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+Scalable Communication for Multi-Agent Reinforcement Learning
+via Transformer-Based Email Mechanism
+Xudong Guo , Daming Shi , Wenhui Fan
+Department of Automation, Tsinghua University
+{gxd20, shidm18}@mails.tsinghua.edu.cn, [email protected]
+Abstract
+Communication can impressively improve co-
+operation in multi-agent reinforcement learning
+(MARL), especially for partially-observed tasks.
+However, existing works either broadcast the mes-
+sages leading to information redundancy, or learn
+targeted communication by modeling all the other
+agents as targets, which is not scalable when the
+number of agents varies. In this work, to tackle
+the scalability problem of MARL communication
+for partially-observed tasks, we propose a novel
+framework Transformer-based Email Mecha-
+nism (TEM). The agents adopt local communica-
+tion to send messages only to the ones that can
+be observed without modeling all the agents. In-
+spired by human cooperation with email forward-
+ing, we design message chains to forward informa-
+tion to cooperate with the agents outside the ob-
+servation range. We introduce Transformer to en-
+code and decode the message chain to choose the
+next receiver selectively. Empirically, TEM outper-
+forms the baselines on multiple cooperative MARL
+benchmarks. When the number of agents varies,
+TEM maintains superior performance without fur-
+ther training.
+1
+Introduction
+Multi-agent reinforcement learning (MARL) has achieved re-
+markable success in many complex challenges, especially in
+game playing [OpenAI et al., 2019; Vinyals et al., 2019].
+MARL shows great potential to solve cooperative multi-agent
+real-world tasks, such as autonomous vehicle teams [Shalev-
+Shwartz et al., 2016], robotics control [Kober et al., 2013]
+and intelligent traffic control [Wei et al., 2019]. However,
+some essential obstacles still exist for MARL to reach satis-
+factory performance. When training the MARL algorithms,
+the agents keep updating their policies and causing dynam-
+ics in the environment, which may hinder the model con-
+vergence. Worse still, in most cooperative multi-agent tasks,
+agents can only observe part of the environment. Partial ob-
+servability and non-stationarity make it harder to success-
+fully cooperate, even though some works employ centralized
+Figure 1:
+Message chain formed by email forwarding in the
+Transformer-based Email Mechanism (TEM). The agents (circles)
+are trying to surround and capture the target (square). The dotted
+circle is the observation range for the agent with the same color.
+The black lines are message chains. ⟨·⟩ demotes concatenating. The
+cross denotes not sending or forwarding after receiver selection. The
+agents a, b and c indirectly cooperate by sending (ma) and forward-
+ing (⟨ma, mb⟩) messages to capture the target T. As they are not in
+the same observation range, forwarding like emails is necessary.
+training and decentralized execution (CTDE) paradigm to im-
+port a critic to coordinate the whole team [Yu et al., 2021;
+Lowe et al., 2020; Son et al., 2019; Rashid et al., 2018].
+Inspired by the ways how humans and animals cooperate,
+communication is introduced to share information between
+agents.
+Some works broadcast the messages to all the
+other agents [Zhang et al., 2019; Sukhbaatar et al., 2016;
+Foerster et al., 2016], and other recent works try to learn
+targeted peer-to-peer communication to reduce the commu-
+nication bandwidth [Ding et al., 2020; Jiang and Lu, 2018;
+Yuan et al., 2022]. Attention mechanism from Transformer
+[Vaswani et al., 2017] is also employed to learn the commu-
+nication [Jiang and Lu, 2018]. However, the existing meth-
+ods rely on modeling every teammate in the environment by
+ID to decide whether to communicate, which will bring huge
+computational overhead when the number of agents is large.
+As the modeling network is trained by a specific amount of
+IDs, the learned communication mechanism is not scalable
+to reuse when the number of agents changes. In fact, the
+agent cannot know the state of an agent outside the observa-
+tion range, and cannot judge whether the information is useful
+for it, so it is unreasonable to directly share information with
+arXiv:2301.01919v1  [cs.MA]  5 Jan 2023
+
+d
+e
+区
+T
+ar
+Csuch an agent. For example, for applications to autonomous
+vehicles, only the vehicles nearby are worth communicating
+with to avoid collisions. Thus, global communication with
+all vehicles is unnecessary. Moreover, communication with
+other vehicles should adapt to different numbers of agents as
+the traffic situation varies a lot.
+In this work, to tackle this new problem - the scalability
+of MARL communication, we propose a scalable multi-agent
+communication mechanism via Transformer-based Email
+Mechanism (TEM) to tackle the abovementioned challenges
+as shown in Fig 1. We adopt local communication to send
+messages only to the agents in the observation range, with-
+out modeling all the agents. The agent will decide whom to
+communicate with by its own intention and by observing the
+agents in the range. Thus, no matter how the overall num-
+ber of the agents changes, the learned communication mech-
+anism is scalable. To better utilize the key information and
+indirectly cooperate with the agents outside the range, we de-
+sign a new communication mechanism like email forward-
+ing to form a message chain. The agents can send and for-
+ward the messages so that the chain connects agents from
+different ranges. For example, the agent a in Fig 1 would
+like to surround and capture the target T with other agents,
+thus the agent a may send a message to the agent b instead of
+d, though d is the nearest. Then the agent b can forward the
+message together with the information from itself to c, so that
+a, b and c can cooperate for the same goal though there is no
+direct communication between them. Similarly, in our daily
+life, cooperation in a big company or organization relies on
+such forwarding emails to share information, as it is always
+hard to directly find the exact contact in another department.
+To suit the unfixed length of the message chain and ensure
+the communication mechanism is scalable, we design a new
+message network and employ Transformer to encode and de-
+code the sequence of messages. Furthermore, augmented by
+the attention mechanism in the Transformer, the communi-
+cation is selective by modeling the correlation between the
+messages and the observation. The message network is inde-
+pendent and can be plugged into any CTDE method. What’s
+more, we design a loss to guide the agent to estimate the im-
+pact of the message on other agents. Note that we do not
+introduce the broadcast mechanism from email to keep the
+communication efficient.
+For evaluation, we test TEM on three partial-observation
+cooperative MARL benchmarks: the Starcraft multi-agent
+challenge (SMAC) [Samvelyan et al., 2019], Predator Prey
+(PP) [Kim et al., 2019] and Cooperative Navigation (CN)
+[Lowe et al., 2020], where TEM reaches better performance
+than the baselines. We also evaluate the scalability of TEM.
+Without extra training, TEM can suit both situations where
+the number of agents increases and decreases, and still out-
+performs the baselines.
+2
+Related Works
+Learning how to communicate is a popular research domain
+in multi-agent reinforcement learning. Researches in this do-
+main focus mainly on cooperative scenarios, where agents
+could communicate with each other explicitly. In the early
+works, RIAL and DIAL [Foerster et al., 2016] are designed
+to learn communication, where messages are delivered from
+one timestep to the next timestep in a broadcast way. Comm-
+Net [Sukhbaatar et al., 2016] proposes a hidden layer as com-
+munication and allows the agents to communicate repeatedly
+in each step. IC3Net [Singh et al., 2018] brings in the gat-
+ing mechanism to control communication based on Comm-
+Net. Both of BiC-Net [Peng et al., 2017] and ATOC [Jiang
+and Lu, 2018] implement the communication layer as bidi-
+rectional RNN, which inputs the observations of all agents
+and outputs the action or the integrated thought of each agent.
+However, these methods either broadcast the messages or rely
+on a centralized communication layer, which is high-cost and
+not stable. Communication should not only serve as an inte-
+gration of information, instead, the agents should share infor-
+mation selectively through peer-to-peer communication.
+To avoid broadcasting messages, recent works try to de-
+sign more intelligent communication mechanisms.
+CTDE
+paradigms are also imported to implement decentralized com-
+munication.
+Some works are based on QMIX [Rashid et
+al., 2018]: VBC [Zhang et al., 2019] proposes a request-
+reply mechanism and a variance-based regularizer to elimi-
+nate the noisy components in messages. NDQ [Wang et al.,
+2020] learns nearly decomposable value functions with com-
+munication. TMC [Zhang et al., 2020] maximizes the mu-
+tual information between the decentralized Q functions and
+the communication messages while minimizing the entropy
+of messages between agents. MAIC [Yuan et al., 2022] al-
+lows each agent to learn to generate incentive messages by
+modeling every teammate and bias other agents’ value func-
+tions directly. Some are based on another CTDE framework
+MADDPG [Lowe et al., 2020]: TarMAC [Das et al., 2019]
+proposes a targeted communication behavior via a signature-
+based soft attention mechanism. Besides the message, the
+sender broadcasts a key used by the receivers to gauge the
+message’s relevance. DAACMP [Mao et al., 2020] adds a
+double attention mechanism in the actor and critic network
+respectively to select and process the important messages.
+I2C [Ding et al., 2020] learns a prior net via causal inference
+for peer-to-peer communication. The influence of one agent
+on another is inferred via the joint action-value function and
+quantified to label the necessity of peer-to-peer communica-
+tion. Nevertheless, the methods above need to model every
+other agent in the environment to achieve individual commu-
+nication, which is not scalable and practical.
+To the best of our knowledge, none of the existing MARL
+communication methods considers the scalability of the com-
+munication mechanism and the forwarding protocol inspired
+by email.
+3
+Background
+3.1
+Policy Gradient (PG) Reinforcement Learning
+Policy Gradient (PG) reinforcement learning has the advan-
+tage of learning a policy network explicitly, in contrast to
+value-based reinforcement learning methods. PG methods
+optimize the policy parameter θ to maximize its objective
+function J(θ) = ES(Vπθ(s)). However, due to the variance
+of environments, it is hard to choose a subtle learning rate in
+
+reinforcement learning. To resolve this problem and ensure
+the safe optimization of policy learning, the Trust Region Pol-
+icy Optimization (TRPO) [Schulman et al., 2015] increases
+the constraint of the parameter difference between policy up-
+dates. Such constraint ensures the parameter changes in a
+small range, so that the collapse of value can be avoided and
+the policy can learn monotonically. The parameter update of
+TRPO is θk+1 = arg maxθL(θk, θ) s.t. ¯DKL(θ||θk) ≤ δ,
+where L(θk, θ) = E[ πθ(a|s)
+πθk (a|s)Aπθk (s, a)] is the approxima-
+tion of the original policy gradient object J(θ) within the
+constraint of KL divergence.
+Based on TRPO, a simplified version Proximal Policy Op-
+timization [Schulman et al., 2017] is carried out, maintaining
+the motivation to constrain the learning step while more ef-
+ficient and easy to be implemented. The object function of
+PPO can be written as:
+L(s, a, θk, θ) =min[ πθ(a|s)
+πθk(a|s)Aπθk (s, a),
+clip( πθ(a|s)
+πθk(a|s)), 1 − ϵ, 1 + ϵ)Aπθk (s, a)],
+(1)
+which forces the ratio of πθ(a|s)
+πθk (a|s) to locate in the interval (1−
+ϵ, 1 + ϵ), so that the new θ is not too far away from old θk.
+3.2
+MAPPO Algorithm
+Multi-agent PPO (MAPPO) introduces PPO into the multi-
+agent scenario [Yu et al., 2021]. MAPPO mainly considers
+decentralized partially observable Markov decision processes
+(DEC-POMDP). In an environment with n agents, s ∈ S de-
+notes the state of the environment. The agent i only has a
+local observation of environment oi = O(s) and chooses its
+action based on its observation and policy ai = πi(ai|oi).
+The joint action A = (a1, ..., an) denotes the set of actions of
+all agents. Then, the environment transits its state based on
+the transition probability P(s′|s, A). In MARL, all the agents
+will get rewards based on the transition of state and their ac-
+tions (or more likely joint action) ri = R(s, A). Each agent is
+supposed to get a higher accumulated reward �
+t rt
+i. There-
+fore, the agents optimize their policy to maximize the dis-
+count accumulated reward J(θ) = Eat,st[�
+t γtR(st, at)],
+where γ ∈ (0, 1] is the discount factor.
+MAPPO utilizes parameter sharing within homogeneous
+agents, i.e., homogeneous agents share the same set of net-
+work structure and parameters during training and testing.
+MAPPO is also a CTDE framework, namely, each PPO agent
+maintains an actor network πθ to learn the policy and a critic
+network Vφ(s) to learn the value function, where θ and φ are
+the parameters of policy network and value network, respec-
+tively. The value function requires the global state and only
+works during training procedures to reduce variance. In our
+work, we take MAPPO as our baseline and backbone, and
+add TEM as the communication mechanism into MAPPO.
+4
+Methods
+In this section, we introduce the detailed structure and de-
+sign of TEM. Before each action decision-making, the agents
+Figure 2: Workflow of TEM during one time step. A denotes the
+actor network, C denotes the critic network, E denotes the environ-
+ment. One training step has three phases: communication, action
+and learning. The execution only includes the first two phases and
+the critic will not work.
+communicate with each other following the designed proto-
+col, sharing the key information efficiently and selectively.
+We design a message module based on Transformer to en-
+code the messages received. At the same time, the module
+is able to decide whether to communicate and whom to com-
+municate with. The message module works together with the
+original action decision module from MAPPO, to form the
+actor network in the CTDE structure. The workflow of TEM
+is illustrated in Fig 2. We design an independent loss to en-
+courage the message module to maximize the messages’ im-
+pact on other agents. The whole model has the scalability to
+transfer from one scenario to another. As the message mod-
+ule is parallel to the action module, our model can be plugged
+into any CTDE structure.
+4.1
+Communication Protocol Design
+We design a communication protocol following the way how
+humans communicate by email. The information flow is like
+a forwarding chain: the chain starts with an agent with key in-
+formation to share, and the following agents merge their own
+information into the chain and then forward the new message
+to the next agent. The chain ends when the final agent finds
+the message useless for others, or there are no more potential
+communication objects.
+When designing the communication protocol, we mainly
+consider the following questions: (1) Whether to communi-
+cate? (2) Whom to communicate with? (3) What to commu-
+nicate? (4) How to utilize the messages?
+(1) Whether to communicate? As shown in Fig 2, in ev-
+ery step of execution and training, the first stage is commu-
+nication. When the communication stage is done, the actor
+networks for each agent will make the action decisions by
+the observations and messages. In the communication stage,
+each agent has the chance to decide whether to start a new
+chain and send a message. And the agents who receive mes-
+sages can decide whether to continue forwarding the mes-
+sages. Multiple message chains are allowed and the informa-
+tion from different chains is merged if there is a shared node
+for the chains.
+
+Communication !
+Action
+Critic & Actor
+Learning
+01'
+A
+a1
+m_a1 = 
+2
+m1
+02
+A
+a2
+m_a2
+m2
+E
++R
+V
+03
+A
+m_a3
+04
+A
+Execution
+TrainingFigure 3: Network structure of TEM. (a) Actor network of agent i, including an action network and a message network. Emb denotes the
+embedding network. (b) Encoder module. (c) Decoder module, where m_dec0
+i = o_fi.
+(2) Whom to communicate with?
+We think that for
+partial-observation (PO) problems, communication with all
+the agents is not reasonable and effective. The direct commu-
+nication with the agent outside the observation range may not
+bring helpful information as the sender does not even know
+the receiver’s state. Therefore, we do not model all the other
+agents to decide whether to communicate with them like in
+some previous works [Ding et al., 2020; Yuan et al., 2022].
+Instead, when the agent i chooses communication objects, we
+only consider the agents in the observation range Oi , and in
+our experiments, Oi includes the nearest several agents of the
+agent i. By training the message module, the agent can pre-
+dict the impact of the message on other agents, and is more
+likely to choose the one with the highest impact to communi-
+cate.
+We combine the two decisions (1) and (2) into one commu-
+nication action. The communication actions of agent i m_ai
+include not sending at all m_ai = 0, and sending to one agent
+j in the observation range m_ai = j, (j ∈ Oi). Namely, we
+have:
+P(m_ai = 0) +
+�
+j∈Oi
+(P(m_ai = j)) = 1.
+(2)
+This way, the agent can decide when and whom to communi-
+cate by one action, simplifying the modeling and learning.
+(3) What to communicate? To keep the information from
+the head nodes in the chain, and merge the information from
+different chains, every agent maintains a message buffer to
+store the messages. In practice, the message buffer is imple-
+mented as a queue, with a fixed storage length, but can flexi-
+bly push in and pop out elements as the communication goes
+(First Input First Output, FIFO). We use m_bi to denote all
+the messages inside the agent i ’s message buffer. When send-
+ing the new message, the agent i merges its own observation
+into the chain, then the message chain expands to ⟨m_bi, oi⟩.
+Here, the operation ⟨·⟩ demotes pushing into the queue. The
+buffer is clear when every step starts.
+(4) How to utilize the messages? Instead of some previ-
+ous works [Zhang et al., 2019; Yuan et al., 2022], we do not
+think the messages directly influence the value estimation of
+other agents is the natural way of communication. The infor-
+mation exchange should be separated from the information
+utilization. And the final effects of the messages should be
+determined by the receiver instead of the sender. Thus, in our
+model, messages are taken as a counterpart of the observa-
+tion, serving as part of the inputs of the actor network.
+4.2
+Network Design
+The schematics of the network design in our model are shown
+as Fig 3. Each agent has an actor network to observe the envi-
+ronment and communicate with other agents. The actor net-
+work of the agent i will output the action to interact with the
+environment ai, the action to communicate m_ai (whether
+to communicate and whom to communicate with), and the
+corresponding message to be sent mi. To better utilize the
+history information and get a smoother action sequence, an
+RNN is employed in the actor network. Thus the agent i also
+keeps a hidden state ht
+i, and updates it every time step.
+The actor network consists of two sub-networks, the ac-
+tion network and the message network. The action network
+mainly concentrates on the task itself and tries to get bet-
+ter rewards by outputting reasonable actions. The message
+network concentrates on the communication to share infor-
+mation with other agents instead. The two sub-networks ex-
+change the representation feature of the observations o_fi and
+that of the messages m_fi to merge the information.
+In the action network, o_fi is learned by a multi-layer
+perceptron (MLP), and then the action network concatenates
+o_fi and m_fi to input into the RNN together with the hidden
+state from the last time step ht−1
+i
+. Another MLP, in the end,
+processes the output of the RNN to generate the final action
+ai.
+On the other hand, the messages from other agents like
+mj · · · mk are stored in the message buffer, like the email
+inbox. The embedding layer (we implement it as a full con-
+nected (FC) layer by practice) converts the messages to fit
+
+Message Buffer
+FC
+FC
+FC
+Action
+Network
+Message
+Emb
+Self-Attention
+Network
+MLP
+m_enci
+Softmax
+o_fi
+Encoder
+× Ne
+m_fi
+MLP
+FC
+Attention
+cat
+m_fi
+Network
+o_f i
+Decoder
+Encoder Φ
+RNN
+m_deci
+MLP
+m_bi
+Decoder
+MLP
+i
+MLPthe input dimensions of the encoders. Ne sequential encoder
+modules and Nd sequential decoder modules are followed by
+the embedding layer. The output of encoder modules m_fi
+serves as the representation of all the messages in the buffer,
+with the key information emphasized by the attention mecha-
+nism. The decoder modules further combine the information
+from both of m_fi and o_fi to get the output m_deci. Fi-
+nally, one MLP produces the communication decision m_ai.
