its class. It indicates that TriNet actually
+arranges intra-class samples, of which the layout gets more
+obvious from z′ of the mid-level embedding space to z′′ for
+the higher-level space, while the inter-class samples scatter
+all over the spaces.
+3.3. Regularization
+Given the predicted latent embedding z′ in the mid-level em-
+bedding space and the contextualized targets zstruc., we use a
+squared L2 norm loss to regress these targets:
+Lstruc. =
+1
+√
+D
+�
+B×T ×D
+(z′
+n − (zstruc.)n)2,
+(1)
+where n is the index of a total of B × T × D elements in a
+batch, and B, T, D are batch, frame, and dimension sizes.
+Given the prediction y′ in the high-level space and the
+pseudo-class targets yregul., we examine and compare two
+kinds of objectives. One is a squared L2 norm for regression:
+Lregre. =
+1
+√
+D
+�
+B×T ×D
+(y′
+n − (yregul.)n)2,
+(2)
+The other is a cross-entropy loss for classification:
+Lregul. =
+1
+√
+D
+CrossEntropy(y′, Softmax(yregul.)),
+(3)
+Our ablation study shows that Lregul. is more effective
+than Lregre.. We consider that is because, in the high-level
+space, Lregul. is a more suitable measure of how well the pre-
+dictions are made by a pseudo-phoneme classification rather
+than a latent embedding regression, again echoing the differ-
+ence of the various-level spaces and their complementary reg-
+ularization effects. Consequently, we adopt L = Lstruc. +
+Lregul. as training objective in our experiments.
+4. EXPERIMENTS
+4.1. Pre-training and Fine-tuning
+We pre-train models on unlabeled data Librispeech [26] that
+contains 960 hours of speech (LS-960h). We fine-tune pre-
+trained models for ASR on the clean 100h (LS-100h) sub-
+set of LS-960h. We evaluate the standard Librispeech dev-
+clean/other and test-clean/other sets.
+We implemented both the proposed model TriNet and the
+reference model Data2vec Base based on Fairseq[23]. For fair
+comparison, both apply the same pre-processing and feature
+extraction: the input 16 kHz waveform is first transformed
+into an 80-dim filter bank to have less memory footprint than
+the raw waveform; it is then processed by a feature extrac-
+tor containing two Convolution-2D subsampling layers with
+576 channels, strides (2,2), and kernel widths (3,3). This re-
+sults in an output sequence of 1/4 of the original length. The
+input is applied with layer norm before sending to the en-
+coder. The dropout and masking strategy for Data2vec and
+the proposed TriNet is identical to [2] and results in approx-
+imately 49% of all time steps masked for a typical training
+sequence. Other hyper-parameters, including annealing rates,
+optimizers, learning rate schedulers, and fine-tuning regimes,
+also follow [2] and otherwise would be described if different.
+The encoder for both Data2vec and TriNet contains 12
+Conformer blocks with 768 hidden dimension and 16 at-
+tention heads. In our experiments, we used the fine-tuned
+
+Fig. 2. Pre-training losses
+Data2vec model on LS-100h as the frozen fine-tuned teacher
+in TriNet to demonstrate that TriNet with no requirement
+for additional data, larger model capacity, or varying model
+structural tricks will surpass the frozen teacher.
+While an
+arbitrary teacher with a heterogeneous architecture may also
+be applicable, we leave the investigation for future work. For
+pre-training, TriNet uses the first 11 Conformer blocks for
+encoding the mid-level latent embedding space as described
+in Sec.3 (blank blocks in Fig. 1), and dedicates the last 1
+Conformer block to the high-level space (blue blocks in Fig.
+1). The overall learnable model size is identical to Data2vec
+Base. Each model was pre-trained for a maximum of 450
+epochs over 2 × 8 GPU V100 cores.
+4.2. Results
+4.2.1. Training Stability and Convergence
+In Fig. 2, the drop of the Data2vec loss from epoch 350 to 450
+actually reflected a slow collapsing case — the downstream
+fine-tuned model has the word error rate (WER, via greedy
+search on the dev-other set) degenerating from 9.949% at
+epoch 300 to 10.25% at epoch 350 and further to 10.432% at
+epoch 400. We chose the best Data2vec checkpoint at epoch
+300 for the comparison in 4.2.2.
+In contrast, we can observe the pre-training loss of TriNet
+is much smoother and stabler than that of Data2vec (light blue
+curve is without smoothing. Note the absolute loss values are
+not directly comparable), indicating that the frozen teacher
+in TriNet works effectively as an “anchor” by providing sta-
+ble and stale regularization and preventing the EMA teacher
+and the student from drifting together toward a collapsed sub-
+space. Meanwhile, TriNet manages to converge within 2/3 of
+the overall epochs of Data2vec.
+4.2.2. Downstream ASR Performance
+Pre-trained models are fine-tuned on LS-100h labeled for
+ASR by mapping the representations via a randomly initial-
+ized linear projection on top of the network into 32 classes
+Table 1. WER (%) on the Librispeech dev/test sets when
+training on the LS-100h labeled.
+Model
+dev
+test
+clean
+other
+clean
+other
+Wav2vec 2.0[3]
+2.7
+7.9
+3.4
+8.0
+Hubert[4]
+2.6
+7.8
+3.4
+8.1
+Data2vec3
+2.61
+6.79
+3.09
+7.07
+TriNet (ablated)
+2.49
+7.16
+2.95
+7.23
+TriNet
+2.45
+6.79
+2.88
+6.48
+representing the vocabulary. Models are optimized by mini-
+mizing a CTC[27] loss.
+Our approach achieves relative word error rate reductions
+(WERRs) of 6.13%/0/6.80%/8.35% (5.32% in average)
+over Data2vec on dev-clean/other and test-clean/other of the
+Librispeech benchmark as shown in Table 1.
+Moreover,
+it achieves WERRs of 35.05%/27.92%/28.14%/25.23%
+(29.09% in average) when training on a 1 hour Libri-light
+(LS-1h) labeled data setup2, reflecting more effectiveness for
+the extremely low-resource setting.
+4.2.3. Ablations
+To exam the different natures of the two spaces, we make
+an ablation study by removing the projectors and spare no
+Conformer layer specific for the high-level target (blue blocks
+in Fig. 1). It turns out the training becomes rather unstable,
+as shown as the orange curve of the training loss in Fig. 2,
+indicating the learning process is dragged zig-zag between the
+two spaces of different natures and can not converge well.
+Another ablation study is on comparing the regulariza-
+tion terms of Lregul. and Lregre.. The second line from the
+bottom of Table 1 indicates the result by replacing Lregul.
+with Lregre..
+Although the result marginally outperforms
+Data2vec, it is much worse than TriNet before ablation. This
+validates the effectiveness of Lregul. being a more suitable
+measure for the high-level space than an MSE loss that is
+suitable for mid-level embedding regression, reflecting the
+complementary nature of the spaces constructed at different
+levels via the triple “legs” in TriNet.
+5. CONCLUSION
+The proposed TriNet addresses challenges of complete or
+slow collapse for joint embedding SSL architectures by em-
+ploying a frozen teacher and a novel architecture. It succeeds
+in stabilizing and accelerating the pre-training and obtaining
+significant WERR gains compared to the SOTA Data2vec
+model in a benchmark ASR task.
+2Using hyper-parameters for Data2vec on LS-100h rather than tuned for
+LS-1h or TriNet.
+3Data2vec Base model reproduced by our implementation based on
+Fairseq
+
+TriNet
+4.6
+TriNet (ablated
+4.2
+Data2vec
+3.8
+3.4
+3
+2.6
+0
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+200
+300
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+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf,len=412
+page_content='TRINET: STABILIZING SELF-SUPERVISED LEARNING FROM COMPLETE OR SLOW  COLLAPSE  Lixin Cao 1†  Jun Wang 1†  Ben Yang 1,2‡  Dan Su1  Dong Yu3  1Tencent AI Lab, China  2 Peking University  3Tencent AI Lab, USA  ABSTRACT  Self-supervised learning (SSL) models confront challenges of  abrupt informational collapse or slow dimensional collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  We propose TriNet, which introduces a novel triple-branch  architecture for preventing collapse and stabilizing the pre-  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Our experimental results show that the proposed  method notably stabilizes and accelerates pre-training and  achieves a relative word error rate reduction (WERR) of  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='32% compared to the state-of-the-art (SOTA) Data2vec for  a downstream benchmark ASR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' We will release our code  at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='com/tencent-ailab/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Index Terms— Self-supervised learning, collapse, pseudo  label, self-learning, bootstrapping  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' INTRODUCTION  Self-supervised learning (SSL) models leverage unlabeled  data, which makes significant advances [1] and reaches per-  formances almost on par with supervised baselines on many  downstream tasks such as speech processing [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Among  these models, state-of-the-art contrastive learning methods  [3, 5, 1, 6] learn to reduce the distance between positive pairs  of a sample and its distorted version, while increasing the dis-  tance between negative pairs of different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' They yield  good performance with large amounts of contrastive pairs[1],  which are difficult to mine and computationally intensive for  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  These challenges motivate alternative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Boot-  strapping approaches [7, 2, 8] emerge to avoid using negative  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Two networks are used to predict the same repre-  sentation from augmented pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' One is the teacher network  with a stop-gradient (SG) operation (otherwise, a complete  informational collapse may happen where the learned rep-  resentations would rapidly collapse towards a single vector  regardless of the inputs), and the other is the student network  updating online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Among these approaches, SimSiam [8] sim-  ply copied the student network’s weights over to the teacher  network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' BYOL [7] updated the teacher network by track-  ing the exponential moving average (EMA) of the student  network’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Data2vec[2] also took EMA to update  † Equal Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  ‡ Contribution made during internship in Tencent  the teacher network, but it used a masking prediction task  similar to Wav2vec2[3] by feeding the student network with  the masked data and the teacher network with the original  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Its objective is to predict the averaged embedding of  several top layers of the teacher network, which is different  from using only the top layer in BYOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  As reported in Data2vec [2], a collapse issue is more pro-  nounced for speech tasks than computer vision or natural lan-  guage processing tasks, due to the very correlated adjacent  targets of the speech modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' It may come from two dif-  ferent natures [9]: 1) the complete collapse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 2) a slow col-  lapse like the observation made in [10] that the architectural  tricks such as BYOL, Data2vec, and SiaSiam are not perfectly  maintaining the variance of the representations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=', very slow  collapse is still happening with these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' In this pa-  per, we propose a novel network structure TriNet, an analogy  with a three-legged stabilizing stand “Trivet”, to address the  main challenge for the above joint embedding architectures  by preventing the complete or slow collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' After introduc-  ing related work in Section 2, we summarize our contribu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Then we describe the method about its detailed network  architecture in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='1, its pre-training process in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2 and  regularization objectives in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Experimental setups and re-  sults are presented in 4, where we demonstrate our proposed  method remarkably stabilizes the pre-training and speeds up  the convergence, and meanwhile achieves a WERR of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='32%  compared to the state-of-the-art (SOTA) Data2vec model in a  benchmark ASR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' RELATED WORK  Aside from contrastive methods for preventing informational  collapse, other main trends are regularization methods for  maximizing the information content of the embedding to pre-  vent collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Recently, various regularization approaches  are proposed to prevent the collapse in which the embed-  ding variables contain highly redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Among  them, W-MSE[11], Barlow-Twinss[12], and VICReg[10] at-  tempt to produce embedding variables that are decorrelated  from each other, whereas CorInfoMax[13] does not constrain  the variables to be uncorrelated but instead avoids covariance  matrix degeneracy by using log-determinant as a regular-  izer loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' However, recent investigations show that  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='00656v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='AS]  12 Dec 2022  these regularization terms worked effectively only if given  specific SSL structural settings [10] and strong data aug-  mentation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Note that all these regularization methods  [10, 13, 11, 12] adopt an SSL-no-SG structure, where “no-  SG” means the branch networks are both learnable with no  stop-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Instead, optimization of some regularization  terms together with SSL-SG structures ([7, 8]) was found  hard[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' We also empirically observed that adding covari-  ance regularization terms was not as effective in an SSL-SG  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Data2vec[2] employs the SSL-SG structural tricks  akin to BYOL[7] and Simsiam [8] that rely on a mechanism  of normalizing the target to prevent collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' This strategy  seems effective but difficult to interpret and may lead to  instabilities during the training[10, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Given the above challenges, we are motivated to study  novel regularization methods that are effective and prac-  tical for SSL models that are susceptible to complete or  slow collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Our idea is also related to a different re-  search area on pseudo-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' BEST-RQ [15] employs a  random-projection quantizer to generate discrete pseudo la-  bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Hubert[4] uses an offline K-means clustering step to  provide discrete pseudo labels for the masked regions, and  takes an iterative re-clustering and re-training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' These  pseudo-labeling methods simplify the SSL targets to the level  of clusters but essentially require the downstream tasks to  be at the appropriate clustering level for the model to learn  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Another related idea is a combination of SSL and self-  training [16, 17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' A fine-tuned SSL model [20, 21] or  a supervised teacher model [22] is used as the initial teacher  model for pseudo-labeling the unlabeled set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Then a student  model is trained on the combined labeled and pseudo-labeled  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Different from these approaches, our proposed TriNet  has contributions mainly as follows:  In contrast to most other pseudo-labeling approaches,  TriNet does not require techniques such as K-means clus-  tering, frame-level alignment, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  For example, unlike  Hubert [4], which builds a fixed set of discrete target units  by clustering, TriNet learns the SSL latent embedding space  and incorporates it to a higher level space for constructing  target vectors with no limitation on the number of target  units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Not requiring to distract from any negative samples like  Wav2vec2 does,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' nor requiring any statistical assumption  as the other advanced regularization approaches do (such  as decorrelation [10] or maximizing log determinant [13]  which may not always be tenable for the sequences and  tasks at hand),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' TriNet instead employs a third branch to  generate stable and stale target vectors from the sequences  themselves in the high-level space to construct regulariza-  tion loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' which acts effectively as barriers against embed-  ding space degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  We show that our regularization method stabilizes and ac-  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Illustration of TriNet with its three-legged networks:  the left and right teacher networks perform in different modes  to produce representations based on the original input, which  are then predicted by the same middle network in student  mode based on a perturbed version of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Bottom  is t-SNE visualization of latent embedding of Data2vec and  TriNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  celerates 1 the pre-training and leads to significant perfor-  mance improvements, with no requirement for more data  augmentation or larger model capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  To the best of our knowledge, TriNet is the first work that of-  fers successful regularization and stabilization for speech pro-  cessing in an SSL-SG framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Meanwhile, we would like  to point out that TriNet achieves the above advances provided  a frozen teacher model, although TriNet will notably surpass  the frozen teacher, as we will demonstrate in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' METHOD  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Network Architecture  As illustrated on top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 1, the proposed TriNet consists of  three supporting networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The middle “leg” represents a stu-  dent network that simultaneously regresses and predicts tar-  gets from the left and right teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The left teacher  tracks the student parameters and generates the regression tar-  get, while the right teacher is a frozen fine-tuned model for  automatic speech recognition (ASR) to generate high-level  target vectors for stabilizing the whole training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Due to the  different nature of the targets, we project both the student and  the right teacher’s embedding to a pseudo-class space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  1Comparing the pretraining time of SSL model with the frozen teacher to  that without, while not counting the training of the frozen teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  original input x  Perturbation  x  x  Encoder  Encoder  Encoder  inEMAteachermode  instudentmode  infrozenteachetmode  trackingstu&entparameters  Projector  Projector  predicting averaged latent  predicting pseudo  embeddingof originalinput  class of original input  Zstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content="  z'  Yreg  Date2vecemb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  TriNet emb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=" z'  TriNet emb." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='z"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='We mask spans of the input sequence x to generate the  perturbed sequence x′ and feed it to a standard Conformer en-  coder [23] of the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The target zstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' is constructed by  encoding the intact input x with the same network but param-  eterized as an EMA teacher, as shown as the left “leg” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  1, and summarizing the teacher’s top-K layer outputs [24, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  This “leg” adopts the same SSL structure as in Data2vec[2]  and BYOL[7] for a straightforward comparison in this paper,  whereas alternative SSL structures should be equally applica-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Meanwhile, TriNet stabilizes the training by introducing  the third “leg”, as shown as the right branch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 1, which  takes the fine-tuned teacher to encode the intact input x and  generates pseudo target yregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' of the original input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The  design prevents the rest joint embedding architectures from  abrupt or very slow collapse, in which output vectors pro-  duced by the branches are identical and constant, or end up  spanning a low-dimension subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Pre-training  In the proposed TriNet, we pretrain the student encoder to  simultaneously learn contextualized representations of differ-  ent levels and structural natures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The EMA teacher relies on  the structural tricks of averaging (including both the moving  average of model weights and the averaging of top-K layer  outputs) to keep the prediction targets relatively stable while  allowing the student to evolve freely and hopefully learn mid-  level contextual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' This freedom is a double-  edged sword, though: the downside is that once the student  starts to collapse, the EMA teacher will end up collapsing  albeit very slowly, which will be demonstrated in our experi-  ment Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  To address either abrupt or slow collapse, the third branch  plays the important role of “anchor” by regularizing and  avoiding cases in which the student and the EMA teacher  degenerate together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' TriNet employs the frozen teacher to  provide high-level targets for regularization in a pseudo-class  space, which is different from the mid-level embedding space  between the student and the EMA teacher that maintains the  SSL property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' At the bottom of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 1, we use t-SNE [25]  to visualize the latent embedding generated by Data2vec and  TriNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Each point denotes a sample in a random batch and  its color denotes its class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' It indicates that TriNet actually  arranges intra-class samples, of which the layout gets more  obvious from z′ of the mid-level embedding space to z′′ for  the higher-level space, while the inter-class samples scatter  all over the spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Regularization  Given the predicted latent embedding z′ in the mid-level em-  bedding space and the contextualized targets zstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=', we use a  squared L2 norm loss to regress these targets:  Lstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' =  1  √  D  �  B×T ×D  (z′  n − (zstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' )n)2,  (1)  where n is the index of a total of B × T × D elements in a  batch, and B, T, D are batch, frame, and dimension sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Given the prediction y′ in the high-level space and the  pseudo-class targets yregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=', we examine and compare two  kinds of objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' One is a squared L2 norm for regression:  Lregre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' =  1  √  D  �  B×T ×D  (y′  n − (yregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' )n)2,  (2)  The other is a cross-entropy loss for classification:  Lregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' =  1  √  D  CrossEntropy(y′, Softmax(yregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' )),  (3)  Our ablation study shows that Lregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' is more effective  than Lregre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='. We consider that is because, in the high-level  space, Lregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' is a more suitable measure of how well the pre-  dictions are made by a pseudo-phoneme classification rather  than a latent embedding regression, again echoing the differ-  ence of the various-level spaces and their complementary reg-  ularization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Consequently, we adopt L = Lstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' +  Lregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' as training objective in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' EXPERIMENTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Pre-training and Fine-tuning  We pre-train models on unlabeled data Librispeech [26] that  contains 960 hours of speech (LS-960h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' We fine-tune pre-  trained models for ASR on the clean 100h (LS-100h) sub-  set of LS-960h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' We evaluate the standard Librispeech dev-  clean/other and test-clean/other sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  We implemented both the proposed model TriNet and the  reference model Data2vec Base based on Fairseq[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' For fair  comparison, both apply the same pre-processing and feature  extraction: the input 16 kHz waveform is first transformed  into an 80-dim filter bank to have less memory footprint than  the raw waveform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' it is then processed by a feature extrac-  tor containing two Convolution-2D subsampling layers with  576 channels, strides (2,2), and kernel widths (3,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' This re-  sults in an output sequence of 1/4 of the original length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The  input is applied with layer norm before sending to the en-  coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The dropout and masking strategy for Data2vec and  the proposed TriNet is identical to [2] and results in approx-  imately 49% of all time steps masked for a typical training  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Other hyper-parameters, including annealing rates,  optimizers, learning rate schedulers, and fine-tuning regimes,  also follow [2] and otherwise would be described if different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  The encoder for both Data2vec and TriNet contains 12  Conformer blocks with 768 hidden dimension and 16 at-  tention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' In our experiments, we used the fine-tuned  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Pre-training losses  Data2vec model on LS-100h as the frozen fine-tuned teacher  in TriNet to demonstrate that TriNet with no requirement  for additional data, larger model capacity, or varying model  structural tricks will surpass the frozen teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  While an  arbitrary teacher with a heterogeneous architecture may also  be applicable, we leave the investigation for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' For  pre-training, TriNet uses the first 11 Conformer blocks for  encoding the mid-level latent embedding space as described  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='3 (blank blocks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 1), and dedicates the last 1  Conformer block to the high-level space (blue blocks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' The overall learnable model size is identical to Data2vec  Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Each model was pre-trained for a maximum of 450  epochs over 2 × 8 GPU V100 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Results  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Training Stability and Convergence  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 2, the drop of the Data2vec loss from epoch 350 to 450  actually reflected a slow collapsing case — the downstream  fine-tuned model has the word error rate (WER, via greedy  search on the dev-other set) degenerating from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='949% at  epoch 300 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='25% at epoch 350 and further to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='432% at  epoch 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' We chose the best Data2vec checkpoint at epoch  300 for the comparison in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  In contrast, we can observe the pre-training loss of TriNet  is much smoother and stabler than that of Data2vec (light blue  curve is without smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Note the absolute loss values are  not directly comparable), indicating that the frozen teacher  in TriNet works effectively as an “anchor” by providing sta-  ble and stale regularization and preventing the EMA teacher  and the student from drifting together toward a collapsed sub-  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Meanwhile, TriNet manages to converge within 2/3 of  the overall epochs of Data2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Downstream ASR Performance  Pre-trained models are fine-tuned on LS-100h labeled for  ASR by mapping the representations via a randomly initial-  ized linear projection on top of the network into 32 classes  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' WER (%) on the Librispeech dev/test sets when  training on the LS-100h labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Model  dev  test  clean  other  clean  other  Wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='0[3]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='0  Hubert[4]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='1  Data2vec3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='61  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='79  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='09  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='07  TriNet (ablated)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='49  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='95  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='23  TriNet  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='45  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='79  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='88  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='48  representing the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Models are optimized by mini-  mizing a CTC[27] loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Our approach achieves relative word error rate reductions  (WERRs) of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='13%/0/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='80%/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='35% (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='32% in average)  over Data2vec on dev-clean/other and test-clean/other of the  Librispeech benchmark as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Moreover,  it achieves WERRs of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='05%/27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='92%/28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='14%/25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='23%  (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='09% in average) when training on a 1 hour Libri-light  (LS-1h) labeled data setup2, reflecting more effectiveness for  the extremely low-resource setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' Ablations  To exam the different natures of the two spaces, we make  an ablation study by removing the projectors and spare no  Conformer layer specific for the high-level target (blue blocks  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' It turns out the training becomes rather unstable,  as shown as the orange curve of the training loss in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' 2,  indicating the learning process is dragged zig-zag between the  two spaces of different natures and can not converge well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  Another ablation study is on comparing the regulariza-  tion terms of Lregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' and Lregre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='. The second line from the  bottom of Table 1 indicates the result by replacing Lregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  with Lregre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='.  Although the result marginally outperforms  Data2vec, it is much worse than TriNet before ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' This  validates the effectiveness of Lregul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' being a more suitable  measure for the high-level space than an MSE loss that is  suitable for mid-level embedding regression, reflecting the  complementary nature of the spaces constructed at different  levels via the triple “legs” in TriNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' CONCLUSION  The proposed TriNet addresses challenges of complete or  slow collapse for joint embedding SSL architectures by em-  ploying a frozen teacher and a novel architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content=' It succeeds  in stabilizing and accelerating the pre-training and obtaining  significant WERR gains compared to the SOTA Data2vec  model in a benchmark ASR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  2Using hyper-parameters for Data2vec on LS-100h rather than tuned for  LS-1h or TriNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='  3Data2vec Base model reproduced by our implementation based on  Fairseq  TriNet  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='6  TriNet (ablated  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='2  Data2vec  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='4  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
+page_content='6  0  100  200  300  4006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAyT4oBgHgl3EQfwvnx/content/2301.00656v1.pdf'}
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etAzT4oBgHgl3EQfaPzl/content/tmp_files/2301.01367v1.pdf.txt

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+The Price of Anarchy in One-Sided Allocation Problems
+with Multi-Unit Demand Agents
+Sissi Jiang
+[email protected]
+Ndiame Ndiaye,
+[email protected]
+Adrian Vetta
+[email protected]
+Eggie Wu
+[email protected]
+January 5, 2023
+Abstract
+We study the one-sided allocation problem with multi-unit demand agents for additive unit-
+sum valuations. The special case of the one-sided matching problem with unit demand agents
+has been studied extensively. The primary focus has been on the folklore Random Priority mech-
+anism and the Probabilistic Serial mechanism, introduced by Bogomolnaia and Moulin [6], with
+emphasis on structural properties, incentives, and performance with respect to social welfare.
+Under the standard assumption of unit-sum valuation functions, Christodoulou et al. [10] proved
+that the price of anarchy is Θ(√n) in the one-sided matching problem for both the Random
+Priority and Probabilistic Serial mechanisms. Whilst both Random Priority and Probabilistic
+Serial are ordinal mechanisms, these approximation guarantees are the best possible even for
+the broader class of cardinal mechanisms.
+To extend these results to the general setting of one-sided allocation problems with multi-
+unit demand agents, we consider a natural cardinal mechanism variant of Probabilistic Serial,
+which we call Cardinal Probabilistic Serial. We present structural theorems for this mechanism
+and use them to show that Cardinal Probabilistic Serial has a price of anarchy of O(√n · log n)
+for the one-sided allocation problem with multi-unit demand agents. This result is near tight.
+1
+Introduction
+In the one-sided matching problem a set of n items must be matched to a set of n agents. This is
+a classical problem in economics and computer science with numerous practical applications, such
+as assigning children to schools, patients to doctors, workers to tasks, social housing to people, etc.
+Consequently, there has been a huge amount of research concerning matching mechanisms, their
+incentive and structural properties, and the social quality of the outcomes they induce. Of course,
+these mechanisms are restricted by the fact that the allocation must be a matching. Equivalently,
+this constraint can be viewed as a restriction to unit demand valuation functions, where each
+agents desires at most one good. Budish et al. [8] highlight the importance of moving beyond unit-
+demand agents. Indeed, in many practical applications the agents have multi-unit demand valuation
+functions. For example, in estate division or the allocation of shifts to employees, university courses
+to students, landing and hanger slots to airlines, etc. This motivates our work: we introduce an
+allocation mechanism for the one-sided allocation problem with multi-unit demand agents and
+analyse the quality of the outcomes it produces with respect to social welfare.
+1
+arXiv:2301.01367v1  [cs.GT]  3 Jan 2023
+
+1.1
+Background
+The one-sided matching problem with indivisible items was formally introduced by Hylland and
+Zeckhauser [13] in 1979, where they studied the competitive equilibrium from equal incomes (CEEI)
+mechanism. This mechanism is envy-free but not strategy-proof and, indeed, early work in the
+economics community focused on the structural and incentive properties of matching mechanisms.
+For example, Zhou [18] gave an impossibility result showing the non-existence of a mechanism that
+is simultaneously strategy-proof, pareto optimal, and symmetric. See [2, 16] for surveys on the
+one-sided matching problem and matching markets more generally.
+Since monetary transfers are typically not allowed in the one-sided matching problem, it be-
+longs to the field of mechanism design without money [15]. A folklore mechanism in this realm
+is Random Priority (RP). Applied to the one-sided matching problem, this mechanism orders the
+agents uniformly at random who then, in turn, select their favourite item that has not previously
+been selected. This mechanism, also popularly known as Random Serial Dictatorship (RSD) [1], is
+strategy-proof.
+Another prominent mechanism is Probabilistic Serial (PS), introduced by Bogomolnaia and
+Moulin [6] in 2001. This is a “consumption” mechanism: to begin, every agent consumes their
+favourite item at the same consumption rate. When the favourite item of an agent is completely
+consumed (that is, together all the agents have consumed exactly one unit of that item) then
+this agent switches to consume its next favorite item, etc. Since its discovery, Probabilistic Serial
+has become the most well-studied mechanism for the one-sided matching problem. It has many
+desirable properties such as envy-freeness and ordinal efficiency when the agents are truthful [6].
+However, unlike Random Priority, it is not strategy-proof and some of its desirable properties fail to
+hold when the agents are strategic [11]. Several extensions to the mechanism have been proposed;
+see, for example, [14, 8, 3]. Aziz et al. studied the manipulability of Probabilistic Serial [5] and
+proved the existence of pure strategy Nash equilibria under the mechanism [4].
+An important recent line of research in the computer science community has been to quantify the
+social welfare of allocations induced by a mechanism in comparison to the optimal obtainable social
+welfare. Two approaches abound in the literature [12, 17]. First is the approximation ratio, where
+agents are assumed to report truthfully to the mechanism. Second, and more interestingly from a
+game-theoretic perspective, is the price of anarchy, where agents are assumed to be strategic [10].
+However, for mechanism design without money, these measures are of little interest without a
+fairness or normalization assumption. As a result, the standard normalization assumption [7, 9,
+10, 12, 17] is that the valuation function of each agent is unit-sum. Specifically, agent i has a
+value v′
+i(j) for item j and �
+j v′
+i(j) = 1. Under the unit-sum assumption, a breakthrough result
+of Christodoulou et al. [10] is that price of anarchy is Θ(√n) for both the Random Priority and
+Probabilistic Serial mechanisms for the one-sided matching problem.
+We remark that both Random Priority and Probabilistic Serial have the characteristic that
+they are ordinal mechanisms. Specifically, rather than requiring the entire valuation function of
+each agent, they need only the preference ordering on the items induced by the valuation function.
+Interestingly, despite being ordinal mechanisms, these bounds are the best possible even for the
+broader class of cardinal mechanisms where agents are required to submit their entire valuation
+function [10].
+2
+
+1.2
+Overview and Results
+The aim of this work is to extend the study of one-sided allocation problems beyond matchings to
+general allocations. Ergo, we allow for agents with multi-unit demand valuation functions rather
+than unit demand valuations.
+In particular, we desire a mechanism with provably good social welfare guarantees. To do this we
+work with cardinal mechanisms rather than ordinal mechanisms. Specifically, we design a cardinal
+variant of the Probabilistic Serial (recall that Probabilistic Serial is an ordinal mechanism): at any
+point in time each agent simultaneously consumes multiple items, with the consumption rates of
+the items weighted in accordance with the cardinal valuation function of the agent. We call this
+the Cardinal Probabilistic Serial (CPS) mechanism and define it formally in Section 3 along with
+examples.
+In Section 4 we present structural theorems for the Cardinal Probabilistic Serial mechanism.
+We use these in Section 5 to prove our main result: the Cardinal Probabilistic Serial mechanism
+has a price of anarchy of O(√n·log n) for the one-sided allocation problem with multi-unit demand
+agents or additive unit-sum valuations. This result is near tight; we show a lower bound of Ω(√n)
+on the price of anarchy holds for fair mechanism1. In Section 6 we show these bounds also apply
+to the price of stability.
+2
+The One-Sided Allocation Problem
+In this section we present the one-sided allocation problem with multi-unit demand agents. There
+is a set I of n agents and a set J of n items. Each agent i ∈ I has a value v′
+i(j) for item j ∈ J.
+The agents have additive multi-unit demands; for any collection S ⊆ J of items, agent i has a
+value v′
+i(S) = �
+j∈S v′
+i(j).2 Furthermore, we assume that valuation functions are unit-sum, that is,
+�
+j∈J v′
+i(j) = 1 for every agent i. Denote by U the set of unit-sum valuation functions.
+Our focus is on direct revelation mechanisms. Given a unit-sum valuation function v′
+i, agent
+i can report to the allocation mechanism M a, possibly non-truthful, unit-sum valuation function
+vi.
+We denote the space of feasible reports the mechanism may receive by V ≡ Un.
+Given a
+set v = {v1, v2, . . . vn} ∈ V of reported valuations, let v−i = {v1, . . . , vi−1, vi+1, . . . vn} be the set
+of reported valuations excluding agent i. We define M(v′
+i; v) = M(v′
+i; {vi, v−i}) to be the true
+value of the bundle of items allocated to agent i by the mechanism M given the report valuations
+v = {vi, v−i}. We say that M(v′
+i; v) is the payoff to agent i. Further, vi is a best response to v−i if it
+maximizes the resultant payoff to agent i, that is, vi = arg maxˆvi∈U M(v′
+i; {ˆvi, v−i}). The reported
+valuation v is a Nash equilibrium if vi is best response to v−i, for every agent i ∈ I.
+Denote
+by NE(v′) the set of valuations which are Nash equilibria with respect to the true valuations
+v′ = {v′
+1, v′
+2, . . . , v′
+n}.
+The social welfare of the allocation given by the mechanism is �
+i∈I M(v′
+i; v). Observe that the
+social welfare is maximized by simply assigning each item to the agent that values it the most.
+Thus the optimal welfare is OPT(v′) = �
+j∈J maxi∈I vi(j). The price of anarchy is the worst-case
+ratio between the optimal welfare and the social welfare of the worst Nash equilibrium, namely
+supv′ supv∈NE(v′)
+OPT(v��)
+�
+i∈I M(v′
+i;v). Similarly, the price of stability is the worst-case ratio between the
+optimal welfare and the social welfare of the best Nash equilibrium,
+1In this context, a mechanism is fair if every agent has the same number of items in expectation.
+2In contrast, for a unit demand agent i, we have v′
+i(S) = maxj∈S v′
+i(j).
+3
+
+3
+The Cardinal Probabilistic Serial Mechanism
+We are now ready to present our allocation mechanism.
+We generalize the ordinal mechanism
+Probabilistic Serial to a cardinal mechanism. In this consumption mechanism, Cardinal Probabilistic
+Serial (CPS), at any time the agents simultaneous consume multiple items rather than just their
+most preferred remaining item. Specifically, at any time, the total consumption rate (speed) of
+an agent over all items is one but this consumption rate is split amongst the remaining items in
+proportion to their value to the agent. Let’s now formalize the mechanism.
+3.1
+A Cardinal Variant of Probabilistic Serial
+Let v = {v1, v2, . . . vn} ∈ V be the reported unit-sum valuations. Each item has a size (quantity) of
+one unit, and each agent i has a consumption rate of 1 at any time. At time t ∈ [0, 1], let rv(t) be
+the set of remaining items, that is, the items that have not yet been entirely consumed. Each agent
+partitions its consumption over the remaining items in proportion to their values. Specifically, at
+time t, the consumption rate of an item j ∈ rv(t) by agent i is denoted cv
+i,j(t). Formally, if j ∈ rv(t)
+and ∃ℓ ∈ rv(t) such that vi(ℓ) > 0 then cv
+i,j(t) =
+vi(j)
+�
+ℓ∈rv(t)
+vi(ℓ).
+If j has already been entirely consumed by time t, that is, j /∈ rv(t), then cv
+i,j(t) = 0. However,
+if the agent has no value for any of the remaining items then any partition of the consumption rate
+over the remaining items is allowed and is consistent. (Indeed, in this case, for our subsequent price
+of anarchy results we may assume the consumption rates are chosen adversarially.)
+Let q: [n] × [0, 1] × V → [0, 1] be the function denoting the quantity of an item j remaining at
+time t given the strategies v. Thus:
+qv
+j (t) = 1 −
+� t
+τ=0
+��
+i∈I
+cv
+i,j(τ)
+�
+dτ
+Observe j ∈ rv(t) ⇔ qv
+j (t) > 0; that is, item j is available at time t if and only if a positive
+quantity of the good remains. In particular, at time 0 we have qv
+j (0) = 1 and rv(0) = J = [n] =
+{1, 2, . . . , n}. Whilst at time 1 we have qv
+j (t) = 0 and rv(t) = ∅. We say that the consumption time
+of item j is the earliest time t at which qv
+j (t) = 0.
+At time 1 we then allocate the items to the agents as follows. Let γi,j to be the total amount
+agent i consumed of item j. Then item j is randomly allocated to agent i with probability γi,j.
+We remark that Cardinal Probabilistic Serial does generalize Probabilistic Serial. Specifically,
+Lemma A.1 in the Appendix formally shows how this cardinal mechanism can simulate the ordinal
+mechanism Probabilistic Serial.
+3.2
+Examples
+The Cardinal Probabilistic Serial mechanism is easy to understand with some examples.
+Example I. First, consider the valuations shown in Figure 1, for three agents (A, B and C) and
+three items (1, 2 and 3).
+At time t = 0 the agents consume the items in proportion to their valuations. For example,
+agent A has a consumption rate of 0.6 for the item 1, 0.3 for the item 2 and 0.1 for the item 3.