+For each encoder module, it takes in m_encne−1
+i
+from the
+embedding layer or the last encoder, then generates m_encne
+i
+as the input for the next layer. The transformer in the mod-
+ule can model the sequential information and is flexible to
+fit message chains with different lengths. Also, the attention
+mechanism will help the agent to pick out the key informa-
+tion from the chain. ne implies the position of the layer in
+the encoder sequence. To prevent gradient vanishing, the en-
+coder module employs the residual connections to link the
+self-attention mechanism and the MLP [Wen et al., 2022].
+The structure of the self-attention mechanism is the same as
+the attention network in the decoder while k, q and v are gen-
+erated from the same input m_encne−1
+i
+.
+In the decoder module, the first m_dec0
+i is the representa-
+tion of the observations o_fi. In the attention network, full
+connected layers generates key k and query q by m_decnd−1
+i
+and m_fi, respectively. Also, the third FC layer generates
+value v from m_decnd−1
+i
+. k and q are used for calculating
+the weights α of the value v as Equation 3.
+α = Softmax(exp(qkT
+√dk
+)).
+(3)
+In fact, the weight α learns the correlations between the
+m_decnd−1
+i
+and m_fi. By multiplying v and α, then we get
+the weighted representation of m_decnd−1
+i
+from the ending
+FC layer. With a similar structure of the residual connections
+and MLP, we get m_decnd
+i
+as the input for the next layer.
+4.3
+Loss Function Design
+The communication among the agents during a collaborative
+task aims to share the key information that one believes is
+useful for some specific other agents. So the learning of the
+message network is driven by the impact of the message to be
+sent.
+As the communication will not change either the action of
+other agents or the loss of the action network, an indepen-
+dent loss to model the influence of the messages on other
+agents’ actions is needed.
+We denote the communication
+loss as L(i)
+m (θ), where θ is the parameters of the actor net-
+work. The action of an agent j is sampled from the categorical
+distribution P(aj|oj, m_bj) learned by the action network.
+Then, when considering the new message from the agent i mi,
+we can estimate the distribution P(aj|oj, ⟨m_bj, mi⟩) as the
+consequence of the communication. Kullback-Leibler (KL)
+divergence is widely used to measure the discrepancy be-
+tween these two conditional probability distributions. Thus,
+the causal effect Γ(i)
+j
+of the message from agent i on agent j
+can be defined as:
+Γ(i)
+j
+= DKL (P(aj|oj, ⟨m_bj, mi⟩)||P(aj|oj, m_bj)) . (4)
+By considering all the possible agents to send the message to
+in the observation range, we can get the expectation of the
+causal effect of the message EΓ(i)(θ) by Equation 5:
+EΓ(i)(θ) =
+�
+j∈Oi
+�
+Pθ(m_ai = j|oi, m_bi)Γ(i)
+j
+�
+,
+(5)
+where Oi denotes the observation range of the agent i. The
+communication decision of agent i is sampled form the cat-
+egorical distribution Pθ(m_ai|oi, m_bi) learned by the mes-
+sage network. Pθ denotes that the gradient of this item should
+be propagated when training.
+The expectation EΓ(i)(θ) represents the overall effect the
+message mi can bring to the whole system, which we should
+maximize in the loss function.
+However, communication
+should also be sparse and efficient.
+If we do not control
+the communication times by the external guidance, the agents
+will tend to send as many messages as possible to get higher
+EΓ(i)(θ).
+Therefore, we also designed another item for
+communication loss to reduce the communication overhead.
+When the agent i chooses not to send the message to any
+agents in the observation range for most of the times, the
+probability Pθ(m_ai = 0|oi, m_bi) should be relatively high.
+So we need to maximize this probability at the same time.
+So far, we can get the final communication loss L(i)
+m (θ) by
+the following equation and maximize it when training.
+L(i)
+m (θ) = EΓ(i)(θ) + δPθ(m_ai = 0|oi, m_bi),
+(6)
+where δ is the weight of the communication reduction.
+The loss of the action network L(i)
+a (θ) is defined followed
+by Equation 1 in MAPPO as:
+L(i)
+a (θ) =min(r(i)
+θ A(i)
+πθold , clip(r(i)
+θ , 1 − ϵ, 1 + ϵ)A(i)
+πθold ),
+(7)
+where r(i)
+θ
+=
+πθ(a(i)|o(i))
+πθold(a(i)|o(i)), A(i)
+πθold is the advantage function.
+What’s more, to encourage more exploration when train-
+ing, we adopt an entropy loss L(i)
+e (θ) as [Yu et al., 2021]:
+L(i)
+e (θ) = S(πθ(oi)).
+(8)
+We can get the overall loss function for the actor network
+when training:
+L(θ) =
+n
+�
+i=1
+�
+L(i)
+a (θ) + λmL(i)
+m (θ) + λeL(i)
+e (θ)
+�
+,
+(9)
+where n is the number of the agents, and λm, λe are the coef-
+ficients to weight the corresponding losses.
+The critic network is trained to minimize the loss function
+L(φ) =
+n
+�
+i=1
+(max[(Vφ(s(i)) − R)2,
+(clip(Vφ(s(i)), Vφold(s(i)) − ϵ, Vφold(s(i)) + ϵ) − R)2]),
+(10)
+where R is the discounted accumulated reword.
+
+Figure 4: Test win rate for the SMAC map 5m vs. 6m, the shaded
+regions represent the 95% confidence intervals. FC: Full Communi-
+cation, RC: Randomly-stop Communication.
+5
+Experiments
+We evaluate the performance of TEM on three widely-used
+partially-observed multi-agent cooperative tasks: the Star-
+craft multi-agent challenge (SMAC), Predator Prey (PP) and
+Cooperative Navigation (CN). We compare the training pro-
+cess of TEM with the baselines and analyze the performance.
+We test the scalability of TEM to scenarios with different
+numbers of agents and targets when zero-shot transferring.
+5.1
+StarCraft II Micromanagement Benchmark
+In the SMAC task, N units controlled by the algorithm try
+to kill all the M enemies. There are usually more enemies
+than agents, or the enemies are more powerful types of units,
+so defeating all the enemies with limited observation range
+is challenging, demanding proper cooperation strategies and
+micro-control of movement and attack. We choose the hard
+map 5m vs 6m to evaluate TEM. TEM controls 5 Marines to
+fight with 6 enemy Marines.
+The baselines include MAPPO, MADDPG, Full Com-
+munication (FC) and Randomly-stop Communication (RC).
+MAPPO is the CTDE backbone we are using in the follow-
+ing experiments, which is proven to have state-of-the-art per-
+formance on several MARL cooperative benchmarks [Yu et
+al., 2021]. MADDPG is another classic CTDE approach for
+multi-agent cooperation tasks [Lowe et al., 2020]. FC and
+RC are two special cases of TEM. We keep the communica-
+tion protocol the same, but disable the decoder in the mes-
+sage module, instead, the agents choose the communication
+targets by pre-defined rules. In FC, the agent will keep ran-
+domly choosing someone to communicate with, to extend the
+message chain until no one is available. In RC, the agent will
+randomly stop the message chain by a probability p, or keep
+forwarding to a random one.
+We run the experiments over 6 seeds. For each seed, we
+compute the win rate over 32 test games after each training
+iteration as shown in Fig. 4.TEM gets the highest win rate
+over the baselines. FC and RC perform worse than MAPPO
+benchmark. One possible reason is that targeted communica-
+tion by TEM could improve cooperation while random com-
+munication by FC and RC may bring redundant information
+for decision-making. The win rate of baseline MADDPG re-
+mains zero, showing that it is hard to defeat an army with
+more units and MADDPG fails to learn such a strategy.
+Figure 5: Reward for Predator Prey (PP) during training, the shaded
+regions represent the 95% confidence intervals.
+5.2
+Predator Prey
+In the Predator Prey (PP) task, N predators try to chase, sur-
+round and finally capture M preys, as shown in Fig 1. The
+predators are the agents to be trained and the preys flee in the
+opposite direction of the closest predator at a faster speed fol-
+lowing pre-defined rules. So the predators have to be grouped
+automatically and cooperate to surround each prey, and it is
+impossible for one predator to capture a prey itself. In prac-
+tice, we set N as 7 and M as 3, denoted as 7-3 scenario.
+Different from the PP task in some previous works, here,
+the agents can only partially observe the teammates and tar-
+gets. The rewards are the sum of the agents’ negative dis-
+tances to their closest preys or landmarks. In addition, the
+agents are penalized for collisions with other agents.
+The baselines include MAPPO, I2C, MADDPG, DDPG,
+FC and RC. I2C proposes an individual communication
+mechanism [Ding et al., 2020]. DDPG is a classic deep re-
+inforcement learning algorithm for continuous control [Lill-
+icrap et al., 2019]. We apply DDPG independently to each
+agent as a baseline without considering cooperation.
+As shown in Fig 5, while other baselines gradually con-
+verge at the last episodes, TEM keeps raising the rewards and
+improves the final reward by 17.2% compared with MAPPO.
+5.3
+Cooperative Navigation
+In the Cooperative Navigation (CN) task, N agents try to
+occupy N stationary landmarks separately, as shown in Fig
+7. The positions of landmarks and agents are randomly ini-
+tialized. The best strategy is that each agent has a different
+target from the beginning through communication instead of
+rescheduling when collisions happen because of choosing the
+same target. In practice, we set N as 7, denoted as 7-7 sce-
+nario. The baselines and reward settings are the same as PP.
+We compare TEM with the baselines on the training perfor-
+mance Fig 6. We can see that TEM converges to the highest
+Figure 6: Reward for Cooperative Navigation (CN) during training,
+the shaded regions represent the 95% confidence intervals.
+
+WinRatefor5mvs6m
+1.0
+TEM
+MAPPO
+0.8
+MADDPG
+RC
+0.6
+FC
+Rate
+Win
+0.4
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Step
+1e7RewardforPredatorPrey(PP)
+TEM
+-25
+MAPPO
+I2C
+-30
+DDPG
+MADDPG
+Reward
+RC
+-35
+FC
+-40
+-45
+-50
+2
+3
+4
+5
+Step
+1e7RewardforCooperative Navigation (CN)
+-40
+TEM
+MAPPO
+-50
+I2C
+DDPG
+MADDPG
+Reward
+-60
+RC
+FC
+-70
+-80
+-90
+3
+5
+Step
+1e7Figure 7: Comparison between (a) TEM and (b) MAPPO on the
+same environment of CN. Five frames are illustrated. Green lines
+are the trajectories and pink lines are message chains.
+reward than all the baselines. FC and RC are only slightly bet-
+ter than MAPPO, suggesting that the communication actions
+m_a learned by TEM are targeted, and the message chain
+brings helpful information to the ones that really need it.
+We compare the illustrations on CN between TEM and
+MAPPO in Fig 7. In (a), the TEM agents Agent 1 and Agent 4
+notice Landmark 1 by communication (pink message chain).
+Thus each agent moves straight forward to the corresponding
+landmark. While in (b), the MAPPO agents miss Landmark
+1, so for Agent 4, there will be nowhere to go. Agent 4 first
+tries to scramble with Agent 1 but fails, then turns to Agent
+2. Agent 2 is forced to leave to avoid collision and turns to
+Agent 3. We can see that communication brought by TEM
+can improve cooperation and reduce internal strife.
+5.4
+Scalability of TEM
+We further examine the scalability of TEM on PP task in Ta-
+ble 1. We take average episode rewards (R), successful cap-
+ture times (S), collision times (C) as the metrics. For R and
+S, the performance is better when the values are higher, while
+for C, the performance is better when the values are lower.
+Note that the existing MARL communication approaches are
+not scalable due to the modeling of each agent, the base-
+lines are the transferred MAPPO and the specifically trained
+MAPPO. We directly transfer the learned model from the 7-
+3 scenario to 9-3 and 3-1 scenarios without further training.
+For 9-3 scenario, two new agents are included, and it will be
+easier to capture the preys. But more agents also increase the
+risks of collision, so the cooperation mode could be different
+and the agents need to communicate to suit the new scenario.
+For TEM, the average episode rewards rise from -40.5 to -
+17.9, and the gain is 55.8%, while for MAPPO, the gain of
+rewards is 52.3%. TEM does not only perform better after
+transferring, but also gains more. For 3-1 scenario, both the
+numbers of the agents and preys change. The results show
+that TEM still keeps a better performance on all the metrics.
+Moreover, it shows that after TEM learns how to commu-
+nicate in a complex scenario, it can successfully transfer to
+simple ones.
+We also train MAPPO from scratch specifically on 9-3
+and 3-1 (denoted as MAPPO (learned)), and the performance
+of transferred TEM (trained on 7-3) is close to MAPPO
+(learned) on 9-3 without training. But the transferred TEM
+works worse on 3-1, and we suggest that cooperation by com-
+munication may not play an essential role in such a simple
+TEM (7-3)
+MAPPO (7-3)
+MAPPO (learned)
+TEM (finetuned)
+7-3
+R
+-40.5±4.7
+-44.9±4.3
+-
+-
+S
+61.6±18.3
+49.0±16.5
+-
+-
+C
+1.4±0.6
+12.6±2.6
+-
+-
+3-1
+R
+-10.6±4.7
+-12.7±5.9
+-7.52±2.6
+-7.0±2.8
+S
+18.7±14.0
+13.5±12.7
+36.5±13.2
+31.7±13.4
+C
+1.2±1.2
+3.6±2.7
+1.8±1.8
+0.8 ±0.6
+9-3
+R
+-17.9±5.0
+-21.3±7.5
+-17.1± 8.2
+-14.0 ±2.6
+S
+107.2±23.6
+69.3±32.0
+109.5±18.7
+127.9±5.9
+C
+16.2±1.8
+30.6±6.7
+7.2±1.9
+5.4±1.6
+Table 1: Scalability of TEM on PP. R: average episode rewards, S:
+successful capture times, C: collision times. TEM (7-3) and MAPPO
+(7-3) are trained on the scenario 7-3: 7 agents to capture 3 preys,
+and tested on ten random environments on 7-3, 3-1, 9-3 scenarios.
+MAPPO (learned) is specifically trained from scratch on the corre-
+sponding test environments. TEM (finetuned) is the TEM model
+trained on 7-3 and tuned on the corresponding test environments.
+environment. We further finetune TEM (7-3) on the new sce-
+narios and the finetuned models even outperform the specially
+learned MAPPO.
+Similar experiments are conducted on CN as shown in Ta-
+ble 2. TEM keeps the scalability when transferred from 7-
+7 scenario to 6-6 and 9-9, and outperforms the transferred
+MAPPO. Surprisingly, the transferred TEM even outper-
+forms the MAPPO trained from scratch (denoted as MAPPO
+(learned)) on most metrics. It suggests that CN requires more
+communication to coordinate the agents to explore the land-
+marks at the corner. And the results also show that the com-
+munication pattern learned from 7-7 still works well in other
+scenarios. Similarly, the finetuned TEM gets even better per-
+formance.
+TEM (7-7)
+MAPPO (7-7)
+MAPPO (learned)
+TEM (finetuned)
+7-7
+R
+-38.8±15.1
+-46.6±14.8
+-
+-
+S
+35.8± 6.2
+23.3±9.0
+-
+-
+C
+2.8±0.3
+4.2±0.2
+-
+-
+6-6
+R
+-39.8±5.3
+-45.0±8.0
+-43.6±10.1
+-36.7 ±5.2
+S
+35.3±6.2
+20.0±5.9
+19.3±6.8
+36.1±5.9
+C
+7.2±0.4
+8.4±0.4
+4.8±0.4
+4.8 ±0.2
+9-9
+R
+-45.8±23.9
+-57.5±25.3
+-50.8±12.9
+-41.1 ±11.4
+S
+39.4±4.9
+26.6±8.7
+29.0±6.6
+38.9±5.2
+C
+9.0±0.3
+28.8±0.4
+10.8±0.2
+8.0±0.2
+Table 2: Scalability of TEM on CN. TEM (7-7) and MAPPO (7-7)
+are trained on the scenario 7-7: 7 agents to occupy 7 landmarks, and
+tested on ten random environments on 7-3, 6-6, 9-9 scenarios.
+6
+Conclusions
+To tackle the scalability problem of MARL communication,
+this paper proposes a novel framework Transformer-based
+Email Mechanism (TEM). The agents adopt local communi-
+cation to send and forward messages like emails to form mes-
+sage chains, which set up bridges among partial-observation
+ranges. We introduce Transformer to encode and decode the
+message chain to choose the next receiver selectively. Em-
+pirical results in diverse multi-agent cooperative tasks show
+that our method outperforms the baselines. Furthermore, we
+can directly apply TEM to a new environment with a different
+number of agents without retraining. Better performance than
+the baselines when zero-shot transferring shows the scalabil-
+ity of TEM. Based on TEM, communication for hundreds of
+agents and further tailored message generation can be devel-
+oped, which may be an important step for MARL applications
+to real-world tasks.
+
+Agent 1
+Agent 1
+Agent 4
+Agent 4
+Agent 2
+Agent 3
+Agent 2
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+

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+IMAGE AND VIDEO COMPRESSION OF FLUID FLOW DATA
+A PREPRINT
+Vishal Anatharaman, Jason Feldkamp, Kai Fukami∗, Kunihiko Taira
+Department of Mechanical and Aerospace Engineering,
+University of California, Los Angeles, CA 90095, USA
+Corresponding author: [email protected]
+January 3, 2023
+ABSTRACT
+We study the compression of spatial and temporal features in fluid flow data using multimedia com-
+pression techniques. The efficacy of spatial compression techniques, including JPEG and JPEG2000
+(JP2), and spatio-temporal video compression techniques, namely H.264, H.265, and AV1, in limiting
+the introduction of compression artifacts and preserving underlying flow physics are considered for
+laminar periodic wake around a cylinder, two-dimensional turbulence, and turbulent channel flow.
+These compression techniques significantly compress flow data while maintaining dominant flow
+features with negligible error. AV1 and H.265 compressions present the best performance across a
+variety of canonical flow regimes and outperform traditional techniques such as proper orthogonal
+decomposition in some cases. These image and video compression algorithms are flexible, scalable,
+and generalizable holding potential for a wide range of applications in fluid dynamics in the context
+of data storage and transfer.
+1
+Introduction
+High-fidelity simulations and experiments within the field of fluid dynamics produce exceedingly large amounts of
+data. As the need for higher fidelity simulations and advanced experimental resources expands, storage and transfer
+requirements for spatio-temporal data from simulations, become a major challenge. To address this issue, spatio-
+temporal redundancies or repeated dominant flow features can be exploited by a variety of compression techniques
+to alleviate memory constraints for fluid flow data storage. A variety of compression techniques, including modal
+analysis [1, 2, 3], sub-sampling and local re-simulation [4], and deep learning [5, 6, 7, 8] have been considered in an
+effort to reduce the size of fluid flow data. Although effective, these techniques can be application-specific and struggle
+to achieve substantial compression ratios without introducing undesirable compression artifacts such as discontinuities
+or deletions of flow features.
+In comparison, multimedia compression techniques are general and simple to use, and have benefited from demand
+for the modern technologies of high-resolution video streaming [9, 10, 11] and video-conferencing [12, 13, 14].
+These compression techniques are classified into two groups: lossless compression and lossy compression [15]. With
+lossless techniques, the data retrieved from or reconstructed from the compressed state is identical to that preceding
+the application of a compression algorithm. Hence, this is preferred for archival purposes and used for medical
+imaging [16] and technical drawings [17]. In contrast, processed data with lossy techniques do not necessarily match the
+original data, enabling a significant data-size reduction in the compressed state. Since this may introduce compression
+artifacts such as discontinuities in image data or the loss of high spatial frequency information, it is suitable for natural
+images such as photographs in applications where imperceptible loss may be acceptable [18]. We consider here the
+impacts of such losses on fluid mechanics simulation data to assess the costs of applying lossy techniques. In 2003,
+Schmalzl [19] considered multimedia data compression for fluid flows with an example of Rayleigh-Bénard convection.
+With multimedia compression technologies having undergone significantly advances in the last two decades, we reassess
+image and video compressions with modern algorithms for applications to fluid flow data.
+Lossy techniques of interest typically involve frequency-domain transformation, filtering, and entropy coding as
+components in the compression process. The development of the discrete cosine transform (DCT) [20, 21] has played a
+arXiv:2301.00078v1  [physics.flu-dyn]  31 Dec 2022
+
+A PREPRINT - JANUARY 3, 2023
+Original data
+Compressed data
+Reconstruction
+…
+Spatial 
+Compression
+Temporal 
+Compression
+t
+Original 
+Data
+Compressed 
+Data
+…
+!(#, %, &)
+!′(#, %, &)
+Original data
+Reconstruction
+(a)
+(b)
+Figure 1: (a) Spatial compression: an example velocity field of flow over a cylinder q(x) is represented as a grayscale
+image, encoded using an image-based technique to a compressed form, and reconstructed as ˜q(x) using a decoder.
+(b) Spatio-temporal compression: multiple snapshots of this flow field data q(x, t) are represented as a grayscale video
+and are compressed to ˜q(x, t) with both spatial and temporal techniques.
+crucial role in image compression, and is the basis of Joint Photographic Experts Group (JPEG) [22]. The emergence of
+JPEG enabled efficient image compression in a wide range of communities and it became a generally accepted format
+for digital images. After the development of DCT, wavelet transforms began to be utilized for image compression
+in such algorithms as JPEG2000 (JP2) [23], which achieves better compression than the DCT of JPEG as a result of
+multi-scale properties of wavelets.
+In tandem with the growth of image compression techniques, advancement in video compression technologies followed
+suit since video data can be characterized as a time series of image frames. Generally, these time frames include both
+spatial and temporal redundancies. In fact, we often see the similarities (redundancies) between temporally adjacent
+frames or spatially adjacent pixels. Video compression algorithms are designed to remove such redundancies and
+obtain a compact form of the original information. Current video compression technologies are generally based on the
+DCT [24]. Although other candidates including fractal compression [25, 26], matching pursuit [27], and discrete wavelet
+transform (DWT) have been investigated as the subject of some studies, these are still not used in practical products.
+Moving Picture Experts Group (MPEG) series have been traditionally used for video compression of high-definition
+television [28, 29, 30]. H.2xx series was then developed and they have achieved significant compression compared to
+the conventional MPEGs [31, 32]. Especially in the recent versions such as H.264 and H.265, motion compensation,
+quantization, and entropy coding are applied for efficient video compression. More recently, AOMedia Video 1 (AV1),
+an open, royalty-free video coding format, was released in 2018, achieving enhanced compression compared to the
+aforementioned techniques [33, 34].
+To meet the demand for these image and video compression tools, significant investment and research have produced
+compression techniques of impressive efficiency and usability in addition to free video encoders [35] to promote
+widespread accessibility. As such, leveraging these multimedia-inspired compression techniques should also be of
+particular interest to the fluid dynamics community given the massive scale of data produced, stored, and transferred.
+2
+
+Original Data
+Compressed Data
+Reconstruction
+nxm
+< nxm
+nxm
+01100111 01101111
+11.
+Encoding
+Decoding
+(qn1
+g(x, y)gfa,tgfr,t2A PREPRINT - JANUARY 3, 2023
+A standardization on one or more multimedia compression formats for storing fluid flow data in a compressed
+representation can yield dividends in research output by allowing greater access to high-fidelity fluid flow data sets and
+by removing memory constraints as a barrier to entry.
+This paper investigates the effectiveness of these image and video compression techniques on fluid flow data. Spatial
+image compression techniques, such as JPEG and JP2, alongside spatio-temporal video compression techniques,
+namely H.264, H.265, and AV1, are examined for various flow fields, including laminar cylinder flow, two-dimensional
+turbulence, and turbulent channel flow. Field variables from simulation data, such as streamwise velocity and vorticity,
+are represented as grayscale images, and multiple snapshots are packaged into a video. These videos are then encoded
+into a compressed form using the aforementioned multimedia compression methods. Modern techniques can compress
+flow data well below 10% of the original file size with negligible error and preserve the underlying physics of the flow.
+Although this paper focuses on applications to canonical fluid flows, the flexibility and scalability of these algorithms
+suggest an expansive potential within this field.
+Compression is a process in which data is compressed (encoded) into a representation that uses less data, and
+decompressed (decoded) into identical data in the case of lossless compression or nearly-identical data in the case
+of lossy compression. Through this procedure, a compression method reduces bits of the original data q(x, t) by
+eliminating statistical redundancies that may be contained within temporally adjacent frames and spatially adjacent
+pixels. In general, a data compression algorithm is referred to as an encoder φ while one that performs the decompression
+is called a decoder ψ,
+γ(x, t) = φ(q(x, t)),
+q(x, t) ≈ ˜q(x, t) = ψ(γ(x, t)),
+(1)
+where γ(x, t) is the compressed data corresponding to the original data q(x, t). Depending on the extent of compression,
+the data, and a choice of encoder/decoder, the reconstruction ˜q(x, t) generally includes some amount of error.
+The data compression process is illustrated in figure 1 for both image and video compressions. Figure 1(a) depicts a lossy
+spatial image compression technique, involving quantization of the image data in a compressed space and producing
+a reconstruction in the image space showing the operations of JPEG and JP2. Figure 1(b) provides a visualization
+of a spatio-temporal compression technique, exploiting a redundant block of a frame that remains consistent across
+subsequent frames, similar to H.264, H.265, and AV1. As these algorithms originated in the multimedia industry, they
+are optimized for human viewers and involve the removal of high-frequency components in the data and down-sampling
+of the color spectrum such that the eyes cannot easily distinguish compressed data from the original data. For the
+purposes of this study, we only consider grayscale images and videos, which are comprised only of a single-component
+field data matrix, denoted as ˜q(x). This is in contrast to full-color data, which requires red, green, and blue components,
+and is unnecessary for the current analysis as we are interested in considering field variables individually. Herein, we
+consider the application of five compression techniques on grayscale images and videos. The encoding schemes, which
+package the data into a compressed binary form, are detailed in what follows.
+2
+Compression techniques
+2.1
+Image Compression
+2.1.1
+JPEG
+Let us first describe JPEG, which is a standard lossy spatial compression used for encoding image data based on the
+discrete cosine transform (DCT). An example of a JPEG compression process with a vorticity field of two-dimensional
+decaying isotropic turbulence is presented in figure 2. The images are partitioned into 8 × 8 blocks in a left-to-right,
+top-to-bottom scan. Pixel values within blocks are quantized to values of [−128, 127] from [0, 255]. The forward DCT
+is individually performed at each block and outputs compressed data. The DCT for 8 × 8 blocks is mathematically
+expressed as
+F(kx, ky) = 1
+4C(kx)C(ky)
+�
+7
+�
+ix=0
+7
+�
+iy=0
+f(ix, iy) cos
+�(2ix + 1)kxπ
+16
+�
+cos
+�(2iy + 1)kyπ
+16
+��
+,
+(2)
+f(ix, iy) = 1
+4
+�
+7
+�
+kx=0
+7
+�
+ky=0
+C(kx)C(ky)F(kx, ky) cos
+�(2ix + 1)kxπ
+16
+�
+cos
+�(2iy + 1)kyπ
+16
+��
+,
+(3)
+where
+C(k) =
+�1/
+√
+2
+for k = 0
+1
+otherwise.
+(4)
+3
+
+A PREPRINT - JANUARY 3, 2023
+Input
+8×8 block
+DCT for 8×8 block
+Reconstructed image
+(Keeping 8.31% of the DCT coefficients)
+Figure 2: JPEG compression process with an example of two-dimensional isotropic turbulent vorticity.
+Here, F(kx, ky) denotes the DCT coefficient corresponding to the horizontal wavelength kx and vertical wavelength ky
+and f(ix, iy) describes the pixel value at the location corresponding to ix and iy. In other words, the forward DCT takes
+as input a discrete signal of 64 points and produces coefficients for a linear combination of 64 unique basis signals, each
+denoting a specific spatial wavelength. Most of the spatial domain information is concentrated across lower wavelength
+because of slow spatial variation from one pixel to the next in image data. This quality permits lossy quantization,
+which refers to constant values in a quantization table Q(kx, ky) with 64 elements. The DCT coefficient is normalized
+by a constant Q(kx, ky) in an element-wise manner,
+F Q(kx, ky) = ⌊F(kx, ky)
+Q(kx, ky)⌋
+(5)
+where F Q(kx, ky) is a normalized coefficient and the operation ⌊·⌋ denotes rounding to the nearest integer. Quantization
+tables are provided by the Joint Photographics Experts Group. Note that dividing the DCT coefficients by values in the
+quantization table reduces high-wavenumber coefficients to 0, which permits efficient entropy coding (explained later)
+to perform the cutoff at high frequencies [36]. The resulting quantized DCT coefficients form a matrix of size 8 × 8
+with low-wavenumber components generally located in the top-left of the matrix and high-wavenumber coefficients at
+the bottom-right, as a consequence of the similar distribution of spatial modes to which these coefficients correspond.
+Subsequently, quantized coefficients are ordered from low to high wavenumbers.
+To reduce the data size, entropy coding [37, 38], a lossless method of compressing bitstreams with redundancies, is then
+performed for the output of DCT. The idea of entropy coding is used not only for JPEG but also other image/video
+compression techniques such as JP2, H.2xx series, and AV1. To express the encoding-based data compression, let us
+consider a message of DAEBCBACBBBC (12 characters). Since this message includes five different characters, it
+needs to prepare 3 bits to convert these characters to bits or binary digits representation. Here, we use the following
+conversion table,
+A
+B
+C
+D
+E
+000
+001
+010
+011
+100
+With this table, the message is expressed as
+D
+A
+E
+B
+C
+B
+A
+C
+B
+B
+B
+C
+011
+000
+100
+001
+010
+001
+000
+010
+001
+001
+001
+010
+As shown, the number of bits is 36. The idea of the encoding-based compression is to prepare an adaptive conversion
+table assigning a shorter bit length for characters that appear in a high probability and a longer bit length for characters
+that barely appear. For example, the following adaptive table can be used:
+A
+B
+C
+D
+E
+110
+0
+10
+1110
+1111
+4
+
+A PREPRINT - JANUARY 3, 2023
+With this new table, the message can be expressed as
+D
+A
+E
+B
+C
+B
+A
+C
+B
+B
+B
+C
+1110
+110
+1111
+0
+10
+0
+110
+10
+0
+0
+0
+10
+The current table can save the total number of bits for the message from 36 to 25. Modern data compression techniques
+efficiently ��nd such an adaptive conversion table for saving image and video sizes.
+The presence of a better adaptive conversion can be proven with the source coding theorem [39]. For any data, the
+expected code length should satisfy the relationship,
+Eβ∼P [l(d(β))] ≥ Eβ∼P [− logb(P(β))],
+(6)
+where l is the number of symbols in a message, d is the coding function, b is the number of symbols in a table, and P is
+the probability of the original symbol. An entropy coding method attempts to approach the lower bound. For JPEG
+compression, Huffman coding [40] is used to determine an adaptable table composed of the estimated probability of
+occurrence for each possible value. Huffman coding uses binary trees [41] for efficient encoding.
+2.1.2
+JPEG2000 (JP2)
+JPEG2000 (JP2) is a successor to JPEG. JP2 operates using a similar four-step process to JPEG, involving image
+partitioning, frequency-domain transformation, quantization, and entropy coding. In contrast to JPEG, JP2 introduces
+more advanced dynamic tiling algorithms based on variable-sized macroblocks. This makes use of the discrete wavelet
+transform (DWT), and performs additional preprocessing prior to entropy coding. Tiling in this context refers to the
+partitioning of the source image into several non-overlapping rectangular blocks, each of which is processed distinctly.
+Whereas JPEG restricts tile sizes to 8 × 8, tiles in JP2 can be of arbitrary size up to the image dimensions. The DWT is
+applied to each tile in a manner similar to DCT, decomposing a signal into a linear combination of wavelet functions.
+The coefficients in this linear combination correspond to a specific wavelet basis function in the signal. A wavelet can
+be defined as a scale and shift of a basis wavelet. Child wavelets [42] are generally considered for DWT, given by
+ψg,r(s) =
+1
+2g/2 ψ
+�s − 2gr
+2g
+�
+(7)
+where g is a scaling factor, r is a shift factor, and s corresponds to the index of the one-dimensional representation of an
+image. In other words, the flow field snapshot is converted to a one-dimensional representation and the independent
+variable upon which this one-dimensional signal f(s). The DWT coefficient given a wavelet of the preceding definition
+is then
+Fg,r =
+� ∞
+−∞
+f(s)ψg,rds
+(8)
+where f(s) is a one-dimensional signal. The signal can be reconstructed through the summation of the product of each
+coefficient with the corresponding wavelet. In a discrete interpretation, this is written as
+f(s) =
+∞
+�
+g=−∞
+∞
+�
+r=−∞
+Fg,rψg,r.
+(9)
+The summation bounds for both the calculation of the coefficients and reconstruction of the signal can be set to finite
+values and can still produce lossless reconstructions assuming that the wavelets contain the maximum and minimum
+wavelengths within the source image.
+An example of the DWT for the vorticity field of two-dimensional decaying turbulence is presented in figure 3. The
+DWT can be applied recursively to one-dimensional signals to produce higher fidelity representations of data. As such,
+successive high-pass filters are applied on down-sampled images, producing a higher fidelity representation of spatial
+frequencies in the source image. The DWT can be extended to higher dimensions by applying the one-dimensional
+DWT on rows and columns. The recursive application of the DWT produces 2n distinct filtered images where n is the
+number of times the DWT is applied.
+Similar to JPEG compression, the DWT coefficients are quantized following the transformation on the wavespace,
+F Q
+a,b = sign(Fa,b)⌊|Fa,b|
+∆b
+⌋,
+(10)
+5
+
+A PREPRINT - JANUARY 3, 2023
+(b)
+(c)
+(d)
+(e)
+(g)
+(f)
+(a)
+0
+50
+100
+-50
+-100
+Figure 3: An example of the two-level discrete wavelet transform for two-dimensional homogeneous turbulent vorticity
+field, used in JP2 compression. High-pass filtering yields three large images. Low-pass filtering and downscaling are
+then performed, producing the three small images. (a) The final approximation image. (b, e) Vertical, (c, f) horizontal,
+and (d, g) diagonal coefficients of the second (b − d) and the first levels (e − g) are shown.
+where ∆b is defined as the quantization step. DWT coefficients within the range (−∆b, ∆b) are quantized to 0.
+Following quantization, the coefficients are processed in preparation for entropy coding. Arithmetic coding [43] is
+used for entropy coding in JPEG2000. While Huffman coding separates the original data into component symbols and
+replaces each with a code in a table, Arithmetic coding encodes the entire message into a single number represented
+with an arbitrary-precision fraction pa, where 0 ≤ pa < 1.
+2.2
+Video Compression
+2.2.1
+H.264 (AVC)
+The H.264 video compression includes a multi-step process, consisting generally of prediction, transformation (a set
+of frequency-domain representation and quantization in image compression), and entropy encoding on the encoder
+side. A similar process for file reconstruction is performed on the decoder side. Prediction in video compression
+amounts to an operation to remove redundancies in the given signal. H.264 supports a range of prediction options
+6
+
+500
+450
+400
+350
+300
+250
+200
+150
+100
+50
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200160
+140
+120
+100
+80
+60
+40
+20
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200160
+140
+120
+100
+80
+60
+40
+20
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200300
+250
+200
+150
+100
+50
+50
+100
+150
+200
+250
+300
+350
+400300
+250
+200
+150
+100
+50
+50
+100
+150
+200
+250
+300
+350
+400100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+Frame 1
+MC Frame 2
+Frame 2 MC Residual
+Figure 4: Motion compensation (MC), used in temporal compression of H.264, H.265, and AV1, is exemplified by the
+calculation of a motion vector field, describing the translation of pixels between successive frames. As an example, a
+streamwise velocity field u of three-dimensional turbulent channel flow is considered. Subtracting this field from the
+original image yields a residual image, which is stored.
+such as intra-prediction used for data within the current frame, inter-prediction for motion compensation, and multiple
+block-size-based predictions. An accurate prediction implies that the residual contains very little information, amounting
+to good compression performance.
+Video data is first partitioned into macroblocks of dimension 16 × 16 pixels. A prediction of the current macroblock
+is formed using 4 × 4 and 16 × 16-sized blocks in the case of intra-frame prediction, referring to predictions from
+surrounding blocks within the same frame. A range of block sizes from 4 × 4 to 16 × 16 are also considered in the
+case of inter-frame prediction, referring to predictions from previously coded frames. Macroblock prediction is further
+discretized into intra-prediction with neighboring blocks in the current frame, blocks in a previously coded frame, and
+blocks from up to two previously coded frames.
+In intra-prediction, the size of prediction macroblocks can be three cases: 16 × 16, 8 × 8, or 4 × 4 pixels [31]. The
+choice of block size is made primarily based on prediction efficiency. One of several prediction modes where each
+prediction mode indicates a direction in which to extrapolate pixel values or to average across all pixels. In the case of
+inter-prediction, the reference frame, where the prediction block is situated, is chosen from several previously decoded
+frames. As shown in figure 4, a motion vector is then obtained for the current macroblock, based on the offset from the
+prediction block or from previously coded motion vectors. These motion vectors are optionally weighted to account for
+temporal proximity between frames, and are sent into the data stream. Generally, a deblocking filter is further applied to
+each frame to store in a decoded format for subsequent inter-frame predictions in smoothing the sharp edges caused by
+the use of block coding [44].
+Transformation involves the conversion of blocks to frequency-domain representations and quantization of coefficients
+corresponding to high wavelength data. The DCT step is given by the transformation of block X by matrix A into DCT
+coefficients Y for each macroblock. For the case of 4 × 4 blocks, these matrices X, Y ∈ R4×4 are expressed as
+Y = AXAT =
+�
+��
+a
+a
+a
+a
+b
+c
+−c
+−b
+a
+−a
+−a
+a
+c
+−b
+b
+−c
+�
+�� X
+�
+��
+a
+b
+a
+c
+a
+c
+−a
+−b
+a
+−c
+−a
+b
+a
+−b
+a
+−c
+�
+�� ,
+(11)
+where a = 1/2, b =
+�
+1/2 cos(π/8), c =
+�
+1/2 cos(3π/8). The rows of A are orthonormal. To calculate equation 11
+on a practical processor, the approximation for b and c is required. This is achieved with a fixed-point approximation,
+which is equivalent to scaling each row of A by 2.5 and rounding to the nearest integer. A core transform Cf4, which
+scales each term of A by 2.5 and rounds to the nearest integer, and a scaling matrix Sf4, which restores norms of row of
+Cf4 to 1 by scaling, are respectively defined as,
+Cf4 =
+�
+��
+c1
+c1
+c1
+c1
+c2
+c1
+−c1
+−c2
+c1
+−c1
+−c1
+c1
+c1
+−c2
+c2
+−c1
+�
+�� ,
+Sf4 =
+�
+��
+s1
+s2
+s1
+s2
+s2
+s3
+s2
+s3
+s1
+s2
+s1
+s2
+s2
+s3
+s2
+s3
+�
+�� ,
+(12)
+where c1 = 1, c2 = 2, s1 = 1/4, s2 = 1/(2
+√
+10), and s3 = 1/10. Matrix Y is then determined as
+Y = [Cf4XCT
+f4]Sf4.