+An important observation is that the consumption rates depend only on the set of remaining items
+4
+
+Figure 1: Valuations for the three agents.
+rv(t). In particular, the consumption rates remain constant until the consumption time of the next
+item to be entirely consumed. In this example, that happens at time t = 2
+3 because that is the
+consumption time of item 2. This is illustrated in Figure 2.
+Figure 2: Situation after item 2 has been entirely consumed.
+Because item 2 is no longer available after t = 2
+3, the consumption rates are updated. These
+consumption rates are constant until the consumption time of item 1 at t = 460/501. At this time
+the amount each agent has consumed is illustrated in Figure 3.
+Figure 3: Situation after item 1 has been entirely consumed.
+Now only item 3 remains so each agent consumes it at rate 1. The consumption time of this
+last item is t = 1 and the algorithm terminates.
+At this point, see Figure 4, the agent A has consumed the item with quantities (0.62, 0.2, 0.18),
+respectively. The agent B has consumed the item with quantities (0.15, 0.47, 0.38). And agent C
+5
+
+1
+2
+3
+A
+0.6
+0.3
+0.1
+B
+0.1
+0.7
+0.2
+C
+0.2
+0.5
+0.3At t1 = 2/3
+Consumption Rates after t1
+item 1
+item 2
+item 3
+1
+2
+3
+Quantities Consumed
+0.6
+0.1
+0.0
+1
+2
+3
+1-0.3
+1- 0.3
+A
+0.4
+0.2
+0.06
+0.1
+0.2
+0.0
+B
+0.07
+0.47
+0.13
+1-0.7
+1-0.7
+C
+0.13
+0.33
+0.2
+0.2
+0.3
+0.0
+C
+1-0.5
+1 - 0.5At t2 = 460/501
+item 1
+item 2
+item 3
+Quantities Consumed
+Consumption Rates after t2
+1
+2
+3
+1
+2
+3
+0.62
+0.2
+0.10
+A
+0.0
+0.0
+1.0
+A
+0.30
+B
+0.0
+0.0
+1.0
+B
+0.15
+0.47
+0.23
+0.33
+0.35
+C 
+0.0
+0.0
+1.0
+CFigure 4: Situation after item 3 has been entirely consumed.
+has consumed the item with quantities (0.23, 0.33, 0.44). Thus, item 1 is assigned to agents A, B
+and C with probabilities 0.62, 0.15 and 0.23, respectively, etc.
+Example II. Let there be two agents (A and B) and two items (1 and 2). Let v′
+A = ( 2
+3, 1
+3) and
+v′
+B = ( 1
+3, 2
+3). Thus agent A prefers item 1 and the agent B prefers item 2. If the agents report
+truthfully v = v′ then agent A obtains the bundle (2
+3, 1
+3) for a payoff of (2
+3)2+( 1
+3)2 = 5
+9. Similarly the
+agent B obtains the bundle (1
+3, 2
+3) for a payoff of (1
+3)2 + ( 2
+3)2 = 5
+9. But this is not an equilibrium.
+In particular, if the agent A reports (1, 0) then it will obtain the bundle (3
+4, 1
+4) for a payoff of
+3
+4 · 2
+3 + 1
+4 · 1
+3 = 7
+12 > 5
+9.
+3.3
+The Social Welfare of Equilibria
+Example II shows that Cardinal Probabilistic Serial is not strategy-proof and motivates studying
+strategic agents and Nash equilibria under this mechanism. We are especially interested in calcu-
+lating the price of anarchy of the mechanism. For the ordinal mechanism Probabilistic Serial the
+price of anarchy is known for the one-sided matching problem due to the work Christodoulou et
+al [10].
+Theorem 3.1. [10] For the one-sided matching problem with unit-sum valuations, the price of
+anarchy of Probabilistic Serial is Θ(√n).
+In fact, this guarantee extends beyond Nash equilibria to coarse correlated equilibria and to
+Bayesian settings. Furthermore, they show this guarantee is the best possible.
+Theorem 3.2. [10] For the one-sided matching problem with unit-sum valuations, the price of
+anarchy of any unit-sum mechanism is Ω(√n).
+The aim of this paper is to extend to results of [10] to general one-sided allocation problems
+with multi-unit demand valuations using the Cardinal Probabilistic Serial mechanism. This we
+achieve in Section 5.1 with our main result:
+Theorem 3.3. For the one-sided allocation problem with multi-unit demand agents, the price of
+anarchy of Cardinal Probabilistic Serial is O(√n · log n).
+This result is nearly tight. In particular, in Section 5.2 we show that the lower bound of Θ(√n)
+of [10] can be extended to this setting. The remainder of the paper is dedicated to proving these
+results.
+6
+
+At t3 = 1
+item 1
+item 2
+item 3
+Quantities Consumed
+1
+2
+3
+A
+0.62
+0.2
+0.18
+B
+0.15
+0.47
+0.38
+C
+0.23
+0.33
+0.444
+Single-Minded & Sequential Bidding
+To quantify the price the anarchy we require an understanding of the allocations and payoffs induced
+at a Nash equilibrium v ∈ V. This is difficult to do directly. So a standard approach is, for each
+agent i, to fix the strategies v−i of the other agents and hypothesize about the payoff obtainable if
+the agent plays an alternative strategy. This lower bounds the payoff obtained by the strategy vi
+because it is a best response to v−i. Summing over all agents then gives a lower bound on social
+welfare.
+But what alternative strategies should be considered? In Sections 4.1 and 4.2, we study two
+simple strategies for each agent: single-minded bidding and sequential bidding. We prove structural
+properties of these strategies and these properties to prove a technical lemma in Section 4.3 that
+can be viewed as a generalization to general allocation problems of the main technical lemma of [10]
+for matching problems.
+4.1
+Single-Minded Bidding
+As stated, a natural approach in trying to quantify the social welfare of a Nash equilibrium v is
+to consider alternate strategies for the agents. Of particular importance is single-minded reporting
+where an agent i reports a value 1 for a specific item j and value 0 for every other item. We denote
+this report by ˆuj.
+To analyze this change of strategy, let tj(v) be the minimum value of t such that qv
+j (t) = 0; recall
+this is the consumption time of item j under the Nash equilibrium v. Now denote by ˜tj = tj(ˆuj, v−i),
+the consumption time of item j when agent i bids single-mindedly for j and the other agents −i
+report v−i.
+Two properties of single-minded reporting will be useful. First, regardless of the strategies of the
+other agents −i, the consumption time of item j will be minimized if agent i bids single-mindedly
+on item j. Second, at a Nash equilibrium, if agent i deviates and bids single-mindedly on item j
+then the consumption time of item j can decrease by at most 75%.
+Before proving these two properties we remark that whilst the first property may seem self-
+evident there is a major subtlety due to dynamic knock-on effects. If agent i bids for an item
+ℓ ̸= j this may decrease the completion time of item ℓ resulting in many other agents then bidding
+(defacto) more strongly for item j thus decreasing the consumption time of item j. The key to the
+proof is showing that these indirect knock-on effects do not outweigh the direct effects of bidding
+single-mindedly.
+Lemma 4.1. Given any v−i, the consumption time of item j is minimized when agent i bids
+single-minded for j. That is, minu∈U tj(u, v−i) = ˜tj and argminu∈U tj(u, v−i) = ˆuj
+Proof. Take any agent i ∈ I = [n], any item j ∈ J = [n] and any v ∈ V.
+We remark that
+throughout the proof v−i will be fixed but starting from an arbitrary vi we will shift towards ˆuj.
+Next, let X = {j′ ∈ [n] : vi(j′) > 0, j′ ̸= j} be the set of items, other than j, for which agent i
+reports a positive value. Now label the items of X by increasing consumption time; that is, xk for
+k = 1, . . . , |X| such that txk is increasing.
+The idea behind the proof is to construct a series of valuations {vi = u|X|, u|X|−1, . . . , u1, u0 =
+ˆuj} such that tj(uk−1, v−i) ≤ tj(uk, v−i), for each k ≤ |X|, and where uk−1 has support of cardinality
+one less than uk. Thus the consumption time of item j is less with the single-minded report ˆuj
+than with the report vi. Because the choice of vi was arbitrary the result will follow.
+7
+
+So we begin with u|X| = vi.
+Then, given uk, we define uk−1 as follows.
+Let uk−1
+xk
+= 0,
+uk−1
+j′
+= uk
+j + uk
+xℓ and uk−1
+j′
+= uk
+j′ for any other item j′. For simplicity, we will use the notation
+uk = (uk, v−i) and tk
+j = tj(uk, v−i). Now consider uk−1 and uk, and let T = min
+�
+tk
+xk, tk−1
+j
+�
+. If
+every item has the same consumption time in both uk−1 and under uk then tk−1
+j
+= tk
+j , which implies
+that tk−1
+j
+≤ tk
+j which is what we wanted.
+Otherwise, let t be the smallest time such that the set of items that have been consumed is
+different under both strategies. We wish to show that t = T. By definition, for any τ ∈ [0, t), we
+have r(uk)(τ) = r(uk−1)(τ) because j′ is the first item to be consumed under one strategy but not
+the other. This implies that if either i′ ̸= i or j′ /∈ {j, xℓ} then we have cuk
+i′,j′(τ) =
+vi′(j′)
+�
+ℓ′∈r(uk)(τ) vi′(ℓ′) =
+cuk−1
+i′,j′ (τ). Thus, if j′ /∈ {j, xk}, the quantity of good j remaining at time t is the same for both
+strategies; that is, quk
+j′ (t) = quk−1
+j′
+(t), since the consumption integrals are identical in both cases.
+Consequently, it must be the case that j′ ∈ {j, xk}.
+But the consumption rate cuk−1
+i,xk (τ) = 0 < cuk
+i,xk(τ). Hence, quk
+xk (t) < quk−1
+xk
+(t). So, if j′ = xk
+then its consumption time must be smaller in uk than in uk−1. Similarly, the consumption rate
+cuk−1
+i,j
+(τ) = cuk
+i,j(τ)+cuk
+i,xk(τ) > cuk
+i,j(τ). Hence, q(uk′)
+j
+(t) > quk′−1
+j
+(t). So, if j′ = j then its consumption
+time must be smaller in uk−1 than in uk. This implies t = T = min
+�
+tk
+xk, tk−1
+j
+�
+as desired. Thus,
+we have two cases to consider.
+Case I: tk−1
+j
+≤ tk
+xk.
+Then j must be the first item to be completed in uk−1 before being
+completed in uk. Hence, tk−1
+j
+< tk
+j and the result holds.
+Case II: tk−1
+j
+> tk
+xk. Observe that r(uk)(t) ⊆ r(uk−1)(t) for any time t ∈ [tk
+xk, tk−1
+j
+]. Thus before
+tk−1
+j
+, no item can finish earlier under uk−1 than under uk. This implies that, unless j′ = j and i′ = i,
+we have cuk
+i′,j′(t) ≥ cuk−1
+i′,j′ (t) until j′ is entirely consumed under uk. In particular, let Y be the set of
+items with consumption time in the interval [tk
+xk, tk
+j ] under uk. (Note that {xk, j} ⊆ Y .) Then, only
+these items in Y have a consumption time in [tk
+xk, tk−1
+j
+] under uk−1. Now set CY,k
+i
+(t) = �
+j′∈Y cuk
+i,j′(t)
+and CY,k−1
+i
+(t) = �
+j′∈Y cuk−1
+i,j′ (t). These are the total consumption rates at time t of agent i for items
+in Y under under uk and uk−1, respectively. We claim that CY,k
+i
+(t) ≤ CY,k−1
+i
+(t) for t ∈ [0, tk−1
+j
+].
+To prove this, recall that, for t ∈ [0, tk
+j ], r(uk)(t) ⊆ r(uk−1)(t). Furthermore, r(uk−1)(t)\r(uk)(t) ⊆ Y .
+Denoting r(uk)(t) = R1 and r(uk−1)(t) = R2, and
+� �
+ℓ∈R2
+vi(j′)
+�� �
+j′∈R1
+vi(j′)
+�
+= R3 we obtain:
+8
+
+CY,k−1
+i
+(t) − CY,k
+i
+(t)
+=
+�
+j′∈Y ∩R2
+vi(j′)
+�
+j∗∈R2 vi(j∗) −
+�
+j′∈Y ∩R1
+vi(j′)
+�
+j∗∈R1 vi(j∗)
+=
+�
+j′∈Y ∩R2
+vi(j′)
+�
+j′∈R2
+vi(j′)
+−
+�
+j′∈Y ∩R1
+vi(j′)
+�
+j′∈R1
+vi(j′)
+=
+�
+�
+j′∈Y ∩R2
+vi(j′)
+� �
+�
+j′∈R1
+vi(j′)
+�
+−
+�
+�
+j′∈Y ∩R1
+vi(j′)
+� �
+�
+j′∈R2
+vi(j′)
+�
+�
+�
+j′∈R2
+vi(j′)
+� �
+�
+j′∈R1
+vi(j′)
+�
+=
+�
+�
+j′∈Y ∩(R2\R1)
+vi(j′) +
+�
+j′∈Y ∩(R1∩R2)
+vi(j′)
+�
+�
+j′∈R1
+vi(j′) −
+�
+j′∈Y ∩R1
+vi(j′)
+�
+�
+j′∈R2\R1
+vi(j′) +
+�
+j′∈R1∩R2
+vi(j′)
+�
+�
+�
+j′∈R2
+vi(j′)
+� �
+�
+j′∈R1
+vi(j′)
+�
+But R1 ⊆ R2 and R2 \ R1 ⊆ Y . So we have
+CY,k−1
+i
+(t) − CY,k
+i
+(t)
+=
+�
+�
+j′∈R2\R1
+vi(j′) +
+�
+j′∈Y ∩R1
+vi(j′)
+�
+�
+j′∈R1
+vi(j′) −
+�
+j′∈Y ∩R1
+vi(j′)
+�
+�
+j′∈R2\R1
+vi(j′) + �
+j′∈R1
+vi(j′)
+�
+��
+�
+j′∈R2
+vi(j′)
+� �
+�
+j′∈R1
+vi(j′)
+�
+=
+�
+�
+j′∈R2\R1
+vi(j′)
+� �
+�
+j′∈R1
+vi(j′)
+�
+−
+�
+�
+j′∈Y ∩R1
+vi(j′)
+� �
+�
+j′∈R2\R1
+vi(j′)
+�
+�
+�
+j′∈R2
+vi(j′)
+� �
+�
+j′∈R1
+vi(j′)
+�
+=
+�
+�
+j′∈R2\R1
+vi(j′)
+� �
+�
+j′∈R1\Y
+vi(j′)
+�
+�
+�
+j′∈R2
+vi(j′)
+� �
+�
+j′∈R1
+vi(j′)
+�
+≥ 0
+By definition of Y , every item in Y has consumption time earlier than j under uk. However, before
+the consumption time tk
+j of j under uk, the total consumption rate of all the items in Y is at least
+as large under uk−1 than under uk. Therefore at consumption time tk−1
+j
+of j under uk−1 not every
+9
+
+item of Y has been consumed under uk. In particular, j itself has not been consumed by then
+under uk. So, j is consumed faster in uk−1 than in uk.
+We iterate this argument with k −1 until we get k = 0 and we have computed u0 = ˆuj in which
+only j has non-0 value. This implies that u0 = ˆuj as desired.
+The consumption time of each item in the Nash equilibrium will be denoted as the time tj =
+tj(v). For convenience we denote t0 = 0. Without loss of generality, we relabel the items so that tj
+is increasing with j.
+Let’s now prove the second property.
+Lemma 4.2. Let v be a pure Nash equilibrium. Take any agent i and any item j. The consumption
+time of item j decreases by at most 75% if agent i switches to the single-minded strategy ˆuj, that
+is ˜tj ≥ 1
+4 · tj.
+Proof. Assume that �
+i′̸=i cˆuj
+i′,j(t) < 1 for any time t.
+Then qˆuj
+i (t)
+� 1
+2
+�
+> 0 which implies that
+˜tj > 1
+2 ≥ 1
+4tj. So we may assume there is a smallest time τ such that �
+i′̸=i cˆuj
+i′,j(τ) ≥ 1. We now
+have two cases.
+Case I: τ ≥ 1
+4tj
+By definition, the total consumption rate of item j is positive at time τ. Thus, item j is still
+available at time 1
+4tj. Consequently ˜tj ≥ 1
+4tj.
+Case II: τ < 1
+4tj
+Agent i has a consumption rate 1 for item j under ˆuj until its consumption time. Note that before
+its consumption time the total consumption rate for j is non-decreasing. In particular, before τ
+(phase 1) the total consumption rate of the other agents for j is at most 1. After τ (phase 2) the
+total consumption rate of the other agents for j is at least 1.
+Then agent i consumes τ units of good j in phase 1. In phase 2, the other agents consume j
+at least as fast as i. Thus agent i consumes at most half the remaining amount of good j, which is
+obviously at most 1
+2. So agent i gets at most 1
+4tj + 1
+2 units of good j. This implies the other agents
+get at least 1
+2 − 1
+4tj units of good j.
+Recall from the proof of Lemma 4.1 that r(v)(t) = r(u|X|)(t) ⊆ r(u0)(t) = r(ˆuj)(t) for any time
+t ≤ tj(ˆuj, v−i), that is before the consumption time of item j under ˆuj. This implies that, at
+each point in time, the total consumption rate at which the agents (excluding i) consume item j
+is smaller under ˆuj than under v. In particular, if it takes time t(α) for the agents excluding i to
+consume α units of item j under v then it will take at least t(α) for them to consume α units of j
+under ˆuj. Moreover, recall cv
+i′,j(t) is non-decreasing. Thus, for any β ∈ (0, 1), the time it takes the
+agents excluding i to consume β · α units of good j under v is at least β · t(α).
+Furthermore, if α is the amount of good j that the agents excluding i consume under v then
+t(α) = tj(v). Now set β =
+1
+2 − 1
+4 tj(v)
+α
+. Then because 1
+α ≥ 1 and 1
+4tj(v) ≤ 1
+4 we have that β ≥ 1
+4.
+Moreover, since α ≥ 1
+2 − 1
+4tj(v), we have β ≤ 1. Hence the agents excluding i consume at least
+α · β = 1
+2 − 1
+4tj(v) units in time tj(ˆuj) under ˆuj and they consume α units in time tj(v) under v.
+Thus
+tj(ˆuj) ≥ t(βα) ≥ βt(α) = βtj(v)
+10
+
+and since β ≥ 1
+4 we have:
+βtj(v) ≥ 1
+4tj(v).
+Thus tj(ˆuj) ≥ 1
+4tj(v) as desired.
+4.2
+Sequential Bidding
+Unfortunately, consideration of deviations to single-minded bidding strategies is insufficient to
+prove a good price of anarchy bound for the Cardinal Probabilistic Serial mechanism. Indeed, this
+is intuitively obvious. If an agent wins many items in the optimal solution to the allocation problem
+then a strategy that targets a single item will likely to do very poorly in comparison.
+To circumvent this problem, we consider a second class of strategies, which we term sequential
+bidding. The idea is that an agent has a target set X = {x1, x2, . . . , xk} and, moreover, requests
+to consume the items one-at-a-time in the given order. However, the Cardinal Probabilistic Serial
+is not perfectly compatible with such a sequential request. But it does allow the agents to mimic
+such a strategy with arbitrary precision. To see this, given a finite sequence X = {x1, . . . , xk} and
+ε ∈
+�
+0, 1
+2
+�
+, define the epsilon-valuation ˆuε
+X to be 1 − �k−1
+ℓ=1 εℓ if j = x1, εℓ if j = xℓ for ℓ > 1, and 0
+if j /∈ X.
+Then, given a finite sequence X, we can define the sequential bidding strategy ˆuX = limε→0 ˆuε
+X.
+Under this sequential bidding strategy, at any time t, the agent will consume the first item in X
+that has not yet been entirely consumed. We defer a formal mathematical justification for the
+validity of this construction to Appendix A.
+4.3
+A Technical Lemma
+Using the two properties we obtained for single-minded bidding, we can analyse the consequences
+of deviating to a sequential bidding strategy. Specifically, we prove the following technical lemma.
+Lemma 4.3. For any agent i, let v′
+i be the true value i has for the items and let v be any pure
+Nash equilibrium with respect to v′. Then, for any sequence of items X = {x1, x2, . . . , xk}, it holds
+that:
+CPS(v′
+i; v) ≥ 1
+4 ·
+k
+�
+ℓ=1
+(txℓ − txℓ−1) · v′
+i(xℓ).
+Proof. Recall that we have:
+• tℓ: consumption time of item xℓ in the Nash Equilibrium v.
+• ˜tℓ: consumption time of item xℓ in (ˆuX, v−i) if i where i makes a sequential bid for X.
+We additionally denote the consumption time of item j when an agent switches to a sequential
+strategy as ˆtj = tj(ˆuX, v−i). Moreover, for convenience we denote ˆt0 = 0.
+By Lemma 4.1 we have ˜tj ≥ ˆtℓ. By Lemma 4.2 we have ˆtℓ ≥ 1
+4tℓ. Therefore, ˜tj ≥ 1
+4tℓ. Now
+recall, under the strategy ˆuX, the items of X = {x1, x2, . . . , xk} are ordered in decreasing order of
+value for agent i. This means that before time ˜tℓ the agent consumes an item whose value is at
+11
+
+least v′
+i(xℓ). In particular, because ˜tj ≥ 1
+4tj, if tℓ−1 ≤ tℓ then during the interval [1
+4tℓ−1, 1
+4tℓ] agent
+i consumes an item of value at least v′
+i(xℓ). Therefore, agent i has a payoff of
+CPS(v′; {ˆuX, v−i}) ≥
+k
+�
+ℓ=1
+max
+�1
+4tℓ − 1
+4tℓ−1 , 0
+�
+· v′
+i(xℓ)
+≥ �k
+ℓ=1
+� 1
+4tℓ − 1
+4tℓ−1
+�
+· v′
+i(xℓ) ≥ 1
+4 · �k
+ℓ=1 (tℓ − tℓ−1) · v′
+i(xℓ)
+As v is a Nash equilibrium, it must also give agent i a payoff of at least 1
+4·�k
+i=1 (tℓ − tℓ−1)·v′
+i(xℓ).
+5
+The Price of Anarchy
+We are now ready to quantify the price of anarchy. We begin with the upper bound. We will then
+present a near-matching lower bound.
+5.1
+Upper Bound on the Price of Anarchy
+For an upper bound we formulate the price of anarchy as an optimization program. This optimiza-
+tion program is very difficult to handle directly. So our basic approach will be to apply a series
+of relaxations and simplifications until we obtain a program we can solve. The task is to ensure
+the transformations are consistent with generating upper bounds and that they do not degrade the
+value of the objective function excessively. So let’s now prove our main result.
+Theorem 3.3. For the one-sided allocation problem with multi-unit demand agents, the price of
+anarchy of Cardinal Probabilistic Serial is O(√n · log n).
+Proof of Theorem 3.3. Let {X1, X2, . . . , Xn} be the optimal allocation, where each agent i receives
+the bundle of items Xi =
+�
+xi
+1, . . . , xi
+ki
+�
+. Here we assume the items in Xi are ordered by increasing
+consumption time in the Nash equilibrium v. We can then use Lemma 4.3 to lower bound the social
+welfare of the Nash equilibrium v. To do this first note that, whilst the items of Xi are ordered by
+consumption time they are not ordered by value. In particular, for the lower bound in Lemma 4.3
+we may use the right-to-left maxima of {v′
+i(xi
+1), v′
+i(xi
+2), . . . , v′
+i(xi
+ki)}. This gives a lower bound on
+the social welfare of the Nash equilibrium v of:
+1
+4
+n
+�
+i=1
+ki
+�
+ℓ=1
+�
+max
+ℓ′=ℓ,...,ki
+�
+v′
+i(xℓ′)
+�
+· (txi
+ℓ(v) − txi
+ℓ−1(v))
+�
+12
+
+We may then bound the price of anarchy using the following optimization program:
+min
+1
+4 · �n
+i=1
+�ki
+ℓ=1
+�
+maxℓ′=ℓ,...,ki {v′
+i(xℓ′)} · (txi
+ℓ(v) − txi
+ℓ(v))
+�
+OPT
+(1)
+s.t.
+n
+�
+i=1
+ki
+�
+ℓ=1
+v′
+i,xi
+ℓ =
+OPT
+(2)
+n�
+i=1
+�
+xi
+ℓ : ℓ ∈ [ki]
+�
+=
+[n]
+(3)
+n
+�
+j=1
+v′
+i(j) =
+1
+∀i ∈ [n]
+(4)
+n
+�
+j=1
+vi(j) =
+1
+∀i ∈ [n]
+(5)
+tj(v) ≤
+tj+1(v)
+∀j ∈ [n − 1]
+(6)
+v ∈
+NE(v′)
+(7)
+Let’s understand this optimization program. Constraints (4) and (5) state that for every agent
+v′
+i and vi are unit-sum valuation functions. Constraints (2) and (3) ensure {X1, X2, . . . , Xn} is a
+partition of the items with optimal social welfare (with respect to the true valuation functions v′).
+Next the constraint (6) forces the items to be ordered by increasing consumption time. Finally,
+the constraint (7) states that v is a Nash equilibrium with respect to the true valuations v′. The
+objective function (1) then gives a worst-case bound on the price of anarchy using Lemma 4.3.
+However, this optimization program is difficult to analyze so our task now is to simplify the
+program without weakening the resultant price of anarchy guarantee. To do this, our first step
+is to fix OPT, the social welfare of the optimal solution. (We will later determine the worst case
+values of OPT.) In doing so we may omit the denominator from the objective function (1). Second,
+observe that the bound can only be worse if we relax or remove some of the constraints from the
+optimization program. In particular, let’s omit the Nash equilibrium constraint (7). This gives:
+min
+1
+4 ·
+n
+�
+i=1
+ki
+�
+ℓ=1
+�
+max
+ℓ′=ℓ,...,ki
+�
+v′
+i(xℓ′)
+�
+· (txi
+ℓ(v) − txi
+ℓ(v))
+�
+(8)
+s.t.
+n
+�
+i=1
+ki
+�
+ℓ=1
+v′
+i,xi
+ℓ =
+OPT
+n�
+i=1
+�
+xi
+ℓ : ℓ ∈ [ki]
+�
+=
+[n]
+n
+�
+j=1
+v′
+i(j) =
+1
+∀i ∈ [n]
+n
+�
+j=1
+vi(j) =
+1
+∀i ∈ [n]
+tj(v) ≤
+tj+1(v)
+∀j ∈ [n − 1]
+13
+
+The reader may ask if removing the Nash equilibrium constraint (7) will then render the opti-
+mization program useless. As we will see, the answer is no because implicitly the Nash equilibrium
+conditions have been used in deriving the objective function. Now note that maxℓ′=ℓ,...,ki {v′
+i(xℓ′)} ≥
+v′
+i(xℓ). Thus after removing the Nash equilibrium constraint we may now assume in the worst case
+that the items of Xi are also ordered in decreasing value. That is, the items of Xi =
+�
+xi
+1, . . . , xi
+ki
+�
+decrease in both consumption time and in value. Adding this new constraint (11) then gives the
+program
+min
+1
+4 ·
+n
+�
+i=1
+ki
+�
+ℓ=1
+�
+v′
+i(xℓ) · (txi
+ℓ(v) − txi
+ℓ(v))
+�
+(9)
+s.t.
+n
+�
+i=1
+ki
+�
+ℓ=1
+v′
+i,xi
+ℓ =
+OPT
+n�
+i=1
+�
+xi
+ℓ : ℓ ∈ [ki]
+�
+=
+[n]
+n
+�
+j=1
+v′
+i(j) =
+1
+∀i ∈ [n]
+ki
+�
+ℓ=1
+vi(xi
+ℓ) ≤
+1
+∀i ∈ [n]
+(10)
+tj(v) ≤
+tj+1(v)
+∀j ∈ [n − 1]
+vi(xi
+ℓ) ≥
+vi(xi
+ℓ+1)
+∀i ∈ [n], ∀ℓ ∈ [ki]
+(11)
+Note above that we may replace the unit-sum condition (5) on vi by a constraint (10) only on the
+values of items in Xi.
+For the next step, for each agent i we use a change of variables to {yi
+1, yi
+2 . . . , yi
+ki}. Specifically,
+set yi
+ki = ki·v′
+i(xi
+ki). Then, recursively, set yi
+ℓ = ℓ·
+�
+v′
+i(xi
+ℓ) − v′
+i(xi
+ℓ+1)
+�
+, for each ℓ = {ki−1, . . . , 2, 1}.
+Observe that
+ki
+�
+ℓ=1
+yi
+ℓ =
+ki
+�
+ℓ=1
+ki · v′
+i(xi
+ki) =
+ki
+�
+ℓ=1
+v′
+i(xi
+ℓ)
+In particular, the yi
+ℓ are non-negative and sum to at most one.
+Moreover, we have v′
+i(x(i)
+ℓ ) =
+�k
+ℓ′=ℓ
+yi
+ℓ′
+ℓ′ . Thus we have
+n
+�
+i=1
+ki
+�
+ℓ=1
+�
+v′
+i,xi
+ℓ ·
+�
+txi
+ℓ(v) − txi
+ℓ−1(v)
+��
+=
+n
+�
+i=1
+ki
+�
+ℓ=1
+� ki
+�
+ℓ′=ℓ
+yi
+ℓ′
+ℓ′ ·
+�
+txi
+ℓ − txi
+ℓ−1
+��
+=
+n
+�
+i=1
+ki
+�
+ℓ′=1
+yi
+ℓ′
+ℓ′
+ℓ′
+�
+ℓ=1
+�
+txi
+ℓ − txi
+ℓ−1
+�
+=
+n
+�
+i=1
+ki
+�
+ℓ′=1
+�yi
+ℓ′
+ℓ′ · txi
+ℓ′
+�
+14
+
+So, relabelling ℓ′ as ℓ, we obtain the optimization program:
+min
+n
+�
+i=1
+ki
+�
+ℓ=1
+�yi
+ℓ
+ℓ · txi
+ℓ
+�
+(12)
+s.t.
+n
+�
+i=1
+ki
+�
+ℓ=1
+yi
+ℓ =
+OPT
+(13)
+n�
+i=1
+�
+xi
+ℓ : ℓ ∈ [ki]
+�
+=
+[n]
+ki
+�
+ℓ=1
+yi
+ℓ ≤
+1
+∀i ∈ [n]
+(14)
+yi
+ℓ ≥
+0
+∀i ∈ [n], ∀ℓ ∈ [ki]
+(15)
+tj(v) ≤
+tj+1(v)
+∀j ∈ [n − 1]
+Observe above that, for simplicity, we have removed the factor 1
+4 from the objective function (12).
+We will reincorporate it later.
+Let’s now investigate the structure of the optimal solution to this program. We claim that,
+for each agent i, only one yi
+ℓ need be positive. To see this, assume yi
+ℓ > 0 and yi
+ℓ′ > 0 for ℓ ̸= ℓ′.
+Without loss of generality, let
+txi
+ℓ
+ℓ
+≤
+txi
+ℓ′
+ℓ′ . Then replacing yi
+ℓ by yi
+ℓ + yi
+ℓ′ and replacing yi
+ℓ′ by 0
+decreases (or keeps constant) the objective function. We may enforce this by adding a constraint
+denoting ℓi to be the index which minimizes
+txi
+ℓ
+ℓ .
+min
+n
+�
+i=1
+�
+yi
+ℓ ·
+txi
+ℓ
+ℓi
+�
+(16)
+s.t.
+n
+�
+i=1
+ki
+�
+ℓ=1
+yi
+ℓ =
+OPT
+n�
+i=1
+�
+xi
+ℓ : ℓ ∈ [ki]
+�
+=
+[n]
+ℓi =
+argmaxℓ∈[ki]
+txi
+ℓ
+ℓ
+∀i
+(17)
+yi
+ℓi ≤
+1
+∀i ∈ [n]
+(18)
+yi
+ℓi ≥
+0
+∀i ∈ [n]
+(19)
+tj(v) ≤
+tj+1(v)
+∀j ∈ [n − 1]
+We may now apply a similar trick over pairs of agents.
+Assume there are two agents i, i′ and
+two indices ℓi, ℓi′ such that both yi
+ℓi and yi′
+ℓi′ are non-integral, that is, yi
+ℓi ∈ (0, 1) and yi′
+ℓi′ ∈ (0, 1).
+Without loss of generality, let
+txi
+ℓi
+ℓi ≤
+txi′
+ℓi′
+ℓ′
+i′ . Then replacing yi
+ℓi by yi
+ℓi′ +δ and replacing yi′
+ℓi′ by yi′
+ℓi′ −δ,
+for some small δ, decreases (or keeps constant) the objective function. Note that this is a feasible
+change because constraint (13) is remains tight and constraints (18) and (19) still hold. Moreover,
+15
+
+setting δ = min
+�
+1 − yi
+ℓi, yi′
+ℓi′
+�
+forces either yi
+ℓi or yi′
+ℓi′ to become integral. But this implies there is
+an optimal solution in which exactly one yi
+ℓi is non-integral.
+In particular, let k = ⌊OPT⌋. Then we may relabel the agents so that yi
+ℓi = 1 for each 1 ≤ i ≤ k,
+yi
+ℓk+1 = OPT − k, and yi
+ℓi = 0 for each k + 2 ≤ i ≤ n. Thus our problem simplifies to:
+min
+k
+�
+i=1
+txi
+ℓi
+ℓi
++ (OPT − k) ·
+txk+1
+ℓk+1
+ℓk+1
+s.t.
+n�
+i=1
+�
+xi
+ℓ : ℓ ∈ [ki]
+�
+=
+[n]
+Of course, we can further reduce the objective function by removing its second term. This gives:
+min
+k
+�
+i=1
+txi
+ℓi
+ℓi
+s.t.
+n�
+i=1
+�
+xi
+ℓ : ℓ ∈ [ki]
+�
+=
+[n]
+To evaluate this, recall that the items are labelled in increasing order of consumption time.
+These consumption times then satisfy the following property.
+Lemma 5.1. The consumption time of item j must satisfy tj ≥ j
+n.
+Proof. Each agent has a total consumption rate of 1. Consequently, the total consumption rate of
+all agents is n. Thus at time t = j
+n the number of units consumed of all goods is exactly j. But the
+quantity of each good is each is exactly 1, so at most j goods can have been completely consumed
+at time t = j
+n. Hence the consumption time of good j is tj ≥ j
+n.
+Next we partition the agents into groups depending upon their ℓi.
+Specifically, let Iτ =
+�
+i ∈ [k] : ℓi ∈ [2τ, 2τ+1)
+�
+for all 0 ≤ τ ≤ ⌈log n⌉.
+Further, for each agent i ≤ k we let Ti be
+the consumption time of xi
+ℓi. We then order the agents of Iτ by increasing Ti. We use the notation
+iτ
+q to denote the qth agent of Iτ in this ordering.
+In particular, by the time xi
+ℓi is consumed for i = iτ
+q, at least 2τ · q items have been consumed.
+Thus, by Lemma 5.1, the consumption time of this item is tx
+ℓ
+iτq
+iτq
+≥ 2τ·q
+n . In particular,
+t
+x
+iτq
+ℓiτq
+ℓiτq
+≥ 2τ · q
+n
+· 1
+ℓiτq
+≥ 2τ · q
+n
+·
+1
+2τ+1 =
+q
+2n
+(20)
+16
+
+We can now obtain a useful bound on the value of the optimization program.
+k
+�
+i=1
+txi
+ℓi
+ℓi
+≥
+⌈log n⌉
+�
+τ=0
+|Iτ|
+�
+q=1
+q
+2n
+=
+⌈log n⌉
+�
+τ=0
+|Iτ| · (|Iτ| + 1)
+4n
+≥
+1
+4n ·
+⌈log n⌉
+�
+τ=0
+|Iτ|2
+≥
+1
+4n ·
+max
+0≤τ≤⌈log n⌉ |Iτ|2
+≥
+1
+4n ·
+�
+k
+log n + 1
+�2
+=
+k2
+4n log2 n + o (n log n)
+We are now ready to prove our price of anarchy upper bound.
+By applying Lemma 4.3, with
+X = [n], at any Nash equilibrium each agent is guaranteed a payoff of at least
+1
+4n. Thus the social
+welfare of any Nash equilibrium is at least 1
+4. The price of anarchy is then at most
+max
+k∈[1,n]
+�
+min{k
+1
+4
+,
+k
+1
+4 ·
+k2
+4n log2 n+o(n log n)
+}
+�
+≤ max
+k∈[1,n]
+�
+min
+�
+4k , (16n log2 n + o (n log n))/k
+��
+= 2√n · log n + o
+��
+n log n
+�
+So, the price of anarchy of the Cardinal Probabilistic Serial mechanism is O (√n · log n).