+(13)
+Similar DCT approximations are specified for other block sizes, involving Cf8, Sf8, and others [31]. A quantization
+mechanism similar to JPEG is applied, with a quantization table specified for various block sizes. The quantized DCT
+coefficients are then traversed in an oscillating, “zig-zag” manner from low- to high-wavenumber components. Entropy
+coding is finally applied to the output of DCT.
+7
+
+Frame 1
+MC Frame 2
+Frame 2 MC ResidualA PREPRINT - JANUARY 3, 2023
+2.2.2
+H.265 (HEVC)
+H.265 was released in 2013 as the successor to H.264. While the fundamental architecture is unchanged from H.264,
+H.265 makes use of coding tree units which are similar to macroblocks but expand the range of possible dimensions,
+with variable dimensions selected by the encoder, allowing coding tree units to be divided into sub-blocks. Predictions
+and reconstructions are performed on coding tree units and supported sub-block sizes range from 64 × 64 to 4 × 4
+pixels. Motion vectors are predicted based on those of adjacent units or blocks in the case of intra prediction, or previous
+encoded frames in the case of inter prediction. As in H.264, the DCT is performed at the coding tree block level, and
+the resultant coefficients are subjected to scalar quantization and entropy coding. Instead of deblocking filter in H.264,
+a sample adaptive offset filter is applied within the prediction loop to improve the quality of the compressed data [45].
+2.2.3
+AV1
+AV1 was released in 2018 in an effort to replace the H.2XX series of video compression algorithms. In AV1, much
+of the fundamental architecture from H.2XX is maintained. Frames are partitioned using a 4-way partition tree with
+dimensions ranging from 128 × 128 to 4 × 4. AV1 supports 56 directional spatial modes for intra-frame prediction with
+finer angle variations than that provided by H.2XX. AV1 extends the number of reference frames that any given frame
+can use to perform predictions to seven references for inter-frame prediction, thus enabling more accurate encoding
+of data with rich temporal characteristics. Motion vector field formation is improved by expanding the spatial search
+domain for vector candidates and through the utilization of a temporal motion field estimation system. AV1 also
+extends frequency-domain transform algorithms to include the asymmetric discrete sine transform with a richer set
+of transform kernels for varying block sizes. Entropy coding and scalar quantization are also used as well as H.2XX
+compression algorithms. To perform deblocking, a constrained directional enhancement filter and loop restoration
+filters are applied [33]. These filters are able to effectively remove artifacts without causing blurring, compared to
+conventional deblocking filters [46].
+3
+Flow Fields
+Let us apply the compression techniques presented above to representative fluid flow data sets from numerical
+simulations. We describe herein the problem setup of the example flow fields we analyze and the simulation approach
+used to generate them.
+3.1
+Two-dimensional laminar cylinder wake
+Bluff body flow forms a large class of problems, such as the vortex shedding around a cylinder. We first apply the
+compression techniques to the two-dimensional cylinder wake obtained by direct numerical simulation (DNS) [47, 48].
+The governing equations are the incompressible Navier–Stokes equations,
+∇ · u = 0,
+(14)
+∂u
+∂t + u · ∇u = −∇p +
+1
+ReD
+∇2u,
+(15)
+where u and p are the non-dimensionalized velocity vector and pressure, respectively.
+All variables are non-
+dimensionalized with the fluid density ρ, the uniform velocity U∞, and the cylinder diameter D. The Reynolds
+number is defined as ReD = U∞D/ν = 100 with ν being the kinematic viscosity. We consider five nested lev-
+els of multi-domains with the finest grid level covering (x, y) = [−1, 15] × [−8, 18] and the largest domain being
+(x, y) = [−5, 75] × [−40, 40]. The time step for DNS is ∆t = 2.50 × 10−3. We extract the domain around a cylinder
+body over (x∗, y∗)/D = [−0.7, 15] × [−5, 5] with (Nx, Ny) = (500, 300) and (∆x, ∆y) = (0.0314, 0.0333). The
+flow exhibits vortex shedding with a single period with 21 snapshots. For the compression analysis, 160 temporal
+snapshots of grayscale images of the streamwise velocity field u are considered.
+3.2
+Two-dimensional decaying homogeneous isotropic turbulence
+To examine the data compression performance by the present techniques for more complex turbulent flows, we also
+consider a two-dimensional decaying homogeneous isotropic turbulence. This time-varying flow can be regarded
+as a canonical fluid flow example for a broad range of turbulent flows. The data set is prepared by DNS using
+the two-dimensional vorticity transport equation [49]. We set the initial Reynolds number Re0 ≡ u∗l∗
+0/ν = 80.4,
+where u∗ is the characteristic velocity obtained by the square root of the spatially averaged initial kinetic energy,
+l∗
+0 = [2u2(t0)/ω2(t0)]1/2 is the initial integral length, and ν is the kinematic viscosity. The computational domain and
+8
+
+A PREPRINT - JANUARY 3, 2023
+the numbers of grid points are set to Lx = Ly = 1 and Nx = Ny = 128, respectively. We use 1000 snapshots in an
+eddy turn-overtime of t ∈ [2, 6] with a time interval of ∆t = 0.004. For the data compression analysis, 128 × 128 grid
+with grayscale contours of the vorticity field ω are used.
+3.3
+Turbulent channel flow
+To further demonstrate the present data compression techniques, we also examine turbulent channel flow at a friction
+Reynolds number of Reτ = uτδ/ν = 180, where uτ is the friction velocity, δ is the half-width of the channel, and ν
+is the kinematic viscosity. This flow involves a broader range of spatio-temporal flow scales and fewer redundancies
+compared to the previous two examples. The data sets are prepared by a three-dimensional DNS [50, 51], numerically
+solving the incompressible Navier–Stokes equations. The present DNS has been validated by comparison with spectral
+DNS data of Moser et al [52]. The streamwise, wall-normal, and spanwise spatial coordinates are denoted by x, y, and z,
+respectively. The size of the computational domain and the number of grid points here are (Lx, Ly, Lz) = (4πδ, 2δ, 2πδ)
+and (Nx, Ny, Nz) = (256, 96, 256), respectively. The grids in the streamwise and spanwise directions are taken to be
+uniform, while that in the y direction is a non-uniform grid. The no-slip boundary condition is imposed on the walls and
+a periodic boundary condition is applied to the x and z directions. The flow is driven by a constant pressure gradient. In
+what follows, we denote wall-unit quantities with the superscript +.
+For the present study, an x − z cross-sectional streamwise velocity u at y+ = 13.2 is analyzed, where representative
+streak structures are present [53]. Fifty temporal snapshots of a 256 × 256 spatial grid over t+ ∈ [0, 63] are formatted
+into grayscale data and are used for compression assessment.
+4
+Results
+For image compression, matrices of flow field data corresponding to specific temporal snapshots are represented as
+uncompressed grayscale images. These images are compressed using JPEG and JP2 encoders. For video compression,
+the matrices of flow field data are represented as grayscale images and concatenated into uncompressed, grayscale
+videos. This raw video file is used as the input to the H.264, H.265, and AV1 encoders. To obtain control over the
+outputted file size for the purposes of this analysis, a two-pass encoding scheme is considered. In the two-pass encoding,
+the encoder runs twice. The first run is used to collect some information and statistics such as how many bytes would
+be needed for data compression and the second run performs the actual encoding. These two processes enable the use
+of the information collected in the first run to achieve enhanced compression.
+This study uses FFmpeg [35], a free and open-source software consisting of various libraries for handling video, audio,
+and other multimedia files. These libraries can be easily used from the command line ffmpeg. Default encoding
+settings for the relevant FFMpeg library are used to maintain consistency across all tests.
+4.1
+Image compression
+To establish a baseline performance for comparison, individual flow snapshots are compressed using singular value
+decomposition (SVD). Individual snapshots are decomposed into left and right singular vector matrices U and V T , and
+a diagonal matrix Σ containing the singular values. Snapshots are reconstructed by retaining the r leading modes. Here,
+the compression ratio η in the SVD context is then defined as
+η = r(m + n + 1)
+mn
+,
+(16)
+where m and n are the snapshot dimensions in the horizontal and vertical directions and a value of η = 1 corresponds
+to the original, uncompressed snapshot. In the present study, the snapshot dimensions for each flow example are set
+as (m, n) = (500, 300) for cylinder wake, (128, 128) for two-dimensional decaying turbulence, and (256, 256) for
+turbulent channel flow, respectively.
+Let us compare the image compression techniques with the cylinder wake example. As for the data attribute, we use
+a streamwise velocity u. The compressed wake fields and the spatial absolute error distributions with η ≈ 0.05 are
+presented in figure 5. The L2 error norm of reconstruction ε = ||fRef − fComp||2/||fRef||2, where fRef and fComp
+are respectively the reference and compressed flow fields, is also shown underneath the decoded fields. The SVD
+produces negligible error for the entire flow field, although slight discontinuities are observed in the wake region. By
+comparison, JPEG compression introduces some compression artifacts including discontinuities of grayscale contours
+and granulated vortical structures. This is due to fixed-size areas upon which the DCT is performed and elementary
+anti-blocking features. This indicates that compression based on the fixed block of 8 × 8 pixels is not appropriate for
+flow fields that include fine-scale spatial variations. In contrast, the compressed flow field with the JP2 algorithm retains
+9
+
+A PREPRINT - JANUARY 3, 2023
+JPEG
+JPEG2000
+SVD
+{ε, η} = {0.0176, 0.0548}
+{ε, η} = {0.00370, 0.0507}
+{ε, η} = {0.00670, 0.0532}
+Compressed flow field
+Spatial error distribution
+0
+50
+100
+-50
+-100
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+Figure 5: Comparison of image compression techniques for cylinder wake at ReD = 100. A streamwise velocity field
+u is considered. The L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow
+field contour. Spatial absolute error distribution for each compression technique is also presented.
+JPEG
+JPEG2000
+SVD
+{ε, η} = {0.0350, 0.282}
+{ε, η} = {0.0233, 0.270}
+{ε, η} = {0.00820, 0.282}
+0
+50
+100
+-50
+-100
+Figure 6: Comparison of image compression techniques for two-dimensional decaying turbulence. A vorticity field ω is
+considered. The L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow
+field contour.
+wake features even while achieving significant data compression, with the L2 error of 0.00370. These results support
+the effectiveness of the adaptive block size of DWT in JP2 compression for bluff body wake data sets.
+Next, we apply the image compression techniques to two-dimensional decaying homogeneous isotropic turbulence,
+as shown in figure 6. We use a vorticity field ω as a quantity of interest and compare the compression results with
+η ≈ 0.280. Similar to the cylinder example, the SVD can provide a smooth field and small error for the entire flow field.
+Although SVD can achieve a reasonable compression for the laminar cylinder wake and two-dimensional turbulence
+that are mainly composed of large vortical structures, we discuss later how the presence of fine-scale turbulent structures
+alters the compression performance. For this two-dimensional turbulence, the effect of 8 × 8 pixel blocks can be clearly
+observed in JPEG compression. Such pixelized artifacts on the flow field can be mitigated by using the JP2 compression
+technique, analogous to the observation with the cylinder example.
+The limitation of the SVD and the efficacy of the DWT-based process in the JP2 algorithm are further emphasized in
+the example of more complex turbulence. Here, the compression techniques are applied to a streamwise velocity u
+10
+
+JPEG
+JPEG2000
+SVD
+(8, n) = {0.0176, 0.0548)
+(8, n) = {0.00370, 0.0507
+(s, n) = {0.00670, 0.0532)
+(8, n) = (0.00490, 0.136)
+(8, n) = (0.00310, 0.120)
+(8, n) = (0.00180, 0.112)100
+ 50
+0
+-50
+-100JPEG
+JPEG2000
+SVD
+ = 0.0350
+8= 0.0233
+8 = 0.00820
+n = 0.282
+n = 0.270
+n = 0.282
+8 = 0.0168
+8 = 0.0134
+8= 0.00310
+n = 0.511
+n = 0.452
+n = 0.502100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0265, 0.234}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+{ε, η} = {0.0122, 0.240}
+{ε, η} = {0.0258 0.235}
+JPEG
+SVD
+JPEG2000
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0265, 0.234}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+{ε, η} = {0.0122, 0.240}
+{ε, η} = {0.0258 0.235}
+JPEG
+SVD
+JPEG2000
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0265, 0.234}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+{ε, η} = {0.0122, 0.240}
+{ε, η} = {0.0258 0.235}
+JPEG
+SVD
+JPEG2000
+JPEG
+JPEG2000
+SVD
+{ε, η} = {0.129, 0.0281}
+{ε, η} = {0.0903, 0.0234}
+{ε, η} = {0.235, 0.0235}
+0
+50
+100
+-50
+-100
+Figure 7: Comparison of image compression techniques for turbulent channel flow at Reτ = 180. A streamwise
+velocity field u is considered. The L2 error norm of the reconstruction ε and the compression ratio η are shown
+underneath each flow field contour.
+(a)
+(b)
+: JPEG
+: JP2
+: SVD
+: Cylinder
+: 2D turbulence
+: Channel
+SVD
+JPEG
+JP2
+SVD
+JP2
+JPEG
+SVD
+JPEG
+JP2
+(c)
+(d)
+(e)
+Figure 8: Relationship between (a) the L2 error norm ε, (b) SSIM, and image compression ratio η. Zoom-in view of
+η-SSIM curve for (c) cylinder wake, (d) two-dimensional turbulence, and (e) turbulent channel flow.
+of the three-dimensional channel flow. The compression results with η ≈ 0.025 are compared in figure 7. As shown,
+the SVD-based compression cannot retain the important features of the streaks. Compared to SVD, JPEG provides a
+better reconstruction although it also introduces discontinuities that obscure small spatial length scales in the flow field.
+Surprisingly, JP2 produces non-negligible artifacts and maintains an L2 error norm less than half that of SVD. The
+channel flow field at this low η remains nearly indistinguishable from the uncompressed flow field, also preserving the
+streak spacing of the reference DNS field [53, 54]. These observations suggest the effectiveness of the JP2 algorithm
+for image compression of complex fluid flow data.
+Building on these assessments, the L2 error between compressed and uncompressed flow fields is evaluated across
+compression ratios, as shown in figure 8(a). The error is averaged over all temporal snapshots of each flow example. In
+general, all compression algorithms produce an asymptotically decaying L2 error. JPEG introduces appreciable error at
+low η, in the same order as SVD compression. It is worth pointing out that JP2 performs especially well at low η while
+SVD compression produces the lowest L2 error for high η for all flow fields.
+11
+
+100
+ 50
+0
+-50
+-100100
+10
+-2
+10
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.00
+0.75
+SSIM
+0.50
+L
+0.25
+0.00
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.005
+1.000
+EE
+0.995
+SSIM
+0.990
+0.985
+0.980
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.000
+0.975
+SSIM
+0.950
+0.925
+0.900
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.00
+SSIM
+0.99
+0.98
+0.97
+0.0
+0.2
+0.4
+0.6
+0.8
+nA PREPRINT - JANUARY 3, 2023
+(a)
+(d)
+(b)
+(e)
+(c)
+(f)
+ηLow = 0.0299
+ηMed = 0.0913
+ηHigh = 0.237
+ηLow = 0.295
+ηMed = 0.539
+ηHigh = 0.885
+ηLow = 0.215
+ηMed = 0.505
+ηHigh = 0.932
+ηLow = 0.0299
+ηMed = 0.0913
+ηHigh = 0.237
+ηLow = 0.0381
+ηMed = 0.0880
+ηHigh = 0.803
+ηLow = 0.0381
+ηMed = 0.0880
+ηHigh = 0.803
+Figure 9: Kinetic energy spectra for two-dimensional decaying homogeneous isotropic turbulence using (a) JPEG and
+(d) JP2. (b, e) Streamwise and (c, f) spanwise kinetic energy spectrum of three-dimensional turbulent channel flow
+compressed with (b, c) JPEG and (e, f) JP2.
+As an additional metric for quantifying the error introduced by each compression method, the localized structural
+similarity index (SSIM) [55] is computed between compressed and uncompressed flow fields. SSIM can capture spatial
+correlation around pixels and is less sensitive against a pixel-wise error caused by translation and rotational difference
+compared to the L2 error. Hence, SSIM is suited for the image and video-based compression analysis. The SSIM χ is
+defined as
+χ = l(ix, iy)c(ix, iy)s(ix, iy)
+(17)
+where
+l(ix, iy) = 2µxµy + C1
+µ2x + µ2y + C1
+,
+c(ix, iy) = 2σxσy + C2
+σ2x + σ2y + C2
+,
+s(ix, iy) = σxy + C3
+σxσy + C3
+(18)
+with µx and σx defined as the mean and standard deviation of ix respectively, σxy being the covariance of ix and iy,
+and c1, c2, and c3 being constants to stabilize division. We set {C1, C2, C3} = {0.16, 1.44, 0.72} following Wang et
+al. [55]. The resultant value lies between 0, representing no similarity, and 1, representing an identical image. The
+relationship between the image compression ratio and the L2 error is depicted in figure 8(b) Generally, JP2 and SVD
+produce a negligible decrease in the SSIM at low compression ratios and asymptotically approach an SSIM value of 1
+at higher compression ratios. The SSIM value of the cylinder flow field with JPEG compression applied decays by
+approximately 10%, as a result of significant discontinuities produced by JPEG.
+We also present the zoom-in view of the relationship between SSIM and the compression ratio η for each flow, in
+figures 8(c) − (e). Similar to the observation in the η − ε curves in figure 8(a), SVD and JP2 provide high SSIM scores
+compared to JPEG. Especially at excessive compression (low η) of three-dimensional turbulent channel flow, JP2 can
+provide better reconstructions than the other two cases. Although scalar metrics such as the L2 error ε and SSIM are
+useful, we note that monitoring not only scalar values but also decoded flow fields is important in assessing how vortical
+structures can be retained through data compression because the influence of local structures are averaged.
+We are additionally interested in whether finer structures in flow images can still be retained through the present
+compression process. To examine this aspect, we consider the kinetic energy spectrum of both two- and three-
+dimensional turbulence examples, as summarized in figure 9. The kinetic energy spectrum E(k) for two-dimensional
+decaying turbulence is
+E(k) = 1
+2(uiui),
+(19)
+12
+
+Uncompressed
+JPEG LoW
+JPEG Medium
+JPEG High
+10~5
+E
+10-10
+101
+102
+103
+k-Uncompressed
+JP2 LoW
+JP2 Medium
+JP2 High
+10-5
+E
+10-10
+101
+102
+103
+k100
+10-4
+10-6
+10-1
+10-2100
+10-4
+10-6
+10~2
+10-1100
+10-4
+10-6
+10-1
+10~2100
+10-4
+10-6
+10-1
+102A PREPRINT - JANUARY 3, 2023
+{ε, η} = {0.0411, 0.0380}
+{ε, η} = {0.00470, 0.148}
+H.265
+{ε, η} = {0.0132, 0.0196}
+{ε, η} = {0.00450, 0.142}
+AV1
+{ε, η} = {0.0171, 0.0309}
+POD
+{ε, η} = {0.00200, 0.130}
+{ε, η} = {0.0578, 0.0300}
+{ε, η} = {0.00770, 0.157}
+H.264
+{ε, η} = {0.0411, 0.0380}
+{ε, η} = {0.00470, 0.148}
+H.265
+{ε, η} = {0.0132, 0.0196}
+{ε, η} = {0.00450, 0.142}
+AV1
+{ε, η} = {0.0171, 0.0309}
+POD
+{ε, η} = {0.00200, 0.130}
+{ε, η} = {0.0578, 0.0300}
+{ε, η} = {0.00770, 0.157}
+H.264
+0
+50
+100
+-50
+-100
+H.264
+{ε, η} = {0.0578, 0.0300}
+H.265
+{ε, η} = {0.0411, 0.0380}
+{ε, η} = {0.0132, 0.0196}
+{ε, η} = {0.0171, 0.0309}
+AV1
+POD
+Figure 10: Comparison of video compression techniques applied on a streamwise velocity field u of cylinder wake
+at ReD = 100, compressed using H.264, H.265, AV1, and POD compression algorithms. The L2 error norm of the
+reconstruction ε and the compression ratio η are shown underneath each flow field contour.
+where ui are the components of the fluctuating velocity and the overbar denotes an averaging operation in space and
+time. For three-dimensional turbulent channel flow, the one-dimensional streamwise and spanwise spectra is evaluated
+Euu(k+
+x ; y+) = ˆu∗ˆu
+z,t, Euu(k+
+z ; y+) = ˆu∗ˆu
+x,t,
+(20)
+where (·)∗ represents the complex conjugate and ˆ(·) denotes the one-dimensional Fourier transformed variable. Here,
+we compare three compression ratios, denoted as low, medium, and high, for each turbulent flow.
+JP2 demonstrates a strong adherence to the kinetic energy spectrum of the uncompressed flow field in both the x
+and z directions while JPEG compression at low η introduces non-negligible errors at higher wave numbers. This is
+a consequence of the quantization step of JPEG compression that removes high wavelength scales from the image.
+Considering the overestimation of Euu(k+
+x ) as seen in figure 9 when using JPEG, this is likely caused by the absence
+of a deblocking filter, producing more high wavelength artifacts in the image than what exists in the uncompressed
+data. Similarly, the underestimation of E(k) by JP2 can be attributed to adaptive block sizes that produce a lower
+peak signal-to-noise ratio, indicative of lower quality. In general, JP2 is more adept at preserving high-wavenumber
+structures.
+4.2
+Video compression
+From the perspective of information, fluid flows are inherently temporally-redundant — as such, video compression
+algorithms that perform temporal compression are a powerful tool, achieving compression performance that outperforms
+the previously analyzed image-based techniques. This section assesses the capabilities of video compression techniques
+such as H.264, H.265, and AV1 compression algorithms for time-varying fluid flow data. Additionally, proper orthogonal
+decomposition (POD) compression [56] is considered to compare this familiar method of compression within the
+fluid dynamics community with those analyzed herein [3, 57]. POD is used to decompose a matrix of vectorized,
+temporally evolving flow field data into a set of basis modes and eigenvalues that contain coherent flow structures
+and can be used for flow field reconstruction. Formally, a flow field q(x, t) − q(x) can be represented as �n
+j=1 ajφj
+where aj is the temporal coefficient for mode φj. The value of aj is the inner product between the mode φj and the
+mean-subtracted flow field, q(x, t) − q(x). This modal representation can be truncated to r modes, such that the flow
+field is approximated by �r
+j=1 ajφj. This study uses the snapshot POD method [58] for comparison to the other video
+13
+
+100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+H.264
+ε = 0.0377
+η =0.0256
+ε = 0.00660
+η =0.222
+ε = 0.0227
+η =0.0324
+ε = 0.00580
+η = 0.230
+ε = 0.0175
+η = 0.0207
+ε = 0.00480
+η = 0.206
+ε = 0.0170
+η = 0.0209
+ε = 0.00300
+η = 0.201
+H.265
+AV1
+POD
+0
+50
+100
+-50
+-100
+H.264
+{ε, η} = {0.0377, 0.0256}
+H.265
+{ε, η} = {0.0227, 0.0324}
+{ε, η} = {0.0175, 0.0207}
+{ε, η} = {0.0170, 0.0209}
+AV1
+POD
+Figure 11: Comparison of video compression techniques applied on two-dimensional isotropic turbulent vorticity field,
+compressed using H.264, H.265, AV1, and POD compression algorithms. The L2 error norm of the reconstruction ε
+and the compression ratio η are shown underneath each flow field contour.
+{ε, η} = {0.231, 0.147}
+{ε, η} = {0.0890, 0.403}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.0588, 0.367}
+{ε, η} = {0.0502, 0.385}
+{ε, η} = {0.255, 0.157}
+{ε, η} = {0.139, 0.412}
+H.264
+H.265
+AV1
+POD
+{ε, η} = {0.231, 0.147}
+{ε, η} = {0.0890, 0.403}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.0588, 0.367}
+{ε, η} = {0.0502, 0.385}
+{ε, η} = {0.255, 0.157}
+{ε, η} = {0.139, 0.412}
+H.264
+H.265
+AV1
+POD
+0
+50
+100
+-50
+-100
+{ε, η} = {0.231, 0.147}
+{ε, η} = {0.0890, 0.403}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.0588, 0.367}
+{ε, η} = {0.0502, 0.385}
+{ε, η} = {0.255, 0.157}
+{ε, η} = {0.139, 0.412}
+H.264
+H.265
+AV1
+POD
+H.264
+{ε, η} = {0.231, 0.147}
+H.265
+{ε, η} = {0.0832, 0.185}
+{ε, η} = {0.151, 0.135}
+{ε, η} = {0.255, 0.157}
+AV1
+POD
+Figure 12: Comparison of video compression techniques applied on a streamwise velocity field u of three-dimensional
+turbulent channel flow at Reτ = 180, compressed using H.264, H.265, AV1, and POD compression algorithms. The
+L2 error norm of the reconstruction ε and the compression ratio η are shown underneath each flow field contour.
+compression techniques. The compression ratio for a reconstructed flow field containing r modes is evaluated as
+η = r(m + n) + m
+n(m + n) + m,
+(21)
+where m is the total number of pixels in the flow field and n is the total number of flow snapshots.
+The results of video compression for laminar cylinder wake at ReD = 100 are shown in figure 10. Here, we compare
+the decoded streamwise velocity field u with η ≈ 0.02 − 0.03. POD compression introduces negligible error, likely
+as a result of the temporally redundant nature of periodic wake and the larger coherent modal structures that POD
+is able to extract. By comparison, H.264 compression produces significant artifacts at low η. H.264 struggles likely
+because inter-frame prediction candidates are chosen from a shallow time range. We also observe that H.265 fails to
+improve in terms of error level over H.264 for the cylinder wake. This is due to the employment of a similar inter-frame
+prediction and selection algorithm to that of H.264. Compared to these H.2XX series, the AV1 algorithm provides much
+14
+
+100
+ 50
+0
+-50
+-100100
+ 50
+0
+-50
+-100A PREPRINT - JANUARY 3, 2023
+(a)
+(b)
+: H264
+: H265
+: POD
+: Cylinder
+: 2D turb.
+: Channel
+POD
+AV1
+(c)
+(d)
+: AV1
+H265 H264
+POD
+AV1
+H265
+H264
+Figure 13: Relationship between (a) the L2 error norm ε, (b) SSIM, and video compression ratio η. Zoom-in view of
+η-SSIM curve for (c) cylinder wake and (d) two-dimensional isotropic turbulence.
+better compression, achieving a lower L2 error than that achieved by H.264 and H.265. This highlights the enhanced
+capability of AV1 to compress laminar and temporally redundant flow fields.
+We next examine the video compression techniques for two-dimensional decaying homogeneous isotropic turbulence,
+as summarized in figure 11. The flow fields are compared for the compression ratios of η ≈ 0.02 − 0.03. Similar to the
+cylinder case, POD compression provides a reasonable reconstruction, likely because large-scale vortical structures
+are dominant at this particular time. It is, however, easily anticipated that the error of this time-varying flow relies
+on the presence of a range of length scales, as the small length scales disappear with the progress of the decay over
+time [59, 60]. The dependence of the compression performance over time for decaying flow will be examined later.
+While H.2xx compression techniques provide a reasonable reconstruction, AV1 provides better compression without
+suffering from pixelized artifacts. These results suggest the powerful capabilities of novel deblocking filters for fluid
+flow applications.
+The video compression techniques are also applied to the x−z sectional streamwise velocity field u of three-dimensional
+turbulent channel flow at Reτ = 180, as depicted in figure 12. The compressed flow fields are compared for η ≈ 0.150.
+In contrast to the other flow examples, POD compression produces significant visible artifacts and a high error value
+for turbulent channel flow because of a complex temporal evolution of the flow field. POD requires a greater number
+of modes for adequate reconstruction [61, 62, 63]. Although H.265 improves over H.264 significantly for turbulent
+channel flow, this still produces few observable discontinuities. This is likely caused by adaptive tiling in macroblocks
+for prediction procedures, allowing lower η with similar flow field representation. AV1 exceeds the performance of
+H.264 and H.265 consistently and POD compression on turbulent channel flow. Flow fields compressed using AV1 are
+indistinguishable from uncompressed flow fields at high η.
+The L2 error and SSIM are evaluated across a range of compression ratios for each type of flow field, as presented in
+figure 13. In general, all video compression algorithms produce asymptotically decaying L2 error values with increasing
+η. AV1 performs well at low η, especially for the cylinder wake. Additionally, all compression algorithms perform
+well for two-dimensional turbulence, likely as a result of the flow field snapshots holding slow changes from one
+frame to the next due to the decaying nature of the flow. Moreover, as observed with samples at various compression
+ratios, the L2 error for all algorithms plateau at non-negligible values for the cylinder wake flow field. SSIM values
+15
+
+n
+10
+-2
+10
+3
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.0
+0.8
+4
+SSIM
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.000
+SSIM
+0.995
+0.990
+0.985
+0.0
+0.2
+0.4
+0.6
+0.8
+n1.000
+SSIM
+0.995
+0.990
+0.985
+0.0
+0.2
+0.4
+0.6
+0.8
+nA PREPRINT - JANUARY 3, 2023
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(a)
+(b)
+(c)
+Figure 14: L2 error norm ε of vorticity field ω for two-dimensional decaying homogeneous isotropic turbulence over
+time. (a) H.264, (b) H.265, and (c) AV1. (i) and (ii) in each case are chosen due to their employment in inter-frame
+prediction in each algorithm.
+generally diverge from the asymptotic limit at low η. The exceptional cases include cylinder wake and two-dimensional
+turbulence compressed using AV1, which introduces negligible error at low η. AV1 outperforms H.264 and H.265 on
+turbulent channel flow as well, due to the improved blocking techniques.
+In addition, the time evolution of the L2 error is examined to gain insight into the performance of video compression
+algorithms for individual snapshots. The temporal evolution of the L2 error norm ε for two-dimensional decaying
+turbulence is shown in figure 14. H.2xx compression techniques exhibit repeated temporal structures in its L2 error
+evolution, likely as a consequence of inter-frame prediction selecting frames to make predictions from at relatively
+similar intervals. AV1 compression provides a distinctive reduction in the L2 error over time for medium and low η,
+indicating improved accuracy as snapshots begin to show redundancies due to the vortex field decaying and exhibiting
+similar large-scale coherent structures from one snapshot to the next. We also observe that H.264 produces a high error
+at low η for early flow field snapshots. This relates to the time-varying flow nature of the present decaying turbulence,
+as mentioned above. The presence of finer structures at the high Taylor Reynolds number Reλ(t) portion of the flow
+likely causes the difficulty in compressing vortical flow data.
+We also examine the L2 error norm ε and the flow fields over time for turbulent channel flow, as depicted in figure 15.
+H.264 generally produces a larger L2 error compared to the other techniques, as we also observed with the visual
+assessments in figure 12. With low η of H.264 compression, the error decreases over time, likely as a result of a later
+snapshot being selected for inter-frame prediction. Compared to H.264, H.265 provides better compression over time.
+Similar to the observation with H.264, the L2 error significantly varies over time at a low η. This is likely due to the
+inter-frame selection of an early frame from which further predictions were made. AV1 produces a negligible error at a
+high η while the errors increase as η decreases.
+We are also interested in the performance of video compression algorithms in preserving high wavenumber structures in
+the compressed state. The general performance of each compression algorithm with regard to kinetic energy spectra of
+each flow field is investigated, as shown in figure 16. H.264 performs well for two-dimensional turbulence, but produces
+a noticeable error at all η in both the stream- and spanwise directions of the kinetic energy spectrum of turbulent channel
+flow. A similar divergence from the expected data can be observed at low η in the spanwise direction as well. H.265
+performs comparatively well for two-dimensional decaying isotropic turbulence, and for turbulent channel flow in the
+spanwise direction. However, it produces a non-negligible error at high wavenumbers when compressed at low η in the
+spanwise direction. This indicates an over-representation of high-wavenumber components due to blocking as a result
+of the adaptive subblock sizes of H.265. Generally, AV1 is the best-suited algorithm for preserving spatial frequency
+information, particularly at high wavenumbers, for both two and three-dimensional turbulent flow fields. At higher η,
+the energy contents at each wavenumber are almost indistinguishable.
+16
+
+0.012
+AV1 Low
+AV1 Medium
+Error
+0.01
+AV1 High
+0.008
+0.006
+0.004
+0.002
+0
+2
+3
+4
+5
+6
+t10
+0
+10
+w
+2
+10
+10
+3
+0.0
+0.2
+0.4
+0.6
+0.8
+n0.015
+H265 L0W
+^H265 Medium
+H265 High
+0.01
+0.005
+2
+3
+4
+5
+6
+t0.025
+H264 L0w
+_H264 Medium
+0.02
+ H264 High
+0.015
+0.01
+0.005
+2
+3
+4
+5
+6
+tA PREPRINT - JANUARY 3, 2023
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(b)
+(c)
+(a)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(i)
+(ii)
+(a)
+(b)
+(c)
+Figure 15: L2 error norm ε of streamwise velocity field u for turbulent channel flow over time. (a) H.264, (b) H.265,
+and (c) AV1. (i) and (ii) in each case are chosen due to their employment in inter-frame prediction in each algorithm.
+At last, we investigate whether the video compression techniques can preserve the temporal evolution of complex
+turbulent flows. Here, let us examine the temporal two-point correlation for three-dimensional channel flow compressed
+using all three video compression algorithms. The temporal two-point correlation coefficient at a given t+ is defined
+as R+
+uu(t+)/R+
+uu(0) [64, 65] and is depicted in figure 17. The assessment of temporal two-point correlation provides
+insight into the relations of flow snapshots to preceding snapshots.
+Consistent with the insights gained from the kinetic energy spectrum, H.264 compression at a η exhibits disagreement
+with the reference curve at t+ values between 5 and 30, and above 50. This is indicative of a de-correlation of the
+velocity field and is likely a result of poor performance in capturing high-wavenumber information. Except for this
+particular case, all compression algorithms generally perform well, with temporal two-point correlation coefficients
+closely following that of the uncompressed flow field. These results suggest that these novel video compression
+techniques capture the spatio-temporal redundancies well even for complex turbulent flows and also significantly reduce
+data size while preserving their physics.
+5
+Conclusion
+We compressed flow field data from canonical flow examples using a number of widely-available multimedia com-
+pression techniques. The performance of the JPEG and JP2 spatial image compression techniques and the H.264,
+H.265, and AV1 spatio-temporal video compression techniques were considered for simulated laminar cylinder flow,
+decaying isotropic turbulence, and turbulent channel flow. Streamwise velocity and vorticity field data were represented
+as grayscale images and videos, and were compressed using the aforementioned techniques.
+All techniques, with the exception of JPEG, were shown to compress flow data below 10% of the original file size while
+introducing negligible error and preserving underlying flow physics. AV1 and H.265 compression were shown to have
+the best performance across a variety of flow regimes. The spatial error distributions were concentrated on the cylinder
+surface and directly behind the cylinder for the streamwise velocity data compression and in the vortex shedding wake
+for the vorticity data. Turbulence statistics in the form of kinetic energy spectra were preserved under compression for
+all methods except JPEG.
+17
+
+0.14
+0.12
+0.1
+0.08
+0.06
+0.04
+0.02
+20
+40
+0
+6010
+0
+10
+w
+2
+10
+10
+3
+0.0
+0.2
+0.4
+0.6
+0.8
+n0.4
+0.3
+0.2
+20
+40
+601.2
+0.8
+0.6
+0.4
+0.2
+20
+40
+60A PREPRINT - JANUARY 3, 2023
+(d)
+(g)
+(e)
+(h)
+(f)
+(i)
+(a)
+(b)
+(c)
+ηLow = 0.0640
+ηMed = 0.585
+ηHigh = 0.880
+ηLow = 0.0724
+ηMed = 0.595
+ηHigh = 0.976
+ηLow = 0.0587
+ηMed = 0.562
+ηHigh = 0.898
+ηLow = 0.119
+ηMed = 0.367
+ηHigh = 0.776
+ηLow = 0.0872
+ηMed = 0.403
+ηHigh = 0.817
+ηLow = 0.0872
+ηMed = 0.403
+ηHigh = 0.817
+ηLow = 0.119
+ηMed = 0.367
+ηHigh = 0.776
+ηLow = 0.187
+ηMed = 0.385
+ηHigh = 0.767
+ηLow = 0.187
+ηMed = 0.385
+ηHigh = 0.767
+Figure 16: Kinetic energy spectra E(k) for (a, d, g) two-dimensional decaying homogeneous isotropic turbulence
+using H.264, H.265, and AV1. (b, e, h) Streamwise Euu(k+
+x ) and (c, f, i) spanwise kinetic energy spectra Euu(k+
+z ) of
+three-dimensional turbulent channel flow.
+For single snapshots of data represented as an image, JP2 compression was shown to far outperform JPEG compression,
+with a tolerable increase in computational complexity. For multiple temporal snapshots of data represented as a
+video, the choice of compression method becomes more nuanced. JP2 compression was shown to achieve the lowest
+compression error as temporal compression adds slight error to the data. The AV1 algorithm maximizes η at the expense
+of computational complexity and non-negligible encoding time. This algorithm is new and emerging from the research
+environment, so future optimizations could bring this encoding time to a manageable level. The H.265 algorithm
+provided excellent compression performance at a fast encoding time, and appears as a promising algorithm for current
+fluid dynamics applications. H.264 provided acceptable compression performance, but was largely triumphed by the
+AV1 and H.265 algorithms.
+We have shown that modern multimedia compression algorithms provide robust performance in a variety of fluid flow
+applications. The implementation of these techniques becomes especially pertinent as simulations within computational
+fluid dynamics become exceedingly data-intensive, a trend that decreases the accessibility to high-fidelity models. These
+methods are free, easily accessible, regularly updated and supported, and provide flexible and scalable compression
+performance. As such, the implementation of these compression techniques has exciting potential across the fluid
+dynamics community for data storage and transfer with minimal loss.