+We remark that this proof is for pure Nash equilibria. However, as we show in Appendix B, the
+result extends to mixed Nash equilibria and to coarse correlated equilibria. Furthermore, mixed
+Nash equilibria and coarse correlated equilibria are guaranteed to exist in this model.
+5.2
+Lower Bound of the Price of Anarchy
+For the lower bound, we verify that Theorem 3.2 extends to the one-side allocation problem with
+multi-unit demand agents.
+Theorem 5.2. For the one-side allocation problem, the pure price of anarchy of any unit-sum
+mechanism is Ω(√n).
+Proof Sketch. Consider the example used by Christodoulou et al. [10] to prove Theorem 3.2 for the
+matching problem. Take the following valuation function:
+vi(j) =
+�
+1
+n + ε if i = j · √n + i′ for i′ = 1, . . . , √n
+1
+n −
+ε
+n−1 otherwise
+17
+
+Now consider a Nash equilibrium for v. Let ij be the index of the agent who has positive value for
+item j but has the smallest probability of being assigned j in the Nash Equilibrium. Next, create
+a new valuation v′
+i(j) which is vi(j) if i ̸= ij′ for any j′ and which is 1 if i = ij and 0 if i = ij′ ̸= ij
+Since the agents get the same number of items in expectation, a Nash equilibrium for v is also a
+Nash equilibrium for v′ where the agents maximize their probability of getting their favorite item.
+The social welfare of the optimal allocation is √n. At the Nash equilibrium, since the agents ij get
+assigned j with probability at most
+1
+√n, the social welfare is at most √n ·
+1
+√n + √n ·
+�
+1 −
+1
+√n
+�
+·
+� 1
+n + 1
+n3
+�
+≤ 3. This gives a lower bound of Ω(√n) on the price of anarchy.
+5.3
+Tightness of Proof Methodology
+We conjecture that the lower bound is tight; that is, that the price of anarchy is Θ(√n). Thus, we
+believe the log n term in Theorem 3.3 is superfluous. However, in Appendix C, we show that the
+tools utilized in this paper are not strong enough to prove such a result. Ergo, to prove a tight
+price of anarchy result the bounds one the value of each agent need to be (slightly) improved.
+6
+The Price of Stability
+We obtain similar bounds for the price of stability.
+Theorem 6.1. For the one-sided allocation problem with multi-unit demand agents, the price of
+stability of of Cardinal Probabilistic Serial is at least Ω(√n) and at most O(√n · log n).
+Here the upper bound follows immediately from our price of anarchy bound. The lower bound
+is given in Appendix D.
+7
+Conclusion
+We studied a cardinal variant of the Probablistic Serial mechanism and provided corresponding
+lower and upper bounds on the price of anarchy for the one-sided allocation problem with multi-unit
+demand agents. Removing the logarithmic gap between these bounds is a natural open problem.
+Another interesting line of research is to study whether these results can be extended to agents
+with non-additive valuation functions.
+References
+[1] A. Abdulkadirogl and T. Sonmez.
+Random serial dictatorship and the core from random
+endowments in house allocation problems. Econometrica, 66(3):689–701, 1998.
+[2] A. Abdulkadirogl and T. Sonmez.
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+Econometric Society Monographs, page 3–47. Cambridge University Press, 2013.
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+Operations Research, 68(2):467–479, 2020.
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+the probabilistic serial rule.
+In Proceedings of 24th International Conference on Artificial
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+priority and beyond.
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+Algorithmic Game Theory (SAGT), pages 1–12, Berlin, Heidelberg, 2014. Springer Berlin
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+A solution to the random assignment problem on the full
+preference domain. Journal of Economic Theory, 131(1):231–250, 2006.
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+ceedings of 10th Conference on Electronic Commerce (EC), pages 177–186, 2009.
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+Theory, 52(1):123–135, 1990.
+19
+
+A
+Appendix: Epsilon Strategies and Sequential Bidding
+In this section we show that an agent can mimic the sequential bidding strategy with arbitrary
+precision using epsilon-valuation strategies. Recall, given a sequence X of length k, the epsilon-
+strategy ˆuε
+X ∈ U is defined by
+ˆuε
+X(j) =
+�
+�
+�
+�
+�
+1 − �k−1
+ℓ=1 εℓ
+if j = x1
+εℓ
+if j = xℓ
+0
+otherwise
+The limit of the epsilon-strategy ˆuε
+X when ε → 0 is the sequential strategy ˆuX.
+Here we will
+formally justify allowing the sequential strategy in the mechanism.
+Lemma A.1. ∀i ∈ [n], ∀v ∈ V, ∀δ > 0, ∃ε > 0 such that CPS(v′
+i, (ˆuε
+X, v−i)) ≥ (1 − ε) ·
+CPS(v′
+i, (ˆuX, v−i)).
+That is, the payoff of the epsilon-strategy is within δ of the payoff of the
+sequential strategy.
+Proof. For convenience, in the proof we assume that the order of the items in the sequential strategy
+is the same as the completion time. If that is not the case for some item j, then compared to the
+case where i is using an epsilon-strategy, the difference between the amount consumed in both cases
+is at most ε while the difference between the amount consumed by the remaining agents is bounded
+by the difference between the consumption times of the items which precede, which is bounded by
+the rest of our proof.
+Let ∆j = tj − tj−1 with ∆1 = t1 in the sequential strategy. Let ∆′
+j = τj − τj−1 with ∆1 = τ1
+where τj is the consumption time of j in the epsilon-strategy. Let i′ be any agent and i be the
+agent that changes strategy.
+Then we have that for the sequential strategy:
+∆j =
+1 − �j−1
+j′=1 ∆j′ · (�n
+i′=1,i′̸=i
+vi′(j)
+1−�j′
+˜j=1 vi′(˜j))
+1 + �
+i′̸=i
+vi′(j)
+1−�j−1
+˜j=1 vi′(˜j)
+.
+For the epsilon-strategy:
+∆′
+j
+=
+1 − �j−1
+j′=1 ∆′
+j′ · (�n
+i′=1,i′̸=i
+vi′(j)
+1−�j′−1
+˜j=1 vi′(˜j)) − �j−1
+j′=1 ∆′
+j′ · (εj−j′fj)
+fj + �
+i′̸=i
+vi′(j)
+1−�j−1
+˜j=1 vi′(˜j)
+,
+where
+fj =
+εj
+1 − 1 + �k
+j′=1 εj′ − �j−1
+j′=1 εj′ =
+1
+�k−j
+j′=0 εj′ = 1 −
+�k−j
+j′=1 εj′
+�k−j
+j′=0 εj′ .
+Now let A(j)
+j′ be the consumption rate of j by agents aside from i when the items up to j′ have
+been consumed. That is:
+A(j)
+j′ =
+n
+�
+i′=1,i′̸=i
+vi′(j)
+1 − �j′
+˜j=1 vi(˜j)
+20
+
+Then we can simplify the expressions for ∆j and ∆′
+j to
+∆j =
+1 − �j−1
+j′=1 ∆j′ · A(j)
+j′
+1 + A(j)
+j
+∆′
+j =
+1 − �j−1
+j′=1 ∆′
+j′ · (A(j)
+j′ + εj−j′ · fj′)
+fj + A(j)
+j
+This gives us the following equation:
+∆j =
+1 − �j−1
+j′=1 ∆j′ · A(j)
+j′
+1 + A(j)
+j
+=
+(fj + A(j)
+j ) · (1 − �j−1
+j′=1 ∆j′ · A(j)
+j′ )
+(1 + A(j)
+j ) · (fj + Aj)
+=
+(fj − 1) · (1 − �j−1
+j′=1 ∆j′ · A(j)
+j′ )
+(1 + A(j)
+j ) · (fj + Aj)
++
+(1 + A(j)
+j ) · (1 − �j−1
+j′=1 ∆j′ · A(j)
+j′ )
+(1 + A(j)
+j ) · (fj + Aj)
+=
+(fj − 1) · (1 − �j−1
+j′=1 ∆j′ · A(j)
+j′ )
+(1 + A(j)
+j ) · (fj + Aj)
++
+1 − �j−1
+j′=1 ∆j′ · A(j)
+j′
+fj + Aj
+So, when taking ∆j − ∆′
+j we get the following:
+∆j − ∆′
+j =
+(fj − 1) · (1 − �j−1
+j′=1 ∆j′ · A(j)
+j′ )
+(1 + A(j)
+j ) · (fj + Aj)
++
+1 − �j−1
+j′=1 ∆j′ · A(j)
+j′
+fj + Aj
+−
+1 − �j−1
+j′=1 ∆′
+j′ · (A(j)
+j′ + εj−j′ · fj′)
+fj + A(j)
+j
+=
+(fj − 1) · (1 − �j−1
+j′=1 ∆j′ · A(j)
+j′ )
+(1 + A(j)
+j ) · (fj + Aj)
++
+�j−1
+j′=1 ∆′
+j′ · (A(j)
+j′ + εj−j′ · fj′) − ∆j′ · A(j)
+j′
+fj + A(j)
+j
+Remark that 1 ≥ fj ≥ 1 − ε and that �j
+j′=1 ∆′
+j < 1 So, by taking the absolute value, we get:
+21
+
+|∆j − ∆′
+j| ≤
+������
+(fj − 1) · (1 − �j−1
+j′=1 ∆j′ · A(j)
+j′ )
+(1 + A(j)
+j ) · (fj + Aj)
+������
++
+������
+�j−1
+j′=1 ∆′
+j′ · (A(j)
+j′ + εj−j′ · fj′) − ∆j′ · A(j)
+j′
+fj + A(j)
+j
+������
+≤
+|fj − 1|
+(1 + A(j)
+j ) · (fj + A(j)
+j )
++
+����j−1
+j′=1(∆′
+j′ − ∆j′) · A(j)
+j′
+���
+fj + A(j)
+j
++
+�����
+�j−1
+j′=1 ∆′
+j′ · εj−j′ · fj′
+fj + A(j)
+j
+�����
+≤ ε
+1
+2
++
+������
+|
+j−1
+�
+j′=1
+(∆′
+j′ − ∆j′)
+������
+·
+A(j)
+j′
+fj + A(j)
+j
++ ε ·
+fj′
+fj + A(j)
+j
+≤ 3ε +
+j−1
+�
+j′=1
+��∆′
+j′ − ∆j′
+��
+So we get |∆j − ∆′
+j| ≤ 3jε.
+So, the set of remaining items only changes for a time of at most �j
+j′=1 |∆j′ − ∆′
+j′| ≤ 3j2ε
+which implies that the payoff for the agents i′ ̸= i changes by at most 3j2ε since they have unit-
+sum valuations.
+On the other hand, for i when both mechanisms agree on the set of remaining items i only
+changes the item they are consuming by less than 2ε, in particular, if we sum the difference between
+what is consumed when the mechanisms disagree and when they agree i’s consumption only changes
+by at most 2ε + 3j2ε ≤ 4j2ε. Since i has a unit-sum valuation, the change in the payoff is at most
+4j2ε.
+By setting ε =
+δ
+4j2 we get the result we wanted.
+B
+Appendix: Mixed Strategies
+Here we show that our main result, the upper bound on the price of anarchy for pure stragey Nash
+Equilibria, also applies to mixed strategy Nash equilibria and coarse correlated equilibria. To show
+this we use the following definitions and notations. A mixed strategy for an agent i is a probability
+distribution of U and is denoted as pi : U → [0, 1]. The mixed strategy used by every agent is
+denoted as p: V → [0, 1] with p(v) = �n
+i=1 pi(vi).
+Theorem B.1. The price of anarchy of coarse correlated equilibria is O(√n·log n) in the one-sided
+allocation problem with multi-unit demand agents.
+Proof. Recall Lemma 4.3states that for any pure Nash equilibrium v and for any sequence of items
+X = {x1, x2, . . . , xk}:
+CPS(v′
+i; v) ≥ 1
+4
+k
+�
+ℓ=1
+(txℓ − txℓ−1) · v′
+i(xℓ).
+22
+
+To prove this, we bounded the payoff i obtained by deviating to the sequential bidding strategy.
+This also applies for mixed strategies. In particular, if the xℓ are ordered by i’s value for them,
+then by deviating to the sequential strategy from any pure strategy agent i can consume item xℓ
+from time 1
+4 maxℓ′=1,...,ℓ−1 txℓ′ to time 1
+4 maxℓ′=1,...,ℓ txℓ′. By the linearity of the expectation, this
+gives the following bound for a mixed strategy:
+E
+� ki
+�
+ℓ=1
+v′
+i(xℓ) ·
+�1
+4
+max
+ℓ′=1,...,ℓ txℓ′ − 1
+4
+max
+ℓ′=1,...,ℓ−1 txℓ′
+��
+≥
+1
+4 ·
+ki
+�
+ℓ=1
+v′
+i(xℓ) ·
+�
+E
+�
+max
+ℓ′=1,...,ℓ txℓ′
+�
+− E
+�
+max
+ℓ′=1,...,ℓ−1 txℓ′
+��
+We can now use this bound and apply the same proof as in Theorem 3.3 to obtain the same upper
+bound on the price of anarchy for mixed equilibria. A similar argument applies for coarse correlated
+equilibria.
+We remark that mixed Nash equilibria and coarse correlated equilibria are guaranteed to exist.
+C
+Appendix: Tightness of Proof Methodology
+As discussed, we conjecture that the price of anarchy of Cardinal Probabilistic Serial is Θ(√n)..
+However, we show here that the tools utilized in this paper are not strong enough to prove such
+a result. In particular, the bound from Lemma 5.1is too loose and so will induce a logarithmic
+term in the upper bound. Namely, assuming that the items are consumed in increasing order, then
+substituting tj by j/n will lead to the appearance of a log factor.
+Lemma C.1. O(√n · logO(1) n) is a tight bound when bounding tj below by j/n.
+Proof. Consider the following example. There are x · k agents. For each z = 0, . . . , x − 1, there are
+exactly k agents who are assigned 2z items in the optimal allocation and have value 1
+2z for each item.
+Hence there are n = (2x −1)·k items. Setting k = (2x −1)/x2, we have n = (xk)2 = ((2x −1)2)/x2.
+Note that each agent will individually consume the items they are meant to be assigned at
+a rate of at most 1/n so unless other agents consume these the consumption time will be 1. In
+particular, this implies that using the remaining n − z · k agents, we can choose the order in which
+the items are consumed. So, assume that the items of higher value, that is those assigned to agents
+with smaller bundles, are consumed faster.
+Let Iz be the set of agents who receive 2z items in the optimal allocation. Let Jz be the set of
+items assigned to agents in Iz. Let prec(j) be the set of items that have been consumed before or
+at the same time as j (including j). Then, for any j ∈ Jz, an upper bound on the number of items
+that have been consumed before j, is the number of items that are assigned to agents with at most
+2z items. That is:
+|prec(j)| ≤
+�����
+z�
+z′=0
+Jz′
+����� =
+z
+�
+z′=0
+2z′ · k = (2z+1 − 1) · k ≤ 2z · k
+In particular, by denoting Xi =
+�
+xi
+1, . . . , xi
+2z
+�
+to be the set of items i ∈ Iz gets, the bound we
+get for the value of the allocation when substituting the time by prec(j)/n is the following:
+23
+
+x
+�
+z=0
+�
+i∈Iz
+1
+2z · supℓ=1,...,2z prec(x(i)
+ℓ )
+n
+≤
+x
+�
+z=0
+|Iz| · 1
+2z · 2z+1 · k
+n
+=
+x
+�
+z=0
+2k2
+n
+=
+2xk2
+(xk)2 = 2
+x
+This means that our bound will only prove O(x√n) and given that n =
+(2x−1)2
+x2
+, we get that
+x = logO(1)(n).
+D
+Appendix: The Price of Stability
+Here we study the price of stability. To do this, we say that a strategy u ∈ U is a safety strategy for
+agent i if ∀v ∈ V the allocation output on input (u, v−i) gives i k
+n of its top k items in expectation.
+For the one-sided matching problem under Random Priority and Probabilistic Serial, truthtelling
+is known to be a safety strategy.
+Similar to the price of anarchy, the price of stability is the worst case ratio between the optimal
+welfare and the social welfare of the best Nash equilibrium, namely:
+sup
+v′
+inf
+v∈NE(v′)
+OPT(v′)
+�
+i∈I M(v′
+i; v)
+Interestingly, the existence of safety strategies induces the following bound on the price of
+stability for the one-sided matching problem:
+Theorem D.1 ([10]). For the one-sided matching problem, the pure price of stability of any mech-
+anism with a safety strategy is Ω(√n).
+As we did for the lower bound on the price of anarchy of general mechanisms, first we show
+that this lower bound extends to our setting.
+Theorem D.2. For the one-sided allocation problem, the pure price of stability of any mechanism
+with a safety strategy is Ω(√n).
+Proof. The example used by Christodoulou et al. [10] to prove Theorem D.1 for matchings suffices.
+Consider the following valuation:
+vi(j) =
+�
+�
+�
+�
+�
+�
+�
+1
+if i = j ≤ √n
+1
+√
+(n)
+if i > √n ≥ j
+0
+otherwise
+Then clearly the optimal allocation is to assign an item to the agent who has value 1 for it, if
+possible, and to assign the remaining items in any way. Denoting pi,j to be the probability assigning
+j to i, then:
+�
+i∈[n]
+�
+j∈[√n]
+pi,j = √n.
+24
+
+However, since the mechanism has a safety strategy, the agents who are matched in the optimal
+solution get their top item with probability at least 1/n so, we get the following bound on the
+contribution of the remaining agents to the social welfare of the Nash equilibrium:
+�
+i∈[n]\[√n]
+�
+j∈[√n]
+pi,j · 1
+√n ≤
+1
+√n ·
+�√n − 1
+√n
+�
+≤ 1
+On the other hand, the agents who do not get matched can get their top √n items with
+probability at least
+1
+√n, so we get the following bound on the contribution of the matched agents
+to the social welfare of the Nash Equilibrium:
+�
+i∈[√n]
+�
+j∈[√n]
+pi,j ≤ √n − (n − √n) · 1
+√n = 1
+So, the contribution of all the agents to the social welfare of the Nash equilibrium is at most 2.
+But the optimal allocation clearly has value √n.
+So, if we can show that Cardinal Probabilistic Serial has a safety strategy then we get a lower
+bound of Ω(√n) on the price of stability. However, interestingly, unlike it’s ordinal counterpart,
+truthtelling is not a safety strategy.
+Lemma D.3. Truthtelling is NOT a safety strategy for Cardinal Probabilistic Serial.
+Proof. Assume that vi(j) = 1, vi(j) = 0, v1(1) = 1 − (n − 1)ε and v1(j) = ε for any i ̸= 1 and
+j ̸= 1. Then if agent 1 is truthful it has a probability less than 1/n of getting item 1, which is its
+top item. So, truthtelling is not a safety strategy.
+Nonetheless, we can find a safety strategy for Cardinal Probabilistic Serial.
+Lemma D.4. Cardinal Probabilistic Serial has a safety strategy.
+Proof. This follows directly from Lemma 5.1by considering the sequential strategy with Xi = [n].
+Before time j/n, at most j − 1 items have been consumed so under the sequential strategy agent i
+is consuming from one of their j favorite items. This is what we need.
+Corollary D.5. For the one-sided allocation problem, the price of stability of Cardinal Probabilistic
+Serial is Ω(√n) and O (√n · log n)
+Proof. So this game has a safety strategy by Corollary D.4 The lower bound then follows Theo-
+rem D.2 which states that any mechanism with a safety strategy has a price of stability of Ω(√n).
+The upper bound follows from Theorem 3.3because the price of anarchy upper bounds the price of
+stability.
+25
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+page_content=' Marquette, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' N’Diaye, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Briot (eds)  COMMISSION FEMMES ET ASTRONOMIE DE LA SF2A :  WOMEN PARTICIPATION IN FRENCH ASTRONOMY  Rhita-Maria Ouazzani1, Caroline Bot2, Sylvie Brau-Nogu´e3, Danielle Briot4, Patrick de Laverny5,  Nad`ege Lagarde6, Nicole Nesvadba7, Julien Malzac8, Isabelle Vauglin9 and Olivia Venot10  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  The Commission Femmes et Astronomie conducted a statistical study that aims at mapping the  presence of women in French professional Astronomy today, and set a starting point for studying its evolution  with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  For the year 2021, we proceeded with a sub-set of 8 astronomy and astrophysics institutes,  hosting a total of 1060 employees, among which PhD students, post-doctoral researchers, and academic,  technical, and administrative staff, representing around 25% of the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' We have investigated how  the percentage of women vary with career stage, level of responsibility, job security, and level of income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The  results of this preliminary study seem to illustrate the leaky pipeline, with one major bottleneck being the  access to permanent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' It appears that the proportion of women steadily decreases with the security  of jobs, with the career stage, with the qualification level and with the income level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  Keywords:  Astronomy & Astrophysics, Gender inequalities, Career  1  Introduction  The Commission Femmes et Astronomie of the SF2A (Soci´et´e Fran¸caise d’Astronomie et d’Astrophysique,  French astronomical Society) was created in 2020 to form an instance where questions related to gender equality  can be addressed within the French astronomical community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The Commission has ten members, of which six  members are currently –or were at some point– also part of the SF2A Council: Caroline Bot, Sylvie Brau-  Nogu´e, Danielle Briot*, Patrick de Laverny*, Nad`ege Lagarde*, Rhita-Maria Ouazzani*, Nicole Nesvadba,  Julien Malzac*, Isabelle Vauglin, and Olivia Venot∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The main goals of the Commission are to promote gender  equality in Astronomy & Astrophysics in France, fight against sexual and gender-based violence, support gender-  focused outreach actions, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  Before the commission was created, different efforts to do a census of the status of women in astronomy in  France were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In particular, a survey was conducted through the SF2A to probe the future of doctors  who obtained their PhD in Astronomy and Astrophysics between 2007 and 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The results presented in Bern´e  & Hilaire (2020) showed, among other points, that women were less likely to be offered permanent positions  than men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' That same year, Bot & Buat (2020) did a census of the percentage of women on permanent positions  in France, finding that 23% percents of permanent positions at that time were held by women.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Looking at two  different age classes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' they found that the number of women seemed to be decreasing for university positions  1 LESIA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Observatoire de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Universit´e PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Sorbonne Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Universit´e Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 5 place Jules Janssen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 92195  Meudon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  2 Universit´e de Strasbourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Observatoire Astronomique de Strasbourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' UMR7550,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' F–67000 Strasbourg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  3 IRAP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Universit´e de Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' UPS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  4 Observatoire de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 61 avenue de l’Observatoire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 75014 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  5 Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Observatoire de la Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Laboratoire Lagrange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='F-06304 Nice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  6 Laboratoire d’Astrophysique de Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Bordeaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' B18N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' all´ee Geoffroy Saint-Hilaire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 33615 Pessac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  7 Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Observatoire de la Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Laboratoire Lagrange,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='F-06304 Nice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  8 IRAP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Universit´e de Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' UPS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  9 Univ Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Universit´e Lyon1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' ENS de Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CRAL UMR5574,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' F-69230 Saint-Genis-Laval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  10 Universit´e de Paris Cit´e and Univ Paris Est Creteil,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' LISA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' F-75013 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' France  ∗members of the SF2A Council  © Soci´et´e Fran¸caise d’Astronomie et d’Astrophysique (SF2A) 2022  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='03658v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='IM]  9 Jan 2023  238  SF2A 2022  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Left: Proportion of women for the overall sample, among permanent staff, and non-permanent staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Right: Pro-  portion of women among the researchers (from PhD to Emeritus, 687 individuals), and among administrative, technical  and engineering staff (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='a ITA, 373 individuals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  while increasing for astronomer positions (CNAP) and that no evidence of a glass ceiling effect was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  While both studies were important and necessary, they gave an instantaneous glimpse of the status of women  in astronomy in 2019-2020, they were limited to the information requested or available and were biased by the  surveyed population (young researchers for Bern´e & Hilaire 2020 or permanent positions for Bot & Buat 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  In this context, the Commission Femmes et Astronomie decided to conduct a statistical study that aims at  mapping the presence of women in French professional Astronomy today, and set a starting point for studying  its evolution with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  As a first step, we would like to address general questions such as:  What is the percentage of women in French Astronomical institutes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  How does their number vary with their level of seniority –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' career stage–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  What is the percentage of women at different levels of responsibility?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  How does the number of women depend on income level?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  The perimeter of this study is restricted to the research units (institutes) depending of the Astronomy &  Astrophysics (AA) section of the National Institute for Universe Sciences (INSU), with the exception of the  LISA institute, which was included in this study although it was not labeled AA, as part of this institute’s  activities are related to astronomy and astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The approach adopted consists in collecting data directly  from the institutes heads, through their administrative services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The data is anonymised upfront, to comply  with privacy policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Once anonymised it is distributed to the members of the committee, who are the only  persons authorized to manipulate them using secured tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  For the year 2021, taken as the starting point for the evolutionary sequence, we proceeded with a sub-set of  institutes of the INSU-AA, as a proof of concept, with the aim of extending the study to all the INSU-AA  institutes in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  2  Participation of women in Astronomy in 2021  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='1  The 2021 study set-up  We were able to collect data from eight research institutes within France: GEPI, IRAP, Lagrange, LESIA, LISA,  LUTh, ObAS and the SYRTE, which include 1060 individuals, representing around 25% of the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  For all these institutes, the data contained the following entries:  Gender,  Date of Birth,  Employer (CNRS/CNAP/University/else),  Status (students/post-doc/researcher/engineers, administrative or technical staff a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' ITA),  Proportionofwomenin2021 (8institutes)  80%  69,4%  71,5%  60%  64,8%  40%  30,6%  35,2%  20%  28,5%  0%  Overall population  Permanent staff  Non-permanent staff  Women  MenDistribution researcher/ITA  80,0%  71,3%  60,0%  66,0%  40,0%  34,0%  28,7%  20,0%  0,0%  Researchers  ITAs  Women  MenCounting women  239  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Proportion of women at each stage of the research career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In solid lines are superimposed linear fits of the  proportion of women (light orange) and men (light green), the quality of the fit is indicated by a value of R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  and for the public servants: the category (known as grade in french: IR/IE/CR/DR/AA/A/MdC/PU/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='2  General results and job security  The first number presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 1 (left, overall population), gives the proportion of women, regardless of  their status, age or position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' This number represents the likelihood of crossing paths with a woman in a corridor  when walking through a French astronomy institute: one in three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  The workforce in French academia is composed of permanent staff, among which we count public servants  –this includes persons in research, engineering, technical or administrative positions–, as well as very few (but  nevertheless growing number of) persons employed on corporate-like permanent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' As for non-permanent  positions, are counted PhD students, post-doctoral researchers, teaching assistants, apprenticeships, and holders  of a short-term contract (on engineering, technical or administrative jobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  Looking at the distribution of  women among permanent and non-permanent positions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='1, left), we see that women (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='2%) are most likely  employed on temporary contracts than men (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' If we restrict this comparison to research positions, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='0%  of researchers on permanent positions are women, whereas it is 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='9% for non-permanent positions (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='7% for  the overall population).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  One potential source of variability in these numbers is expected to come from the type of position (research  or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' That is what is explored on the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' We are aware that persons hired on ITA positions  also contribute to the research that is produced in these institutes, but different social and economical values  are attributed to research and ITA positions, and that is what, we believe, is determining here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Among women  working in the 8 institutes included in the study, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='2% are hired as ITA, while for men the percentage is 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  For the following discussions, we consider separately the population of researchers (in the broad sense: from  PhD to Emeritus) on the one hand, and the population of Engineering, Technical and Administrative staff  altogether (ITA) on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='3  Research career  Concerning researchers, we have sorted the population (687 individuals) according to the category of their  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' From the youngest to the most seniors, we have listed: PhD students, post-doctoral researchers, and  positions equivalent to associate professors (CR, ASTA, PHYSA, MC), to second-class professors (DR2, PU2,  AST2), to first-class professors (DR1, PU1, AST1), to professor of exceptional class (DRCE, PUCE, ASTCE),  Distribution of researchers according to category  76,6%  76,9%  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='8%  80,0%  70,6%  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='7%  64,9%  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='3%  60,0%  40,0%  35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='1%  35,7%  29,4%  27,3%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='4%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='1%  21,2%  20,0%  0,0%  PhD students  Post-docs  CR -ASTA - PHYSA DR2 -PU2-AST2 DR1 - PU1 -AST1  DRCE- PUCE -  Emeritus  MC  ASTCE  Women  Linear trend for women R² = 0,486  Men Linear trend for Men R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' = 0,486240  SF2A 2022  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Proportion of women in each category of ITA jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' From left to right they are ordered by qualification and  income level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The solid lines give the optimal linear fits of these distributions, with a value of R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  and Emeritus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The result is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In order to emphasize the general trend, a linear fit has been  performed (R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='486, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='08), which shows that the proportion of women decreases with the career stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  This overall trend seems to be mostly caused by the first drop in the distribution: when women represent around  35% of the population in non-permanent positions (PhD students and post-doctoral researchers), the proportion  decreases to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='4% for the first level of permanent employment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' After this first bottleneck, the number varies  slightly between around 23% and around 29%, with two noticeable increases: one at the second-class professor  level, and another one at the Emeritus level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  It is important to bear in mind that this study gives a snapshot of the distribution for 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Temporal or  causal relations between one stage to another are delicate to establish, and could properly be addressed only if  this snapshot is renewed every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' However, concerning the clear increase of proportion of women between the  associate professor level and the second-class professor level, one can wonder if it is not a stellar-main-sequence  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Astronomers know that the large majority of observable stars are currently in their main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  That is because the main sequence, during which they burn hydrogen in their core, is the longest of all stellar  evolution stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Hence, we are led to wonder if this increase of women is not due to the fact that once they are  promoted to second-class professorship, they spend a particularly long time in that stage before being promoted  further up, if at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='4  ITA careers  Concerning the population of ITA, composed here of 373 individuals, their distribution by career level is illus-  trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' It is worth mentioning that the histogram presents broad categories of ITA (known as corps in  French) ordered by levels of income and qualification, but contrary to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 2, the progression from one category  to the other is very little or none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Furthermore, the sources of variability are much more numerous than in the  research career case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Firstly because of the variety of jobs it encompasses (administrative, technical, R&D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  The jobs that fall into the ATR-T and AI categories have an dominant administrative component, whereas  IE and IR are mostly scientific and technical jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Another source of variability are the qualifications needed  to apply for these different kinds of positions: some require a PhD (IR), and others the High School Leaving  Certificate (Baccalaur´eat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In general, it is safe to assert that women are present in higher proportion in jobs  that have a clear administrative component, and have lower levels of qualifications and lower income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' As for  men, they dominate in more technical jobs, where the level of qualification can be higher, and access higher  income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In summary, we can conclude from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 3 that as the qualification and the level of income increases,  the number of women decreases (illustrated by the linear fit, with a p value of p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' A finer analysis would  DistributionofITAsbycategory  80%  74,7%  60%  64,8%  56,7%  51,4%  48,6%  40%  43,3%  35,2%  20%  25,3%  0%  ATR-T  AI  IE  IR  Women  MenCounting women  241  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Proportion of women and men at each income level give in Table 1, increasing from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' The quality of  the fit is given by a value of R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  level  category  L1  AJT, TCN, TCS  L2  TCE, AI  L3  IECN, IR2  L4  IEHC, IR1, CRCN, MC, PRAGCN, ASTA, PHYSA  L5  IRHC, CR1, CRHC, MCHC, DR2, PU2, AST2  L6  DR1, DRCE, PU1, PUCE, AST1, ASTCE, Emerites  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Scale of income for all the workers in the sample, researchers and ITA altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In terms of gross salary, it  starts from about 1590 euros and reaches around 6200 euros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  require to blow up each of the histogram stick into finer category (grade in French), but with the current data  at hand, we risk ending up with the small numbers statistics issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='5  The sinews of war  We also addressed the crux of the matter: salary levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' For this purpose, we bring together again the whole  sample (1060 individuals), and sort them out simply by level of income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' To do so, we chose to use a scale of  income set up by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Brau-Nogu´e (see https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='irap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='omp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='eu/egalite/bilan-social-et-parite-2022/)  in her work to document gender inequalities in her institute (IRAP), work which has largely inspired this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  The scale is presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Far from being a motivation for working in the academic sector, we believe  that the level of income is a good indicator of the social value associated to a given position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Concerning the first  two levels, they encompass mainly technical and administrative positions that are known to be positions which  are very gender specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' But once we reach level L3, we observe that there is a clear and regular decrease of the  proportion of women as the income increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In solid lines are given the optimal linear fits of the distributions  for men and women, with a p value of p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  3  Preliminary conclusions and perspectives  We present the results of our first statistical study on the participation of women in French Astronomical  Institutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Our sample was composed of 1060 individuals, belonging to 8 institutes, making up for around 25%  Distributionbylevelofincome  74,6%  76,7%  80,0%  71,4%  72,9%  58,5%  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='0%  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='1%  47,9%  41,5%  40,0%  28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='6%  27,1%  25,4%  23,3%  20,0%  0,0%  L1  L2  L3  L4  L5  L6  Incomelevel  omer  LMen242  SF2A 2022  of the targeted population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Although the results are still preliminary, it appears that the proportion of women  steadily decreases with the security of jobs / the career stage / the qualification level / the income level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' This  seems to illustrate the well known leaky pipeline issue, but needs further confirmation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  In particular, the sample for this study suffered from a number of shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Some institutes included  in the study cover topics which go beyond Astrophysics, such as the LISA and the SYRTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Looking at the p  values associated to the trends determined, we wish to improve the statistical robustness of the inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' We  aim at solving this issue by extending the sample to all the French Astronomical Institutes in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  Moreover, the snap-shot nature of this study prevents from drawing strong conclusions about the evolutionary  trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' Even if it can be very tempting to get a sense of evolution by looking at different generations of workers,  one should keep in mind that state policies, or even the culture can change from one generation to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  Appendix: Index of jobs in A&A  Here we give some elements for understanding the zoo of jobs that one can find in the astronomical institutes  in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' In general all the positions are divided into category (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' corps in french), and class (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' grade  in french).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  Academic careers: The three main job providers are the CNRS (French national institute for research), Univer-  sities, and the CNAP (Conseil National des Astronomes et Physiciens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' According to the hiring institution, we  can define different career paths, all of which contain 2 categories and each category can be split into 2 or 3  classes:  CNRS: Charg´es de Recherche (research associates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 2 classes:  CRCN and CRHC) → Directeurs de  Recherche (research directors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 3 classes: DR2, DR1, DRCE)  CNAP: Astronomes Associ´es (associate astronomers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 2 classes: ASTA, PHYSA) → Astronomes (as-  tronomers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 3 classes: AST2, AST1, ASTCE)  University: Maitres de Conf´erence (associate professors, 2 classes: MC, MCHC) → Professeurs (professors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  3 classes: PU2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' PU1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' PUCE)  Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' technical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' administrative careers (ITA): For ITA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' the positions are divided into 5 categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' which  are themselves divided into several classes (ordered by level of qualification):  Adjoints techniques de la recherche (AJT)  Techniciens de la recherche (TCN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' TCS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' TCE)  Assistants ing´enieurs (AI)  Ing´enieurs d’´etudes (IECN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' IEHC)  Ingenieurs de recherche (IR2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' IR1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' IRHC)  The members of the Commission Femmes et Astronomie would like to thank all institutes directors and colleagues who kindly  agreed to providing the data that made this study possible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' and the administrative staff who worked on their compilation and  anonymisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='  References  Bern´e, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' & Hilaire, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 2020, Nature Astronomy, 4, 296  Bot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' & Buat, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content=' 2020, http://sf2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='eu/Bot_Buat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}
+page_content='pdf' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE2T4oBgHgl3EQfGgZ8/content/2301.03658v1.pdf'}

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@@ -0,0 +1,2668 @@
+Non-arithmetic uniformization of metric
+spaces attached to unitary Shimura varieties
+Olivier de Gaay Fortman, January 5, 2023
+1
+Introduction
+In various cases, Hodge theory provides a bridge between hyperbolic geometry and
+algebraic geometry: there are moduli spaces of complex varieties whose period map
+identifies it with a quotient of complex hyperbolic space.
+The ball quotients that
+arise in this way are often (but not always) arithmetic [Pic83; Shi64; DM86; Kon00;
+ACT02b]. Similar constructions can be carried out to uniformize moduli of real vari-
+eties [AY98; ACT06; ACT10; Chu11; HR18]. A striking result in this direction was
+given by Allcock–Carlson–Toledo, who identified the space of stable real cubic surfaces
+with a four-dimensional non-arithmetic real ball quotient [ACT10]. Their space is as-
+sembled from various pieces, each of which an arithmetic quotient containing an open
+subset that parametrizes moduli of smooth real cubic surfaces of one topological type.