+18
+
+-Uncompressed
+1H265 L0W
+^H265 Medium
+H265 High
+10~5
+10-10
+101
+102
+103
+k-Uncompressed
+AV1 Low
+AV1 Medium
+·AV1 High
+10~5
+10-10
+101
+102
+103
+k100
+10-4
+10-2
+10-1100
+10-4
+10-2
+10-1
+Y100
+10-4
+10-2
+10-1100
+10-4
+10-2
+10-1-Uncompressed
+H264 LoW
+H264 Medium
+H264 High
+10~5
+10-10
+101
+102
+103
+k100
+10-4
+10-6
+10-2
+10-1100
+10
+10-4
+10-6
+10-2
+10-1A PREPRINT - JANUARY 3, 2023
+(a)
+(b)
+(c)
+ηLow = 0.187
+ηMed = 0.385
+ηHigh = 0.767
+ηLow = 0.0872
+ηMed = 0.403
+ηHigh = 0.817
+ηLow = 0.119
+ηMed = 0.367
+ηHigh = 0.776
+Figure 17: Normalized temporal two-point correlation coefficients Ruu(t+)/Ruu(0) for three-dimensional turbulent
+channel flow using H.264, H.265, and AV1.
+JPEG
+JP2
+H.264
+H.265
+AV1
+u, Cylinder Flow
+0.58
+2.27
+0.94
+2.85
+61.21
+ω, Cylinder Flow
+0.59
+1.53
+0.92
+2.39
+43.83
+u, Channel Flow
+0.29
+1.42
+0.45
+1.96
+49.59
+Table 1: Encoding time (s) for different compression algorithms and flow regimes, compressed at 100 KB/s bitrate.
+Acknowledgements
+KT acknowledges the support from the US Army Research Office (W911NF-21-1-0060), the US Air Force Office
+of Scientific Research (FA9550-21-1-0178), and the US Department of Defense Vannevar Bush Faculty Fellowship
+(N00014-22-1-2798). We also thank Professor Koji Fukagata (Keio University) for sharing his DNS code.
+Appendix: Encoding time
+The increased performance of new compression algorithms comes at a cost; non-negligible increases in computational
+complexity should be considered when implementing these algorithms. In fact, in a paper from 2000 on compressing
+three-dimensional flows with the JPEG and JP2 algorithms [66], the added complexity of the JP2 algorithm caused
+JPEG to be recommended over JP2, despite losing clear performance benefits. The recommendation of the present
+study reverses that statement. As such, it is important to quantify the encoding time of these algorithms at the time of
+writing this study.
+The decoding time is observed to be negligibly small for all compression codecs; thus, this appendix focuses on
+encoding. The streamwise velocity and vorticity data are encoded for both the laminar cylinder flow and turbulent
+channel flow cases at the same bitrate (100 KB/s) for all compression algorithms and the encoding time is measured.
+The encoding is performed with a 2.5GHz i7 Intel Core processor and 8 GB RAM. The results are summarized in
+table 1. Encoding time per frame is observed to be larger for the turbulent channel flow than the laminar cylinder flow,
+indicating that the algorithms struggle to encode multiscale turbulent flow data. Across encoding algorithms, JPEG and
+H.264 compression are the fastest, a testament to the maturity and low complexity of these methods. JP2 and H.265
+encoding are generally several times slower, but still relatively fast, justifying their added compression performance.
+AV1 is observed to be far slower in encoding than the other methods: over 100 times slower than JPEG and H.264, and
+over 25 times slower than JP2 and H.265. This severe encoding time increase limits the practicality of implementing this
+algorithm in large-scale applications, and perhaps justifies the use of H.265 over AV1. As the algorithm was released
+only a few years prior to the writing of this paper, advances in computing and algorithm development could increase its
+practicality in the near future.
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+

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+1 
+ 
+SFI-Swin: Symmetric Face Inpainting with Swin 
+Transformer by Distinctly Learning Face Components 
+Distributions 
+ 
+MohammadReza Naderi1*, MohammadHossein Givkashi1*, Nader Karimi1, Shahram Shirani2, Shadrokh Samavi1,2,3 
+1Department of Electrical and Computer Engineering, Isfahan University of Technology, 84156-83111, Iran 
+2Department of Electrical and Computer Engineering, McMaster University, L8S 4L8, Canada 
+3Computer Science Department, Seattle University, Seattle 98122 USA  
+Abstract 
+Image inpainting consists of filling holes or missing parts of an image. Inpainting face images with symmetric 
+characteristics is more challenging than inpainting a natural scene. None of the powerful existing models can fill 
+out the missing parts of an image while considering the symmetry and homogeneity of the picture. Moreover, the 
+metrics that assess a repaired face image quality cannot measure the preservation of symmetry between the rebuilt 
+and existing parts of a face. In this paper, we intend to solve the symmetry problem in the face inpainting task by 
+using multiple discriminators that check each face organ's reality separately and a transformer-based network. 
+We also propose "symmetry concentration score" as a new metric for measuring the symmetry of a repaired face 
+image. The quantitative and qualitative results show the superiority of our proposed method compared to some 
+of the recently proposed algorithms in terms of the reality, symmetry, and homogeneity of the inpainted parts.  
+The code for the proposed method is available at https://github.com/mohammadrezanaderi4/SFI-Swin   
+1. Introduction 
+Removing objects from an image or filling in holes is a typical application of computer vision. With 
+the image inpainting technique, it is possible to either fill in empty regions or remove a few elements 
+from the image. New deep learning models with convolutional neural networks or transformer models 
+are intended to produce realistic-looking inpainted images. Face inpainting is a subset of image 
+inpainting. Its purpose is to fill the missing regions of a face image. Two major concerns should be 
+considered carefully during the inpainting of missing parts of a face: First, the inpainted regions must be 
+homogeneous with the other parts of the face and highly correlated to the available surrounding areas of 
+the input image. Second, facial symmetry must be preserved between the left and right sides. Many 
+inpainting methods have been proposed, and some achieved excellent results in repairing missing areas 
+of natural images. But almost all have difficulty repairing a face image symmetrically and 
+homogeneously. This shortcoming is because the network losses do not convey a general understanding 
+of the facial features to the generator. To further illustrate the main issues of previous works, we will 
+discuss the effect of usual losses that have been used in references [1] to [6]. 
+The loss functions that are mainly used in inpainting are pixel-wise, adversarial, feature-matching, 
+and perceptual loss. We will discuss the effect of each loss on the model training in this section. 
+1.1 Pixel-Wise Loss: As shown in Equation 1, Pixel-wise loss is computed between the inpainted 
+image and ground truth. Its goal is to lead the model to inpaint the missing regions similar to ground 
+truth by considering available parts of the face. However, the available regions cannot completely 
+describe the missing parts of the image. Therefore, this loss only can lead the inpainting network to 
+understand the low-level features of the missing parts. 
+ 
+ℒ������������������������ (������������, �������������) = ‖������������ − ������������� ‖1 or 2 
+ 
+(1) 
+ 
+In Equation 1, ������������ is the inpainted image, �������������  is the ground truth, and ‖ ‖1 or 2 stands for L1 or L2 norm 
+computation. 
+                                                 
+* The first two authors contributed equally to this work. 
+
+2 
+ 
+ 
+1.2 Adversarial and Feature Matching Losses: The adversarial loss [7] (Equations 2, 3, 4) attempts 
+to check the reality of an inpainted image based on the distribution of ground truths and generated images 
+using an extra network called a discriminator. In Equations 2, 3, and 4, ������������, �������������, ������������, ������������, ������������, and ������������, represent 
+the inpainted image, ground truth, discriminator, generator, parameters of the discriminator, and 
+parameters of the generator. Also, ������������������������ is the stop gradient, indicating that the backpropagated gradient 
+stops when it reaches specific parameters. Finally, the feature matching loss (Equation 5) is computed 
+between the features extracted from the discriminator's middle layers for the inpainted image and ground 
+truth. In Equation 5, ������������������������������������ presents middle features of the discriminator. Adversarial and feature 
+matching losses came from the idea that although we cannot reconstruct the missing regions exactly 
+similar to ground truth, at least we can be sure that the inpainted regions look realistic.  
+ 
+ℒ������������ = −������������������������[log ������������ℰ(������������)] − �������������������������log  �1 − ������������������������(�������������)�� 
+(2) 
+ℒ������������ = −������������������������[log ������������������������(�������������)] 
+(3) 
+ℒ������������������������������������ = ������������������������������������(ℒ������������) + ������������������������������������(ℒ������������) 
+(4) 
+ℒ������������������������(������������, �������������) = ‖ ������������������������������������������������(������������) − ������������������������������������������������(�������������)‖1 or 2 
+(5) 
+ 
+The architecture of the discriminator is patch-based. The discriminator tests the reality of the patches 
+of the whole image [7]. 
+1.3 Perceptual Loss: The perceptual loss is shown in Equation 6.  This loss is computed using the 
+encoder of a pre-trained segmentation network to compare the high-level features of the generated and 
+ground truth images. In Equation 6, ������������, �������������, ������������, present inpainted image, ground truth, and encoder of a pre-
+trained segmentation network, respectively.  This loss mostly considered high-level features such as 
+edges in the image. 
+ 
+ℒ������������������������(������������, �������������) = ‖ ������������(������������) −  ������������(�������������)‖1 or 2 
+(6) 
+ 
+The adversarial, feature matching, and perceptual losses mainly focus on the reality of the patches 
+and edges' smoothness. Therefore, these losses do not represent the importance of the homogeneity and 
+symmetry of the image. The perceptual loss causes the inpainting network to occasionally prefer the 
+reality of the patches over the validity of the whole picture. In other words, due to the pressure of the 
+mentioned losses, the generator sacrifices the symmetry of the face to get more realistic patches. This 
+patch-based behavior is one of the apparent drawbacks of available image synthesizing and image 
+inpainting methods. The mentioned shortcoming motivates us to add a new term to the loss function of 
+image inpainting networks to consider symmetry and global features of each part of the face. We can 
+get more realistic faces by assessing an image's symmetry and global facial features. Meanwhile, we use 
+a base inpainting network that balances global and patch-based losses to get the maximum benefit from 
+this additional loss.  
+A large receptive field is used in Swin transformer blocks [8]. We used Swin transformer blocks in 
+this paper and applied a new loss focusing on distinct facial feature distribution. Hence, we increase the 
+inpainting model's concentration on the symmetry problem. Furthermore, current metrics cannot 
+measure the symmetry of the inpainted face. Therefore, we propose a new metric to resolve this 
+shortcoming. 
+ 
+ 
+
+3 
+ 
+Our main contributions are as follows: 
+• Proposing a homogeneity-aware loss to solve the heterogeneity and asymmetricity problems 
+in inpainting methods. 
+• Using transformer base architecture to achieve a wider receptive field and balance the usage 
+of global and local features and input and output image gradients. 
+• Increasing the generalizability of the inpainting model by upgrading its understanding of the 
+semantic face parts using separate discriminators for each part of the face. 
+• Proposing a new metric, symmetry concentration score (SCS), to assess the symmetry and 
+homogeneity of facial organs.  
+The structure of this paper is as follows: In section 2, we will present some previous relevant methods. 
+Then, section 3 offers the proposed method. Finally, experimental results are detailed in section 4, and 
+the conclusion is presented in section 5. 
+2. Related Works 
+Several methods for image inpainting have been proposed, including deep generative base and 
+transformer base methods. This section will discuss these two types of methods in more detail. 
+2.1 Deep Generative Methods 
+In recent years, the generative adversarial network has been used for image completion and 
+inpainting [9]–[12]. In [1], a new approach has been proposed based on conditional and unconditional 
+generative networks. The authors present new metrics for measuring image completeness. These metrics 
+are based on the linear separability of features in a feature space. These metrics demonstrate a measure 
+of the perceptual fidelity of the inpainted image compared to the real image. A method for generating 
+high-quality inpainted image based on aggregated contextual transformations was presented in [2]. 
+Authors of [2] also employed distant contexts, not in the masked region's neighborhood. Inpainting 
+methods are evaluated on different tasks, such as removing logos, editing faces, and removing objects 
+[13]–[16]. In [3], the authors focused on a large missing region and proposed an adversarial method for 
+image inpainting. The model can generate visually realistic images for contiguous and separated large 
+missing areas. Also, a new loss was proposed to measure the non-local correlation between patches and 
+help the model get a better inference result.  
+A method is proposed in [4] that uses edges and masked images to generate inpainted images. They 
+used an edge generator that completed edges in missing regions and a completion network that used an 
+edge map and input image to generate the result. With this approach, they achieved a good result with a 
+focus on the edges of the image. In [17], they introduced a method for human body completion. They 
+used three steps in their work: prior encoding, segmentation completion, and texture completion. A 
+segmentation map can help capture important information from the human body. They also designed a 
+memory module, a dictionary for storing learned latent vectors. These segmentation maps are typically 
+used to segment each part of the human body and assist in extracting high-level information. As a result, 
+it helps the model to learn better. Multi-scale structure discriminators have been used to generate 
+segmentation maps with good information, so the results were interesting in human body completion. In 
+[18], the authors used a generative adversarial network that contains dilated convolution for increasing 
+receptive fields. They tried to achieve a reasonable structure without blurriness in the output. Further, 
+they developed a new self-guided regression loss technique for enhancing semantic features and 
+concentrating on uncertain areas. When a model fills holes in an image, it's essential to generate a good 
+result and have color consistency in all pictures.  
+In [19], the authors proposed a method for image inpainting with attention to color consistency in 
+images. The output was reliable without any artifacts, and the stable colors in different parts of the 
+images provided the ability to create realistic images. In [5], the authors used gated convolutions in the 
+
+4 
+ 
+model to solve the problem of using vanilla convolutions that understand the pixels differently. They 
+also proposed a new loss function named SN-PatchGAN that helps the model generate a high-quality 
+and flexible image compared to previous methods.    
+2.2 Transformer Methods 
+Since transformers are highly effective for tasks related to natural language processing, researchers 
+have used them in vision tasks. A transformer has a global view and uses an attention mechanism to 
+extract information [20–23]. In [24], a transformer was used for image completion. They use the global 
+view of the transformer and achieve better results for large masks than other methods. Recently a 
+combination of transformer and convolution has been used. In [25], the authors proposed a model 
+containing a transformer and convolution for image inpainting. The global view of the transformer and 
+feature extraction in convolution helped the model generate good results in image completion. In [26], 
+a novel inpainting framework has been proposed for high-resolution images. They also proposed 
+modifying transformer blocks to stabilize the training of large masks.  
+With the long-range interaction modeling in the transformer, the model generated high-fidelity 
+images. Therefore, the architecture can understand critical dependencies between different regions on 
+the image with attention-based models. In [27], they proposed an inpainting method that used a 
+transformer and CNN model. The model gets additional input that contains lines and edges on the 
+masked image. As a result, the model can reconstruct edges in a better way. With this approach, the 
+proposed model can learn to reconstruct masked images focusing on the edges and lines. In [28], the 
+blind face inpainting method has been proposed. They used two stages for the blind face inpainting 
+process due to the difficulty in detecting masks of different shapes and sizes, as well as the difficulty of 
+restoring masked images that are realistic. They used a transformer model during the first stage to detect 
+corrupted regions. In the next step, a network has been proposed for restoring features at different levels 
+in a hierarchical manner, thereby producing semantically coherent content based on unmasked regions 
+of the face. 
+3. Proposed method 
+Our proposed method is called Symmetric Face Inpainting with Swin Transformer (SFI-Swin). We 
+will discuss our method from two perspectives. First, we will discuss the generator architecture based 
+on a transformer in subsection 3.1. Then in subsection 3.2, we will concentrate on the loss functions and 
+propose a loss that focuses on the symmetry and homogeneity of the face features. We use six additional 
+discriminators with the same architecture to compute this loss.  
+3.1 Generator architecture 
+As discussed in [6], using a generator architecture with a wider receptive field produces more 
+homogeneous outputs while considering the entire facial features. Thus, they added fast Fourier 
+convolution layers [29] to their models to make a generator with a wider receptive field. Although their 
+work seems to be effective, it sometimes fails to make a balance between using global and local features 
+[6]. We use the Swin-Unet [30] architecture as our generator to create a balance between local and global 
+features. Swin-Unet has a large receptive field because of its self-attention mechanism and could balance 
+local and global feature usage well. As discussed in [31], skip connections downgrade the image 
+inpainting ability of the generator model because it allows the generator to copy the available parts of 
+the input to the output.  
+Skip connections prevent the model from constructing the whole available and missed parts of the 
+input based on high-level features that are extracted in the middle of the generator. Such a network 
+results in totally heterogeneous face features in output. Therefore, we omit the skip connections from 
+the Swin-Unet model, and our final generator architecture is shown in Fig. 1. The patch discriminator 
+[7] architecture is also shown in Fig. 1. 
+ 
+
+5 
+ 
+ 
+Using the Swin transformer [8] layers, which could balance the utilization of local and global 
+features, leads to more homogeneous and realistic facial features. In addition, Swin transformer blocks 
+do not add much computational complexity compared to convolutional models with a similar number of 
+parameters. This behavior is because the Swin transformer computes the correlation between the patches 
+in certain parts of the image. 
+3.2 Homogeneity Loss 
+In this section, we propose a new loss that we will use in addition to the losses that were discussed 
+in Section 1. The idea behind this new loss is that we compute the realness of each part of the face 
+compared to the distribution of that specific part in the whole dataset. This will oblige the generator to 
+pay more attention to symmetric and global features. We consider six additional discriminators with the 
+same architecture to compute the homogeneity loss, as shown in Figure 1. Each discriminator is for a 
+specific part of the face, such as skin, eyes, lips, clothes, and hair. It is noteworthy that each of these 
+parts is extracted from a face image using a pre-trained semantic segmentation module [32]. The 
+segmentation module parameters are frozen during training. While the generator intends to inpaint the 
+missed regions, computing the realness of each facial organ will conclude more symmetry in the output 
+image. The architecture of these discriminators is designed to assess the total realness of a specific face. 
+ 
+Fig. 1. Block diagram of our proposed method. First, the generator takes the masked image as input and attempts to inpaint it. Then 
+the inpainted image is fed to the patch discriminator to check the overall reality of the patches. Meanwhile, the inpainted image is 
+also fed into a semantic segmentation network [32] to separate the semantic parts of the face, such as eyes, and ears. The architecture 
+of these six semantic discriminators is the same. In the next step, six distinct discriminators calculate the total realness of each 
+semantic part of the face. Then, the generator parameters will be updated based on these seven discriminators and the pixel-wise loss 
+gradient signals, which are backpropagated to the generator. 
+
+Patch Partition
+Patch Merging
+Linear Projection
+Linear Embedding
+Swin Transformer
+Swin Transformer
+Block x 2
+Patch Expanding
+Block x 1
+Encoder
+Bottlenec
+Decoder
+Skin
+Discriminator
+F××48
+Wx兴xC
+Wx×2C
+WH
+WH
+WxH
+xC
+W×H×C(4x) W×H×Clas
+Eye
+Discriminator
+Masked
+Hair
+Image
+Discriminator
+Semantic
+Generator
+Segmentor
+Lip
+Discriminator
+Patch
+Discriminator
+Cloth
+Discriminator
+Conv 4 × 4, stride=2
+Linear
+Input Conv
+Conv 4 x 4, stride=2
+Input Conv
+projection
+Ear
+Conv 4 × 4, stride = 1
+Conv 1 × 1, stride = 1
+Conv 4 x 4, stride = 1
+Conv 1 × 1, stride = 1
+Discriminator
+Discriminator
+Discriminator
+Output =
+Vector with
+10 elements6 
+ 
+This characteristic is unlike the patch discriminator, which does not completely understand the face 
+image. One of the main features of the face is its symmetry which the related discriminator will consider. 
+Using these semantic discriminators to check the realness of each face organ increases the generator's 
+knowledge about different semantic parts of the image. Considering facial organs helps the network to 
+repair portraits more realistically. The block diagram of SFI-Swin is shown in Figure 1.  