+The starting point of this paper is the question whether similar glueing construc-
+tions apply to real loci of arbitrary unitary Shimura varieties, a priori not depending
+on real moduli theory. Our first goal is to prove that this can indeed be done, leading
+to a construction which generalizes [ACT07; ACT10] and seems an orbifold analogue
+of the construction of Gromov–Piatetski-Shapiro [GPS87]. For indeed, our second goal
+is to show that many of the lattices that arise in this way, are in fact non-arithmetic.
+To explain this, let n ∈ N and consider a CM field K and a hermitian OK-lattice
+Λ of signature (n, 1) for one infinite place of K and definite for others. To Λ one can
+associate a quotient Γ \ Bn(C), and each anti-unitary involution α: Λ → Λ defines a
+real arithmetic ball quotient Γα\Bn
+α(R) := StabΓ(Bn(C)α)\Bn(C)α. We glue these real
+quotients together along a certain orthogonal hyperplane arrangement H ⊂ Bn(C).
+The resulting space Xn is a complete metric space (c.f. Proposition 3.7). Our first
+main result says that moreover, Xn carries a complete hyperbolic orbifold structure:
+Theorem 1.1 (c.f. Theorem 3.1). For each connected component X+
+n ⊂ Xn there
+exists a lattice Γ+
+n ⊂ PO(n, 1) and a canonical isometry X+
+n ∼= Γ+
+n \ Bn(R).
+For suitable choices of K and Λ, the space X2 is connected (c.f. Lemma 5.1), immerses
+totally geodesically into a connected component X+
+n of Xn for each n ∈ Z≥2 (c.f.
+Theorem 5.8), and the lattice Γ+
+2 ⊂ PO(2, 1) underlying X2 is non-arithmetic. By
+[BC05, Proposition 15.2.2], this forces Γ+
+n to be non-arithmetic. As a result, we obtain:
+1
+arXiv:2301.01598v1  [math.GT]  4 Jan 2023
+
+Theorem 1.2 (c.f. Theorem 5.18). For n ∈ Z≥2 and Λ = (Z[ζ3]n+1, diag(−1, 1, . . . , 1))
+or Λ =
+�
+Z[ζ5]n+1, diag( 1−
+√
+5
+2
+, 1, . . . , 1)
+�
+, there is a connected component X+
+n ⊂ Xn such
+that the lattice Γ+
+n ⊂ PO(n, 1) underlying X+
+n is non-arithmetic.
+Remark 1.3. For some moduli stacks of GIT stable hypersurfaces Ms, one can con-
+sider the hermitian lattice Λ that arises as the cohomology of the cover of projec-
+tive space ramified along a member of the moduli space, and define an isomorphism
+Ms(R) ∼= Xn for n = dim(Ms). By applying Theorem 1.1, one obtains a uniformiza-
+tion of Ms(R) by real hyperbolic space. For cubic surfaces and binary sextics, this is
+the content of [ACT10; ACT06]. In [GF21], we prove this for real binary quintics.
+We now explain the glueing construction in more detail. Let K be a CM field of
+degree 2g over Q with ring of integers OK, and let Λ be a free OK-module of rank
+n + 1 equipped with a hermitian form h : Λ × Λ → OK. Suppose that h has signature
+(n, 1) with respect to an embedding τ : K → C and is definite other infinite places of
+K. Let CHn be the space of negative lines in Λ ⊗OK,τ C and PΓn = Aut(Λn, h)/µK
+where µK ⊂ O∗
+K is the group of finite units in OK. Let PAn be the quotient of the
+set of anti-unitary involutions α : Λ → Λ by µK. Attached to the hermitian lattice
+(Λ, h) there is a Shimura variety ShK(G, X) (c.f. [Ach20, Section 5.3]) with complex
+uniformization ShK(G, X)(C) = Γn \ CHn, see Corollary 4.9 (or [Shi63, Theorem 2]).
+Consider the hyperplane arrangement
+H =
+�
+r∈Λ:
+h(r,r)=1
+⟨rC⟩⊥ ⊂ CHn
+and assume that the following holds :
+Condition 1.4 (c.f. [ACT02a]). Any two different hyperplanes in ⟨rC⟩⊥, ⟨tC⟩⊥ ⊂ H
+intersect orthogonally or not at all, i.e. either h(r, t) = 0 or ⟨rC⟩⊥ ∩ ⟨tC⟩⊥ = ∅.
+In many cases, this condition is automatically satisfied:
+Theorem 1.5 (c.f. Theorem 4.12). If the CM type is primitive and the different ideal
+DK ⊂ OK is generated by a purely imaginary element, then Condition 1.4 holds.
+This condition on the different ideal DK ⊂ OK is satisfied when the field K is
+cyclotomic or quadratic, see Proposition 4.14.
+The glueing construction is then carried out by assembling the different copies
+RHn
+α := (CHn)α ⊂ CHn,
+α ∈ PAn
+of real hyperbolic space RHn along the hyperplane arrangement H . See Definition
+2.14 and Remark 2.15 for the formulation of the equivalence relation. This gives a
+topological space Yn, acted upon by PΓn. Define PΓα ⊂ PΓn to be stabilizer of RHn
+α.
+2
+
+Theorem 1.6 (c.f. Theorem 3.1). The space Xn = PΓn \ Yn admits a complete path
+metric such that the natural map Xn → PΓn \CHn is a local isometry. Moreover, this
+metric on Xn extends to a real hyperbolic orbifold structure, such that
+�
+α∈PΓn\PAn
+PΓα \ (RHn
+α − H ) ⊂ Xn
+is an open suborbifold, and such that for each connected component X+
+n ⊂ Xn there is
+a lattice Γ+
+n ⊂ PO(n, 1) and an isomorphism of hyperbolic orbifolds X+
+n ∼= Γ+
+n \ RHn.
+Remark 1.7. Our construction relies on Condition 1.4, saying that the hyperplane
+arrangement H ⊂ CHn is an orthogonal arrangement in the sense of [ACT02a]. In
+fact, there is always a canonical orthogonal arrangement H ⊂ CHn attached to h in
+such a way that H = H when H is orthogonal (c.f. Remark 4.15). Moreover, one
+can glue the different copies RHn
+α of real hyperbolic space along the arrangement H to
+obtain a complete hyperbolic orbifold as in Theorem 1.6, but we will not prove this.
+1.1
+Acknowledgements. This research was undertaken partly during my PhD at the
+ENS in Paris, supported by the European Union’s Horizon 2020 research and inno-
+vation programme under the Marie Skłodowska-Curie grant agreement No754362
+,
+and partly at the Leibniz University Hannover, where I am supported by the RationAl-
+gic ERC Starting Grant. I thank Olivier Benoist, Nicolas Bergeron, Samuel Bronstein,
+Frans Oort, Pierre Py, Nicolas Tholozan and Domenico Valloni for stimulating con-
+versations and useful comments on earlier versions of this paper.
+2
+The glueing construction
+2.1
+The complex arithmetic ball quotient and anti-unitary involutions. Let g
+and n be positive integers. Let K be a CM field of degree 2g over Q and let F ⊂ K
+be its totally real subfield. Let OK (resp. OF) be the ring of integers of K (resp. F)
+and let σ ∈ Gal(K/F) be the non-trivial element. We will often write
+σ: K → K,
+σ(x) = x.
+Fix a set of embeddings
+Ψ = {τi : K → C}1≤i≤g
+|
+Ψ ∪ Ψσ = {τi, τiσ}1≤i≤g = Hom(K, C).
+(1)
+Let Λ be a free OK-module of rank n + 1 equipped with a hermitian form
+h: Λ × Λ → OK
+of signature (ri, si) with respect to τi. In other words, h is linear in its first argument
+and σ-linear in its second, and the complex vector space Λ⊗OK,τi C admits a basis {ei}
+such that (hτi(ei, ej))ij is a diagonal matrix with ri diagonal entries equal to 1 and si
+diagonal entries equal to −1. Here
+hτi : Λ ⊗OK,τi C × Λ ⊗OK,τi C → C
+3
+
+is the hermitian form attached to h and the embedding τi. Define
+τ = τ1 : K → C,
+and
+V = Λ ⊗OK,τ C,
+and assume that
+(ri, si)
+=
+�
+(n, 1)
+if i = 1,
+(n + 1, 0)
+if 2 ≤ i ≤ g.
+Let m be the largest positive integer for which the m-th cyclotomic field Q(ζm) can
+be embedded in K, where ζm = e2πi/m ∈ C. Let ζ ∈ K be a primitive m-th root of
+unity in K, and define
+µK = ⟨ζ⟩ ⊂ O∗
+K ⊂ OK.
+Moreover, define Γ to be the unitary group of Λ, and PΓ as its quotient by µK:
+Γ = U(Λ)(OK) = AutOK(Λ, h)
+and
+PΓ = Γ/µK.
+A norm one vector r ∈ Λ is called a short root. Let R ⊂ Λ be the set of short roots.
+For r ∈ R, define isometries φi
+r : V → V as follows:
+φr(x) = x − (1 − ζ)h(x, r) · r,
+φi
+r(x) = x − (1 − ζi)h(x, r) · r,
+i ∈ (Z/m)∗.
+Note that φi
+r ∈ Γ for r ∈ R, and that φi
+r = φr ◦ · · · ◦ φr (i times). In particular,
+φm
+r = id. Let P(V ) be the projective space of lines in V , and let
+CHn = {ℓ = [v] ∈ P(V ) | h(v, v) < 0} ⊂ P(V )
+be the space of negative lines in V . Define
+Hr = {x ∈ CHn : h(x, r) = 0} for r ∈ R,
+and
+H =
+�
+r∈R
+Hr
+⊂
+CHn.
+Lemma 2.1. The family of hyperplanes (Hr)r∈R is locally finite, so that the hyperplane
+arrangement H ⊂ CHn is a divisor of CHn.
+Proof. See [Bea09, Lemma 5.3].
+Define an OF-linear map α : Λ → Λ to be anti-unitary if for all x, y ∈ Λ and λ ∈ OK,
+one has α(λx) = σ(λ) · α(x) and h(α(x), α(y)) = σ(h(x, y)) ∈ OK. Define Γ′ to be the
+group of unitary and anti-unitary OF-linear bijections Λ
+∼−→ Λ. Let A ⊂ Γ′ be the set
+of anti-unitary involutions α: Λ → Λ. Then
+µK ⊂ Γ ⊂ Γ′
+−
+define
+PΓ′ = Γ′/µK.
+(2)
+Let λ ∈ K∗. Observe that
+�
+λ ∈ O∗
+K and |λ|2 = λ · σ(λ) = 1
+�
+⇐⇒
+λ ∈ µK.
+(3)
+Indeed, we have, for any embedding ϕ: K → C, that
+|ϕ(λ)|2 = ϕ(λ) · ϕ(λ) = ϕ(λ) · ϕ(σ(λ)) = ϕ(λ · σ(λ)),
+where ϕ(λ) = ϕ(σ(λ)) by [Mil20, Proposition 1.4]. Moreover, we have |ϕ(λ)| = 1 for
+each ϕ: K → C if and only if λ is a root of 1, see [Mil08, Corollary 5.6].
+4
+
+Lemma 2.2. Let Isom(CHn) be the group of isometries f : CHn
+∼−→ CHn. The natural
+homomorphism PΓ′ → Isom(CHn) is injective.
+Proof. This is easy and left to the reader.
+The group µK acts on A by multiplication; define
+PA = µK \ A ,
+and
+CA = PΓ \ PA ,
+where PΓ acts on PA by conjugation. Any α ∈ PA defines an anti-holomorphic
+involution
+α : CHn → CHn;
+define
+RHn
+α = (CHn)α ⊂ CHn.
+For any element α ∈ A , the quadratic form h|V α on the real vector space
+V α = Λα ⊗OF ,τ|F C
+has hyperbolic signature. The following lemma is readily proved:
+Lemma 2.3. For α ∈ A , let P(V α) be the real projective space of lines in V α, and
+let RH(V α) ⊂ P(V α) be the space of negative lines in V α. The canonical isomorphism
+P(V α) ∼= P(V )α restricts to an isomorphism RH(V α) ∼= RHn
+α.
+We conclude that RHn
+α ⊂ CHn is isometric to the real hyperbolic space of dimension
+n. Finally, we define
+PΓα = StabPΓ(RHn
+α) ⊂ PΓ
+(the stabilizer of RHn
+α in PΓ).
+2.2
+Orthogonal hyperplanes and complex reflections. We assume, in the entire
+Section 2, that the following condition is satisfied:
+Condition 2.4. If r, t ∈ R are such that Hr ̸= Ht and Hr ∩ Ht ̸= ∅, then h(r, t) = 0.
+Example 2.5. Theorem 4.12 of Section 4 shows that Condition 2.4 holds if the fol-
+lowing conditions are satisfied:
+1. The different ideal DK ⊂ OK is generated by a single element η ∈ OK such that
+σ(η) = −η and ℑ(τi(η)) > 0 for every i.
+2. The CM type (K, Ψ) is primitive.
+We will see that condition 1 is automatically satisfied for quadratic and cyclotomic
+CM fields K, see Proposition 4.14.
+Note that Condition 2.4 implies that if Hr1, . . . , Hrk for ri ∈ R are mutually distinct,
+and if their common intersection is non-empty, then ∩k
+i=1Hri ⊂ CHn is a totally
+geodesic subspace of codimension k. Note also that for any r ∈ R, the element φr ∈ Γ
+generates a finite subgroup ⟨φr⟩ ⊂ Γ of order m, and that the restriction of the quotient
+map π: Γ → PΓ to this subgroup ⟨φr⟩ ⊂ Γ is injective. We will abuse notation, by
+letting φr ∈ PΓ denote the image of φr ∈ Γ in PΓ.
+5
+
+Definition 2.6. Let H = {Hr | r ∈ R}. For x ∈ CHn, define
+H(x) = {H ∈ H | x ∈ H} ,
+G(x) = ⟨φi
+r ∈ PΓ with r ∈ R, i ∈ Z/m | x ∈ Hr⟩.
+The hyperplanes H ∈ H(x) are called the nodes of x. We say that x has k nodes if
+the cardinality of H(x) is k.
+Proposition 2.7. Let r, t ∈ R. The following are equivalent:
+1. One has φr = φt ∈ Γ.
+2. There exist i, j ∈ Z/m − {0} such that φi
+r = φj
+t ∈ Γ.
+3. There exist a, b ∈ OK − {0} with |a|2 = |b|2 such that a · r = b · t ∈ OK.
+Proof. [1 =⇒ 2] : This is clear.
+[2 =⇒ 3] : Suppose for contradiction that there are no such a, b ∈ OK − {0}. Since
+K = Frac(OK), this implies that r, t ∈ V are linearly independent over K, and hence
+over C, which contradicts the equality
+ζi · r = φi
+r(r) = φj
+t(r) = r − (1 − ζj)h(r, t) · t ∈ V.
+Thus, there exist a, b ∈ OK − {0} with a · t = b · r. We have |a|2 = h(a · t, a · t) =
+h(b · r, b · r) = |b|2.
+[3 =⇒ 1] : Write λ = b/a ∈ K∗; then t = λ · r with |λ|2 = 1. For x ∈ V , one gets
+φt(x) = φλ·r(x) = x − (1 − ζ) · h(x, λ · r) · λ · r = x − (1 − ζ) · h(x, r) · r = φr(x).
+Lemma 2.8. Let x ∈ CHn and suppose that x has k nodes. Then G(x) ∼= (Z/m)k.
+Proof. Let r, t ∈ R. Then, for z ∈ Λ, one has
+φi
+r(φj
+t(z)) = φi
+r
+�
+z − (1 − ζj)h(z, t) · t
+�
+= z − (1 − ζj)h(z, t) · t − (1 − ζi)h
+��
+z − (1 − ζj)h(z, t) · t
+�
+, r
+�
+· r
+= z − (1 − ζj)h(z, t) · t − (1 − ζi)h(z, r) · r + (1 − ζi)(1 − ζj)h(z, t)h(t, r) · r.
+(4)
+Now suppose that Hr, Ht ∈ H(x), with Hr ̸= Ht. By Condition 2.4, we have h(r, t) = 0;
+by (4), this implies that φi
+r ◦ φj
+t = φj
+t ◦ φi
+r for each i, j ∈ Z/m. We conclude that the
+group G(x) is abelian.
+Next, suppose that Hr = Ht ∈ H(x). To finish the proof, it suffices to show that
+φt = λ · φi
+r for some i ∈ Z/m and λ ∈ µK. This follows from Lemma 2.9 below.
+Lemma 2.9. Let r ∈
+R. Let φ : CHn → CHn be an isometry of order m that
+restricts to the identity on Hr ⊂ CHn. Then φ = φi
+r for some i ∈ Z/m.
+6
+
+Proof. Let Hn
+C be the hyperbolic space attached to the standard hermitian space Cn,1
+of dimension n + 1. It is classical that
+StabU(n,1)(Hn−1
+C
+) = U(n − 1, 1) × U(1).
+Thus, any φ ∈ U(n, 1) that fixes Hn−1
+C
+pointwise lies in C∗ × U(1), where U(1) denotes
+{z ∈ C∗ : |z|2 = 1}. If φm ∈ C∗ × {1}, then φ ∈ C∗ × ⟨ζ⟩ ⊂ U(n − 1, 1) × U(1).
+We will also consider G(x) as a subgroup of Γ sometimes. Lemma 2.2 allows us to
+view PA as a subset of Isom(CHn), and also to view the groups G(x) ⊂ PΓ ⊂ PΓ′
+(for any x ∈ CHn) as subgroups of Isom(CHn). Define
+�Y =
+�
+α∈PA
+RHn
+α.
+We will glue the different hyperbolic spaces RHn
+α, by defining an equivalence relation
+∼ on �Y . Before we define it, we state and prove a couple of trivial results.
+Lemma 2.10. Let α ∈ A and r ∈ R. Then α ◦ φi
+r = φ−i
+α(r) ◦ α.
+Proof. Indeed, for x ∈ Λ, we have
+α(φi
+r(x)) = α
+�
+x − (1 − ζi)h(x, r) · r
+�
+= α(x) − (1 − ζ−i)h(x, r) · α(r)
+= α(x) − (1 − ζ−i)h(α(x), α(r)) · α(r) = φ−i
+α(r)(α(x)).
+Lemma 2.11. Let x ∈ RHn
+α and write H(x) = {Hr1, . . . , Hrk} for some ri ∈ R. Then
+for each i ∈ {1, . . . , k} there is a unique j ∈ {1, . . . , k} such that α(Hri) = Hα(ri) = Hrj.
+Proof. Indeed, we have, for any β ∈ A and r ∈ R, that β(Hr) = Hβ(r). Since x ∈ Hri,
+we have x = α(x) ∈ α(Hri) = Hα(ri) for every i. In particular, we have Hα(ri) ∈ H(x)
+(see Definition 2.6), so that Hα(ri) = Hrj for some j.
+Definition 2.12. Let α ∈ PA and x ∈ RHn
+α. Write H(x) = {Hr1, . . . , Hrk}, see
+Definition 2.6. By Lemma 2.11, the involution α induces an involution on the set
+H(x). Define α: I → I as the resulting involution on the set I = {1, . . . , k}.
+Proposition 2.13. Let α ∈ PA and x ∈ RHn
+α. Write H(x) = {Hr1, . . . , Hrk} (Def-
+inition 2.6) and let g = φi1
+r1 ◦ · · · ◦ φik
+rk ∈ G(x) for some iν ∈ Z/m. The following are
+equivalent:
+1. We have g ◦ α ∈ PA . (In other words, g ◦ α is an involution.)
+2. For each ν ∈ I, we have iν ≡ iα(ν) mod m.
+7
+
+Proof. Lift α ∈ PA to an element α ∈ A . We claim that, for each i ∈ I, we have
+φα(ri) = φrα(i). Indeed, by Lemma 2.11, for each i ∈ I, we have α(Hri) = Hα(ri) = Hrj ∈
+H(x) for some j ∈ I. By definition of the involution α: I → I, we have j = α(i).
+Therefore, φα(ri) = φbirα(i) for some bi ∈ Z/m, see Lemma 2.9. By Proposition 2.7, we
+have bi = 1. By Lemma 2.10 and by this claim, we obtain φiν
+rν ◦α = α◦φ−iν
+α(rν) = α◦φ−iν
+rα(ν)
+for each ν ∈ I, which implies that φi1
+r1 ◦ · · · ◦ φik
+ik ◦ α = α ◦ φ−i1
+rα(1) ◦ · · · ◦ φ−ik
+iα(k). Therefore,
+�
+φi1
+r1 ◦ · · · ◦ φik
+ik ◦ α
+�2 = φi1
+r1 ◦ · · · ◦ φik
+ik ◦ φ−i1
+rα(1) ◦ · · · ◦ φ−ik
+iα(k)
+= φ
+iα(1)
+rα(1) ◦ · · · ◦ φ
+iα(k)
+iα(k) ◦ φ−i1
+rα(1) ◦ · · · ◦ φ−ik
+iα(k)
+= φ
+iα(1)−i1
+rα(1)
+◦ · · · ◦ φ
+iα(k)−ik
+iα(k)
+∈ G(x),
+and this is the identity in G(x) if and only if iν ≡ iα(ν) mod m.
+2.3
+The glueing construction.
+Definition 2.14. Define a relation R ⊂ �Y × �Y as follows. An element
+(xα, yβ) ∈ RHn
+α × RHn
+β ⊂ �Y × �Y
+is an element of R if the following conditions are satisfied:
+1. The images of xα and yβ in CHn agree.
+2. If α ̸= β, then
+(a) xα = yβ lies in H ; and
+(b) β = g ◦ α ∈ PA for some g ∈ G(xα) = G(yβ) (c.f. Lemma 2.2).
+Remark 2.15. Conditions 1 and 2 in Definition 2.14 say that we are identifying
+points of RHn
+α ∩ H and RHn
+β ∩ H that have the same image in CHn. But we do
+not glue all such points: the real structures α and β are required to differ by complex
+reflections in the hyperplanes that pass through x. In fact, we will see below (see
+Lemma 3.2) that the glueing can be rephrased as follows: we glue RHn
+α and RHn
+β
+along their intersection, provided that this intersection is contained in H in such a
+way that for some (equivalently, any) x ∈ RHn
+α ∩ RHn
+β , the real structures α and β
+differ by reflections in hyperplanes that pass through x.
+Lemma 2.16. R is an equivalence relation.
+Proof. Consider three elements xα, yβ, zγ ∈ �Y . The fact that xα ∼ xα is clear.
+Suppose that xα ∼ yβ. If α = β then xα = yβ ∈ �Y hence yβ ∼ xα. If α ̸= β then
+xα = yβ ∈ H ⊂ CHn, and β = g ◦ α for g ∈ G(xα) = G(yβ) as in Definition 2.14.
+Since α = g−1 ◦ β with g−1 ∈ G(xα), this shows that yβ ∼ xα.
+Suppose that xα ∼ yβ and yβ ∼ zγ; we claim that xα ∼ zγ. We may and do assume
+that α, β and γ are different, which implies that xα = yβ = zγ ∈ H , that γ = h◦β for
+some h ∈ G(yβ), and that β = g◦α for some g ∈ G(xα). We obtain γ = h◦β = h◦g◦α
+for h ◦ g ∈ G(xα) = G(yβ) = G(zγ).
+8
+
+Definition 2.17. Define Y to be the quotient of �Y by the equivalence relation R, and
+equip it with the quotient topology. We shall prove (Lemma 2.18) that the group PΓ
+acts on Y . We call PΓ\Y the glued space attached to the hermitian OK-lattice (Λ, h).
+Lemma 2.18. The action of PΓ on CHn induces an action of PΓ on �Y , compat-
+ible with the equivalence relation R, so that PΓ acts on Y .
+Moreover, PΓ \ �Y =
+�
+α∈CA PΓα \ RHn
+α.
+Proof. If φ ∈ PΓ, then φ (RHn
+α) = RHn
+φαφ−1 hence PΓ acts on �Y = �
+α∈PA RHn
+α, and
+PΓ \ �Y = PΓ \
+�
+α∈PA
+RHn
+α =
+�
+α∈CA
+PΓα \ RHn
+α.
+Now suppose that xα ∼ yβ ∈ �Y and f ∈ PΓ. Then f(xα) ∈ RHn
+fαf−1 and f(yβ) ∈
+RHn
+fβf−1. We claim that
+f(xα)fαf−1 ∼ f(yβ)fβf−1.
+For this, we may and do assume that xα ̸= yβ, hence xα = yβ ∈ H and β = g ◦ α for
+some g ∈ G(xα) as in Definition 2.14. In particular, f(xα) = f(yβ). Since f ◦φi
+r◦f −1 =
+φi
+f(r) for each r ∈ R and i ∈ Z/m, and h(x, r) = 0 if and only if h(f(x), f(r)) = 0, we
+have fG(x)f −1 = G(f(x)) for each x ∈ CHn. We are done:
+fβf −1 = f(g ◦ α)f −1 = fgf −1 ◦ (fαf −1),
+fgf −1 ∈ G(f(xα)).
+3
+The hyperbolic orbifold structure on the glued space
+The following is the main result of Section 3:
+Theorem 3.1.
+1. The glued space PΓ\Y admits a metric that makes it a complete
+path metric space. The natural map PΓ \ Y → PΓ \ CHn is a local isometry.
+2. Each point x ∈ PΓ\Y admits an open neighborhood U ⊂ PΓ\Y which is isomet-
+ric to the quotient of an open subset V ⊂ RHn by a finite group of isometries.
+Therefore, the glued space PΓ \ Y has a real hyperbolic orbifold structure.
+3. One has �
+α∈CA PΓα \ (RHn
+α − H ) ⊂ PΓ \ Y as an open suborbifold.
+4. The connected components of the real hyperbolic orbifold PΓ \ Y are uniformized
+by RHn: for each component C ⊂ PΓ \ Y there exists a lattice PΓC ⊂ PO(n, 1)
+and an isomorphism of real hyperbolic orbifolds C ∼= PΓC \ RHn. Consequently,
+PΓ \ Y ∼=
+�
+C∈π0(PΓ\K)
+PΓC \ RHn.
+It can happen that PΓ \ Y is connected: such is the case when K = Q(ζ3) and
+h = diag(−1, 1, 1, 1, 1), see [ACT10]. When K = Q(ζ3) and h = diag(−1, 1, 1, 1), then
+PΓ \ Y has two components, see [ACT07, Remark 6].
+9
+
+3.1
+The path metric on the glued space. We start with a lemma. We will need it in
+the proof of Lemma 3.4 below, which will be used to define a path metric on PΓ \ Y
+making it locally isometric to quotients of RHn by finite groups of isometries. It also
+serves as a sanity check: if there exists x ∈ RHn
+α ∩ RHn
+β such that xα ∼ xβ, then one
+glues the entire copy RHn
+α to the copy RHn
+β along their intersection in CHn.
+Lemma 3.2.
+1. Let g = �k
+ν=1 φiν
+rν ∈ Γ for some set {rν} ⊂ R of mutually orthog-
+onal short roots rν, where iν ̸≡ 0 mod m for each ν. Then (CHn)g = ∩k
+ν=1Hrν.
+2. Let α, β ∈ PA and x ∈ RHn
+α ∩ RHn
+β such that xα ∼ xβ. Then yα ∼ yβ for every
+y ∈ RHn
+α ∩ RHn
+β .
+3. The natural map �Y → CHn descends to a PΓ-equivariant map P : Y → CHn.
+Proof. 1. Let y ∈ V be representing an element in (CHn)φ. Since the ri are orthogonal,
+and g(y) = λ for some λ ∈ C∗, we have
+g(y) =
+k
+�
+ν=1
+φiν
+rν(y) = y −
+k
+�
+ν=1
+�
+1 − ζiν�
+h(y, rν)rν = λy,
+(5)
+hence (1 − λ)y = �ℓ
+i=1 (1 − ζiν) h(y, rν)rν ∈ V .
+But y spans a negative definite
+subspace of V while the rν span a positive definite subspace, so that we must have
+1 − λ = 0 = �k
+ν=1 (1 − ζiν) h(y, rν)rν. Since the rν are mutually orthogonal, they
+are linearly independent; since ζiν ̸= 1 we find h(y, rν) = 0 for each ν. Conversely, if
+x ∈ ∩Hrν, then φiν
+rν(x) = x for each ν.
+2. Since xα ∼ xβ, there exists g ∈ G(x) such that β = g ◦ α. Write H(x) =
+{Hr1, . . . , Hrk}. Let y ∈ RHn
+α ∩ RHn
+β . Then α(y) = β(y) = y implies that g(y) = y.
+In particular, y ∈ ∩νHrν by Part 1, which implies that H(x) ⊂ H(y), which in turn
+implies that G(x) ⊂ G(y). We conclude that g ∈ G(y). Hence yα ∼ yβ.
+3. If xα ∼ yβ, then x = y ∈ CHn.
+By Lemma 3.2, we obtain continuous maps
+P : K → CHn,
+and
+P : PΓ \ Y → PΓ \ CHn.
+Our next goal is to prove that each point x ∈ Y has a neighbourhood V ⊂ Y that
+maps homeomorphically onto a finite union ∪ℓ
+i=1RHn
+αi ⊂ CHn. Hence x has an open
+neighourhood x ∈ U ⊂ V that identifies with an open set in a union of copies of RHn
+in CHn under the map P. This allows us to define a metric on Y by pulling back the
+metric on CHn.
+Lemma 3.3. Each compact set Z ⊂ CHn meets only finitely many RHn
+α, α ∈ PA .
+Proof. Recall the subgroup PΓ′ ⊂ Isom(CHn) (see (2) and Lemma 2.2). We have that
+PΓ′ acts properly discontinuously on CHn. So if S is the set of α ∈ PA such that
+αZ ∩ Z ̸= ∅, then S is finite. In particular, Z meets only finitely many RHn
+α.
+10
+
+Fix a point f ∈ Y and a point xα ∈ �Y lying above f. Let α1, . . . , αℓ be the elements
+in PA such that xαi ∼ xα for each i ∈ I := {1, . . . , ℓ} (since the group G(x) is finite
+by Lemma 2.8, these are finite in number).
+Let p : �Y → Y be the quotient map, and define
+Yf = p
+� ℓ�
+i=1
+RHn
+αi
+�
+⊂ Y.
+(6)
+We prove that Y is locally isometric to opens in unions of real hyperbolic subspaces
+of CHn. Indeed, we have the following:
+Lemma 3.4.
+1. The set Yf is closed in Y .
+2. We have P (Yf) = ∪ℓ
+i=1RHn
+αi ⊂ CHn, and the map
+Pf : Yf → ∪ℓ
+i=1RHn
+αi
+induced by P is a homeomorphism.
+3. The set Yf ⊂ Y contains an open neighborhood Uf of f in Y .
+Proof. 1. One has
+p−1 (Yf) = p−1
+�
+p
+� ℓ�
+i=1
+RHn
+αi
+��
+=
+ℓ�
+i=1
+p−1 �
+p
+�
+RHn
+αi
+��
+⊂ �Y .
+Therefore, it suffices to show that p−1 �
+p
+�
+RHn
+αi
+��
+is closed in �Y .
+But notice that
+xβ ∈ p−1 (p (RHn
+α)) if and only if x ∈ RHn
+α and xα ∼ xβ, which implies (Lemma 3.2)
+that RHn
+α ∩ RHn
+β ⊂ p−1 (p (RHn
+α)). Hence for any α ∈ PA , one has
+p−1 (p (RHn
+α)) =
+�
+β∼α
+RHn
+α ∩ RHn
+β ,
+where β ∼ α if and only if there exists x ∈ RHn
+α ∩ RHn
+β such that xα ∼ xβ. It follows
+that p−1 (p (RHn
+α)) ∩ RHn
+β is closed in RHn
+β for every β ∈ PA . But the RHn
+β are open
+in �Y and cover �Y , so that p−1 (p (RHn
+α)) is closed in �Y .
+2. We have
+Pf(Yf) = P
+�
+p
+� ℓ�
+i=1
+RHn
+αi
+��
+= �
+P
+� ℓ�
+i=1
+RHn
+αi
+�
+=
+ℓ�
+i=1
+RHn
+αi ⊂ CHn.
+To prove injectivity, let xαi, yαj ∈ �Y and suppose that x = y ∈ CHn. Then indeed,
+xαi ∼ yαj because ∼ is an equivalence relation by Lemma 2.16.
+Let Z ⊂ CHn be a compact set. Write
+�
+P : �Y → CHn
+11
+
+for the canonical map.
+Remark that Z meets only finitely many of the RHn
+α for
+α ∈ PA , see Lemma 3.3. Each Z ∩ RHn
+α is closed in Z since RHn
+α is closed in CHn,
+so each Z ∩ RHn
+α is compact. We conclude that �
+P−1(Z) = � Z ∩ RHn
+α is compact.
+In particular, �
+P is closed [Lee13, Theorem A.57].
+Finally, we prove that Pf is closed. Let Z ⊂ Yf be a closed set. Then Z is closed
+in Y by Part 1, hence p−1(Z) is closed in �Y , hence �
+P (p−1(Z))) is closed in CHn, so
+that
+Pf(Z) = P(Z) = �
+P
+�
+p−1 (Z)
+�
+=
+�
+�
+P
+�
+p−1 (Z)
+��
+∩
+�
+∪ℓ
+i=1RHn
+αi
+�
+is closed in ∪ℓ
+i=1RHn
+αi.
+3. Let x = P(f) ∈ CHn. Since CHn is locally compact, there exists a compact
+set Z ⊂ CHn and an open set U ⊂ CHn with x ∈ U ⊂ Z. Since Z is compact, it
+meets only finitely many of the RHn
+β ⊂ CHn (Lemma 3.3). Consequently, the same
+holds for U; define V = P−1(U) ⊂ Y . Define
+B = {β ∈ PA : U ∩ RHn
+β ̸= ∅}.
+Also define, for β ∈ PA , Zβ = p
+�
+RHn
+β
+�
+⊂ Y . Then
+f ∈ V ⊂
+�
+β∈B
+Zβ =
+�
+β∈B
+β(x)=x
+Zβ
+�
+β∈B
+β(x)̸=x
+Zβ.
+Since each Zβ is closed in Y by the proof of part 1, there is an open V ′ ⊂ V with
+f ∈ V ′ ⊂
+�
+β∈B
+β(x)=x
+Zβ =
+�
+β∈B
+β(x)=x
+xβ∼xα
+Zβ
+�
+β∈B
+β(x)=x
+xβ̸∼xα
+Zβ
+Hence again there exists an open subset V ′′ ⊂ V ′ with
+f ∈ V ′′ ⊂
+�
+β∈B
+β(x)=x
+xβ∼xα
+Zβ ⊂
+�
+β∈PA
+β(x)=x
+xβ∼xα
+Zβ = Yf.
+Therefore, Uf := V ′′ ⊂ Y satisfies the requirements.
+We need one further lemma:
+Lemma 3.5. The topological space Y is Hausdorff.
+Proof. Let f, f ′ ∈ Y be elements such that f ̸= f ′. First suppose that f ̸∈ Yf′. Since
+Yf′ is closed in Y by Lemma 3.4, there is an open neighbourhood U of f such that
+U ∩ Uf′ ⊂ U ∩ Yf′ = ∅.
+Next, suppose that f ∈ Yf′. Lift f and f ′ to elements xα, yβ ∈ �Y . Assume first
+that x = y. This means that P(f) = P(f ′). Since P : Yf′ → CHn is injective,
+this implies that f = f ′, contradiction. So we have x ̸= y ∈ CHn. But CHn is
+Hausdorff, so there are open subsets (U ⊂ CHn, V ⊂ CHn) such that x ∈ U, y ∈ V
+and U ∩ V = ∅. Then P−1(U) ∩ P−1(V ) = ∅.
+12
+
+We then obtain:
+Proposition 3.6. Y is naturally a path metric space, piecewise isometric to RHn.
+Proof. From Lemma 3.4, we deduce that for each f ∈ Y there exists an open neighbor-
+hood f ∈ Uf ⊂ Y such that P induces a homeomorphism Y ⊃ Uf
+∼−→ P(Uf) ⊂ CHn.
+Pull back the metric on P(Uf) to obtain a metric on Uf. Then define a metric on
+Y as the largest metric which is compatible with the metric on each open set Uf and
+which preserves the lengths of paths.
+Proposition 3.7. The path metric on Y descends to a path metric on X = PΓ \ Y .