+For each of these discriminators and the corresponding facial organ, the adversarial and feature-
+matching loss is computed using Equations 4 and 5. Therefore the overall homogeneity loss could be 
+presented as Equations 7 and 8. 
+ 
+⎩
+⎪⎪
+⎨
+⎪⎪
+⎧ ℒ������������������������������������������������(������������, �������������) = ℒ������������������������������������(������������������������������������������������������������, �������������������������������������������������������������) + ℒ������������������������(������������������������������������������������������������, �������������������������������������������������������������)
+ℒ������������������������������������(������������, �������������) = ℒ�������������������������������������������������������������������������������������, �������������������������������������������������� + ℒ�������������������������������������������������������������������������, ��������������������������������������������������
+ℒℎ������������������������������������(������������, �������������) = ℒ������������������������������������(������������ℎ������������������������������������, �������������ℎ������������������������������������) + ℒ������������������������(������������ℎ������������������������������������, �������������ℎ������������������������������������)
+ℒ������������������������������������(������������, �������������) = ℒ�������������������������������������������������������������������������������������, �������������������������������������������������� + ℒ�������������������������������������������������������������������������, ��������������������������������������������������
+ℒ������������������������������������������������ℎ(������������, �������������) = ℒ������������������������������������(������������������������������������������������������������ℎ, �������������������������������������������������������������ℎ) + ℒ������������������������(������������������������������������������������������������ℎ, �������������������������������������������������������������ℎ)
+ℒ������������������������������������(������������, �������������) = ℒ������������������������������������(������������������������������������������������, �������������������������������������������������) + ℒ������������������������(������������������������������������������������, �������������������������������������������������)
+⎭
+⎪⎪
+⎬
+⎪⎪
+⎫
+ 
+ 
+(7) 
+ℒ������������������������(������������, �������������) = ������������1ℒ������������������������������������������������(������������, �������������) + ������������2ℒ������������������������������������(������������, �������������) + ������������3ℒℎ������������������������������������(������������, �������������) + ������������4ℒ������������������������������������(������������, �������������)
++ ������������5ℒ������������������������������������������������ℎ(������������, �������������) + ������������6ℒ������������������������������������(������������, �������������) 
+(8) 
+ 
+ 
+where ������������1, ������������2, ������������3, ������������4, ������������5, and ������������6 are set to be 0.083, 0.25, 0.083, 0.25, 0.083, 0.083, and 0.25, 
+respectively. The coefficients of eyes, ears, and lips are set three times greater than other parts of the 
+face. This is because, during the experiments, we realized that the model is mostly incapable of 
+maintaining the symmetry of these face organs.  
+4. Experimental results 
+In this section, we present the experiment setups and results accomplished using these setups.  
+4.1 Experiments setups 
+In the following, we will discuss the dataset, metrics, hyperparameters, and the total loss function 
+we used to train our models. 
+Data and metrics: We use CelebAHQ [33] dataset. This dataset contains 28k images. We split the 
+data to train, validate, and test, similar to [6]. Learned Perceptual Image Patch Similarity (LPIPS) [34] 
+and Fr'echet inception distance (FID) [35] metrics are used to evaluate our proposed method 
+performance. Compared to pixel-level L1 and L2 distances, LPIPS and FID are more suitable for 
+measuring the performance of large masks when multiple natural completions are plausible. The 
+experimentation pipeline is implemented using PyTorch [36].  
+Training hyperparameters: The learning rate of the generator is 0.001, and that of the seven 
+discriminators is 0.0001. The discrimination process is more straightforward than the generation process, 
+especially in the initial epochs of the training [6]. Hence, a lower initial learning rate for the 
+discriminators allows the generator to converge faster during the initial epochs. This will balance the 
+training procedures between the discriminators and the generator. This balance prevents the training 
+procedure from collapsing. These learning rates also are optimized using a beam search algorithm to get 
+the best results [6]. The batch size is also set to 20, and we train our model for 40 epochs using the Adam 
+optimizer [37]. We trained our model with an NVIDIA GeForce RTX 3090 with 24GB RAM GPU. 
+
+7 
+ 
+Loss function: besides the proposed homogeneity loss, we used the same losses as [6]. Therefore, 
+the total loss function of our model is shown in Equation 9. 
+ℒ(������������, �������������) =  ������������ℒ������������������������(������������, �������������) + ������������ℒ������������������������������������(������������, �������������) +  ������������ℒ������������������������(������������, �������������) + ������������ℒ������������������������(������������, �������������) 
+ 
+(9) 
+ 
+where ������������, ������������, ������������, and ������������ are set to 10, 10, 100, and 20. These values are based on our experiments to get the 
+best results on the test data.  
+ 
+4.2 Qualitative results: Fig. 2 shows SFI-Swin performance to inpaint for different mask sizes, in 
+comparison with [6]. To further illustrate our proposed method's capability to inpaint the face images 
+symmetrically, we generate masks that omit one of the eyes and slowly grow around the omitted region 
+to cover half of the face. We compare our method with [6], and the results are shown in Figure 3. Further, 
+we randomly chose ten images from the test set. Then, we masked (eliminated) one eye and a K×K block 
+of the image. We tested with K set to 16, 32, and 64.  
+We repaired the image with a missing eye and a K×K block. We then calculated the difference 
+between the repaired eye while a K×K patch was masked with the inpainted image that only the eye was 
+missing. We then find the mean of the absolute difference image and assign this value to the 
+corresponding K×K block. The average absolute difference will be large if a block is essential to repair 
+the missing eye. Finally, we built a heatmap to show the blocks that play an important role in the 
+inpainting of the missing eye. The results are shown in Figure 4. As we can see, the non-missing eye 
+and the chick area around the missing eye are the most important patches for reconstructing the missing 
+area. We repeated this experiment for half of the face. The effects of each missing K×K patch on the 
+inpaint eye and half face are shown in Figure 5. The lighter the color of the shown patch, the more effect 
+it has on reconstructing the missing right-side eye.  
+ 
+Fig. 2. a) Original images, b) masks, inpainted images by c) LaMa [6], d) Swin, and e) SFI-Swin. 
+
+Narrow masks
+Medium masks
+Wide masks
+a)
+b)
+c)
+(p
+e)8 
+ 
+ 
+While inpainting a facial part from one side of the face, our method focuses on the same organ on the 
+other side.  This consideration causes symmetry in the repaired image.  
+4.3 Quantitative results: Table 1 shows the performance of our proposed method to repair the 
+missing parts of the face compared with the best recently proposed methods in this field in terms of FID 
+and LPIPS scores. Because previous works trained multiple models for different mask sizes, therefore 
+to compare fairly, we show their performance based on the mask sizes which they used in training. 
+However, our method follows the aggressive mask generation proposed by [6] that helps it handle 
+narrow, medium, and wide masks simultaneously. As a result, our proposed methods, Swin and Swin 
+ 
+Fig. 3. Performances of b) LaMa, c) Swin, d) SFI-Swin in comparison with e) ground-truth to inpaint a) masked images starting from an 
+eye and grown to cover half of the face. A white border is drawn to show the reconstructed region.  
+
+D=0
+D = 12
+D = 24
+D = 36
+D = 48
+D=0
+D = 12
+D = 24
+D= 36
+D = 48
+a)
+b)
+c)
+(p
+e)
+a)
+b)
+d)
+e)9 
+ 
+with multiple semantic discriminators (SFI-Swin), achieved the best results in medium and wide mask 
+inpainting. 
+ 
+FID and LPIPS are patch-based metrics and do not consider homogeneity and symmetricity in the 
+face. Thus, we propose the symmetry concentration score (SCS) to assess the symmetry of the left and 
+right sides of the face. In Figure 5, we presented heatmaps that depict the influence the model takes from 
+a K×K block of the face during the inpainting of an eye or half of the face. Our symmetry concentration 
+score measures the amount of attention the network pays to a part of the face while repairing the same 
+semantic part on the other side. To acquire this metric, we calculate the mean of the overlapping K × K 
+patches with the desired organs while considering three different patch sizes (16×16, 32×32, 64×64). 
+The results are shown in Table 2. Our method using multiple semantic discriminators performed better 
+than the Swin network without these discriminators. Also, we achieved better symmetric results than 
+[6].  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 4. Steps to compute symmetry concentration score (SCS). First, we mask an eye and a K×K patch of the face, and reconstruct the 
+missing eye and the K×K patch. Then the absolute difference between the image with inpainted eye with the image with a missing K×K 
+patch and the missing eye is computed. The difference shows the effect of that K×K patch on inpainting result of the missing eye. The 
+effect of all K×K patches is computed and shown as a heatmap. The face borders are also depicted to investigate the impact of each part of 
+the face on inpainting the missed eye. 
+
+eye mask
+Absolute value of
+Difference between
+generated eye when
+only that eye is masked
+Mask
+Output
+Output x eye mask
+and when an additional
+patch is also masked
+from the Original
+X
+Image
+Overlay the mean of the
+difference image on the
+X
+corresponding patch
+absO, mean
+Original
+X
+Image
+absO, mean
+Effect of each
+patch on
+filling the
+abs(), mean
+empty eye
+X
+X
+absO, mean10 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 5. Influence that the generator gets from each K×K block of the face receives to repair the missing parts of the face. a) Lama, b) 
+Swin, c) SFI-Swin. The missing parts are shown as empty black regions (an eye or half of the face), while the background is not 
+considered in these experiments. 
+
+Eye
+Half face
+Block size = 16x16
+a)
+b)
+c)
+a)
+Block size = 32x32
+n
+C
+a)
+Block size = 64x64
+b)
+C11 
+ 
+Table 1. Presenting the performance of our method SFI-Swin to inpaint different types of masks comparing to powerful methods in 
+this field using two popular metrics, FID and LPIPS. The three best models in each column are shown in red, orange, and green. 
+Train 
+masks type 
+Methods 
+CelebA-HQ (256×256) 
+Narrow masks 
+Medium masks 
+Wide masks 
+40-50% masked 
+All samples 
+40-50% masked 
+All samples 
+40-50% masked 
+All samples 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+FID 
+LPIPS 
+Aggressive 
+train 
+masks 
+Ours (SFI-Swin) 
+23.7 
+0.157 
+7.44 
+0.101 
+33.43 
+0.161 
+5.54 
+0.088 
+26.81 
+0.102 
+5.97 
+0.104 
+Swin [30] 
+23.2 
+0.151 
+7.31 
+0.098 
+32.88 
+0.160 
+5.60 
+0.086 
+26.69 
+0.101 
+5.98 
+0.102 
+LaMa-Fourier [6] 
+22.7 
+0.132 
+7.26 
+0.085 
+34.1 
+0.145 
+6.13 
+0.080 
+27.8 
+0.168 
+6.96 
+0.098 
+Narrow 
+train 
+masks 
+CoModGAN [1] 
+35.9 
+0.139 
+16.8 
+0.079 
+48.4 
+0.169 
+19.4 
+0.092 
+64.4 
+0.191 
+24.4 
+0.102 
+AOT GAN [2] 
+21.0 
+0.127 
+6.67 
+0.081 
+39.1 
+0.162 
+7.28 
+0.089 
+40.4 
+0.204 
+10.3 
+0.118 
+RegionWise [3] 
+32.5 
+0.188 
+11.1 
+0.124 
+40.4 
+0.179 
+7.52 
+0.101 
+33.9 
+0.205 
+8.54 
+0.121 
+DeepFill v2 [5] 
+37.0 
+0.201 
+12.5 
+0.190 
+45.3 
+0.189 
+9.05 
+0.105 
+43.0 
+0.214 
+11.2 
+0.126 
+EdgeConnect [4] 
+29.2 
+0.156 
+9.61 
+0.099 
+40.5 
+0.174 
+7.56 
+0.095 
+34.7 
+0.205 
+9.02 
+0.120 
+Wide 
+ train 
+masks 
+RegionWise [3] 
+47.5 
+0.246 
+17.9 
+0.164 
+50.9 
+0.220 
+10.3 
+0.124 
+42.6 
+0.233 
+11.2 
+0.140 
+DeepFill v2 [5] 
+30.4 
+0.169 
+9.99 
+0.108 
+40.3 
+0.173 
+7.65 
+0.095 
+34.6 
+0.196 
+8.95 
+0.115 
+EdgeConnect [4] 
+55.5 
+0.248 
+18.3 
+0.152 
+40.2 
+0.174 
+7.79 
+0.097 
+32.7 
+0.196 
+8.43 
+0.116 
+ 
+ 
+Table 2. Comparing the symmetry concentration score (SCS) of our method SFI-Swin to 
+inpaint certain parts of the face compared to Swin and LaMa [6]. 
+Method | Metric + face parts 
+SCS for eye 
+SCS for half face 
+Ours (SFI-Swin) 
+0.7177 
+0.4233 
+Swin [30] 
+0.6319 
+0.3948 
+LaMa [6] 
+0.6225 
+0.3740 
+ 
+5. Conclusion 
+ 
+This paper discussed the effect of using multiple semantic discriminators incorporated with the Swin 
+transformer-based architecture to repair face images. Our proposed method preserved the symmetry and 
+homogeneity of the face parts. Our experimental results show the proposed method's superiority over 
+powerful rivals, especially on the medium and wide masks.  
+We also proposed a new method to assess the concentration of the inpainting network while inpainting 
+a specific face organ.  
+By using multiple discriminators to compute the reality of each facial organ, the generator was guided 
+to preserve the symmetry and homogeneity of the face. This resulted in a generator that resolved one of 
+the most critical inpainting shortcomings.  
+ 
+ 
+ 
+ 
+ 
+
+12 
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+ 
+

INE0T4oBgHgl3EQfhwHS/content/tmp_files/2301.02437v1.pdf.txt

@@ -0,0 +1,1353 @@
+Valid P-Value for Deep Learning-Driven Salient Region
+Daiki Miwa∗
+Nagoya Institute of Technology
+[email protected]
+Vo Nguyen Le Duy∗
+RIKEN
+[email protected]
+Ichiro Takeuchi†
+Nagoya University and RIKEN
+[email protected]
+January 9, 2023
+Abstract
+Various saliency map methods have been proposed to interpret and explain predictions of
+deep learning models. Saliency maps allow us to interpret which parts of the input signals have a
+strong influence on the prediction results. However, since a saliency map is obtained by complex
+computations in deep learning models, it is often difficult to know how reliable the saliency map
+itself is. In this study, we propose a method to quantify the reliability of a salient region in the
+form of p-values. Our idea is to consider a salient region as a selected hypothesis by the trained
+deep learning model and employ the selective inference framework. The proposed method can
+provably control the probability of false positive detections of salient regions. We demonstrate
+the validity of the proposed method through numerical examples in synthetic and real datasets.
+Furthermore, we develop a Keras-based framework for conducting the proposed selective inference
+for a wide class of CNNs without additional implementation cost.
+∗Equal contribution
+†Corresponding author
+1
+arXiv:2301.02437v1  [stat.ML]  6 Jan 2023
+
+1
+Introduction
+Deep neural networks (DNNs) have exhibited remarkable predictive performance in numerous practical
+applications in various domains owing to their ability to automatically discover the representations
+needed for prediction tasks from the provided data. To ensure that the decision-making process of
+DNNs is transparent and easy to understand, it is crucial to effectively explain and interpret DNN
+representations.
+For example, in image classification tasks, obtaining salient regions allows us to
+explain which parts of the input image strongly influence the classification results.
+Several saliency map methods have been proposed to explain and interpret the predictions of DNN
+models [Ribeiro et al., 2016, Bach et al., 2015, Doshi-Velez and Kim, 2017, Lundberg and Lee, 2017,
+Zhou et al., 2016, Selvaraju et al., 2017]. However, the results obtained from saliency methods are
+fragile [Kindermans et al., 2017, Ghorbani et al., 2019, Melis and Jaakkola, 2018, Zhang et al., 2020,
+Dombrowski et al., 2019, Heo et al., 2019]. It is important to develop a method for quantifying the
+reliability of DNN-driven salient regions.
+Our idea is to interpret salient regions as hypotheses driven by a trained DNN model and employ
+a statistical hypothesis testing framework. We use the p-value as a criterion to quantify the statistical
+reliability of the DNN-driven hypotheses. Unfortunately, constructing a valid statistical test for DNN-
+driven salient regions is challenging because of the selection bias. In other words, because the trained
+DNN selects the salient region based on the provided data, the post-selection assessment of importance
+is biased upwards.
+To correct the selection bias and compute valid p-values for DNN-driven salient regions, we intro-
+duce a conditional selective inference (SI) approach. The selection bias is corrected by conditional SI
+in which the test statistic conditional on the event that the hypotheses (salient regions) are selected
+using the trained DNNs. Our main technical contribution is to develop a computational method for
+explicitly deriving the exact (non-asymptotic) conditional sampling distribution of the salient region
+for a wide class convolutional neural networks (CNNs), which enables us to conduct conditional SI
+and compute valid p-values. Figure 1 presents an example of the problem setup.
+Related works.
+In this study, we focus on statistical hypothesis testing for post-hoc analysis, i.e.,
+quantifying the statistical significance of the salient regions identified in a trained DNN model when
+a test input instance is fed into the model. Several methods have been developed to visualize and
+understand trained DNNs. Many of these post-hoc approaches [Mahendran and Vedaldi, 2015, Zeiler
+and Fergus, 2014, Dosovitskiy and Brox, 2016, Simonyan et al., 2013] have focused on developing
+visualization tools for saliency maps given a trained DNN. Other methods have aimed to identify
+the discriminative regions in an input image given a trained network [Selvaraju et al., 2017, Fong
+2
+
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Two-sample 
+Test
+(a) Image without tumor region. The naive-p = 0.00 (wrong detection) and selective-p = 0.43 (true negative)
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Two-sample 
+Test
+(b) Image with tumor region. The naive-p = 0.00 (true positive) and selective-p = 0.00 (true positive)
+Figure 1: Examples of the problem setup and the proposed method on brain tumor dataset. By
+applying a saliency method called CAM [Zhou et al., 2016] on a query input image, we obtain the
+salient region. Our goal is to provide the statistical significance of the salient region in the form of
+p-value by considering two-sample test between the salient region and the corresponding region in
+the a reference image. Note that, since the salient region is selected based on the data, the degree
+of saliency in the selected region is biased upward. In the upper image where there is no true brain
+tumor, the naive p-value which is obtained without caring the selection bias is nearly zero, indicating
+the false positive finding of the salient region.
+On the other hand, the selective p-value which is
+obtained by the proposed conditional SI approach is 0.43, indicating that the selected saliency region
+is not statistically significant. In the lower figure where there is a true brain tumor, both the naive p-
+value and the selective p-value are very small, indicating true positive finding. These results illustrate
+that naive p-value cannot be used to quantify the reliability of DNN-based salient region. In contrast,
+with the selective p-values, we can successfully identify false positive and true positive detections with
+a desired error rate.
+and Vedaldi, 2017, Zhou et al., 2016, Lundberg and Lee, 2017]. Furthermore, some recent studies
+have shown that many popular methods for explanation and interpretation are not stable against a
+perturbation or adversarial attack on the input data and model [Kindermans et al., 2017, Ghorbani
+et al., 2019, Melis and Jaakkola, 2018, Zhang et al., 2020, Dombrowski et al., 2019, Heo et al.,
+2019]. However, to the best of our knowledge, no study to date has quantitatively evaluated and
+reproducibility of DNN-driven salient regions with a rigorous statistical inference framework.