+Proof. The metric on Y descends in any case to a pseudo-metric on PΓ \ Y , and
+by [Gro07, Chapter 1], this is a metric if PΓ acts by isometries on Y with closed
+orbits. This is true: the fact that PΓ acts isometrically on Y comes from the PΓ-
+equivariance of P : Y → CHn (Lemma 3.2) together with the construction of the
+metric on Y (Proposition 3.6). To check that the PΓ-orbits are closed in Y , let f ∈ Y
+with representative xα ∈ �Y . By equivariance of p : �Y → Y , we have p−1 (PΓ · f) =
+PΓ · (p−1f), so since p is a quotient map, it suffices to show that
+PΓ ·
+�
+p−1f
+�
+= PΓ · ∪xβ∼xαxβ = ∪xβ∼xαPΓ · xβ
+is closed in �Y , thus that each orbit PΓ · xβ is closed in �Y . Since PΓ is discrete, it
+suffices to show that PΓ acts properly on �Y . So let Z ⊂ �Y be any compact set: we
+claim that {g ∈ PΓ : gZ ∩ Z ̸= ∅} is a finite set. Indeed, for each g ∈ PΓ, one has
+�
+P (gZ ∩ Z) ⊂ g �
+P(Z) ∩ �
+P(Z), and the latter is non-empty for only finitely many
+g ∈ PΓ, by properness of the action of PΓ on CHn.
+Since the metric on Y is a path metric, so is the metric on PΓ \ Y [Gro07].
+3.2
+The orbifold structure on the glued space. The next step is to prove that the
+glued space PΓ \ Y (see Definition 2.17) is locally isometric to quotients of open sets
+in RHn by finite groups of isometries.
+Definition 3.8. Let f ∈ Y with representative xα ∈ �Y . Thus, x is an element in
+CHn, and α ∈ PA is the class of an anti-unitary involution such that α(x) = x.
+1. The nodes of f are by definition the nodes of xα (see Definition 2.6). Thus, these
+are the hyperplanes H ∈ H(x), i.e. the hyperplanes Hr ∈ H defined by short
+roots r ∈ R such that x ∈ Hr (equivalently, such that h(x, r) = 0).
+2. The number of nodes of f is the cardinality of H(x).
+3. The anti-unitary involution α ∈ PA induces an involution on the set H(x) by
+Lemma 2.11. Let H ∈ H(x) be a node. We call H a real node of f if α(H) = H.
+We call (H, α(H)) a pair of complex conjugate nodes of f if α(H) ̸= H.
+4. If k is the number of nodes of f, we generally write k = 2a+b, with a the number
+of pairs of complex conjugate nodes of f, and b the number of real nodes of f.
+13
+
+Fix again a point f ∈ Y and a point xα ∈ �Y lying above f. Let k = 2a + b be the
+number of nodes of f. Thus x ∈ RHn
+α, and there exist r1, . . . , rk ∈ R such that
+H(x) = {Hr1, . . . , Hrk} ,
+G(x) = ⟨φr1, . . . , φrℓ⟩ ∼= (Z/m)k.
+For β ∈ PA , observe that xβ ∼ xα if and only if α ◦ β ∈ G(x). We relabel the ri so
+that they satisfy the following condition:
+α(Hri) = Hri+1 for i odd and i ≤ 2a,
+α(Hri) = Hri−1 for i even and i ≤ 2a, and
+α(Hri) = Hri for i ∈ {2a + 1, . . . , k} .
+(7)
+In other words, Hri is a real root if and only if i > 2a, and
+�
+Hri, Hri+1
+�
+is a pair of
+complex conjugate roots if and only if i < 2a is odd.
+Lemma 3.9. Continue with the notation from above.
+1. Let β ∈ PA be such that xβ ∼ xα. Then
+β =
+a
+�
+i=1
+�
+φr2i−1 ◦ φr2i
+�ji ◦
+k
+�
+i=2a+1
+φji
+ri ◦ α
+for some j1, . . . , ja, j2a+1, . . . , jk ∈ Z/m. In particular, there are ma+b such β.
+2. There is an isometry CHn
+∼−→ Bn(C) identifying x with the origin, φri with the
+map
+Bn(C) → Bn(C),
+(t1, . . . , ti, . . . , tn) �→ (t1, . . . , ζti, . . . , tn),
+and α with the map defined by
+ti �→
+�
+�
+�
+�
+�
+¯ti+1
+for i odd and i ≤ 2a
+¯ti−1
+for i even and i ≤ 2a
+¯ti
+for i > 2a.
+(8)
+Proof. 1. This follows readily from Proposition 2.13.
+2. Since the Hri are orthogonal by Condition 2.4, and their intersection contains
+x, we can find coordinates t1, . . . , tn+1 on V that induce an identification (V, h) ∼=
+Cn,1 := (Cn+1, H) with H(x, x) = |x1|2 + · · · + |xn|2 − |xn+1|2, in such a way that
+Hri ⊂ V is identified with the hyperplane {ti = 0} ⊂ Cn+1 and x ∈ ∩iHri with the
+point (0, 0, . . . , 0, 1). We will do this in the following way. Define
+T = ⟨x⟩ ⊕ ⟨r1⟩ ⊕ · · · ⊕ ⟨rk⟩ ⊂ V,
+W = T ⊥ = {w ∈ V | h(w, t) = 0 ∀t ∈ T}.
+For each i ∈ I = {1, . . . , k}, we have α(ri) = λi · rα(i) for some λi ∈ K (see Lemma
+2.11, Definition 2.12, and Lemmas 2.9 and 2.8). Observe that α(W) = W. Since
+W ⊂ ⟨x⟩⊥, the hermitian space (W, h|W) is positive definite. Let {w1, . . . , wn−k} ⊂ W
+be an orthonormal basis such that α(wi) = wi, which exists by the elementary
+14
+
+Lemma 3.10. Let (W, h) a non-degenerate hermitian vector space of dimension n ≥ 1
+and let α: W → W be an anti-linear involution with h(α(x), α(y)) = h(x, y) for
+x, y ∈ W. For each positive integer m ≤ n, there exists a linearly independent set
+{wi}m
+i=1 ⊂ W such that h(wi, wj) = ±δij and α(wi) = wi for each i = 1, . . . , m.
+Let {ei}n+1
+i=1 be the standard basis of Cn+1, and define a C-linear isomorphism
+Φ: V
+∼−→ Cn+1,
+�
+x
+h(x, x) �→ en+1,
+ri �→ ei,
+wi �→ ei
+�
+.
+(9)
+By (7), we have that α(ri) = λi · ri+1 for i odd and i ≤ 2a, that α(ri) = λi · ri−1 for i
+even and i ≤ 2a, and that α(ri) = λi · ri for i > 2a. We conclude that the anti-linear
+involution on Cn+1 induced by α and (9) corresponds to the matrix
+α =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+α1
+. . .
+0
+. . .
+. . .
+0
+α2
+0
+0
+0
+. . .
+...
+0
+0
+0
+α4
+0
+0
+α3
+0
+...
+0
+...
+...
+...
+...
+...
+...
+0
+...
+αn
+0
+0
+0
+. . .
+0
+. . .
+0
+αn+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+where each αi is an anti-linear involution C → C, and αi = αi+1 for i < 2a odd. If
+αi(1) = µi ∈ C∗, then µ−1
+i
+· αi = conj: C → C (complex conjugation). Since |µi| = 1,
+there exists ρi ∈ C such that µi = ρi/ρi and |ρi| = 1. This gives µ−1
+i
+·αi = ρi ·αi ·ρ−1
+i
+=
+conj: C → C. The composition
+V
+Φ � Cn+1diag(ρi)� Cn+1
+induces an isomorphism CHn ∼= Bn(C) with the required properties.
+Definition 3.11.
+• Define Af = StabPΓ(f) to be the subgroup of PΓ fixing f ∈ Y .
+This contains the group G(x) ∼= (Z/m)k.
+• Define Bf as the subgroup of G(x) generated by the order m complex re-
+flections associated to the real nodes of f, rather than all the nodes. Hence
+Bf = ⟨φri⟩i>2a ∼= (Z/m)b.
+Recall the quotient map p: �Y → Y , the definition (6) of Yf, and Lemma 3.4.
+Lemma 3.12. The stabilizer Af of f ∈ Y preserves the subset Yf ⊂ Y .
+Proof. Let ψ ∈ Af, with f = p(xα) ∈ Y , x ∈ ∩iHri. Then ψ(x)ψαψ−1 ∼ xα. Now let
+p(yβ) ∈ Yf. Then β(x) = x and xα ∼ xβ. Hence xα ∼ ψ · xα ∼ ψ · xβ = ψ(x)ψβψ−1.
+This implies that ψβψ−1 ◦ α ∈ G(x), so that p (ψ(y)ψβψ−1) ∈ Yf.
+We also need the following lemma. Write m = 2ak with k ̸= 0 mod 2.
+15
+
+Lemma 3.13. Let T = {t ∈ C : tm ∈ R}. Then G = ⟨ζm⟩ acts on T by multiplication.
+Each element in T/G has a unique representative of the form ζϵ
+2a+1 · r for r ≥ 0 and
+ϵ ∈ {0, 1}.
+Proof. Therefore, we have a ≥ 1. Next, observe that t = rζj
+2m for some j ∈ Z and
+r ∈ R if and only if tm ∈ R. One easily shows that since gcd(2, k) = 1, we have
+ζ2a+1 ·ζ2ak = (ζ2a+1k)k+2. Raising both sides to the power b = (k +2)−1 ∈ (Z/m)∗ gives
+ζ2m = ζb
+2a+1 · ζb
+m. Consequently, tm ∈ R if and only if t = r · ζbj
+2a+1 · ζbj
+m for some r ∈ R.
+Finally, ζu
+2a+1 · ζv
+2a = ζu+2v
+2a+1 hence ⟨ζ2a+1⟩/⟨ζ2a⟩ ∼= Z/2.
+We obtain the key to Theorem 3.1.
+Proposition 3.14. Keep the above notations, and consider the set Yf ⊂ Y (see (6)).
+1. If f has no nodes, then G(x) = Bf is trivial, and Yf = RHn
+α ∼= Bn(R).
+2. If f has only real nodes, then Bf \ Yf is isometric to Bn(R).
+3. If f has a pairs of complex conjugate nodes (k = 2a), and no other nodes, then
+Bf \ Yf = Yf is the union of ma copies of Bn(R), any two of which meet along a
+B2c(R) for some integer c with 0 ≤ c ≤ a.
+4. If f has 2a complex conjugate nodes and b real nodes, then there is an isometry
+between Bf \ Yf and the union of ma copies of Bn(R) identified along common
+B2c(R)′s, that is, the set Yf of case 3 above.
+5. In each case, Af acts transitively on the indicated copies of Bn(R). If Bn(R) is
+any one of them, and Γf = (Af/Bf)Bn(R) its stabilizer, then the natural map
+Γf \ Bn(R) → (Af/Bf) \ (Bf \ Yf) = Af \ Yf
+is an isometry of path metrics.
+Proof. 1. This is clear.
+2. Suppose then that f has k real nodes. Then in the local coordinates ti of Lemma
+3.9.2, we have that α : Bn(C) → Bn(C) is defined by α(ti) = ¯ti. Part 1 of the same
+lemma shows that any β ∈ PA fixing x such that xα ∼ xβ is of the form
+Bn(C) → Bn(C),
+(t1, . . . , ti, . . . , tn) �→ (¯t1ζj1, . . . , ¯tkζjk, ¯tk+1, . . . , ¯tn).
+Since f has k real nodes and no complex conjugate nodes, we have (writing j =
+(j1, . . . , jk) and αj = �k
+i=1 φjiri ◦ α):
+Yf ∼=
+m
+�
+j1,...,jk=1
+RHn
+αj ∼= {(t1, . . . , tn) ∈ Bn(C) : tm
+1 , . . . , tm
+k , tk+1, . . . , tn ∈ R} .
+Each of the 2k subsets
+Kf,ϵ1,...,ϵk :=
+�
+(t1, . . . , tn) ∈ Bn(C) : ζ−ϵ1
+2a+1t1, . . . , ζ−ϵk
+2a+1tk ∈ R≥0 and tk+1, . . . , tn ∈ R
+�
+,
+16
+
+indexed by ϵ1, . . . , ϵk ∈ {0, 1}, is isometric to the closed region in Bn(R) bounded by
+k mutually orthogonal hyperplanes. By Lemma 3.13, their union U is a fundamental
+domain for Bf, in the sense that it maps homeomorphically and piecewise-isometrically
+onto Bf \ Yf. Under its path metric, U = ∪Kf,ϵ1,...,ϵk is isometric to Bn(R) by the
+following map:
+U → Bn(R),
+(t1, . . . , tk) �→
+�
+(−ζ2a+1)−ϵ1t1, . . . , (−ζ2a+1)−ϵktk, tk+1, . . . , tn
+�
+.
+This identifies Bf \ Yf with the standard Bn(R) ⊂ Bn(C).
+3. Now suppose f has k = 2a nodes Hr1, . . . , Hr2a. There are now ma anti-isometric
+involutions αji fixing x and such that xαji ∼ xα: they are given in the coordinates ti
+as follows, taking j = (j1, . . . , ja) ∈ (Z/m)a:
+αj : (t1, . . . , tn) �→ (¯t2ζj1, ¯t1ζj1, . . . , ¯t2aζja, ¯t2a−1ζja, ¯t2a+1, . . . , ¯tn).
+So any fixed-point set RHn
+αj is identified with
+Bn(R)αj :=
+�
+(t1, . . . , tn) ∈ Bn(C) : ti = ¯ti−1ζji for 1 ≤ i ≤ 2a even, ti ∈ R for i > 2a
+�
+.
+All these ma copies of Bn(R) meet at the origin of Bn(C); in fact, for j ̸= j′, the space
+Bn(R)αj meets the space Bn(R)αj′ in a B2c(R) if c is the number of pairs (ji, j′
+i) with
+ji = j′
+i.
+4. Now we treat the general case. In the local coordinates ti, any anti-unitary
+involutions fixing x and equivalent to α is of the form
+αj :(t1, . . . , tn) �→
+(¯t2ζj1, ¯t1ζj1, . . . , ¯t2aζja, ¯t2a−1ζja, ¯t2a+1ζj2a+1, . . . , ¯tkζjk, ¯tk+1, . . . , ¯tn)
+for some j = (j1, . . . , ja, j2a+1, . . . , jk) ∈ (Z/m)a+b. We now have Bf ∼= (Z/m)b acting
+by multiplying the ti for 2a+1 ≤ i ≤ k by powers of ζ, and there are ma+b anti-unitary
+involutions αj. We have
+Yf ∼=
+m
+�
+j1,...,jk=1
+RHn
+αj ∼=
+�
+(t1, . . . , tn) ∈ Bn(C) | tm
+2 = ¯tm
+1 , . . . , tm
+2a = ¯tm
+2a−1, tm
+2a+1, . . . , tm
+k , tk+1, . . . , tn ∈ R
+�
+.
+We look at subsets Kf,ϵ1,...,ϵk ⊂ Yf again, this time defined as
+Kf,ϵ =Kf,ϵ1,...,ϵk
+= {(t1, . . . , tn) ∈ Bn(C) |
+tm
+i = ¯tm
+i−1 i ≤ 2a even, ζ−ϵi
+2a+1ti ∈ R≥0 2a < i ≤ k, ti ∈ R, i > k
+�
+.
+As before, we have that the natural map U := �
+ϵ Kf,ϵ → Bf \Yf is an isometry. Define
+�Yf =
+�
+(t1, . . . , tn) ∈ Bn(C) : tm
+i = ¯tm
+i−1 for i ≤ 2a even, ti ∈ R, for i > 2a
+�
+.
+17
+
+Under its path metric, U = ∪ϵKf,ϵ1,...,ϵk is isometric to �Yf by the following map:
+U → �Yf,
+(t1, . . . , tk) �→
+�
+t1, . . . , t2a, (−ζ2a+1)−ϵ1t2a+1, . . . , (−ζ2a+1)−ϵktk, tk+1, . . . , tn
+�
+.
+Hence Bf \ Yf ∼= �Yf; but since �Yf is what Yf was in case 3, we are done.
+5.
+The transitivity of Af on the copies of Bn(R) follows from the fact that
+G(x) ⊂ Af contains transformations multiplying t1, . . . , t2a by powers of ζ, hence
+ti �→ ζuti, ti−1 �→ ti−1 maps those ti−1, ti with ti = ¯ti−1ζji to those ti−1, ti with
+ti = ¯ti−1ζj1+u.
+So if B is any one of the copies of Bn(R), and G = (Af/Bf)H is
+its stabilizer, then it remains to prove that G \ B → Af \ Yf is an isometry. Surjec-
+tivity follows from the transitivity of Af on the Bn(R)
+′s. It is a piecewise isometry so
+we only need to prove injectivity. This will follow from an elementary lemma.
+Lemma 3.15. Let a group G act on a set X, let Y and I be sets, and let {φi : Y �→ X}i∈I
+be a set of embeddings. Write Yi = φi(Y ) and suppose that X = ∪iYi. Fix 0 ∈ I. Let
+H ⊂ G be the stabilizer of Y0. Suppose that for all y ∈ X, the stabilizer of y in G acts
+transitively on the sets Yi containing y. Then H \ Y0 → G \ X is injective.
+Proof. Let x, y ∈ Y0 and g ∈ G such that g · x = y. Then y = gx ∈ gY0. Since also
+y ∈ Y0, there is an element h ∈ StabG(y) such that hgY0 = Y0 and hg(x) = h(y) = y.
+Let f = hg; then f ∈ H and f · x = y, which proves what we want.
+Now let us use the lemma: suppose that y ∈ Bf \ Yf.
+We need to prove that
+StabAf/Bf(y) acts transtivitely on the copies of Bn(R) containing y. There exists
+j = (j1, . . . , ja, j2a+1, . . . , jk) ∈ (Z/m)a+b
+such that y = (t1, . . . , tn) with ti = ¯ti−1ζji for i ≤ 2a even, ti = ¯ti−1ζji for 2a < i ≤ k,
+and ti ∈ R for i > k. If all ti are non-zero, then y ∈ ∪j′RHn
+αj′ is only contained in
+RHn
+αj, so there is nothing to prove. Let us suppose that t1 = t2 = 0 and the other
+ti are non-zero. Then y is contained in all the RHn
+αj′ with j′
+i = ji for i ≥ 2; there
+are m of them. The stabilizer of y multiplies t1 and t2 by powers of ζ and leaves the
+other ti invariant; it acts transitively on the RHn
+αj′ containing y for if t2 = ¯t1ζj′
+1 then
+ζ(j′′
+1 −j′
+1)t2 = ¯t1ζj′′
+1 . The general case is similar.
+We need one more lemma before we can prove Theorem 3.1:
+Lemma 3.16. The maps P : Y → CHn and P : PΓ \ Y → PΓ \ CHn are proper.
+Proof. The map P : Y → CHn is proper because any compact set in CHn meets only
+finitely many (RHn
+α)
+′s, α ∈ PA (Lemma 3.3), and P carries each Hα = p (RHn
+α)
+homeomorphically onto RHn
+α.
+To prove that P is proper, let π (resp.
+q) be the
+quotient map CHn → PΓ \ CHn (resp. Y → PΓ \ Y ), consider the commutative
+diagram
+Y
+q
+�
+P
+� CHn
+π
+�
+PΓ \ Y
+P � PΓ \ CHn
+18
+
+and let Z ⊂ PΓ \ Y be a closed subset. Its inverse image W = q−1(Z) ⊂ Y in Y is a
+closed, PΓ-invariant subset. Since P is proper and PΓ-equivariant, P(W) ⊂ CHn is
+a closed, PΓ-invariant subset of CHn and we have
+P(W) = π−1π (P(W)) = π−1 �
+P (Z)
+�
+.
+Hence P(Z) is closed in PΓ \ CHn, which proves that P is closed. Therefore, to
+prove that P is proper, it suffices to show that it has finite fibers.
+Let y ∈ PΓ \ CHn and let V = P
+−1(y) ⊂ PΓ \ Y . For each v ∈ V , choose an
+element uv ∈ Y such that q(uv) = v. This gives
+q−1(V ) =
+�
+v∈V
+PΓ · uv ⊂ Y.
+Moreover, we have that
+P(PΓ · uv) = PΓ · P(uv) = π−1π (P(uv)) = π−1(q(v)) = π−1(y)
+for each v ∈ V . This gives a map
+P :
+�
+v∈V
+PΓ · uv → π−1(y)
+which is surjective on each PΓ · uv. By properness of P, for any z ∈ π−1(y), the
+inverse image
+P−1(z) ⊂
+�
+v∈V
+PΓ · uv ⊂ Y
+is finite. Since P−1(z) meets every orbit PΓ · uv, it follows that V is finite.
+Proof of Theorem 3.1. 1. The path metric on PΓ \ Y is given by Proposition 3.7.
+Note that the map P : Y → CHn is a local embedding by Lemma 3.4, which was
+used to define the metric on Y (Proposition 3.6). Thus, almost by definition, P is
+a local isometry. For each f ∈ Y we can find a PΓf-invariant open neighborhood
+Uf ⊂ Yf ⊂ Y such that PΓf \ Uf ⊂ PΓ \ Y , with Uf mapping bijectively onto an open
+subset Vf in the closed subset P(Yf) = ∪iRHn
+αi ⊂ CHn. By PΓ-equivariance of P,
+the set Vf is PΓf-invariant, and we have PΓf \ Vf ⊂ PΓ \ CHn. Thus
+P : PΓ \ Y → PΓ \ CHn
+is also a local isometry. The space PΓ \ CHn is complete, and P is proper by Lemma
+3.16, so PΓ \ Y is complete as well.
+2. [f] ∈ PΓ \ Y be the image of f ∈ Y . Then [f] has an open neighborhood
+isometric to the quotient of an open set W in RHn by a finite group of isometries Γf.
+Indeed, take Yf ⊂ Y as in Equation (6), and f ∈ Uf ⊂ Yf as in Lemma 3.4.2. We
+let Af = PΓf be the stabilizer of f in PΓ as before, and take an Af-equivariant open
+neighborhood Vf ⊂ Uf such that Af \ Vf ⊂ PΓ \ Y . By Proposition 3.14.5, we know
+that Af \ Yf is isometric to Γf \ RHn for some finite group of isometries of RHn. This
+19
+
+implies that Af \ Vf is isometric to some open set W ′ in Γf \ RHn. Take W ⊂ RHn
+to be the preimage of W ′.
+Claim: For any path metric space X locally isometric to quotients of RHn by finite
+groups of isometries, there is a unique real-hyperbolic orbifold structure on X whose
+path metric is the given one.
+Proof of the Claim: If U and U ′ are connected open subsets of RHn and Γ and Γ′
+finite groups of isometries of RHn preserving U and U ′ respectively, then any isometry
+¯φ : Γ \ U → Γ′ \ U ′ extends to an isometry φ : RHn → RHn such that φ(U) = U ′ and
+φΓφ−1 = Γ′ ⊂ Isom(RHn).
+We conclude that PΓ \ Y is naturally a real hyperbolic orbifold.
+3. Let us show that
+O :=
+�
+α∈CA
+[PΓα \ (RHn
+α − H )] ⊂ PΓ \ Y
+as hyperbolic orbifolds. It suffices to show the following
+Claim: For those f = p(xα) ∈ Y that have no nodes, the stabilizer Af = PΓf ⊂ PΓ
+of f ∈ Y and the stabilizer PΓα,x ⊂ PΓα of x in RHn
+α agree as subgroups of PΓ.
+Proof of the Claim: To prove that Af = PΓα,x, we first observe that p : �Y → Y
+induces an isomorphism between PΓxα, the stabilizer of xα ∈ �Y and PΓf, the stabilizer
+of f = [x, α] ∈ Y . So it suffices to show that PΓxα = PΓα,x. For this we use that the
+normalizer NPΓ(α) and the stabilizer PΓα ⊂ PΓ of α in PΓ are equal, which implies
+that PΓα,x = PΓxα because
+{g ∈ PΓα : gx = x} = {g ∈ NPΓ(α) : gx = x}
+=
+�
+g ∈ PΓ : g · xα = (g(x), gαg−1) = xα
+�
+.
+The claim is proved. Part 3 of the theorem can be deduced from it as follows. Let
+f = p(xα) ∈ Y have no nodes. We have Yf = RHn
+α, hence
+Af \ RHn
+α = Af \ Yf = Γf \ RHn
+with
+Γf = Af \ Bf = Af.
+By construction, an orbifold chart of the glued space PΓ \ Y is given by
+W → Af \ W ⊂ PΓα \ RHn
+α ⊂ Y
+for an invariant open subset W of RHn
+α containing x. Because Af = PΓα,x by the
+claim, this is also an orbifold chart for O at the point xα.
+4. The real-hyperbolic orbifold PΓ \ Y is complete by Part 1, so the uniformiza-
+tion of the connected components of PΓ \ Y follows from the Ehresmann–Thurston
+uniformization theorem for (G, X)-orbifolds, see [Thu80, Proposition 13.3.2].
+This
+concludes the proof of Theorem 3.1, and thereby also of Theorem 1.6.
+4
+Unitary Shimura varieties
+The goal of this section is to prove Proposition 4.7, which describes the complex ball
+quotient PΓ\CHn in terms of moduli of abelian varieties with OK-action of hyperbolic
+20
+
+signature, and Proposition 4.10, which interprets the divisor PΓ \ H as the locus
+of abelian varieties A that admit an OK-linear homomorphism Cg/Ψ(OK) → A of
+polarized abelian varieties with OK-action. This has two applications:
+1. Consider a relative uniform cyclic cover (see e.g. [AV04])
+X → P → S,
+where P = P1
+S (resp. P3
+S), the fibers of X → S are curves (resp. threefolds with H0,3 =
+0) and the induced hermitian form on middle cohomology has hyperbolic signature.
+Since the image I = P(S(C)) ⊂ PΓ \ CHn of the period map P : S(C) → PΓ \ CHn
+is contained in the locus of abelian varieties whose theta divisor is irreducible, one has
+I ⊂ PΓ \ (CHn − H ) .
+2.
+If the different ideal DK ⊂ OK is generated by some η ∈ OK − OF with
+η2 ∈ OF, then the hyperplanes in the arrangement H ⊂ CHn are orthogonal along
+their intersection (see Theorem 4.12).
+4.1
+Alternating and hermitian forms on the lattice. The goal of this subsection is
+to prove two lemmas. They will later be used to show that PΓ\CHn is a moduli space
+of abelian varieties, and to give a modular interpretation of the divisor PΓ \ H ⊂
+PΓ \ CHn. We continue with the notation of Section 2.1. In particular, Λ is a free
+OK-module of rank n + 1.
+Lemma 4.1. The assignment T �→ TrK/Q ◦ T defines a bijection between:
+1. The set of skew-hermitian forms T : ΛQ × ΛQ → K.
+2. The set of alternating forms E : ΛQ×ΛQ → Q such that E(a·x, y) = E(x, aσ ·y).
+Under this correspondence, T(Λ, Λ) ⊂ D−1
+K if and only if E(Λ, Λ) ⊂ Z.
+Proof. Let
+T : ΛQ × ΛQ → K
+be as in 1. Define ET = TrK/Q ◦ T. Since T is skew-hermitian, we have, for each
+x, y ∈ ΛQ, that
+TrK/QT(x, y) = −TrK/QT(y, x).
+Since K/Q is separable, for any x ∈ K, we have [Ste08, (7-1)]:
+TrK/Q(x) =
+�
+1≤i≤g
+(τi(x) + τiσ(x)) .
+Thus, we have TrK/Q(σ(x)) = TrK/Q(x), so that ET(x, y) = −ET(y, x) for any x, y ∈
+ΛQ. The property in 2 is easily checked.
+21
+
+Conversely, let E : ΛQ × ΛQ → Q be as in 2. Choose a basis {b1, . . . , bn+1} ⊂ Λ for
+Λ over OK. Define Q to be the induced map Kn+1 × Kn+1 → Q and consider the map
+K → Q, a �→ Q(a · ei, ej). Since the trace pairing
+K × K → Q,
+(x, y) �→ TrK/Q(xy)
+is non-degenerate [Stacks, Tag 0BIE], there is a unique tij ∈ K such that Q(a·ei, ej) =
+TrK/Q(a · tij) for every a ∈ K.
+This gives a matrix (tij)ij ∈ Mn+1(K) such that
+σ(tij) = −tji, and the basis {bi} induces a skew-hermitian form TE : ΛQ × ΛQ → K.
+The last claim from the definition of D−1
+K ⊂ K as the trace dual of OK, see [Ser79,
+Chapter III, §3].
+Examples 4.2.
+1. Suppose K = Q(
+√
+∆) is imaginary quadratic with discriminant
+∆ and non-trivial Galois automorphism a �→ aσ. Let E : Λ × Λ → Z be an
+alternating form with E(a · x, y) = E(x, aσ · y). The form T : Λ × Λ → D−1
+K =
+(
+√
+∆)−1 is defined as
+T(x, y) = E(
+√
+∆ · x, y) + E(x, y)
+√
+∆
+2
+√
+∆
+.
+2. Let K = Q(ζ) where ζ = ζp = e2πi/p ∈ C for some prime number p > 2. Let
+E : Λ × Λ → Z be an alternating form with E(a · x, y) = E(x, aσ · y). Then
+DK = (p/(ζ − ζ−1)) and
+T : Λ × Λ → D−1
+K ,
+T(x, y) = 1
+p
+p−1
+�
+j=0
+ζjE
+�
+x, ζj · y
+�
+.
+Now consider a corresponding pair
+(E : ΛQ × ΛQ → Q,
+T : ΛQ × ΛQ → K)
+as in Lemma 4.1, and suppose that E is non-degenerate.
+Let ϕ : K → C be an
+embedding. Define a skew-hermitian form T ϕ as
+T ϕ : Λ ⊗OK,ϕ C × Λ ⊗OK,ϕ C → C,
+T ϕ(
+�
+i
+xi ⊗ λi,
+�
+j
+yj ⊗ µj) =
+�
+ij
+λiµj · ϕ (T(xi, yj)) .
+On ΛC, we also have the skew-hermitian form A(x, y) = EC(x, ¯y). The composition
+(Λ ⊗Z C)ϕ → Λ ⊗Z C → Λ ⊗OK,ϕ C
+is an isomorphism. Define Aϕ to be the restriction of A to the subspace (Λ ⊗Z C)ϕ =
+Λ ⊗OK,ϕ C ⊂ ΛC. Note that
+Λ ⊗Z C ∼= ⊕φ:K→C (Λ ⊗Z C)φ .
+For x ∈ Λ ⊗Z C, let xφ be the image of x under Λ ⊗Z C → (Λ ⊗Z C)φ.
+22
+
+Lemma 4.3. Let ϕ: K → C be an embedding. We have an equality of skew-hermitian
+forms:
+T ϕ = Aϕ : (Λ ⊗Z C)ϕ × (Λ ⊗Z C)ϕ → C.
+More precisely, we have A(x, y) = �
+φ:K→C T φ(xφ, yφ) for every x, y ∈ Λ ⊗Z C.
+Proof. Write V = ΛQ. The lemma follows from the fact that the following diagram
+commutes:
+V × V
+� �
+��
+T
+� K� �
+�
+TrK/Q
+� Q� �
+�
+V ⊗Q C × V ⊗Q C
+TC
+�
+A(x,y)
+�
+K ⊗Q C
+⊕φ (V ⊗Q C)φ × (V ⊗Q C)φ
+⊕T φ
+� ⊕φCφ
+�
+� C.
+Here, φ ranges over the set of embeddings K → C, Cφ is the K-module C where K
+acts via φ, and
+TC : V ⊗Q C × V ⊗Q C → K ⊗Q C
+is the map that sends (v ⊗ λ, x ⊗ µ) to λ¯µT(v, w).
+4.2
+Moduli of abelian varieties with an action by a CM field.
+Notation 4.4. In the rest of Section 4, we fix:
+1. a non-degenerate hermitian form h : Λ × Λ → D−1
+K ; and
+2. an element ξ ∈ D−1
+K such that σ(ξ) = −ξ and ℑ (τi(ξ)) < 0 for 1 ≤ i ≤ g and
+write η = ξ−1. Here, the embeddings τi : K → C are those introduced in (1).
+These data define a skew-hermitian form
+T : Λ × Λ → D−1
+K ,
+T := ξ · h.
+The form T is in turn attached to a symplectic form (see Lemma 4.1)
+E : Λ × Λ → Z
+such that E(ax, y) = E(x, aσy) for all a ∈ OK, x, y ∈ Λ.
+Write Vi = ΛQ ⊗K,τi C and define
+hτi : Vi × Vi → C
+to be the hermitian form restricting to τi ◦ h on Λ. Let (ri, si) be the signature of the
+hermitian form hτi.
+Let A be a complex abelian variety, ι a homomorphism OK → End(A), and λ a
+polarization A → A∨, satisfying the following (c.f. [KR14, Part I, §2.1]):
+23
+
+Conditions 4.5.
+1. We have ι(a)† = i(aσ) for the Rosati involution
+†: End(A)Q → End(A)Q,
+and
+2. char(t, ι(a)|Lie(A)) = �g
+ν=1(t − aτi)ri · (t − aτiσ)si ∈ C[t]
+(the characteristic polynomial of ι(a)).
+Note that dim A = g(n + 1). Define EA : H1(A, Z) × H1(A, Z) → Z to be the alter-
+nating form corresponding to λ. The condition on the Rosati involution implies that
+EA(ι(a)x, y) = EA(x, ι(aσ)y) for x, y ∈ H1(A, Q). Define a hermitian form hA on the
+OK-module H1(A, Z) as follows:
+hA = ηTA : H1(A, Z) × H1(A, Z) → D−1
+K .
+Here, TA : H1(A, Z) × H1(A, Z) → D−1
+K is the skew-hermitian form attached to EA via
+Lemma 4.1.
+Definition 4.6.
+1. Let �
+ShK(h) be the set of isomorphism classes of four-tuples
+(A, i, λ, j), where (A, i, λ) is as above and satisfies Conditions 4.5, and where
+j : H1(A, Z) → Λ is a symplectic isomorphism of OK-modules.
+2. Let D(Vi) be the space of negative si-planes in the hermitian space (Vi, hτi).
+We have the following proposition which is due to Shimura, see [Shi63, Theorem 2]
+or [Shi64, §1]. We give a different proof since it will imply Proposition 4.10 below,
+whereas we did not know how to deduce Proposition 4.10 from loc.cit. We remark
+that Shimura assumes Λ to be an R-module for any order R ⊂ OK; our proof carries
+over, but we do not need this generalization.
+Proposition 4.7. There is a canonical bijection
+�
+ShK(h) ∼= D(V1) × · · · × D(Vg).
+Proof. Let (A, i, λ, j) be a representative of an isomorphism class in �
+ShK(h).
+Let
+H1(A, C) = H−1,0 ⊕ H0,−1 be the Hodge decomposition of A. For 1 ≤ i ≤ g there is a
+decomposition
+H1(A, C)τi = H−1,0
+τi
+⊕ H0,−1
+τi
+,
+(10)
+with dim H−1,0
+τi
+= ri and dim H0,−1
+τi
+= si. The latter holds because
+H−1,0
+τiσ = H0,−1
+τi
+.
+By Lemma 4.3, τi(η)EA,C(x, ¯y) and hτi
+A,C(x, y) agree as hermitian forms on the complex
+vector space H1(A, Z) ⊗OK,τi C. Since ℑτi(η) > 0 for every i, the decomposition of
+H1(A, C)τi in (10) is a decomposition into a positive definite ri-dimensional subspace
+and a negative definite si-dimensional subspace. The isomorphism j : H1(A, Q) → ΛQ
+induces an isometry ji : H1(A, C)τi → Vi for every i, and so we obtain a negative
+si-plane j(H0,−1
+τi
+) in the hermitian space Vi for all i.
+Reversing the argument shows that given a negative si-plane Xi ⊂ Vi for every i,
+there is a canonical polarized abelian variety A = H−1,0/Λ, acted upon by OK and
+inducing the planes Xi ⊂ Vi.
+24
+
+Definition 4.8.
+1. Let ShK(h) be the set of isomorphism classes of polarized OK-
+linear abelian varieties (A, i, λ), satisfying Conditions 4.5, such that H1(A, Z) is
+isometric to Λ as hermitian OK-modules.
+2. Let Γ(h) = AutOK(Λ, h); this is the group of OK-linear automorphisms of Λ
+preserving our form h : Λ × Λ → D−1
+K .
+The bijection in Proposition 4.7 being Γ(h)-equivariant, we obtain the following:
+Corollary 4.9. There is a canonical bijection
+ShK(h) ∼= Γ(h) \ D(V1) × · · · × D(Vg).
+4.3
+Abelian varieties with moduli in the hyperplane arrangement. The set of
+embeddings Ψ defined in (1) defines a map Ψ : OK → Cg, giving a complex torus
+Cg/Ψ(OK). The map
+Q : K × K → Q,
+Q(x, y) = TrK/Q(ξx¯y)
+is a non-degenerate Q-bilinear form such that Q(ax, y) = Q(x, aσy) for every a, x, y ∈
+K. Moreover, Q(OK, OK) ⊂ Z because ξ ∈ D−1
+K . By [Mil20, Example 2.9 & Footnote
+16], Q defines a Riemann form on the complex torus Cg/Ψ(OK).
+As in Section 2.1, let CHn be the set of negative lines in Λ ⊗OK,τ1 C, and define
+H = ∪h(r,r)=1⟨rC⟩⊥ ⊂ CHn.
+Proposition 4.10. Suppose that (r1, s1) = (n, 1), (ri, si) = (n + 1, 0) ∀2 ≤ i ≤ g.
+Under the bijection �
+ShK(h) ∼= CHn of Proposition 4.7, points in H ⊂ CHn correspond
+to isomorphism classes of those polarized marked OK-linear abelian varieties A that
+admit a OK-linear homomorphism Cg/Ψ(OK) → A of polarized abelian varieties.
+Proof. Consider an isomorphism class [(A, i, λ, y)] ∈ �
+ShK(h) corresponding to a point
+[x] ∈ CHn. We may assume that A = H−1,0/Λ with Λ ⊗Z C = H−1,0 ⊕ H0,−1, and that
+TA = T. Let
+φ : Cg/Ψ(OK) → A
+be a homomorphism as in the proposition. We obtain a homomorphism
+OK → Ψ(OK) → H1(A, Z) = Λ
+which, for simplicity, we also denote by φ : OK → Λ. Let r ∈ Λ be the image of
+1 ∈ OK. The fact that Q = φ∗EA implies that TQ = φ∗TA = φ∗T. Therefore, we have
+η−1 = TQ(1, 1) = TA(φ(1), φ(1)) = T(φ(1), φ(1)) = T(r, r),
+25
+
+so that h(r, r) = η · T(r, r) = 1.
+We claim that h(x, rτ) = 0, where the element
+rτ ∈ (Λ ⊗Z C)τ is the image of r ∈ Λ. To see this, write
+Ψ(OK) = L,
+L ⊗ C = W −1,0 ⊕ W 0,−1,
+and let α ∈ L correspond to 1 ∈ OK. Notice that (L ⊗Z C)τ = W −1,0
+τ
+. Consequently,
+since the composition
+W −1,0
+τ
+= (L ⊗Z C)τ → (Λ ⊗Z C)τ = H−1,0
+τ
+⊕ H0,−1
+τ
+factors through the inclusion of H−1,0
+τ
+into (L ⊗Z C)τ, we see that
+rτ = r−1,0
+τ
+∈ H−1,0
+τ
+=
+�
+H0,−1
+τ
+�⊥ = ⟨x⟩⊥,
+and the claim follows.
+Conversely, let [x] ∈ ⟨rC⟩⊥ ⊂ H with r ∈ Λ such that h(r, r) = 1 and consider the
+marked abelian variety A = H−1,0/Λ corresponding to [x]. Define a homomorphism
+φ : OK → Λ by φ(1) = r. Then φ can be shown to be a morphism of Hodge structures
+using the fact that its C-linear extension preserves the eigenspace decompositions. We
+obtain an OK-linear homorphism φ : Cg/Ψ(OK) → A.
+The fact that h(r, r) = 1
+implies that φ preserves the polarizations on both sides.
+Observe that if the different DK ⊂ OK is a principal ideal (η) ⊂ OK, then we have
+{x ∈ K : TrK/Q
+�
+xη−1OK
+�
+⊂ Z} = {x ∈ K : x · η−1OK ⊂ η−1OK}
+= {x ∈ K : xOK ⊂ OK} = OK.
+Thus, Q : Ψ(OK)×Ψ(OK) → Z defines a principal polarization on the torus Cg/Ψ(OK)
+in this case. In fact, for β ∈ K, the rational Riemann form
+Ψ(K) × Ψ(K) → Q,
+(Ψ(x), Ψ(y)) �→ TrK/Q(β−1x¯y)
+defines a principal polarization on Cg/Ψ(OK) if and only if (i) we have that β generates
+the different ideal DK, (ii) we have that σ(β) = −β, and (iii) we have that ℑ(ϕ(β)) > 0
+for every ϕ ∈ Ψ. This follows from the above; see also [Wam99].
+Consider the following:
+Conditions 4.11.
+1. The CM type (K, Ψ) is primitive.
+2. We have DK = (η) for some η ∈ OK such that σ(η) = −η.
+3. The signature of hτi is (n, 1) for i = 1 and (n + 1, 0) for i ̸= 1.
+Theorem 4.12. Suppose that Conditions 4.11 hold. Let r1, r2 ∈ Λ satisfy Hr1∩Hr2 ̸= ∅
+and Hr1 ̸= Hr2 ⊂ CHn for Hri = ⟨ri,C⟩⊥ ⊂ CHn. Then h(r1, r2) = 0.
+26
+
+Proof. Let [x] ∈ Hr ∩ Ht ⊂ CHn(V ), and let A be an abelian variety whose iso-
+morphism class gives [x].
+Define B to be the principally polarized abelian variety
+Cg/Ψ(OK). By Proposition 4.10, the roots r and t induce OK-linear embeddings
+φ1 : B �→ A
+and
+φ2 : B �→ A
+of polarized abelian varieties. By Lemma 4.13 below, the φi induce decompositions
+A ∼= B × C1
+and
+A ∼= B × C2
+as polarized abelian varieties. Note that B is non-decomposable as an abelian vari-
+ety because End(B) ⊗Z Q = K is a field (here we use that the CM type (K, Ψ) is
+primitive). By [Deb96], the decomposition of (A, λ) into non-decomposable polarized
+abelian subvarieties is unique, in the strong sense that if (Ai, λi), i ∈ {1, . . . , r} and
+(Bj, µj), j ∈ {1, . . . , m} are polarized abelian subvarieties such that the natural ho-
+momorphisms �
+i(Ai, λi) → (A, λ) and �
+j(Bj, λj) → (A, λ) are isomorphisms, then
+r = m and there exists a permutation σ on {1, . . . , r} such that Bj and Aσ(j) are equal
+as polarized abelian subvarieties of (A, λ), for every j ∈ {1, . . . , r}. Consequently, for
+the two abelian subvarieties
+Bi = φi(B) ⊂ A,
+we have either that
+B1 = B2 ⊂ A
+or that
+B1 ∩ B2 = {0}.
+Suppose first that B1 = B2. Then
+OK · r = φ1(OK) = φ2(OK) = OK · t ⊂ Λ.
+Therefore, r = λt for some λ ∈ O∗
+K; but then Hr = Ht which is absurd. Thus, we
+must have
+A ∼= B1 × B2 × C
+as polarized abelian varieties, for some polarized abelian subvariety C of A.
+This
+implies that
+H−1,0 = Lie(A) ∼= Lie(B1) × Lie(B2) × Lie(C),
+which is orthogonal for the positive definite hermitian form iEC(x, ¯y) on H−1,0.
+Observe that rτ = r−1,0
+τ
+∈ H−1,0
+τ
+and tτ = t−1,0
+τ
+∈ H−1,0
+τ
+: see the proof of Proposition
+4.10. By Lemma 4.3, we have
+h(r, t) = hτ(rτ, tτ) = τ(η) · T τ
+C(rτ, tτ)
+= τ(η) · EC(rτ, ¯tτ) = τ(η) · EC(r−1,0
+τ
+, t−1,0
+τ
+).
+Since r−1,0
+τ
+∈ Lie(B1) and t−1,0
+τ
+∈ Lie(B2), we have iEC(r−1,0
+τ
+, t−1,0
+τ
+) = 0.
+Lemma 4.13. Let A be an abelian variety over a field k, with polarization λ : A → �A.
+Let B ⊂ A be an abelian subvariety such that the polarization µ = λ|B is principal.
+There is a polarized abelian subvariety Z ⊂ A such that A ∼= B×Z as polarized abelian
+varieties.
+27
+
+Proof. Let W = Ker(A
+λ−→ �A → �B). Let Z = W 0
+red. Then Z is an abelian subvariety of
+dimension dim(A)−dim(B) of A. The kernel of the natural homomorphism B×Z → A
+is contained in (B ∩Z)×(B ∩Z). However, B ∩Z ⊂ B ∩W = (0) because µ : B → �B
+is an isomorphism. Thus, B × Z → A is an isomorphism.
+Finally, we remark that the condition on the different ideal DK
+⊂ OK in Theorem
+4.12 (see Conditions 4.11) is satisfied in two interesting cases:
+Proposition 4.14. Suppose that K/Q is an imaginary quadratic extension, or that
+K = Q(ζn) is a cyclotomic field for some integer n ≥ 3. Then Condition 4.11.2 is
+satisfied, i.e. we have DK = (η) ⊂ OK for some element η ∈ OK such that σ(η) = −η.
+Proof. If K/Q is imaginary quadratic with discriminant ∆, then DK = (
+√
+∆) and the
+assertion is immediate. Let n ≥ 3 be an integer, and consider the fields
+K = Q(ζn) ⊃ F = Q(α),
+with
+α = ζn + ζ−1
+n .
+Since OK = Z[ζn] by [Neu99, I, Proposition 10.2], we have OK = OF[ζn]. Notice
+that f(x) = x2 − αx + 1 ∈ OF[x] is the minimal polynomial of ζn over F. We have
+f ′(ζn) = 2ζn − αζn = ζn − ζ−1
+n . Therefore,
+DK/F = (f ′(ζn)) =
+�
+ζn − ζ−1
+n
+�
+,
+see [Neu99, III, Proposition 2.4].
+By [Lia76], we know that OF = Z[α]. Moreover, if g(x) ∈ Z[x] is the minimal polyno-
+mial of α over Q, then DF/Q = (g′(α)). By [Neu99, III, Proposition 2.2], we have that
+DK/Q = DK/FDF/Q. Combining all this yields
+DK/Q = DK/FDF/Q =
+�
+ζn − ζ−1
+n
+�
+· (g′(α)) =
+�
+(ζn − ζ−1
+n )g′(α)
+�
+.
+Remark 4.15. It would be more natural to attach an orthogonal hyperplane arrange-
+ment H ⊂ CHn to every primitive CM field K and integral hermitian form h of
+hyperbolic signature, such that H = H = ∪h(r,r)=1⟨rC⟩⊥ if DK = (η) for some η ∈ OK
+such that σ(η) = −η. This turns out to be possible. The idea is as follows.
+Consider our CM field K. Choose β ∈ OK − OF such that β2 ∈ OF; then choose
+a CM type Ψ = {τi : K → C}1≤i≤g such that ℑ(τi(β)) > 0 for all i. Let h be a non-
+degenerate hermitian form Λ × Λ → D−1
+K such that sign(hτ) = (n, 1) and sign(hτi) =
+(n + 1, 0) for i ̸= 1. Let S be the set of fractional ideals a ⊂ K for which there exist
+an element b ∈ OF such that DKaa = (bβ). By [Wam99, Theorem 4], S is not empty.
+For a ∈ S , define η = bβ ∈ OK and consider the complex torus B = Cg/Ψ(a). It
+is equipped with the Riemann form Q : Ψ(a) × Ψ(a) → Z, (x, y) �→ TrK/Q(η−1x¯y),
+and Q defines a principal polarization on B [Wam99, Theorem 3]. Let R be the set
+of embeddings φ : a → Λ, a ∈ S , such that h(φ(x), φ(y)) = x¯y for all x, y ∈ a. For
+φ ∈ R, one obtains a hyperplane Hφ = {x ∈ CHn : hτ(x, φ(a)) = 0} ⊂ CHn. The
+sought-for hyperplane arrangement H ⊂ CHn is defined as H = ∪φ∈RHφ. Indeed, if
+the CM type (K, Ψ) is primitive, then H is an orthogonal arrangement by arguments
+similar to those used to prove Proposition 4.10 and Theorem 4.12.
+28
+
+5
+Non-arithmeticity of the lattice underlying the glued space
+5.1
+The glued space attached to the standard hermitian lattice over the Eisen-
+stein integers. Let K = Q(ζ3). For an integer n ≥ 2, let Λn = Z[ζ3]n+1 and define
+Hn : Λn ⊗Z Λn → OK,
+Hn(x, y) = −x0¯y0 +
+n
+�
+i=1
+xi¯yi.
+Let Rn = {r ∈ Λn | Hn(r, r) = 1}. By Proposition 4.14 and Theorem 4.12, the hyper-
+plane arrangement H = ∪r∈RnHr is an orthogonal arrangement, i.e. Condition 2.4
+is satisfied. Thus, we can perform the glueing construction of Section 2 to obtain a
+metric space
+Xn = PΓn \ Yn.
+Moreover, this metric on Xn extends to a complete real hyperbolic orbifold structure,
+hence its connected components are quotients of RHn by discrete groups of isometries
+(see Theorem 3.1.4).
+The goal of Section 5 is to prove that for every n ≥ 2, there exists a connected
+component X+
+n of Xn such that the lattice Γ+
+n ⊂ PO(n, 1) underlying X+
+n is non-
+arithmetic. Moreover, we will prove the analogous statement for the field K′ = Q(ζ5)
+and the lattice Λ′
+n =
+�
+Z[ζ5]n+1, diag((1 −
+√
+5)/2, 1, . . . , 1)
+�
+. See Theorem 5.18 below.
+5.2
+The case n = 2. Define four anti-unitary involutions αi : Λ2 → Λ2 as follows:
+α0 : (x0, x1, x2) �→ (¯x0,
+¯x1,
+¯x2)
+α1 : (x0, x1, x2) �→ (¯x0, −¯x1,
+¯x2)
+α2 : (x0, x1, x2) �→ (¯x0, −¯x1, −¯x2).
+(11)
+Lemma 5.1. The anti-unitary involutions α0, α1, α2 are pairwise non-conjugate, each
+anti-unitary involution of Λ2 is Γ2-conjugate to exactly one of the ±αi for i = 0, 1, 2,
+the composition
+2
+�
+i=0
+RH2
+αi → Y2 → PΓ2 \ Y2 = X2
+is surjective, and X2 is connected.
+Proof. For the first statement, see [ACT06; ACT10].
+The idea is as follows.
+Let
+θ = ζ3 − ζ−1
+3
+= √−3 ∈ Z[ζ3] and consider the vector space W2 = Λ2/θΛ2. For α ∈ A ,
+let α: W2 → W2 be the induced involution, and let q2 : V → F3 be the quadratic form
+q2(x) = H2(x, x) mod θ. Define
+D(α) = dim(V α),
+T(α) = det(q2|V α) ∈ F∗
+3/(F∗
+3)2 = {±1}.
+(12)
+The restrictions of q2 := H2 mod θ to the fixed spaces V αi ⊂ V for α0, α1, α2 have
+pairwise distinct conjugacy invariant (D(αi), T(αi)) (see Lemma 5.2 below), which
+proves the first statement. Using moduli of real binary sextics with one marked real
+root, one can then show that each anti-unitary involution of Λ2 must be Γ2-conjugate
+to exactly one of the ±αi for i = 0, 1, 2. The third statement follows from this. The
+fourth statement (i.e. the connectivity of X2) follows from the third statement.
+29
+
+5.3
+Conjugacy classes of anti-unitary involutions. Define θ = ζ3 − ζ−1
+3
+as before,
+and define
+Wn = Λn/θΛn.
+Define anti-unitary involutions as follows:
+β0 : Λn → Λn,
+β0(x0, . . . , xn) = (¯x0, . . . , ¯xn) ,
+βi : Λn → Λn,
+βi(x0, . . . , xn) = (¯x0, −¯x1, . . . , −¯xi, ¯xi+1, . . . , ¯xn) .
+(13)
+These involutions induce the following involutions on the F3-vector spaces Wn:
+β0 : Wn → Wn,
+β0(x0, . . . , xn) = (x0, . . . , xn) ,
+βi : Wn → Wn,
+βi(x0, . . . , xn) = (x0, −x1, . . . , −xi, xi+1, . . . , xn) .
+We also consider the quadratic form
+qn : Wn → F3,
+qn(x) = Hn(x, x)
+mod θ.
+Note that, for i ≥ 1 and x = (x0, xi+1 . . . , xn) ∈ W βi
+n , one has qn(x) = −x2
+0 + x2
+i+1 +
+· · · + x2
+n. Similarly, for x = (x1, . . . , xi) ∈ V −βi, one has qn(x) = x2
+1 + · · · + x2
+i . For an
+anti-unitary involution α: Λn → Λn, define the following Γn-conjugacy invariants:
+D(α) = dim(V α),
+T(α) = det(qn|V α).
+The above shows that
+(D(βi), T(βi)) = (n − i + 1, −1) ,
+(D(−βi), T(−βi)) = (i, 1) ,
+i = 0, 1, . . . , n.
+Therefore, we have shown:
+Lemma 5.2. The classes ±βi ∈ A for i = 0, . . . , n are all pairwise non Γn-conjugate.
+In particular, the classes βi ∈ PAn are all pairwise non PΓn-conjugate.
+5.4
+The map between the glued spaces. Consider the canonical embedding of her-
+mitian OK-lattices
+Ψ: Λ2 → Λn,
+Ψ(x0, x1, x2) = (x0, x1, x2, 0, 0, . . . , 0) .
+We view Λ2 as a sublattice of Λn via Ψ, and write
+Λn = Λ2 ⊕ (Λ2)⊥ .
+(14)
+Using the canonical basis of Λn, we may view Γn = AutOK(Λn, Hn) as a subgroup of
+GLn+1(OK). This gives an embedding
+j : Γ2 → Γn,
+M �→ (M, In−2).
+(15)
+The natural totally geodesic embedding
+ν : CH2 �→ CHn,
+[x0 : x1 : x2] �→ [x0 : x1 : x2 : 0 : . . . : 0]
+(16)
+30
+
+induces a totally geodesic embedding
+ν :
+2
+�
+i=0
+RH2
+αi �→
+2
+�
+i=0
+RHn
+βi ⊂
+�
+α∈PAn
+RHn
+α = �Yn.
+(17)
+Since the composition SO(2, 1) → O(2, 1) → PO(2, 1) = Isom(RH2) is an isomor-
+phism, there is a natural embedding PO(2, 1) �→ PO(n, 1). Moreover, since we have
+PΓαi := (PΓ2)αi = PO(Λαi
+2 ) by [ACT06, (5.1)], the map (15) induces embeddings
+j : PΓαi �→ PΓβi,
+i = 0, 1, 2
+(18)
+that make the map ν in (17) equivariant.
+By Lemma 5.2, the classes of β0, β1 and β2 in PAn are pairwise non-conjugate
+under the action of PΓn on PAn, so that the induced map
+O2 :=
+2
+�
+i=0
+PΓαi \
+�
+RH2
+αi − H2
+�
+→
+�
+α∈CAn
+PΓα \ (RHn
+α − Hn) =: On
+(19)
+induces an injective map on sets of connected components π0(O2) → π0(On).
+Lemma 5.3. For each integer n ≥ 2, there exists a natural map of metric spaces
+ι: X2 → Xn.
+Proof. We claim that (19) extends to a commutative diagram of metric spaces
+PΓ2 \ Y2
+ι
+� PΓn \ Yn
+O2
+��
+�
+� On.
+��
+�
+Define
+�Y #
+2 =
+2
+�
+i=0
+RH2
+αi ⊂ �Y2.
+It suffices to prove the following assertions:
+1. The map �Y #
+2 → �Yn defined in (17) is compatible with the equivalence relations
+on both sides (see Definition 2.14).
+2. The resulting map of metric spaces
+φ: Y #
+2 → Yn
+(20)
+is equivariant for the actions of PΓ2 and PΓn. Thus, φ descends to a map
+ι: X2 = PΓ2 \ Y2 = PΓ2 \ �Y #
+2 → PΓn \ Yn = Xn,
+(21)
+where the equality on the left of (21) follows from Lemma 5.1.
+31
+
+As for 1, let r ∈ R2 be a short root in Λ2, which we also view as a short root in
+Λn. Let k ∈ Z/m and let x = y + z ∈ Λn be any element in Λn, where we decomposed
+x using (14). Consider the reflection φk
+r : Λ2 → Λ2. We have
+j
+�
+φk
+r
+�
+(x) = j(φk
+r)(y + z) = x − (1 − ζk
+3)Hn(x, r) · r.
+Thus j(φk
+r) = φk
+r. Next, let (x, αi), (x, αj) ∈ �Y #
+2
+be such that (x, αi) ∼ (x, αj). We
+want to show that φ2(x, αi) ∼ φ2(x, αj). We may assume that αi ̸= αj. Therefore,
+x ∈ H2 and αj = g ◦ αi ∈ PA2 for some g ∈ G2(x), where G2(x) is as in Definition
+2.6. By the above, we have j(G2(x)) = Gn(ν(x)), where ν : CH2 → CHn is as in (16).
+Since αi ◦ αj = g ∈ G2(x), we have f(αi) ◦ f(αj) = j(g) ∈ Gn(ν(x)), which proves 1.
+To prove 2, let (x, αi), (y, αj) ∈ �Y #
+2
+with images v = p2(x, αi), w = p2(y, αj) ∈ Y #
+2 .
+Suppose that there exists an element g ∈ StabPΓ2(�Y #
+2 ) ⊂ PΓ2 such that
+g · v = p2(g · x, gαig−1) = w = p2(y, αj).
+We want to show that j2(g)·φ(v) = φ(w). Since g ∈ StabPΓ2(�Y #
+2 ), we have gαig−1 = αi
+by Lemma 5.1. We thus get that p2(g · x, αi) = p2(y, αi). Since the map p2 : �Y2 → Y2
+is injective when restricted to RH2
+αi, it follows that
+g · (x, αi) = (g · x, gαig−1) = (g · x, αi) = (y, αi),
+which implies that
+j(g) · (ν(x), f(αi)) = ν(g · (x, αi)) = ν(y, αi) = (ν(y), f(αi)).
+Since φ(v) = pn(ν(x), f(αi)) and φ(w) = pn(ν(y), f(αj)), we obtain j(g) · φ(v) = φ(w)
+as desired.
+5.5
+The orbifold map between the glued spaces. Let ¯f ∈ X2 and lift ¯f to an element
+f ∈ Y #
+2
+⊂ Y2 (this is possible by Lemma 5.1). In turn, we can lift f to an element
+(x, αj) ∈ �Y #
+2 . If f has zero nodes (see Definition 2.6), then ¯f lies in the open subset
+O2 ⊂ X2 and the map O2 → On of (19) is a morphism of orbifolds.
+So suppose that f has one real node, defined by a short root r ∈ R2 such that
+x ∈ Hr. Consider the image g = φ(f) ∈ Yn of f in Yn by the map φ: Y #
+2
+→ Yn
+defined in (20). It admits the lift (ν(x), βj) ∈ �Yn. For p ∈ {2, . . . , n − 1}, let rp =
+(0, 0, 0, . . . , 1, . . . , 0) ∈ Rn, where the 1 is on the (p + 1)-th coordinate. Note that ν(x)
+has n − 1 nodes. Define
+r1 = Ψ(r) ∈ Rn,
+so that
+ν(x) ∈
+n−1
+�
+i=1
+Hri ⊂ CHn,
+and
+{r1, r2, . . . , rn−1} ⊂ Rn
+is a set of short roots of maximal cardinality such that ν(x) ∈ Ht for t ∈ Rx.
+32
+
+Lemma 5.4. Consider x ∈ CH2, ν(x) ∈ CHn, r ∈ R2 and Ψ(r) ∈ Rn as above.
+There are isometries
+κ2 : CH2
+∼−→ B2(C),
+κn : CHn
+∼−→ Bn(C),
+identifying x (resp. ν(x)) with the origin, φr (resp. φr1, resp. φri for i ≥ 2) with the
+respective maps
+B2(C) → B2(C),
+(t1, t2) �→ (ζ6 · t1, t2),
+Bn(C) → Bn(C),
+(t1, . . . , tn) �→ (ζ6 · t1, . . . , tn),
+Bn(C) → Bn(C),
+(t1, . . . , tn) �→ (t1, . . . , ζ6 · ti+1, . . . , tn),
+and αj (resp. βj) with the respective maps
+B2(C) → B2(C),
+(t1, t2) �→ (¯t1, ¯t2),
+Bn(C) → Bn(C),
+(t1, . . . , tn) �→ (¯t1, . . . , ¯tn),
+and such that, for the canonical embedding ρ: B2(C) �→ Bn(C), the following diagram
+commutes:
+CH2
+κ2 �
+ν
+�
+B2(C)
+ρ
+�
+CHn
+κn � Bn(C).
+Proof. Define Vn as the complex n+1-dimensional hermitian space Vn = (Λ ⊗ C, (Hn)C) .
+Let x ∈ V2 be a lift of x ∈ CH2 = {negative lines in V2}. Defining ν : V2 → Vn as the
+natural embedding allows us to consider x and r an elements of Vn. Then define
+T2 = ⟨x⟩ ⊕ ⟨r⟩ ⊂ V2,
+R2 = T ⊥
+2 ⊂ V2,
+Tn = ⟨x⟩ ⊕ ⟨r⟩ ⊂ Vn,
+Rn = T ⊥
+n ⊂ Vn.
+Choose appropriate bases for R2 and Rn scale appropriately as in Lemma 3.9.
+We are now ready to prove:
+Proposition 5.5. The map ι defined in Lemma 5.3 is a map of hyperbolic orbifolds.
+Proof. We continue with the notation from above. Consider the coordinates CH2
+∼−→
+B2(C) of Lemma 5.4: they make the involution αj correspond to (t1, t2) �→ (¯t1, ¯t2), and
+any ξ ∈ PA2 with (x, α) ∼ (x, ξ) to an involution of the form (t1, t2) �→ (ζi
+6·¯t1, ¯t2). Thus
+if the ξi for i = 1, . . . , 6 are the six anti-unitary involutions such that (x, α) ∼ (x, ξi),
+then for the set Yf defined in (6), one has
+Yf ∼=
+6�
+i=1
+RH2
+ξi ∼=
+�
+(t1, t2) ∈ B2(C) | t6
+1, t2 ∈ R
+�
+.
+The union of two subsets
+Kf,ϵ =
+�
+(t1, t2) ∈ B2(C) | i−ϵ · t1 ∈ R≥0, t2 ∈ R
+�
+,
+33
+
+indexed by ϵ ∈ {0, 1} is a fundamental domain for the action of Bf on Yf, and under
+its path metric, U = Kf,0 ∪ Kf,1 is isometric to B2(R) by the following map:
+U → B2(R),
+(t1, t2) �→
+�
+(−i)−ϵ · t1, t2
+�
+.
+Therefore, the following composition identifies Bf \ Yf with B2(R):
+Bf \ Yf ∼= Bf \
+6�
+i=1
+RHn
+ξi ∼= ⟨ζ6⟩ \
+�
+(t1, t2) ∈ B2(C) | t6
+1, t2 ∈ R
+� ∼=
+�
+ϵ∈{0,1}
+Kf,ϵ ∼= B2(R).
+Similarly, we get, using the coordinates of Lemma 5.4, writing i = (i1, . . . , in−1) and
+χi = �n−1
+p=1 φip
+rp ◦ βj, that
+Yg ∼=
+6�
+i1,...,in−1=1
+RHn
+χi ∼=
+�
+(t1, . . . , tn) ∈ Bn(C) | t6
+1, t2, t6
+3, . . . , t6
+n ∈ R
+�
+.
+Moreover, if we define
+Kg,ϵ1,...,ϵn−1 =
+�
+(t1, . . . , tn) ∈ Bn(C) | iϵ1t1, i−ϵ2t3, . . . , i−ϵn−1tn ∈ R≥0, t2 ∈ R
+�
+,
+then Ug = �1
+ϵ1,...,ϵn−1=0 Kg,ϵ1,...,ϵn−1 is a fundamental domain for the action of Bg on Yg,
+and there is an isometry
+Ug → Bn(R),
+(t1, . . . , tn) �→
+�
+(−i)−ϵ1 · t1, t2, (−i)−ϵ3 · t3, . . . , (−i)−ϵn · tn
+�
+.
+Together, this gives
+Bg \ Yg ∼= Bg \
+6�
+i1,...,in−1=1
+RHn
+χi ∼= ⟨ζ6⟩n−1 \
+�
+(t1, . . . , tn) | t6
+1, t2, t6
+3, . . . , t6
+n ∈ R
+�
+∼= Ug =
+1�
+ϵ1,...,ϵn−1=0
+Kg,ϵ1,...,ϵn−1 ∼= Bn(R).
+Since ξi = φi
+r ◦ αj and ξ(i,6,...,6) = φi
+r1 ◦ βj for i ∈ {1, . . . , 6}, one has
+x ∈ RH2
+ξi ⇐⇒ (φi
+r ◦ αj)(x) = x =⇒ (φi
+r1 ◦ βj)(ν(x)) = ν(x) ⇐⇒ ν(x) ∈ RHn
+χi,6,...,6.
+Therefore, with respect to the natural embedding ν : CH2(C) → CHn(C), one has
+for each i = 1, . . . , 6, that ν
+�
+RHn
+ξi
+�
+⊂ RHn
+χi,6,...,6. Thus, the maps defined above fit
+34
+
+together in the following commutative diagram of metric spaces:
+Bf \ Yf
+≀
+�
+� �
+� Bg \ Yg
+≀
+�
+Bf \ �6
+i=1 RHn
+ξi
+� �
+�
+≀
+�
+Bg \ �6
+i1,...,in−1=1 RH2
+χi
+≀
+�
+⟨ζ6⟩ \ {(t1, t2) ∈ B2(C) | t6
+1, t2 ∈ R}
+≀
+�
+� �
+� ⟨ζ6⟩n−1 \ {(t1, . . . , tn) | t6
+1, t2, t6
+3, . . . , t6
+n ∈ R}
+≀
+�
+�
+ϵ∈{0,1} Kf,ϵ
+≀
+�
+� �
+� �1
+ϵ1,...,ϵn−1=0 Kg,ϵ1,...,ϵn−1
+≀
+�
+B2(R)� �
+� Bn(R).
+(22)
+Since the map Y #
+2
+→ Yn defined in (20) is equivariant with respect to Γ2 �→ Γn, we
+obtain a map StabΓ2(f) �→ StabΓn(g). Finally, via the commutative diagram
+PO(2, 1) = Isom(B2(R))� �
+� Isom(Bn(R)) = PO(n, 1)
+StabΓ2(f)� �
+�
+�
+StabΓn(g)
+�
+Bf
+�
+� �
+� Bg
+�
+we obtain a well-defined embedding
+Af/Bf = StabPΓ2(f)/Bf �→ StabPΓn(g)/Bg = Ag/Bg.
+(23)
+The embedding Bf \Yf �→ Bg\Yg is equivariant with respect to (23). As in the proof of
+Theorem 3.1.2, we choose an Af (resp. Ag)-equivariant open neighbourhood f ∈ Vf ⊂
+Yf (resp. g ∈ Vg ⊂ Yg) such that Af \ Vf ⊂ PΓf \ Y2 (resp. Ag \ Vg ⊂ PΓg \ Yn). Using
+the above diagram (22), we get open neighbourhoods Wf ⊂ B2(R) and Wg ⊂ Bn(R),
+acted upon by Af/Bf and Ag/Bg respectively, such that Af \ Vf = (Af/Bf) \ Wf and
+Ag \ Vg = (Ag/Bg) \ Wg, and such that there exists a totally geodesic embedding
+B2(R) ⊃ Wf � �
+ρ
+� Wg ⊂ Bn(R)
+(24)
+35
+
+which is equivariant for (23) and makes the diagram
+Wf � �
+ρ
+�
+�
+Wg
+�
+(Af/Bf) \ Wf
+� �
+�
+ρ
+� (Ag/Bg) \ Wg
+� �
+�
+PΓ2 \ Y2
+ι
+� PΓn \ Yn
+commute. We conclude that ι: X2 → Xn is an orbifold map at each point ¯f ∈ X2
+such that any lift f ∈ Y #
+2
+of ¯f has either no nodes or one real node. The proof of the
+case when f has two real nodes or one pair of complex conjugate nodes is similar.
+5.6
+Totally geodesic immersions. The goal of Section 5.6 is to prove Theorem 5.8,
+using the results of Sections 5.7 - 5.5. Recall the following (see e.g. [Bel+21, §2.3.1]):
+Definition 5.6. Let n, m ∈ N. A map i: RHm/Λ → RHn/Γ of hyperbolic orbifolds
+is a totally geodesic immersion if there exists a totally geodesic subspace U ⊂ RHn
+and a lift ˜i: RHm → RHn of i that factors through an isometry ˜i: RHm
+∼−→ U. In this
+setting, the orbifold RHm/Λ is called a totally geodesic suborbifold of RHn/Γ.
+Definition 5.7. Consider the map ι: X2 → Xn, recall that X2 is connected (Lemma
+5.14) and let X+
+n be the connected component containing ι(X2). Let Γ+
+n ⊂ PO(n, 1)
+be the lattice underlying the complete connected hyperbolic orbifold X+
+n .
+The goal of Section 5.6 is to prove the following theorem.
+Theorem 5.8. For each integer n ≥ 2, there exists a canonical proper totally geodesic
+immersion of complete connected hyperbolic orbifolds ι: X2 → X+
+n .
+To prove this, we need several results.
+Lemma 5.9. The map ι: X2 → Xn defined in Lemma 5.3 is proper with finite fibers.
+Proof. Observe that for each α ∈ PAn, the canonical map fα : PΓα \ RHn → Xn is
+a closed immersion. Since CAn = PAn/PΓn is finite, �
+α∈CAn fα (PΓα \ RHn) = Xn
+forms a finite closed covering of Xn. Moreover, the restriction of ι to the closed subset
+fαi (PΓαi \ RH2) ⊂ X2 is induced by the canonical map
+PΓαi \ RH2
+αi → PΓβi \ RHn
+βi.
+(25)
+Here, αi and βi are as in (11) and (13). To prove that ι is proper with finite fibers, it
+therefore suffices to prove that (25) is proper with finite fibers for each i ∈ {0, 1, 2}.
+Define quadratic forms
+Qn
+0(x0, . . . , xn) = −x2
+0 + x2
+1 + · · · + x2
+n,
+Qn
+1(x0, . . . , xn) = −x2
+0 + 3x2
+1 + · · · + x2
+n,
+Qn
+2(x0, . . . , xn) = −x2
+0 + 3x2
+1 + 3x2
+2 + x2
+3 + · · · + x2
+n.
+(26)
+36
+
+For each i = 0, 1, 2, the canonical homomorphism PO(Qn
+i , Z) → PΓβi is an isomor-
+phism by [ACT10, Theorem 5.1], thus (25) is proper with finite fibers.
+Lemma 5.10. For any base point of X2, the induced map Γ+
+2 → Γ+
+n is injective.
+Proof. The natural map of orbifolds X+
+n → PΓn \ CHn induces a homomorphism
+Γ+
+n → PΓn = PU(n, 1)(Z[ζ3]). In case n = 2, this map is injective and factors as
+Γ+
+2 �→ PO(2, 1)(Z[
+√
+3]) ⊂ PU(2, 1)(Z[ζ3]),
+see [ACT06, p. 167-168]. Since SO(2, 1) = PO(2, 1), this map Γ+
+2 → PU(2, 1)(Z[ζ3])
+factors through an embedding Γ+
+2 �→ U(2, 1)(Z[ζ3]). The commutative diagram of
+orbifolds
+Γ+
+2 \ RH2
+�
+�
+Γ+
+n \ RHn
+�
+U(2, 1)(Z[ζ3]) \ CH2
+� PU(n, 1)(Z[ζ3]) \ CHn
+gives rise to a commutative diagram of orbifold fundamental groups
+Γ+
+2
+�
+�
+Γ+
+n
+�
+U(2, 1)(Z[ζ3])
+� PU(n, 1)(Z[ζ3]).
+The composition Γ+
+2 → U(2, 1)(Z[ζ3]) → PU(n, 1)(Z[ζ3]) is injective, hence the map
+Γ+
+2 → Γ+
+n is injective as desired.
+Lemma 5.11. Let X/Λ and Y/L be quotient orbifolds, where X and Y are simply
+connected manifolds, and Λ and L discrete groups acting smoothly, properly discontin-
+uously on X and Y . Let f : X/Λ → Y/L be a proper map with finite fibers and suppose
+that f∗ : Λ → L is injective. Then any lift ˜f : X → Y of f is proper with finite fibers.
+Proof. The map ˜f is locally given by maps U/G → V/H where U (resp.
+V ) is
+connected open in X (resp. Y ) and G and H are finite groups. Thus ˜f is closed. To
+see that ˜f has finite fibers, observe that Λ·x → L· ˜f(x) is injective for each x ∈ X.
+We obtain:
+Proposition 5.12. The morphism of hyperbolic orbifolds
+ι: X2 = Γ+
+2 \ RH2 → Γ+
+n \ RHn = X+
+n
+is proper with finite fibers, and the same holds for any lift ˜ι: RH2 → RHn of ι.
+Proof. This follows from Lemma’s 5.9, 5.10 and 5.11.
+The proof of Theorem 5.8 rests on the above results, together with the following:
+37
+
+Lemma 5.13. Let N = RHm/Λ and M = RHn/Γ be real hyperbolic orbifolds. Let
+i: N → M
+be a proper map of hyperbolic orbifolds, induced by a homomorphism φ: Λ → Γ and
+an equivariant proper map ˜i: RHm → RHn. Suppose that, locally around each point
+of N, the morphism i is given by a diagram of the form
+U
+�
+˜i
+� V
+�
+U/G
+i
+� V/H,
+G
+φ
+� H,
+(U ⊂ RHm, V ⊂ RHn connected open)
+(27)
+where the map ˜i: U → V is a proper totally geodesic immersion. Then i: N → M is
+a totally geodesic immersion of hyperbolic orbifolds.
+Proof. Since ˜i is proper and locally a totally geodesic immersion, it is a totally geodesic
+embedding. Thus i: N → M is a totally geodesic immersion, see Definition 5.6.
+Proof of Theorem 5.8. Consider the map of metric spaces ι: X2 → X+
+n ⊂ Xn defined
+in Lemma 5.3, where X+
+n is the connected component of Xn containing ι(X2). By
+Proposition 5.5, the map ι is a morphism of hyperbolic orbifolds. Moreover, locally ι
+lifts to a proper totally geodesic embedding ρ: Wf �→ Wg as in (24) for some connected
+open subsets Wf ⊂ B2(R) and Wg ⊂ Bn(R), equivariant for the inclusion of stabilizer
+groups Af/Bf �→ Ag/Bg defined in (23). By Proposition 5.12, the map ι: X2 → X+
+n
+is proper with finite fibers, and the same holds for any lift ˜ι: RH2 → RHn of ι. By
+Lemma 5.13, it follows that ι is a totally geodesic immersion of hyperbolic orbifolds.
+5.7
+Fifth roots of unity. Let K′ = Q(ζ5). For an integer n ≥ 2, let Λ′
+n = Z[ζ5]n+1, and
+define
+H′
+n : Λ′
+n ⊗Z[α] Λ′
+n → OK,
+H′
+n(x, y) = −
+�√
+5 − 1
+2
+�
+· x0¯y0 +
+n
+�
+i=1
+xi¯yi.
+Let R′
+n = {r ∈ Λ′
+n | H′
+n(r, r) = 1}. By Proposition 4.14 and Theorem 4.12, the hy-
+perplane arrangement H ′ = ∪r∈R′nHr ⊂ CHn is an orthogonal arrangement, i.e.
+Condition 2.4 is satisfied. The glueing construction of Section 2 provides us with a
+metric space
+Zn = PΓ′
+n \ Y ′
+n,
+and the metric on Zn extends to a complete real hyperbolic orbifold structure by
+Theorem 3.1. Thus, each connected components of Zn is the quotient of RHn by a
+discrete group of isometries.
+Define three anti-unitary involutions αi : Λ′
+2 → Λ′
+2 as follows:
+α0 : (x0, x1, x2) �→ (¯x0,
+¯x1,
+¯x2)
+α1 : (x0, x1, x2) �→ (¯x0, −¯x1,
+¯x2)
+α2 : (x0, x1, x2) �→ (¯x0, −¯x1, −¯x2)
+(28)
+38
+
+Lemma 5.14. The involutions α0, α1, α2 are pairwise non-conjugate, the composition
+2
+�
+i=0
+RH2
+αi → Y ′
+2 → PΓ′
+2 \ Y ′
+2 = Z2
+is surjective, and Z2 is connected and compact.
+Proof. See [GF21].
+Similar to the the Eisenstein case above, we have the following result.
+Theorem 5.15. For each n ≥ 2, there exists a connected component Z+
+n ⊂ Zn and a
+natural proper totally geodesic immersion of complete connected hyperbolic orbifolds
+ι: Z2 → Z+
+n .
+Proof. The proof is completely analogous to the proof of Theorem 5.8.
+Definition 5.16. Let Λ+
+n ⊂ PO(n, 1) be the lattice underlying the complete connected
+hyperbolic orbifold Z+
+n .
+5.8
+Non-arithmetic real hyperbolic lattices. Consider the lattices Γ+
+n ⊂ PO(n, 1)
+and Λ+
+n ⊂ PO(n, 1), see Definitions 5.7 and 5.16. We are now in position to prove that
+Γ+
+n and Λ+
+n are non-arithmetic. The key is to use the following result.
+Theorem 5.17 (Bergeron–Clozel). Let RHm/Λ → RHn/Γ be a totally geodesic im-
+mersion of real hyperbolic orbifolds of finite volume. If the lattice Γ ⊂ PO(n, 1) is
+arithmetic, then the lattice Λ ⊂ PO(m, 1) is arithmetic as well.
+Proof. See [BC05, Proposition 15.2.2] (compare also [Bel+21, Theorem 1.4]).
+Theorem 5.18. For each n ∈ Z≥2, the lattices Γ+
+n ⊂ PO(n, 1) and Λ+
+n ⊂ PO(n, 1)
+are non-arithmetic.
+Proof. By Theorem 5.8 (resp. Theorem 5.15), there exists a proper totally geodesic
+immersion of hyperbolic orbifolds X2 → X+
+n (resp. Z2 → Z+
+n ). Therefore, by Theorem
+5.17, it suffices to show that Γ+
+2 ⊂ PO(2, 1) and Λ+
+2 ⊂ PO(2, 1) are non-arithmetic.
+For Γ+
+2 , this is shown in [ACT06, Section 5], and for Λ+
+2 in [GF21].
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+41
+

jtE5T4oBgHgl3EQfGQ7H/content/tmp_files/2301.05430v1.pdf.txt

@@ -0,0 +1,2611 @@
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+1
+HS-GCN: Hamming Spatial Graph Convolutional
+Networks for Recommendation
+Han Liu, Yinwei Wei, Member, IEEE, Jianhua Yin, Member, IEEE,
+and Liqiang Nie, Senior Member, IEEE
+Abstract—An efficient solution to the large-scale recommender system is to represent users and items as binary hash codes in the
+Hamming space. Towards this end, existing methods tend to code users by modeling their Hamming similarities with the items they
+historically interact with, which are termed as the first-order similarities in this work. Despite their efficiency, these methods suffer from
+the suboptimal representative capacity, since they forgo the correlation established by connecting multiple first-order similarities, i.e.,
+the relation among the indirect instances, which could be defined as the high-order similarity. To tackle this drawback, we propose to
+model both the first- and the high-order similarities in the Hamming space through the user-item bipartite graph. Therefore, we develop
+a novel learning to hash framework, namely Hamming Spatial Graph Convolutional Networks (HS-GCN), which explicitly models the
+Hamming similarity and embeds it into the codes of users and items. Extensive experiments on three public benchmark datasets
+demonstrate that our proposed model significantly outperforms several state-of-the-art hashing models, and obtains performance
+comparable with the real-valued recommendation models.
+Index Terms—Hashing, Efficient Recommendation, High-order Similarity, Graph Convolutional Network, Hamming Space.
+!
+1
+INTRODUCTION
+T
+HE recommender system is developed to locate the
+interested items from the overwhelming information
+according to users’ preferences. Hence, how to measure the
+similarities between users and items is at the core of the per-
+sonalized recommendation. Towards this end, existing stud-
+ies [1], [2], [3] tend to follow a two-stage pipeline: represent-
+ing the users and items with vectors [4], and then predicting
+their interactions by measuring the similarities between
+vectors. Despite the remarkable performance, these methods
+still face an inevitable problem that the computation grows
+exponentially with increasing users and items [5]. Theoreti-
+cally, for recommending top-k preferred items for each user,
+the time complexity is O(NMK + NMlogk) when there
+are N users and M items, represented by K-dimensional
+embeddings in the latent space.
+Diving into these methods, we can easily find that the
+problem mainly comes from the user-item similarity com-
+putations [6]. However, it is virtually impossible to design
+a new algorithm that not only can compute the similarity
+between two vectors but is more efficient than the inner-
+product. Therefore, some efforts have been dedicated to
+learning a new kind of representation—hash code—for the
+user and item, so as to alleviate the complexity [7], [8]. In
+particular, benefiting from such a code consisting of ±1
+bits, the measurement can be accelerated via XOR bit op-
+eration [9]. Hence, the time complexity of recommendation
+is significantly reduced and even constant time search is
+made possible by exploiting lookup tables [6]. Since the
+•
+Han Liu, Jianhua Yin, and Liqiang Nie are with School of Computer
+Science and Technology, Shandong University, Qingdao 266200, China.
+E-mail: [email protected], [email protected], [email protected].
+•
+Yinwei Wei is with School of Computing, National University of Singa-
+pore, Singapore.
+E-mail: [email protected].
+•
+Liqiang Nie is the corresponding author.
+𝑢3
+𝑖2
+𝑖3
+𝑖5
+1
+1
+1
+1
+-1
+1
+-1
+-1
+-1
+1
+-1
+1
+𝑢2
+𝑖1
+𝑖3
+𝑖4
+1
+1
+1
+1
+-1
+-1
+-1
+1
+1
+1
+1
+1
+𝑢1
+𝑖1
+𝑖2
+1
+-1
+1
+1
+-1
+-1
+1
+-1
+？
+(a) First-order Hamming similarity 
+1
+-1
+1
+1
+-1
+-1
+1
+1
+1
+-1
+1
+1
+-1
+-1
+-1
+1
+1
+1
+1
+-1
+1
+1
+-1
+1
+𝑢1
+𝑖1
+𝑖2
+𝑢2
+𝑢3
+𝑖4
+𝑖5
+𝒍 = 𝟏
+𝒍 = 𝟐
+𝒍 = 𝟑
+𝑖3
+(b) High-order Hamming similarity for 𝒖𝟏
+Fig. 1. Illustration of the high-order and the first-order Hamming similarity
+in learning to hash.
+computation efficiency of the user-item similarity is su-
+percharged, the large-scale recommendation could be con-
+ducted efficiently [10], [11], especially on mobile application
+where the computational resource is very limited [12], [13].
+Nevertheless, since users and items are approximately rep-
+resented as the binary vectors, the performance of hashing-
+based recommendation models tends to be suboptimal. To
+deal with this drawback, several methods are developed to
+enhance the representation ability of the hash codes [14],
+[15], [16]. For instance, HashNet [17] utilizes the deep
+neural networks to learn the user and item embeddings
+and then map them into the Hamming space to calculate
+the Hamming distance between two vectors. More recently,
+inspired by the success of graph convolutional networks in
+representation learning, HashGNN [18] has been proposed
+to encode the local structural information into each node in
+the user-item interaction graph, and binarize their enhanced
+representations via a hash layer.
+Although these methods enrich the user and item repre-
+sentations, we argue that they learn the hash codes merely
+with the similarities of interacted user-item pairs, but ig-
+nore the similarities hidden in the indirect interactions. To
+distinguish these similarities with the similarity computing
+arXiv:2301.05430v1  [cs.IR]  13 Jan 2023
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+2
+in Euclidean space [2], we term them as the first-order and
+high-order Hamming similarities, respectively. To illustrate
+our argument, we connect the direct interactions to model
+the indirect interactions, which builds a user-item bipartite
+graph shown in Figure 1(b), and compare it with the direct
+interactions shown in Figure 1(a) on hash code learning. In
+particular, the users (i.e., u1, u2, and u3) and items (i.e., i1, i2,
+and i3) are represented by binary vectors, and the presence
+of an edge signifies that the user interacts with the item. To
+learn the hash codes for the users, both the deep learning
+based and GNN-based models code their interacted items
+and minimize their first-order Hamming similarities (i.e.,
+⟨u1, i1⟩ and ⟨u1, i2⟩), as shown in Figure 1(a). However, for
+u1, we argue that its code is hard to determine merely by the
+first-order Hamming similarity, because it lies right in the
+middle of two items. On the contrary, by incorporating the
+high-order Hamming similarity, as illustrated in Figure 1(b),
+the hidden cues can be captured by measuring its high-order
+Hamming similarities, such as ⟨u1, u2⟩ and ⟨u1, u3⟩, largely
+facilitating the hash code learning.
+Therefore, we propose to code the users and items by
+explicitly modeling the high-order Hamming similarity on
+the user-item bipartite graph. It is worth noticing that this
+is different with HashGNN that learns the nodes’ repre-
+sentations in Euclidean space and then transforms them
+into hash codes. We argue that the supervision signal for
+the hash code optimization merely comes from the first-
+order interactions in the history, which may aggravate the
+information loss of high-order Hamming similarity during
+the transformation.
+However, conducting the graph convolutional opera-
+tions in the Hamming space to capture the Hamming sim-
+ilarity is non-trivial. Following the terms in graph con-
+volutional networks, we attribute the challenges into two
+aspects:
+• Different from the operations on the continuous vectors,
+each element in hash codes is restricted to the binary
+values. Therefore, for each node, how to aggregate the
+information from its neighbors in the Hamming space is the
+first challenge we face.
+• In order to represent the nodes in Hamming space, it is
+essential to preserve their Hamming similarities with the
+codes of their neighbors. Hence, how to explicitly encode
+the Hamming similarities into each node is another technical
+challenge in this work.
+In order to address the outlined challenges, we develop a
+novel hashing-based recommendation model, named Ham-
+ming Spatial Graph Convolutional Networks (HS-GCN),
+consisting of the initial, propagation, and prediction layers.
+Specifically, in the initial layer, we recognize the user-item
+interactions as a bipartite graph and initialize the nodes
+with a hash code generation method proposed in [17], which
+guarantees the feasibility of end-to-end optimization. Upon
+the constructed graph, we devise a code propagation layer
+to implement the graph convolutional operations in Ham-
+ming space. More specifically, the layer could be divided
+into two components — hash code aggregation and hash
+code encoding. The former aggregates information from
+first-order similar neighbors by counting the bit-wise signs
+(i.e., finding out the dominated bit value in each dimension)
+of their corresponding hash codes. And the latter injects the
+Hamming similarities into the node hash codes by refining
+their bits consistent with the bit-wise signs of their neigh-
+bors. With the stacked propagation layers, we iteratively
+embed the learned hash codes into the nodes. And then,
+the interactions of user-item pairs could be predicted by
+scoring their affinities at the prediction layer. To evaluate
+our proposed model, we conduct extensive experiments on
+three publicly accessible datasets and compare the perfor-
+mance with several state-of-the-art baselines. In addition to
+outperforming the hash-based models (e.g., HashNet and
+HashGNN), our proposed method achieves the results com-
+parable with the real-valued based models, such as PinSage
+and GraphSAGE.
+Overall, the main contributions of our work are summa-
+rized in three-folds:
+• To the best of our knowledge, this is the first attempt
+to explicitly model the high-order Hamming similarity
+between users and items in the recommender system,
+which enhances the representation ability of hash codes
+and accordingly optimizes the user-item interaction pre-
+diction.
+• We
+present
+a
+GCN-based
+hashing
+recommendation
+model, named Hamming Spatial Graph Convolutional
+Network (HS-GCN), which explicitly captures the first-
+and high-order Hamming similarities. In particular, we
+develop the novel graph convolutions operation—hash
+code aggregation and hash code encoding—in the Ham-
+ming space.
+• Extensive
+experimental
+results
+on
+three
+real-world
+datasets have demonstrated that the proposed model
+yields a substantial performance improvement compared
+with several state-of-the-art baselines. As a side contri-
+bution, we have released the codes to facilitate other
+researchers: https://github.com/hanliu95/HS-GCN.
+2
+RELATED WORK
+In this section, we briefly review graph-based methods for
+recommendation and learning to hash for recommendation.
+2.1
+Graph-based Methods for Recommendation
+Machine learning on graphs is an important task with the
+advantage of incorporating structural information. As one
+of the primary applications, representation learning on the
+user-item graph structure has been widely-studied in rec-
+ommendation scenarios by utilizing information propaga-
+tion. In this line of research, ItemRank [19] and BiRank [20]
+make early efforts for label propagation. In particular, these
+methods directly propagate the user preference scores (i.e.,
+labels) on the graph, scoring connected items with similar
+labels for a user. However, these methods are essentially
+neighbor-based and insufficient to encode structural infor-
+mation of a graph.
+Recently, GNNs have received focused attention [21],
+[22], since GNNs have special advantage on modeling
+the graph structure, especially information propagation, to
+guide the representation learning. However, early GNN-
+based methods suffer from expensive computation costs
+as the graph convolution on the spectral domain. Subse-
+quently, GCNs exploit a graph convolution operation on the
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+3
+spatial domain, which aggregates the embeddings of neigh-
+bors to refine the embedding of the target node [3], [23]. Mo-
+tivated by the efficiency of graph convolution, GC-MC [24],
+PinSage [25], and NGCF [2] employ GCNs to capture the
+propagation of latent Collaborative Filtering (CF) signals in
+the user-item interaction graph for recommendation. How-
+ever, later studies argue the excessive complexity of GCNs.
+For example, LightGCN [26] develops a linear model by
+removing all redundant parameters. Experimental results
+demonstrate that the simplified design outperforms original
+GCNs by a large margin. This shows that the nonlinearity
+of GCNs is unnecessary for CF recommendation.
+As aforementioned, we summarized two key points: 1)
+graph-based methods shed light on modeling the relational
+information propagation like the high-order similarity in
+the Hamming space, based on a user-item graph; 2) graph
+convolution has better performance after being simplified,
+verifying that the nonlinearity is not necessary in graph-
+based recommendation. Our work is highly inspired by
+these analyses, since the high-order Hamming similarity
+can be effectively modeled by the graph-based mechanism,
+and all the operations are simply linear in the Hamming
+space. Moreover, Heterogeneous GNN is proposed to be
+adapted upon structurally complex graphs to utilize richer
+information within them. Based on this technology, some
+methods effectively learn representations by aggregating
+different types of neighboring information in heterogeneous
+information network, such as HERec [27], MCRec [28],
+MEIRec [29], and ie-HGCN [30]. On contrary, our method
+works with simple interaction information which can be
+easily collected.
+2.2
+Learning to Hash for Recommendation
+Another relevant research line is learning to hash for recom-
+mendation, which proceeds along two directions: unsuper-
+vised hashing and supervised hashing [31]. The former [32],
+[33] learns hash functions that encode objects to binary
+codes by training from unlabeled data, while the latter [34],
+[35], [36], [37] aims to learn more discriminative hash codes
+by exploring labeled signals, such as the feedback between
+users and items in recommendation scenarios. In general,
+early learning to hash for recommendation methods are es-
+sentially two-stage approaches [38]. However, the Hamming
+similarity between learned hash codes might not correspond
+to the original relevance between a user and an item, since
+there are quantization deviations when thresholding real
+values to binary bits during the binarization step. DCF [6]
+tackles the challenging discrete optimization problem and
+learns user and item hash codes directly. Therefore, the
+learned hash codes are able to model the intrinsic user-item
+relevance.
+A prevailing trend is to leverage deep learning for
+recommendation. For instance, NFM [39] successfully in-
+troduces deep learning to enhance representation learning
+and matching function modeling in recommendation. In the
+light of this, deep learning to hash is subsequently devel-
+oped and yields promising recommendation performance.
+Early efforts of deep learning to hash also adopt a two-stage
+strategy: the first stage employs deep networks to learn con-
+tinuous representations and the second one uses the sign(x)
+function to binarize the learned representations into binary
+hash codes. This category of methods, such as CNNH [40],
+DNNH [41], and DHN [15], also suffer from the quantiza-
+tion deviation. To alleviate it, HashNet [17] is proposed to
+devise a one-stage learning to hash technique to decrease
+the quantization deviation of binarization. It approximates
+the sign(x) function with function tanh(βx), where β is a
+scaled parameter that increases during training. The infinite
+approximation makes the deviation negligible, and thus
+contributes to better recommendation performance.
+In a sense, the aforementioned hashing techniques only
+consider the first-order similarity between hash codes.
+Therefore, we introduced graph-based techniques in hash
+learning to capture the high-order Hamming similarity. It
+is worth mentioning that several recent efforts have in-
+corporated GNN insights into hashing, such as DGCN-
+BinCF [42], GCNH [43], and HashGNN [18]. Particularly,
+HashGNN [18] sets up a GNN in the continuous space,
+followed by a straight through estimator [44] for generating
+hash codes in the Hamming space. However, the hashing
+step leads to information loss, which impedes the capturing
+of intrinsic high-order Hamming similarity. To bridge this
+gap, we directly constructed a GCN in the Hamming space
+to model the high-order similarity.
+Different from the diffusion models [45], [46], [47] which
+combine different similarity measures by multiple similar-
+ity graphs [48], [49], [50], [51], our method extends the
+Hamming similarity from first-order to high-order via a
+single graph, and the high-order similarity is exploited to
+learn hash codes with better representative capacity. Also
+different from hypergraph convolution networks that ex-
+ploit multi-modal data in hypergraph for representation
+learning [52], our method is simply based on the single-
+modal interaction information in user-item graph, which
+can be obtained more easily.
+3
+PRELIMINARIES
+We first give the definition of Hamming similarity, a similar-
+ity measurement for the hash codes in the Hamming space,
+highlighting that it is related to the number of the same bits.
+We then formulate the problem to be solved in our work.
+3.1
+Hamming Similarity
+In the Hamming space, the user-item similarity is equivalent
+to the similarity between their corresponding hash codes,
+called Hamming similarity. Given user u and item i, we
+denote their hash codes as hu ∈ {±1}K and hi ∈ {±1}K
+respectively, where K represents the length of the codes. Ac-
+cordingly, the Hamming similarity between them is defined
+as:
+sim(u, i) = 1
+K
+K
+�
+k=1
+I(huk = hik),
+(1)
+where huk and hik are the k-th bits of hu and hi, respec-
+tively, and I(·) denotes the indicator function that returns
+1 if the statement is true and 0 otherwise. Furthermore,
+it could be proven that sim(u, i) is proportional to the
+number of same bits in hu and hi. As such, we rewrite the
+formulation [6] as:
+sim(u, i) = 1
+2 +
+1
+2K hu
+⊤hi ∝ hu
+⊤hi,
+(2)
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+4
+Prediction layer
+-ReLU
+clamp
+1
+1
+1
+𝐡𝑢1
+(𝑙−1)
+𝐡𝑖1
+(𝑙−1)
+𝐡𝑖2
+(𝑙−1)
+𝒍 = 𝟏
+𝑢1
+𝑖1
+𝑖2
+𝒍 = 𝟑
+𝒍 = 𝟐
+1
+1
+1
+1
+-1
+1
+-ReLU
+clamp
+1
+1
+1
+𝐡𝑖3
+(𝑙−1)
+𝐡𝑢2
+(𝑙−1)
+𝐡𝑢3
+(𝑙−1)
+𝒍 = 𝟏
+𝑖3
+𝑢2
+𝑢3
+𝒍 = 𝟑
+𝒍 = 𝟐
+𝐡𝑢1
+(3)
+𝐡𝑖3
+(3)
+Propagation layers
+ො𝑦 𝑢1, 𝑖3
+× −2
+× −2
+Fig. 2. Illustration of our proposed HS-GCN model (the arrow lines
+present the flow of information). The hash codes of user u1 (left) and
+item i3 (right) are refined with multiple hash code propagation layers.
+where the derivation process is omitted. Based on this
+formulation, we directly employ the inner product of hu
+and hi to measure the similarity between u and i.
+3.2
+Problem Formulation
+In this work, we aim to tackle the problem of mapping
+nodes in a user-item graph to hash codes for recommen-
+dation. We formulate the problem to be addressed in this
+paper as the following:
+• Input: the user-item bipartite graph G = (V, E) that
+records historical user-item interactions in a graph struc-
+ture. Whereinto, the node set consists of the user set U
+of N users and the item set I of M items, formally
+V = U ∪ I. And, the edge set E contains the observed
+user-item interactions in the training phase.
+• Output: the hash codes of all users and items, defined as
+{h∗
+u|u ∈ U} ∪ {h∗
+i |i ∈ I}. Since h∗
+u
+⊤h∗
+i denotes the simi-
+larity score between user u and item i, and on top of that
+we can generate a ranking list of items for a given user and
+hence solve the practical problem of recommendation.
+To accurately measure the similarity score, we focus on
+devising a GCN for hash code propagation, which is
+able to learn node hash codes with high-order Hamming
+similarities latent in the user-item bipartite graph.
+4
+METHODOLOGY
+In this section, we detail our proposed HS-GCN model, as
+illustrated in Figure 2. Specifically, the model consists of
+three components: 1) the initial layer that yields the trainable
+hash codes for the nodes in the user-item bipartite graph; 2)
+the propagation layer that explicitly model the Hamming
+similarities of node pairs and inject them into their codes
+for enhancement of the representation ability; and 3) the
+prediction layer that scores the similarity between the user
+and item by conducting the inner product of their hash
+codes. Moreover, we provide the optimization of HS-GCN
+in the form of matrix and compare HS-GCN with a graph-
+based learning to hash method HashGNN [18].
+4.1
+Initial Layer
+The initial layer serves to generate the initial hash codes
+hu ∈ {±1}K (hi ∈ {±1}K) for user u (item i), where K is
+the size of hash codes. Obviously, due to its discrete value,
+the hash code cannot be trained by the gradient descent.
+Hence, we alternatively employ a proxy to represent the
+users and items during the training phase. To achieve this
+goal, we follow the existing work [17] to reformulate the
+hash codes as,
+hu = lim
+β→∞ tanh(βeu), hi = lim
+β→∞ tanh(βei),
+(3)
+where eu ∈ RK and ei ∈ RK denote the parameter vector of
+user u and item i, respectively. By scaling the value of β, the
+outputs can approximate the desired codes. With the help
+of such a proxy, it is capable of addressing the ill-posed gra-
+dient issue and optimizing the representations during the
+training process. When the model reaches the convergence,
+we could infer the interactions between users and items via
+the learned hash codes, instead of their approximations.
+In what follows, we take the approximate codes of users
+and items as the initial representations of the nodes in the
+user-item graph and detail our devised graph convolutional
+operations in Hamming space. For notational convenience,
+we denote them as h(0)
+u
+and h(0)
+i
+, identifying the nodes’
+codes at the 0-th layer.
+4.2
+Propagation Layer
+To perform the graph convolutional operations in Hamming
+space, we devise a novel propagation layer, which disentan-
+gles the operations into the hash code aggregation and hash
+code encoding. With the stacked propagation layers, we
+could explicitly model the first- and high-order Hamming
+similarities hidden in the user-item graph and optimize the
+nodes’ representations by injecting the learned Hamming
+similarities into the codes.
+In this section, without particular clarification, we elabo-
+rate our proposed model on the user nodes and conduct the
+same operations on item nodes.
+4.2.1
+Hash Code Aggregation
+Regarding the graph convolutional operations, it is essential
+to capture the local structure information for each centric-
+node. To the end, we are inspired by the prior studies [2], [3]
+and develop a new aggregation to explicitly model the local
+structure information in Hamming space, which reveals the
+first-order Hamming similarity. Formally, the aggregation at
+(l + 1)-th propagation layer can be defined as,
+m(l)
+u = f
+�h(l)
+u , {h(l)
+i |i ∈ Nu}
+�,
+(4)
+where h(l)
+u
+∈ {±1}K and h(l)
+i
+∈ {±1}K recursively denote
+the hash codes of user u and item i at the l-th layer
+propagation, respectively. And, Nu denotes the set of item
+neighbors of user u, which u directly interacted with. Using
+the function f(·), we could capture the informative signal
+m(l)
+u distilled by incorporating the node with its neighbors.
+However, the implementation of this function is not
+straightforward in Hamming space. In this work, we imple-
+ment it in two steps. In particular, considering the fact that
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+5
+the transformation of the weighted matrix is inapplicable
+in Hamming space, we first adopt the non-weighted sum to
+integrate the bit-wise signals of the hash codes. Accordingly,
+we rewrite the aggregation in Hamming space:
+m(l)
+u = f(h(l)
+u +
+�
+i∈Nu
+h(l)
+i ).
+(5)
+Then, since the hash codes constrain binary values (i.e., +1
+and -1), we capture the bit-wise dominate value in the hash
+codes of centric node and its neighbors, which is different
+with the graph operations in Euclidean space, like mean-
+and max-pooling. We attribute it to the binomial distribution
+of the hash code. As such, we aim at designing a bit-wise
+function f(·) for aggregated codes and use each bit in m(l)
+u
+as the sign to reflect the dominate value at the correspond-
+ing bit of codes of u associated with its neighbors. Thereby,
+jointly considering the continuity of function, we implement
+f(·) with the interval limited clamp function as follows:
+m(l)
+u = clamp(x) =
+�
+�
+�
+�
+�
+1, if x > 1
+x, if − 1 ≤ x ≤ 1
+−1, if x < −1
+,
+(6)
+where x = (h(l)
+u + �
+i∈Nu h(l)
+i ). Owing to the characteristic
+of the clamp function, we can guarantee m(l)
+u ∈ {−1, 0, 1}K.
+For each node, the function is expected to reflect the dis-
+tribution of its neighbors’ hash codes. It is able to deter-
+mine the value of centric node, making it similar with its
+neighbors. Taking one bit in hash code as an example, the
+function will output −1 on this bit when the majority of
+corresponding bits in neighbors’ hash codes are −1, and vice
+versa. In a nutshell, we propose an aggregation operation in
+Hamming space, which integrates the codes from the centric
+node associated with its neighbors and distills the signal to
+reflect the bit-wise value distributions of hash codes.
+4.2.2
+Hash Code Encoding
+Obtaining the aggregated codes from the local structure, we
+then encode the information into the nodes to preserve their
+first-order Hamming similarities towards their connected
+nodes. As for each node, it is of importance to minimize
+the total of bits which are different from the corresponding
+bits of its neighbors’ codes. Therefore, we first compare the
+code of user u (i.e., h(l)
+u ) with the obtained signs (i.e., m(l)
+u )
+to capture the number of different bits, as,
+d(l)
+u = h(l)
+u ⊙ m(l)
+u ,
+(7)
+where ⊙ denotes the element-wise product, and d(l)
+u
+∈
+{−1, 0, 1}K is a vector consisting of the bit-wise comparison
+result. More specifically, the value −1 in the output indi-
+cates the corresponding bit differs from most of neighbors’;
+whereas, the value 1 means the node has the same bit value
+with the majority of nodes around it. Besides, when the
+value equals to 0, it implies that the corresponding bit is
+caught in the middle.
+Then, we could refine the codes of user u according to
+each value in the obtained vector. Considering the element 0
+existing in the vector, this operation is hardly implemented
+by conducting the element-wise product of two vectors,
+which probably outputs zero in addition to the binary
+value (i.e., +1 and −1). Therefore, we design a new function
+to avoid the negative influence of the value 0. It is formally
+defined as,
+c(l)
+u = −ReLU(−2 × d(l)
+u ) + ones,
+(8)
+where ones = {1}K is the vector that adjusts the outputs
+following the binary constraint. Based on this transform
+function, the values are tuned within {−1, 1}. Note that
+we convert the values 0 to 1 rather than −1, making the
+−1 values in c(l)
+u
+exactly indicate the unquestioned bit
+differences between the node and its neighbors. This will
+improve the reliability of the following bit-wise refinement.
+With the tuned vector, we could refine the hash code h(l)
+u
+by maximizing the similarity with its neighbors. Since c(l)
+u
+indicates the bits which are distinct with the majority of its
+neighbors, we utilize c(l)
+u
+to refine the bits of h(l)
+u
+with the
+element-wise product, formally,
+h(l+1)
+u
+= c(l)
+u ⊙ h(l)
+u ,
+(9)
+where h(l+1)
+u
+denotes the hash code of user u learned at the
+(l+1)-th propagation layer. Analogously, we can obtain the
+hash code h(l+1)
+i
+of item i by propagating hash codes from
+itself and its connected users. In summary, the designed
+hash code propagation layer contributes to exploit the first-
+order similarity information to relate user and item hash
+codes in the Hamming space.
+4.2.3
+Hash Code Propagation Rule in the Matrix Form
+To offer a holistic view of the hash code propagation archi-
+tecture and facilitate the batch implementation, we provide
+the matrix form of the layer-wise propagation rule as:
+H(l+1) =
+�
+− ReLU
+� − 2 × clamp
+�(A + I)H(l)� ⊙ H(l)�
++ Ones
+�
+⊙ H(l),
+(10)
+where H(l) ∈ {±1}(N+M)×K are the hash codes for users
+and items obtained after l steps of propagation. H(0) is
+the matrix of the initial hash codes after the initial layer,
+where h(0)
+u
+= hu and h(0)
+i
+= hi. I denotes the identity
+matrix, and Ones ∈ {1}(N+M)×K is a matrix of all ones. A
+denotes the adjacency matrix for the user-item graph, which
+is formulated as:
+A =
+�
+0
+R
+R⊤
+0
+�
+,
+(11)
+where R ∈ {0, 1}N×M is the user-item interaction matrix,
+and 0 is a all zero matrix. By implementing the matrix
+form propagation rule, we can simultaneously update the
+hash codes for all users and items in a rather efficient way.
+Moreover, it assists us to discard the node sampling proce-
+dure, which is commonly used to make graph convolution
+networks applicable on large-scale graph.
+4.3
+Prediction Layer
+We stack multiple propagation layers to explore the high-
+order similarities between hash codes, which are vital for
+upgrading the hash codes to finally estimate the relevance
+between the user-item pair. In particular, we define L as the
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+6
+depth of the final propagation layer, and the hash code of
+user u is recursively defined as:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+h(1)
+u
+= c(0)
+u
+⊙ hu,
+h(2)
+u
+= c(1)
+u
+⊙ h(1)
+u
+= c(1)
+u
+⊙ c(0)
+u
+⊙ hu,
+· · · · · ·
+h(L)
+u
+= c(L−1)
+u
+⊙ h(L−1)
+u
+= c(L−1)
+u
+⊙ · · · ⊙ c(0)
+u
+⊙ hu.
+(12)
+It shows that our model obtains the final hash codes by
+element-wise multiplying the initial ones with a series of
+refinement vectors. These refinement vectors then modulate
+the bit signals of user u’s hash code layer by layer. As
+thus, the high-order Hamming similarity is injected into the
+hash code learning process. Note that the length of hash
+code is identically K in each layer according to Eqn.(12).
+Analogously, we can obtain the hash code for item i at the
+L-th layer.
+After performing L propagation layers, we obtain mul-
+tiple hash codes for user u, termed {h(1)
+u , · · · , h(L)
+u }. It is
+worth noting that we choose not to concatenate them as the
+final representation for a user, which is widely used in the
+graph-based recommendation models [2], [26]. The reason is
+that the length of concatenation codes probably harms the
+computational efficiency and memory space. Alternately, we
+take the hash codes at the last layer (i.e., h(L)
+u
+and h(L)
+i
+)
+as the representations of user u and item i to predict their
+interaction, formally,
+h∗
+u = h(L)
+u ,
+h∗
+i = h(L)
+i
+.
+(13)
+Finally, we apply the inner product to estimate the
+similarity between hash codes of the target user and item,
+which can be equivalently treated as the user’s preference
+towards the item:
+ˆyui = h∗
+u
+⊤h∗
+i .
+(14)
+4.4
+Optimization
+In this work, we optimize the proposed model based on
+the users’ implicit feedback, such as observations and pur-
+chases [53], [54]. Compared to explicit ratings, the implicit
+feedback is easier to collect but more challenging for ex-
+ploring the user preference, due to its scarcity of nega-
+tive feedback. To optimize the model by the maximization
+likelihood estimation (MLE), we formulate the likelihood
+function P(rui|h∗
+u, h∗
+i ) as the probability of the interaction
+between user u and item i equals rui given their trainable
+codes h∗
+u and h∗
+i . As such, its optimal objective can be
+instantiated as the pairwise logistic function as follows:
+P(rui|h∗
+u, h∗
+i ) =
+�
+σ(ˆyui), rui = 1,
+1 − σ(ˆyui), rui = 0,
+(15)
+where σ(·) is the standard sigmoid function. And, ˆyui =
+h∗
+u
+⊤h∗
+i is the estimated Hamming similarity between h∗
+u
+and h∗
+i . In this case, the more similar the hash codes are,
+the larger the conditional probability P(1|h∗
+u, h∗
+i ) will be,
+and vice versa. Therefore, Eqn.(15) is a reasonable extension
+of the logistic regression classifier for the pairwise classifica-
+tion problem based on hash codes. Further, we can optimize
+the parameter by minimizing the following loss function:
+Lcross = −
+�
+rui∈R
+ruilog(σ(ˆyui)) + (1 − rui)log(1 − σ(ˆyui)),
+(16)
+where R is the aforementioned user-item interaction matrix
+that records the ground-truth implicit feedback. With this
+loss, the optimal object turns to reconstruct the observed
+interactions in the training phase.
+In general, Eqn.(16) is effective in learning useful hash
+codes for users and items, and guarantees that the inter-
+acted user-item pairs have similar hash codes. However, the
+ranking among candidate items tends to be more important
+than their absolute scores, since recommender systems sug-
+gest items to users mainly according to their rankings. For
+implicit feedback data, the ranking relation can be obtained
+by sampling from the interaction matrix R [1]. Specifically,
+we assume that the observed interactions, which are more
+reflective of a user’s preference, should be assigned with
+higher estimated values than the unobserved ones. Inspired
+by this, we introduce a ranking order reinforced loss func-
+tion as:
+Lrank =
+�
+(u,i,j)∈D
+max(0, −σ(ˆyui) + σ(ˆyuj) + α),
+(17)
+where D = {(u, i, j)|(u, i) ∈ R+, (u, j) ∈ R−} denotes the
+triplet training data, R+ indicates the observed interactions,
+and R− is the unobserved interactions; α is the margin pa-
+rameter that helps to control the observed and unobserved
+interactions. By minimizing Lrank, the interacted items will
+gather around the user u more closely than items that are
+not interacted with u in the Hamming space.
+By combining the aforementioned factors, the overall
+loss function to be minimized is formulated as:
+L = Lcross + λ1Lrank + λ2∥E∥2
+2
+= −
+�
+rui∈R
+ruilog(σ(ˆyui)) + (1 − rui)log(1 − σ(ˆyui))
++ λ1
+�
+(u,i,j)∈D
+max(0, −σ(ˆyui) + σ(ˆyuj) + α) + λ2∥E∥2
+2,
+(18)
+where λ1 is the trade-off parameter to balance the pro-
+portion between the entropy and the ranking loss. Note
+that both the pair-wise ranking loss and the point-wise
+reconstruction loss are necessary for the model optimiza-
+tion; E ∈ R(N+M)×K denotes the real-valued parameter
+matrix in the initial layer as trainable model parameters,
+and λ2 controls the L2 regularization strenghth to prevent
+overfitting. In addition, we adopt mini-batch Adam [55]
+to optimize our model and update the model parame-
+ters. Particularly, for a batch of randomly sampled triplets
+(u, i, j) ∈ D, we obtain their final hash codes h∗
+u, h∗
+i , and h∗
+j,
+and then update model parameters by using the gradients
+of the loss function.
+4.5
+Comparison with HashGNN
+We
+compare
+our
+hash
+code
+propagation
+layer
+with
+HashGNN [18], which adopts the embedding propagation
+in Euclidean space to learn hash codes. HashGNN designs
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+7
+the graph convolution operation not on the hash code h(l)
+u
+(or h(l)
+i ) but on its real-valued parameter approximation
+e(l)
+u (or e(l)
+i ). There obviously exists a quantization deviation
+between them, i.e., e(l)
+u = h(l)
+u + ϵ(l)
+u . Hence, the propagation
+layer of HashGNN can be formulated as:
+e(l+1)
+u
+= tanh
+�W(l) · mean
+�h(l)
+u + ϵ(l)
+u +
+�
+i∈Nu
+(h(l)
+i
++ ϵ(l)
+i )
+��,
+(19)
+where the quantization deviations are accumulated with
+the aggregation. The nonlinear tanh function and weight
+matrix W(l) ∈ RK×K cause a new deviation between e(l+1)
+u
+and h(l+1)
+u
+, which will make the final hash codes unable to
+obtain the high-order Hamming similarity. Compared with
+HashGNN, our proposed HS-GCN model implements the
+propagation of hash codes without introducing quantization
+deviations, effectively capturing the high-order Hamming
+similarity between hash codes.
+5
+EXPERIMENTAL SETUP
+In this section, we first present the evaluation datasets,
+and then introduce our experimental settings, followed by
+elaboration of baseline methods.
+5.1
+Datasets
+To justify the effectiveness of our proposed model, we
+conducted experiments on six publicly accessible datasets:
+MovieLens1, Yelp2, Amazon3, Gowalla, Pinterest, and Net-
+flix4. Table 1 summarizes the detailed statistics of the evalu-
+ated datasets.
+MovieLens: This is a widely used movie rating dataset,
+and we adopted the 1M version in our experiments. Sim-
+ilar to [56], we transformed the rating scores into implicit
+feedback, so that the label of each user-item pair is either 1
+or 0, indicating whether the user rated the movie. The other
+datasets are processed in a similar way.
+Yelp: Yelp is a famous online review platform for business,
+such as restaurants, bars, and spas. We selected the dataset
+from the latest version and used a 20-core setting to provide
+a denser dataset.
+Amazon: Amazon-review is a popular dataset for commod-
+ity recommendation [57]. We selected Amazon-book from
+the collection. Similarly, we used the 20-core setting to
+ensure that each user and item have 20 interactions at least.
+Gowalla: This is the check-in dataset [58] obtained from
+Gowalla, where users share their locations by checking-
+in. To ensure the quality of the dataset, we used the 10-
+core setting, i.e., retaining users and items with at least ten
+interactions similar to [2].
+Pinterest: This implicit feedback dataset is constructed
+by [59] for evaluating image recommendation. We adopted
+its processed version shared by [60].
+Netflix: This is the large-scale movie rating dataset used
+in the Netflix challenge. We applied the 20-core setting to
+obtain a denser dataset.
+1. https://grouplens.org/datasets/movielens/.
+2. https://www.yelp.com/dataset.
+3. http://jmcauley.ucsd.edu/data/amazon/.
+4. http://www.netflixprize.com/.
+TABLE 1
+Statistics of the datasets.
+Dataset
+#Users
+#Items
+#Interactions
+Density
+MovieLens
+6,040
+3,706
+1,000,209
+4.47%
+Yelp
+40,500
+58,755
+2,024,283
+0.09%
+Amazon
+36,783
+77,379
+2,402,416
+0.08%
+Gowalla
+29,585
+40,981
+1,027,370
+0.09%
+Pinterest
+55,187
+9,916
+1,500,809
+0.27%
+Netflix
+429,584
+17,764
+99,884,887
+1.31%
+For each dataset, we randomly split its implicit feedback
+into two parts: 70% for training and the rest 30% for testing.
+Moreover, 10% interactions in the training set are randomly
+selected as the validation set for hyper-parameter tuning.
+For each observed user-item interaction in the training set,
+we treated it as a positive instance, and then adopted the
+negative sampling strategy to pair it with five negative items
+that the user did not interact with in the training set.
+5.2
+Experimental Settings
+Evaluation Metric. For each user in the testing set, we
+regarded all items with no interaction with her/him as
+negative ones. Then the recommendation method predicts
+the user’s preference scores (e.g., Hamming similarity) over
+all the items, except the ones used in the training set. To
+measure the performance of top-K recommendation and
+preference ranking, we adopted HR@K (Hit Ratio) [60] and
+NDCG@K (Normalized Discounted Cumulative Gain) as
+the evaluation metrics. Whereinto, NDCG@K are formally
+defined as:
+NDCG@K = DCGK
+IDCGK
+, and DCGK =
+K
+�
+i=1
+2rui − 1
+log2(1 + i),
+where IDCG is the ideal DCG, and rui denotes the interac-
+tion status of the i-th recommended item. By default, we set
+K = 50 and 100. We reported the average metrics for all
+users in the testing set.
+Model Implementation. We implemented our HS-GCN
+model via the development tool Pytorch5 and Pytorch Ge-
+ometric6. We set the depth of HS-GCN L as two to model
+the second-order Hamming similarity. The hash code size is
+set to be 64. We optimized our model by Adam optimizer
+with a batch size of 3,000. Besides, we utilized the popular
+approach of Xavier [61] to initialize all the parameters in
+our model. And we applied the grid search for tuning the
+hyper-parameters based on the results from the validation
+set: the learning rate is tuned amongst {1e-4, 3e-4, 1e-3, 3e-
+3}, and finally set to be 3e-4; the trade-off coefficient λ1 is
+tuned from 0.1 to 1 with step size of 0.1, and finally set
+to be 0.1; the coefficient λ2 of L2 normalization is searched
+within {1e-8, 1e-7, · · · , 1}, and 1e-7 is the optimal value. The
+margin parameter α is fixed to 0.2. Moreover, early stopping
+is adopted if HR@K on the validation data does not increase
+for 10 successive epochs.
+5. https://pytorch.org/.
+6. https://pytorch-geometric.readthedocs.io/en/latest/.
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+8
+5.3
+Baseline Comparison
+To demonstrate the effectiveness, we compared the perfor-
+mance of our proposed method with a series of state-of-
+the-art hash learning based models (i.e., DHN, Hash ste,
+HashNet, and HashGNN). Beyond these methods, we also
+introduce several real-valued recommendation models (i.e.,
+MF, BiNE, PinSage, and GraphSAGE) to justify that HS-
+GCN could achieve comparable results.
+• DCF [6]: It is the first method to directly optimize hash
+codes based on the rating matrix. In our experiments, we
+adapted DCF for addressing the interaction matrix.
+• DFM [5]: This is a discretely parameterized factorization
+machine for rating prediction. We adapted it to be trained
+on implicit data.
+• DHN [15]: This is a popular deep hashing method for
+similar image recommendation. It learns high-quality
+hash codes by controlling the quantization deviation. We
+adapted it for user-item recommendation by replacing the
+original AlexNet [62] framework with Graph Convolution
+Networks (GCNs).
+• Hash ste: This is an effective end-to-end hash learning
+method based on straight through estimator [44], which
+enables the optimization of discrete variables by directly
+copying approximated gradients of them.
+• HashNet [17]: This is a state-of-the-art deep hashing
+method, which devises a continuous and differentiable
+function to approximate the sign function. Similarly, we
+used GCN to replace the AlexNet in HashNet.
+• GCNH [43]: It is a graph-based hashing method, which
+introduces a novel and efficient asymmetric graph convo-
+lution network, followed by a binarization step to learn
+similarity-preserving hash codes. We adapted it for rec-
+ommendation via the user and item ID information as its
+input feature.
+• DGCN-BinCF [42]: It is a graph-based hashing method.
+This method relaxes the binary constraint and makes
+continuous optimization possible to distill the ranking
+information from GCN into hash codes.
+• HashGNN [18]: It is a graph-based hashing model, where
+GNN is utilized towards high quality hash codes.
+• MF [63]: Matrix Factorization is the most common embed-
+ding model for recommendation, which only exploits the
+user-item direct interactions for recommendation.
+• BiNE [64]: This embedding model adopts biased random
+walks for representation learning based on the bipartite
+graph. It applies an optimization framework for both
+explicit ratings and implicit feedback.
+• PinSage [25]: PinSage is a graph embedding model. It
+combines efficient random walks and graph convolutions
+on the item-item graph, which incorporates both the
+graph structure and the node feature information. Hereon,
+we applied it on the user-item interaction graph.
+• GraphSage [65]: As a famous graph embedding model, it
+is a general inductive framework that learns embeddings
+by sampling and aggregating features from a node’s local
+neighborhood.
+Baseline Implementation. For fair comparison, all base-
+line methods are implemented in Pytorch and Pytorch Ge-
+ometric, except the publicly available implementation of
+DCF7, DFM8, GCNH9, and BiNE10. Without specification,
+the default size of embedding or hash code is 64. For
+all graph-based baselines, we adopted a two-layer GCN
+initialized from Xavier, and used the Adam optimizer with
+a well-chosen mini-batch size for model optimization. The
+learning rate is tuned amongst {1e-4, 3e-4, 1e-3, 3e-3}. For
+HashGNN, we set the architecture of the graph layer, and
+the hyper-parameters, following the original paper [18]. For
+HashNet, we initialized the scale parameter β for tanh(βx)
+as 1, and exponentially increased it after constant epochs as
+suggested in [17]. For DGCN-BinCF, PinSage, and Graph-
+Sage, we implemented them according to the default archi-
+tectures in the original papers [25], [42], [64], [65], and tuned
+their hyper-parameters based upon the model performance
+on validation set.
+6
+EXPERIMENTAL RESULTS
+In order to validate the effectiveness of our proposed
+method, we conducted overall experiments to compare our
+proposed HS-GCN model with baseline methods in perfor-
+mance, efficiency, and sparsity issue. Moreover, in order to
+evaluate the effectiveness of components in HS-GCN, we
+performed ablation experiments on the key components
+including: propagation layers, ranking loss, and dropout
+technique.
+6.1
+Overall Experiments
+We started by comparing the performance of all the meth-
+ods, and then analyzed the efficiency of these methods.
+Finally, we explored the effectiveness of modeling high-
+order similarity under the sparse settings.
+6.1.1
+Performance Comparison
+The results of our method and baselines over the exper-
+imented datasets are presented in Table 2. Besides, we
+reported the improvements and statistical significance test
+in Table 2, which are calculated between our proposed
+method and the strongest hashing baseline (highlighted
+with underline). Observing the results shown in Table 2
+from top to bottom, we obtained the following key findings:
+• DCF and DFM underperform on the implicit feedback
+datasets. This reflects the limitation of hashing methods
+that are designed on the basis of explicit feedback. DHN
+obtains poor performance on all the datasets. This indi-
+cates that the two-stage hashing method is insufficient to
+capture first-order similarities between users and items in
+the Hamming space, which is caused by the quantization
+deviation. Hash ste consistently exceeds DHN across all
+cases, demonstrating the advantage of directly learning to
+hash strategy. However, Hash ste fails to completely solve
+the quantization deviation between the straight through
+estimator and the sign function.
+• Compared with DHN and Hash ste, HashNet effectively
+eliminates the quantization deviation, and the better per-
+formance verifies that HashNet captures the first-order
+7. https://github.com/hanwangzhang/Discrete-Collaborative-
+Filtering.
+8. https://github.com/hanliu95/DFM.
+9. https://github.com/zxJohnFly/GCN.
+10. https://github.com/clhchtcjj/BiNE.
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+9
+TABLE 2
+Overall performance comparison on six datasets. % Improv. and p-value denote the relative improvements (%) and t-test results of HS-GCN
+compared with HashGNN, respectively.
+Methods
+MovieLens
+Yelp
+Amazon
+Gowalla
+Pinterest
+Netflix
+HR@50
+HR@100
+HR@50
+HR@100
+HR@50
+HR@100
+HR@50
+HR@100
+HR@50
+HR@100
+HR@50
+HR@100
+DCF
+0.0469
+0.0910
+0.0143
+0.0281
+0.0145
+0.0288
+0.1376
+0.1471
+0.0723
+0.0840
+0.0667
+0.1143
+DFM
+0.0549
+0.1068
+0.0163
+0.0310
+0.0171
+0.0336
+0.1753
+0.1820
+0.0849
+0.0931
+0.0819
+0.1330
+DHN
+0.0782
+0.1393
+0.0228
+0.0430
+0.0238
+0.0448
+0.2281
+0.2415
+0.1200
+0.1328
+0.1084
+0.1813
+Hash ste
+0.0955
+0.1684
+0.0349
+0.0620
+0.0327
+0.0557
+0.3268
+0.3606
+0.1125
+0.2032
+0.1211
+0.2007
+HashNet
+0.1359
+0.2232
+0.0364
+0.0651
+0.0353
+0.0599
+0.3370
+0.3830
+0.1539
+0.2730
+0.1463
+0.2337
+GCNH
+0.1552
+0.2488
+0.0389
+0.0714
+0.0394
+0.0636
+0.3979
+0.4209
+0.1847
+0.3129
+0.1515
+0.2402
+DGCN-BinCF
+0.1554
+0.2546
+0.0358
+0.0666
+0.0373
+0.0623
+0.3420
+0.3853
+0.1834
+0.3112
+0.1744
+0.2883
+HashGNN
+0.1598
+0.2557
+0.0447
+0.0774
+0.0363
+0.0611
+0.3932
+0.4329
+0.1947
+0.3214
+0.2058
+0.3185
+HS-GCN
+0.2052
+0.3169
+0.0497
+0.0883
+0.0523
+0.0885
+0.4084
+0.4480
+0.2066
+0.3322
+0.2192
+0.3432
+MF
+0.1332
+0.2042
+0.0319
+0.0513
+0.0349
+0.0566
+0.3360
+0.3675
+0.1400
+0.2472
+0.1981
+0.3042
+BiNE
+0.1308
+0.1868
+0.0345
+0.0546
+0.0335
+0.0560
+0.3603
+0.3812
+0.1526
+0.2582
+0.2097
+0.3346
+PinSage
+0.1587
+0.2501
+0.0503
+0.0812
+0.0402
+0.0656
+0.3791
+0.4131
+0.1837
+0.3008
+0.2279
+0.3585
+GraphSage
+0.2132
+0.3326
+0.0581
+0.0942
+0.0408
+0.0672
+0.3894
+0.4305
+0.1919
+0.3162
+0.2334
+0.3616
+% Improv.
+28.41%
+23.93%
+11.19%
+14.08%
+44.08%
+44.84%
+3.87%
+3.49%
+6.11%
+3.36%
+6.51%
+7.76%
+p-value
+2.77e-4
+7.01e-4
+5.62e-3
+1.95e-4
+1.04e-3
+1.27e-4
+1.81e-4
+1.39e-4
+1.37e-3
+5.08e-4
+1.35e-4
+2.04e-4
+Methods
+MovieLens
+Yelp
+Amazon
+Gowalla
+Pinterest
+Netflix
+N@50
+N@100
+N@50
+N@100
+N@50
+N@100
+N@50
+N@100
+N@50
+N@100
+N@50
+N@100
+DCF
+0.0641
+0.0785
+0.0097
+0.0137
+0.0113
+0.0173
+0.2066
+0.2599
+0.0756
+0.1109
+0.0862
+0.1014
+DFM
+0.0767
+0.0937
+0.0105
+0.0165
+0.0131
+0.0187
+0.2616
+0.3041
+0.0912
+0.1242
+0.1019
+0.1294
+DHN
+0.1038
+0.1251
+0.0149
+0.0226
+0.0176
+0.0264
+0.3440
+0.4080
+0.1212
+0.1710
+0.1373
+0.1660
+Hash ste
+0.1305
+0.1608
+0.0225
+0.0322
+0.0263
+0.0354
+0.4692
+0.5114
+0.1248
+0.1748
+0.1515
+0.1817
+HashNet
+0.1940
+0.2233
+0.0232
+0.0335
+0.0281
+0.0378
+0.4825
+0.5245
+0.1441
+0.1877
+0.1710
+0.1900
+GCNH
+0.1443
+0.2283
+0.0294
+0.0450
+0.0319
+0.0408
+0.4957
+0.5490
+0.1616
+0.2105
+0.1723
+0.2049
+DGCN-BinCF
+0.2298
+0.2688
+0.0268
+0.0345
+0.0302
+0.0390
+0.4646
+0.5234
+0.1436
+0.2072
+0.2230
+0.2745
+HashGNN
+0.2335
+0.2630
+0.0308
+0.0430
+0.0273
+0.0374
+0.5155
+0.5512
+0.1693
+0.2198
+0.3056
+0.3502
+HS-GCN
+0.3081
+0.3376
+0.0336
+0.0483
+0.0436
+0.0585
+0.5351
+0.5696
+0.1786
+0.2325
+0.3883
+0.4501
+MF
+0.2023
+0.2216
+0.0240
+0.0312
+0.0303
+0.0392
+0.4977
+0.5221
+0.1029
+0.1514
+0.2469
+0.2883
+BiNE
+0.1370
+0.1765
+0.0231
+0.0320
+0.0278
+0.0370
+0.5104
+0.5346
+0.1169
+0.1858
+0.2584
+0.2999
+PinSage
+0.2316
+0.2517
+0.0380
+0.0496
+0.0337
+0.0438
+0.5136
+0.5447
+0.1536
+0.2063
+0.2923
+0.3261
+GraphSage
+0.3195
+0.3490
+0.0434
+0.0570
+0.0349
+0.0457
+0.5368
+0.5581
+0.1643
+0.2124
+0.3703
+0.4114
+% Improv.
+31.95%
+28.37%
+9.09%
+12.33%
+59.71%
+56.42%
+3.80%
+3.34%
+5.49%
+5.78%
+27.1%
+28.5%
+p-value
+3.28e-3
+2.87e-3
+1.41e-2
+8.47e-4
+3.51e-4
+1.44e-4
+2.21e-3
+4.09e-4
+4.11e-3
+1.69e-3
+4.15e-3
+3.72e-3
+Hamming similarity more accurately. However, none of
+these methods explicitly models the high-order Hamming
+similarity in the hash code learning process, which leads
+to suboptimal results.
+• GNN-based hashing methods significantly outperform
+the hashing methods based on deep learning. It makes
+sense since GNN improves the quality of user and item
+continuous representations prior to the binarization step,
+which indirectly enriches the final hash codes. Within
+GNN-based methods, HashGNN is superior since it is
+a real sense of end-to-end hash learning method, due
+to its ability of solving the back-propagation issue of
+sign function by gradient copy. In contrast, DGCN-BinCF
+and GCNH both use sign function as an extra step for
+binarization, inevitably resulting in quantization loss.
+• It is clear that HS-GCN yields the best performance
+among hashing methods. In particular, the average im-
+provements of HS-GCN over the strongest hashing base-
+line HashGNN w.r.t. HR@K are 26.17%, 12.64%, 44.46%,
+3.68%, 4.74%, and 7.14%, and w.r.t. NDCG@K are 30.16%,
+10.71%, 58.07%, 3.57%, 5.64%, and 27.8% in MovieLens,
+Yelp, Amazon, Gowalla, Pinterest, and Netflix, respec-
+tively. The reason is that HS-GCN is capable of capturing
+the high-order similarity of hash codes by directly con-
+structing GCN in the Hamming space, while HashGNN
+only uses the first-order similarity to guide the hash
+learning. Additionally, we conducted the cross validation
+t-test, and p-value < 0.05 indicates that the improvements
+of HS-GCN over the strongest hashing baseline are statis-
+tically significant.
+• HS-GCN significantly outperforms the embedding based
+baselines MF and BiNE. Moreover, compared with the
+graph embedding models GraphSage and PinSage, our
+model yields comparable performance. GraphSage per-
+forms better than most hash-based methods, since it uti-
+lizes GCN to learn real-valued embeddings for users and
+items, which have better representative ability than binary
+codes. It is worth mentioning that our performance on
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
+10
+Amazon is superior than the best graph embedding model
+GraphSage. This also verifies the importance of capturing
+the high-order similarity for hash learning.
+6.1.2
+Efficiency Comparison
+In this section, we analyzed the efficiency of HS-GCN
+compared with baselines. In particular, we focused on both
+the training and the testing efficiency of these models, and
+recorded their running time. For fair comparison, all the
+models are trained on Ubuntu 16.04.5 with NVIDIA TITAN
+Xp, 12GB frame buffer and Python3.7 until the convergence,
+and tested on Windows 10 with Intel Core i7, 16GB RAM
+and Python3.7. During the training process, the hyper-
+parameters (e.g., the training epoch, the batch size, and
+the learning rate) of all models follow the optimal settings.
+Though we adopted linear scan as the testing technique to
+recommend items for all methods, it is worth mentioning
+some techniques like multi-level indexing [66] can further
+accelerate the testing of hashing methods. Table 3 summa-
+rizes the results on different-scale Amazon dataset, where
+“1/8Amazon” is constructed by randomly selecting one in
+eight interactions from the complete dataset, and the rest
+can be done in the same manner. There are similar efficiency
+comparisons on other datasets, and thus we omitted them
+for space saving. Regarding the experiments, we found that:
+• Our model costs much less training time than other
+methods. Compared to MF, HS-GCN and other graph-
+based models are more efficient in training since they
+quickly converge in experiments. This is reasonable since
+indirectly connected users and items are involved through
+the graph structure when optimizing the interaction pairs
+in mini-batch. Compared to other graph-based models,
+HS-GCN performs more efficiently during the training
+process, since it has the most concise model size that
+is the same as MF. Specifically, HS-GCN introduces no
+additional parameters except for the initial real-valued
+matrix E, while other models introduce the trainable
+weight matrix as transfer parameters in each propagation
+layer.
+• Similar with most hashing models, HS-GCN achieves
+significant speedup compared with continuous recom-
+mendation models regarding the testing time. By jointly
+analyzing Table 2 and Table 3, we can see that HS-GCN
+not only can achieve competitive performance compared
+to state-of-the-art continuous models, but also is time-
+efficient. This verifies that HS-GCN is an operable solution
+for large-scale Web services to substantially reduce the
+computation cost of their recommendation systems [67].
+• Across the different-scale datasets, the training efficiency
+ratio of HS-GCN is stable around 2.4 times based on
+HashGNN, and the testing efficiency ratio is stable around
+4.9 times based on GraphSage. From Table 3, we can
+observe that both training and testing efficiency ratios of
+our method show upward trends with the scale of the
+dataset gradually increasing.
+6.1.3
+Performance Comparison w.r.t. Interaction Sparsity
+Levels
+The sparsity issue usually limits the expressiveness of
+recommender systems, since inactive users lack sufficient
+TABLE 3
+Training/Testing time on different-scale Amazon dataset with 64 bits.
+The results are reported in hours and seconds, respectively.
+Training/Testing efficiency ratio is computed between HS-GCN and
+HashGNN/GraphSage, which is the best graph-based
+hashing/embedding baseline.
+Methods
+1/8Amazon
+1/4Amazon
+1/2Amazon
+Amazon
+Train
+Test
+Train
+Test
+Train
+Test
+Train
+Test
+HS-GCN
+1.3
+3.0
+2.4
+6.1
+4.6
+13.3
+9.1
+24.8
+DHN
+1.4
+3.0
+2.7
+6.1
+5.4
+13.2
+11.2
+24.9
+Hash ste
+1.5
+3.0
+3.0
+6.0
+6.1
+13.3
+12.6
+24.7
+HashNet
+2.1
+3.0
+4.3
+6.2
+9.0
+13.1
+18.3
+24.8
+GCNH
+1.9
+3.1
+3.6
+6.1
+7.2
+13.3
+14.8
+25.0
+DGCN-BinCF
+1.6
+3.1
+3.2
+6.1
+6.2
+13.3
+12.6
+24.8
+HashGNN
+2.7
+3.0
+5.6
+6.1
+11.1
+13.2
+23.3
+24.9
+MF
+2.3
+14.1
+4.3
+30.5
+9.1
+66.2
+18.9
+127.3
+BiNE
+1.4
+14.2
+2.6
+30.2
+5.4
+66.2
+10.9
+127.0
+PinSage
+1.6
+14.1
+3.2
+30.4
+6.5
+66.0
+13.1
+127.2
+GraphSage
+1.5
+14.1
+2.9
+30.1
+6.0
+65.9
+12.1
+127.0
+Efficiency Ratio
+2.1
+4.7
+2.3
+4.9
+2.4
+5.0
+2.6
+5.1
+interactions to generate high-quality representations. This
+especially impedes the learning to hash methods only based
+on the first-order Hamming similarity. In this section, we
+attempted to answer whether exploring the high-order
+Hamming similarity is useful to alleviate this issue.
+Towards this end, we first divided users into different
+sparsity-level groups. In particular, according to the inter-
+action number per user in the training set, we divided the
+testing set into four groups, each of which has the same
+interaction sum. Taking Yelp dataset as an example, the
+interaction numbers per user are less than 23, 42, 89, and
+1,713, respectively. Figure 3 shows the experiment results
+w.r.t. NDCG@50 on different user groups in MovieLens,
+Yelp, and Amazon. There is a similar performance trend
+w.r.t. HR@50, and thus we omitted this part for space saving.
+Regarding the results, we found that:
+• HS-GCN consistently outperforms all other hashing base-
+lines on all user groups. This demonstrates that exploiting
+the high-order Hamming similarity can improve the hash
+code learning for both active and inactive users.
+• After analyzing Figures 3, we observed that HS-GCN
+achieves larger improvements in the first two groups (e.g.,
+9.31% and 6.88% over the best baseline separately for
+< 23 and < 42 in Yelp) than that of the others (e.g., 0.52%
+for <1,713 Yelp group). It verifies that capturing the high-
+order Hamming similarity is especially beneficial to the
+inactive users, since their hash codes are learned from
+more sufficient similar information besides first-order
+similarities. Hence, exploiting the high-order similarity
+is promising to solve the sparsity issue in hashing-based
+recommender systems.
+6.2
+Ablation Experiments
+In this section, we studied the effect of key components in
+our proposed model, including propagation layers, ranking
+loss, and dropout. As the hash code propagation layer plays
+an important role in our model, we started by investigating
+the influence of layer numbers on the performance. Then
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
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+Fig. 3. Performance comparison over the sparsity distribution of user groups on three different datasets. Wherein, the background histograms
+indicate the number of users involved in each group, and the lines demonstrate the performance w.r.t. NDCG@50.
+TABLE 4
+Effect of hash codes propagation layer numbers (L).
+Methods
+MovieLens
+Yelp
+Amazon
+HR@50
+N@50
+HR@50
+N@50
+HR@50
+N@50
+HS-GCN-1
+0.1822
+0.2774
+0.0452
+0.0316
+0.0407
+0.0340
+HS-GCN-2
+0.2052
+0.3081
+0.0497
+0.0336
+0.0523
+0.0436
+HS-GCN-3
+0.1840
+0.2798
+0.0482
+0.0329
+0.0500
+0.0421
+we analyzed the ranking reinforced loss in the model opti-
+mization. Moreover, we jointly analyzed the affects of node
+dropout and bit dropout ratios.
+6.2.1
+Effect of Layer Numbers
+To investigate the optimal number of hash code propagation
+layers, we varied the model depth. Particularly, we searched
+the layer numbers within {1, 2, 3}. Table 4 summarizes the
+experimental results, where HS-GCN-2 denotes the model
+with two propagation layers, and similar notations for the
+others. Jointly analyzing Table 2 and Table 4, we have the
+following observations:
+• Obviously, HS-GCN-2 and HS-GCN-3 consistently out-
+perform HS-GCN-1 across all datasets. The reason is
+that HS-GCN-1 only considers the first-order Hamming
+similarity, while HS-GCN-2 and HS-GCN-3 both utilize
+the high-order Hamming similarity.
+• HS-GCN-2 achieves the best performance. When further
+stacking the propagation layer, we found that HS-GCN-
+3 leads to overfitting on all the datasets. The optimal
+layer number of our model is consistent with the prior
+HashGNN [18]. Therefore, the second-order similarity is
+sufficient for hash learning.
+• When varying the number of propagation layers, HS-
+GCN is consistently superior to the learning to hash
+baselines across three datasets. This verifies that directly
+capturing the high-order Hamming similarity can facili-
+tate the quality of hash codes.
+6.2.2
+Effect of the Ranking Loss
+For capturing the relative item ranking in hash learning, we
+introduced the ranking reinforced loss in model optimiza-
+tion. During the experiments, we found that the ranking loss
+can contribute to both the initial layer and the prediction
+layer, where the ranking losses are computed by the initial
+state {hu, hi, hj} and the final hash codes {h∗
+u, h∗
+i , h∗
+j},
+respectively. To study the influence of these two types of
+TABLE 5
+Performance comparison of the models with and without different
+losses.
+Methods
+MovieLens
+Yelp
+Amazon
+HR@50 N@50 HR@50 N@50 HR@50 N@50
+HS-GCN
+0.2052
+0.3081
+0.0497
+0.0336
+0.0523
+0.0436
+W/o Initial Rank
+0.1773
+0.2747
+0.0468
+0.0313
+0.0509
+0.0428
+W/o Final Rank
+0.1769
+0.2566
+0.0418
+0.0281
+0.0460
+0.0385
+W/o CR-Entropy
+0.1564
+0.2426
+0.0156
+0.0100
+0.0289
+0.0239
+ranking losses, we conducted ablation study to validate how
+each of them contributes to the overall performance of our
+model.
+Table 5 records the results of the ablation study on
+three datasets, where HS-GCN is our intact model with two
+types of ranking losses, which are computed upon triplets
+of the initial state {hu, hi, hj} and the final hash codes
+{h∗
+u, h∗
+i , h∗
+j}, respectively. W/o Initial Rank denotes the
+ablation variant of HS-GCN via removing the ranking loss
+on the triplets of initial hash codes, while W/o Final Rank
+is the variant via removing the ranking loss on the final
+hash codes. From the table, we can observe that HS-GCN
+performs worse when ignoring the ranking loss for pre-
+diction. This demonstrates the capacity of ranking loss for
+more effective hashing. Moreover, we found that the ranking
+loss can improve the model performance more significantly
+when acting on the initial layer, since it injects the order
+information into the initial state. This also demonstrates that
+initialization enhances the quality of final hash codes, which
+is consistent with the findings of prior efforts [6], [11].
+Following the operations in HashGNN [18], we intro-
+duce the cross-entropy loss to supervise the interactions
+between users and items they consumed before. We suggest
+that the cross-entropy loss maximization provides the signal
+to preserve the bipartite graph structure, which is the base of
+graph neural networks. Since this is not originally presented
+by us, we did not claim this design in our paper. To evaluate
+the effectiveness of each design, we do ablation study on
+the several datasets. In particular, we removed the binary
+cross-entropy loss from the final loss function, denoted as
+W/o CR-Entropy in Table 5, and observed that the variant
+is suboptimal compared with the intact one. The possible
+reason is that the binary cross-entropy loss is capable of
+supervising the hash codes to reconstruct the interactions
+between users and items, and thus optimizes their hash
+coding.
+
+IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING
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+Fig. 5. Effect of the length of hash codes.
+6.2.3
+Effect of Dropout
+Although GCNs have strong representation ability, they
+usually suffer from the overfitting problem. Dropout is an
+effective solution to prevent models from overfitting [68].
+Following the prior GCN-based researches and consid-
+ering our model architecture, we attempted to employ
+two dropout techniques: node dropout and bit dropout
+to improve the performance of HS-GCN. Node dropout
+randomly discards particular neighbor nodes during the
+aggregation of hash codes. For the l-th propagation layer,
+we randomly dropped p1 ×(N +M) nodes of the adjacency
+matrix, where p1 is the dropout ratio. We also conducted bit
+dropout that randomly drops a few bits of the final user and
+item hash codes before the prediction, with a probability p2.
+Note that dropout is only used during the training
+process, and should be disabled in the testing process.
+Figure 4 plots the effect of node dropout ratio p1 and
+bit dropout ratio p2 under HR@50 evaluation metric on
+different datasets. Regarding the two dropout strategies,
+bit dropout delivers much better performance especially on
+sparse datasets. Node dropout even has a negative effect on
+Yelp dataset. One reason might be that dropping neighbor
+nodes damages the hash code aggregation operation in
+the propagation process of HS-GCN, especially when the
+neighbor nodes are sparse. Bit dropout is more effective,
+which means that it can be an effective strategy to address
+the overfitting of learning to hash on GCNs.
+6.2.4
+Effect of the Length of Hash Codes
+In order to explore how the length of hash code affects HS-
+GCN and other baselines, we changed the length of their
+hash codes within {8, 16, 32, 64}, and repeated the experi-
+ments. Figure 5 shows the performance of these models with
+different length K on MovieLens and Gowalla datasets.
+It can be observed that the hash code length has positive
+influence on hashing models, while the influence weakens
+as length K increases. Similar trends are observed on other
+datasets.
+7
+CONCLUSION AND FUTURE WORK
+In this work, we explicitly incorporate the high-order Ham-
+ming similarity into the hash code learning. Particularly, we
+devise a novel framework HS-GCN, which yields binary
+hash codes by propagating the similarity information on
+the user-item graph structure. HS-GCN is the first proposed
+graph convolutional network in the Hamming space, where
+the propagation layers are built upon Hamming space op-
+erations to directly capture the high-order Hamming sim-
+ilarity. Extensive experiments on three real-world datasets
+demonstrate the rationality and effectiveness of capturing
+the high-order similarity for learning to hash.
+In future, we plan to strengthen HS-GCN by incor-
+porating multi-form knowledge [69], like attributes in ta-
+bles [70], and celebrities in matrices [71], [72]. Moreover,
+we are interested in building recommender systems for
+micro videos [73]. Considering the micro videos constantly
+emerge in large numbers, they thus require more efficient
+and accurate recommendation. Another emerging research
+direction is to explore the interpretable hashing-based rec-
+ommendation [74].
+ACKNOWLEDGMENTS
+This work is supported by the National Natural Science
+Foundation of China, No.:U1936203; the Key R&D Program
+of Shandong (Major scientific and technological innovation
+projects), No.:2020CXGC010111; and Beijing Academy of
+Artificial Intelligence(BAAI).
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