+Recently, conditional SI has been recognized as a promising new approach for evaluating the
+3
+
+27statistical significance of data-driven hypotheses. Conditional SI has been mainly studied for inference
+of linear model features selected by a feature selection method such as Lasso [Lee et al., 2016, Liu
+et al., 2018, Hyun et al., 2018, Le Duy and Takeuchi, 2021] and stepwise feature selection [Tibshirani
+et al., 2016, Sugiyama et al., 2021a]. The main idea of conditional SI study is to make inferences
+conditional on selection events, which allows us to derive exact sampling distributions of test statistics.
+In addition, conditional SI has been applied to various problems [Fithian et al., 2015, Tian and Taylor,
+2018, Yang et al., 2016, Hyun et al., 2021, Duy et al., 2020, Sugiyama et al., 2021b, Chen and Bien,
+2019, Panigrahi et al., 2016, Tsukurimichi et al., 2021, Hyun et al., 2018, Tanizaki et al., 2020, Duy
+and Takeuchi, 2021, Tibshirani et al., 2016, Sugiyama et al., 2021a, Suzumura et al., 2017, Das et al.,
+2021, Duy and Takeuchi, 2022].
+Most relevant existing work of this study is Duy et al. [2022], where the authors provide a framework
+for computing valid p-values for DNN-based image segmentation results. In this paper, we generalized
+this work so that hypotheses characterized by any internal nodes of the network can be considered,
+enabling us to quanfity the statistical significance of salient regions.
+This is in contrast to Duy
+et al. [2022]’s work, which only considered the inference of the DNN’s output in a segmentation task.
+Furthermore, we introduce a Keras-based implementation framework that enables us to conduct SI for
+a wide class of CNNs without additional implementation costs. This is in contrast to Duy et al. [2022]’s
+work, where the selection event must be implemented whenever the network architecture is changed.
+In another direction, Burns et al. [2020] considered the black box model interpretability as a multiple-
+hypothesis testing problem. They aimed to deduce important features by testing the significance of the
+difference between the model prediction and what would be expected when replacing the features with
+their counterfactuals. The difficulty of this multiple-hypothesis testing approach is that the number
+of hypotheses to be considered is large (e.g., in the case of an image with n pixels, the number of
+possible salient regions is 2n). Multiple testing correction methods, such as the Bonferroni correction,
+are highly conservative when the number of hypotheses is large. To circumvent this difficulty, they
+only considered a tractable number of regions selected by a human expert or object detector, which
+causes selection bias because these candidate regions are selected based on the data.
+Contribution.
+Our main contributions are as follows:
+• We provide an exact (non-asymptotic) inference method for salient regions based on the SI
+concept. To the best of our knowledge, this is the first method that proposes to provide valid p-values
+to statistically quantify the reliability of DNN-driven salient regions.
+• We propose a novel algorithm and its implementation. Specifically, we propose Keras-based
+implementation enables us to conduct conditional SI for a wide class of CNNs without additional
+implementation costs.
+4
+
+• We conducted experiments on both synthetic and real-world datasets, through which we show
+that our proposed method can successfully control the false positive rate, has good performance in
+terms of computational efficiency, and provides good results in practical applications. We provide the
+detailed description of our implementation in the supplementary document. Our code is available at
+https://github.com/takeuchi-lab/selective inference dnn salient region.
+2
+Problem Formulation
+In this paper, we consider the problem of quantifying the statistical significance of the salient regions
+identified by a trained DNN model when a test input instance is fed into the model. Consider an
+n-dimensional query input vector
+X = (X1, ..., Xn)⊤ = s + ε,
+ε ∼ N(0, σ2In)
+and an n-dimensional reference input vector,
+Xref = (Xref
+1 , ..., Xref
+n )⊤ = sref + εref,
+εref ∼ N(0, σ2In),
+where s, sref ∈ Rn are the signals and ε, εref ∈ Rn are the noises for query and reference input vectors,
+respectively. We assume that the signals, s and sref are unknown, whereas the distribution of noises
+ε and εref are known (or can be estimated from external independent data) to follow N(0, σ2In),
+an n-dimensional normal distribution with a mean vector 0 and covariance matrix σ2In, which are
+mutually independent. In the illustrative example presented in §1, X is a query brain image for a
+potential patient (we do not know whether she/he has a brain tumor), whereas Xref is a brain image
+of a healthy person known to be without brain tumors.
+Consider a saliency method for a trained CNN. We denote the saliency method as a function
+A : Rn → Rn that takes a query input vector X ∈ Rn and returns the saliency map A(X) ∈ Rn. We
+define a salient region MX for the query input vector X as the set of elements whose saliency map
+value is greater than a threshold
+MX = {i ∈ [n] : Ai(X) ≥ τ} ,
+(1)
+where τ ∈ R denotes the given threshold. In this study, we consider CAM [Zhou et al., 2016] as an
+example of saliency method and threshold-based definition of the salient region. Our method can be
+applied to other saliency methods and other definition of salient region.
+Statistical inference.
+To quantify the statistical significance of the saliency region MX, we con-
+sider such two-sample test to quantify the statistical significance of the difference between the salient
+5
+
+regions of the query input vector XMX and corresponding region of the reference input vector Xref
+MX.
+As concrete examples of the two-sample test, we consider the mean null test:
+H0 :
+1
+|MX|
+�
+i∈MX
+si =
+1
+|MX|
+�
+i∈MX
+sref
+i
+v.s.
+H1 :
+1
+|MX|
+�
+i∈MX
+si ̸=
+1
+|MX|
+�
+i∈MX
+sref
+i .
+(2)
+and global null test:
+H0 : si = sref
+i , ∀i ∈ MX,
+v.s.
+H1 : si ̸= sref
+i , ∃i ∈ MX,
+(3)
+In the mean null test depicted in Eq. (2), we consider a null hypothesis that the average signals in the
+salient region MX are the same between X and Xref. In contrast, in the global null test in Eq. (3),
+we consider a null hypothesis that all elements of the signals in the salient region MX are the same
+between X and Xref. The p-values for these two-sample tests can be used to quantify the statistical
+significance of the salient region MX.
+Test-statistic.
+For a two-sample test conducted between XMX and Xref
+MX, we consider a class of
+test statistics called conditionally linear test-statistic, which is expressed as
+T(X, Xref) = η⊤
+MX
+� X
+Xref
+�
+,
+(4)
+and conditionally χ test-statistic, which is expressed as
+T(X, Xref) = σ−1
+����PMX
+� X
+Xref
+����� ,
+(5)
+where ηMX ∈ R2n is a vector and PMX ∈ R2n×2n is a projection matrix that depends on saliency
+region MX.
+The test statistics for the mean null tests and the global null test can be written in the form of Eq.
+(4) and (5), respectivery. For the mean null test in Eq. (2), we consider the following test-statistic
+T(X, Xref) = η⊤
+MX
+� X
+Xref
+�
+=
+1
+|MX|
+�
+i∈MX
+Xi −
+1
+|MX|
+�
+i∈MX
+Xref
+i
+,
+where ηMX =
+1
+|MX|
+�
+� 1n
+MX
+−1n
+MX
+�
+� ∈ R2n. For the gloabl null test in Eq. (3), we consider the following
+test-statistic
+T(X, Xref) = σ−1
+����PMX
+� X
+Xref
+����� =
+�
+�
+�
+� �
+i∈MX
+�Xi − Xref
+i
+√
+2σ
+�2
+,
+where
+PMX = 1
+2
+�
+� diag(1n
+MX)
+−diag(1n
+MX)
+−diag(1n
+MX)
+diag(1n
+MX)
+�
+� .
+(6)
+6
+
+To obtain p-values for these two-sample tests we need to know the sampling distribution of the
+test-statistics. Unfortunately, it is challenging to derive the sampling distributions of test-statistics
+because they depend on the salient region MX, which is obtained through a complicated calculation
+in the trained CNN.
+3
+Computing Valid p-value by Conditional Selective Inference
+In this section, we introduce an approach to compute the valid p-values for the two-sample tests for
+the salient region MX between the query input vector X and the reference input vector Xref based
+on the concept of conditional SI [Lee et al., 2016].
+3.1
+Conditional Distribution and Selective p-value
+Conditional distribution.
+The basic idea of conditional SI is to consider the sampling distribution
+of the test-statistic conditional on a selection event. Specifically, we consider the sampling property
+of the following conditional distribution
+T(X, Xref)
+��� {MX = MXobs} ,
+(7)
+where Xobs is the observation (realization) of random vector X. The condition in Eq.(7) indicates the
+randomness of X conditional on the event that the same salient region MX as the observed MXobs
+is obtained. By conditioning on the salient region MX, derivation of the sampling distribution of the
+conditionally linear and χ test-statistic T(X, Xref) is reduced to a derivation of the distribution of
+linear function and quadratic function of (X, Xref), respectively.
+Selective p-value.
+After considering the conditional sampling distribution in (7), we introduce the
+following selective p-value:
+pselective = PH0
+� ��T(X, Xref)
+�� ≥
+��T(Xobs, Xref
+obs)
+��
+��� MX = MXobs, QX,Xref = Qobs
+�
+,
+(8)
+where
+QX,Xref = ΩX,Xref,
+Qobs = QXobs,Xref
+obs
+with
+ΩX,Xref =
+�
+I2n − ηMXη⊤
+MX
+∥ηMX∥2
+� � X
+Xref
+�
+∈ R2n
+in the case of mean null test, and
+QX,Xref =
+�
+VX,Xref, UX,Xref
+�
+,
+Qobs = QXobs,Xref
+obs
+7
+
+with
+VX,Xref = σPMX
+� X
+Xref
+������PMX
+� X
+Xref
+����� ∈ R2n,
+UX,Xref = P ⊥
+MX
+� X
+Xref
+�
+∈ R2n
+in the case of global null test. The QX,Xref is the sufficient statistic of the nuisance parameter that
+needs to be conditioned on in order to tractably conduct the inference 1.
+The selective p-value in Eq.(8) has the following desired sampling property
+PH0
+�
+pselective ≤ α | MX = MXobs
+�
+= α,
+∀α ∈ [0, 1].
+(9)
+This means that the selective p-values pselective can be used as a valid statistical significance measure
+for the salient region MX.
+3.2
+Characterization of the Conditional Data Space
+To compute the selective p-value in (8), we need to characterize the conditional data space whose
+characterization is described introduced in the next section. We define the set of (X Xref)⊤ ∈ R2n
+that satisfies the conditions in Eq. (8) as
+D =
+�
+(X Xref)⊤ ∈ R2n �� MX = MXobs, QX,Xref = Qobs
+�
+.
+(10)
+According to the second condition, the data in D is restricted to a line in R2n as stated in the following
+Lemma.
+Lemma 1. Let us define let us define,
+a = ΩXobs,Xref
+obs
+and
+b =
+ηMX
+∥ηMX∥2 ∈ R2n.
+(11)
+in the mean null test, and
+a = UXobs,Xref
+obs
+and
+b = VXobs,Xref
+obs
+(12)
+in the case of global null test. Then, the set D in (10) can be rewritten as D =
+��
+X Xref�⊤ = a+bz |
+z ∈ Z
+�
+by using the scalar parameter z ∈ R, where
+Z = {z ∈ R | Ma1:n+b1:nz = MXobs} .
+(13)
+x1:n represents a vector of elements 1 through n of x.
+1This nuisance parameter QX,Xref corresponds to the component z in the seminal conditional SI paper [Lee et al.,
+2016] (see Sec. 5, Eq. 5.2 and Theorem 5.2) and z, w in [Chen and Bien, 2019](see Sec. 3, Theorem 3.7). We note that
+additional conditioning on QX,Xref is a standard approach in the conditional SI literature and is used in almost all
+the conditional SI-related studies. Here, we would like to note that the selective p-value depend on QX,Xref , but the
+property in (9) is satisfied without this additional condition because we can marginalize over all values of QX,Xref (see
+the lower part of the proof of Theorem 5.2 in Lee et al. [2016] and the proof of Theorem 3.7 in Chen and Bien [2019] ).
+8
+
+Proof. The proof is deferred to Appendix A.1
+■
+Lemma 1 indicates that we do not need to consider the 2n-dimensional data space. Instead, we
+only need to consider the one-dimensional projected data space Z in (13).
+Now, let us consider
+a random variable Z ∈ R and its observation Zobs ∈ R that satisfies (X Xref)⊤ = a + bZ and
+(Xobs Xref
+obs)⊤ = a + bZobs. The selective p-value (8) is rewritten as
+pselective = PH0 (|Z| ≥ |Zobs| | Z ∈ Z) .
+(14)
+Because the variable Z ∼ N(0, σ2∥η∥2) in the case of mean null test and Z ∼ χ (Trace(P)) in the case
+of global null test under the null hypothesis, Z | Z ∈ Z follows a truncated normal distribution and a
+truncated χ distribution, respectively. Once the truncation region Z is identified, computation of the
+selective p-value in (14) is straightforward. Therefore, the remaining task is to identify Z.
+In general, computation of Z in (13) is difficult because we need to identify the selection event
+Ma1:n+b1:nz for all values of z ∈ R, which is computationally challenging. In the next section, we
+show that the challenge can be resolved under a wide class of problems.
+4
+Piecewise Linear Network
+The problem of computing selective p-values for the selected salient region is casted into the problem
+of identifying a set of intervals Z = {z ∈ R | MX(z) = MXobs}. Given the complexity of saliency
+computation in a trained DNN, it seems difficult to obtain Z. In this section, however, we explain
+that this is feasible for a wide class of CNNs.
+Piecewise linear components in CNN
+The key idea is to note that most of basic operations and
+common activation functions used in a trained CNN can be represented as piecewise linear functions
+in the following form:
+Definition 1. (Piecewise Linear Function) A piecewise linear function f : Rn �→ Rm is defined as:
+f(X) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ψf
+1X + ψf
+1 ,
+if X ∈ Pf
+1 := {X′ ∈ Rn | ∆f
+1X′ ≤ δf
+1 },
+Ψf
+2X + ψf
+2 ,
+if X ∈ Pf
+2 := {X′ ∈ Rn | ∆f
+2X′ ≤ δf
+2 },
+...
+Ψf
+K(f)X + ψf
+K(f),
+if X ∈ Pf
+K(f) := {X′ ∈ Rn | ∆f
+K(f)X′ ≤ δf
+K(f)}
+where Ψf
+k, ψf
+k, ∆f
+k and δf
+k for k ∈ [K(f)] are certain matrices and vectors with appropriate dimensions,
+Pf
+k := {x ∈ Rn | ∆f
+kx ≤ δf
+k} is a polytope in Rn for k ∈ [K(f)], and K(f) is the number of polytopes
+for the function f.
+9
+
+Examples of piecewise linear components in a trained CNN are shown in Appendix A.2.
+Piecewise Linear Network
+Definition 2. (Piecewise Linear Network) A network obtained by concatenations and compositions
+of piecewise linear functions is called piecewise linear network.
+Since the concatenation and the composition of piecewise linear functions is clearly piecewise linear
+function, the output of any node in the piecewise linear network is written as a piecewise linear function
+of an input vector X. This is also true for the saliency map function Ai(X), i ∈ [n]. Furthermore,
+as discussed in §4, we can focus on the input vector in the form of X(z) = a1:n + b1:nz which is
+parametrized by a scalar parameter z ∈ R. Therefore, the saliency map value for each element is
+written as a piecewise linear function of the scalar parameter z, i.e.,
+Ai(X(z)) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+κAi
+1 z + ρAi
+1 ,
+if z ∈ [LAi
+1 , U Ai
+1 ],
+κAi
+2 z + ρAi
+2 ,
+if z ∈ [LAi
+2 , U Ai
+2 ],
+...
+κAi
+K(Ai)z + ρf
+K(Ai),
+if z ∈ [LAi
+K(Ai), U Ai
+K(Ai)],
+(15)
+where K(Ai) is the number of linear pieces of the piecewise linear function, κAi
+k , ρAi
+k
+are certain scalar
+parameters, [LAi
+k , U Ai
+k ] are intervals for k ∈ [K(Ai)] (note that a polytope in Rn is reduced to an
+interval when it is projected onto one-dimensional space).
+This means that, for each piece of the piecewise linear function, we can identify the interval of z
+such that Ai(X(z)) ≥ τ as follows 2
+z ∈
+�
+�
+�
+�
+max
+�
+LAi
+k ,
+�
+τ − ρAi
+k
+�
+/κAi
+k
+�
+, U Ai
+k
+�
+if κAi
+k
+> 0
+�
+LAi
+k , min
+�
+U Ai
+k ,
+�
+τ − ρAi
+k
+�
+/κAi
+k
+�
+,
+�
+if κAi
+k
+< 0
+⇒
+Ai(X(z)) ≥ τ.
+(16)
+With a slight abuse of notation, let us collectively denote the finite number of intervals on z ∈ R that
+are defined by LAi
+k , U Ai
+k , (τ − ρAi
+i /κAi
+k ) for all (k, i) ∈ [K(Ai)] × [n] as
+[z0, z1], [z1, z2], . . . , [zt−1, zt], [zt, zt+1], . . . , [zT −1, zT ],
+where zmin = z0 and zmax = zT are defined such that the probability mass of z < zmin and z > zmax
+are negligibly small.
+2For simplicity, we omit the description for the case of κAi
+k
+= 0. In this case, if ρAi
+k
+≥ τ, then z ∈ [LAi
+k , UAi
+k
+] ⇒ i ∈
+MX(z).
+10
+
+Algorithm 1 SI DNN Saliency
+Input: Xobs, zmin, zmax, T ← ∅
+1: Obtain Eobs, compute η as well as a and b ← Eq. (12), and initialize: t = 1, zt = zmin
+2: for t ≤ T do
+3:
+Compute zt+1 by Auto-Conditioning (see §5)
+4:
+if EX(z),Xref (z) = Eobs in z ∈ [zt, zt+1] (by using Eq.(16)) then
+5:
+T ← T + {t}
+6:
+end if
+7:
+t = t + 1
+8: end for
+9: Identify Z ← �
+t∈T [zt, zt+1]
+10: pselective ← Eq. (14)
+Output: pselective
+Algorithm
+Algorithm 1 shows how we identify Z = {z ∈ R | MX(z),Xref(z) = Mobs}. We simply
+check the intervals of z in the order of [z0, z1], [z1, z2], ..., [zT −1, zT ] to see whether MX(z) = MX(zobs)
+or not in the interval by using Eq.(16). Then, the truncation region Z in Eq.(13) is given as Z =
+�
+t∈[T ]|EX(z),Xref (z)=Eobs for z∈[zt,zt+1][zt, zt+1].
+5
+Implementation: Auto-Conditioning
+The bottleneck of our algorithm is Line 3 in Algorithm 1, where zt+1 must be found by considering
+all relevant piecewise linear components in a complicated trained CNN. The difficulty lies not only in
+the computational cost but also in the implementation cost. To implement conditional SI in DNNs
+naively, it is necessary to characterize all operations at each layer of the network as selection events and
+implement each of the specifically[Duy et al., 2022] To circumvent this difficulty, we introduce a mod-
+ular implementation scheme called auto-conditioning, which is similar to auto-differentiation [Baydin
+et al., 2018] in concept. This enables us to conduct conditional SI for a wide class of CNNs without
+additional implementation cost.
+The basic idea in auto-conditioning is to add a mechanism to compute and maintain the interval
+z ∈ [Lf
+k, U f
+k ] for each piecewise linear component f in the network (e.g., layer API in the Keras
+framework).
+This enables us to automatically compute the interval [Lf
+k, U f
+k ] of a piecewise linear
+function f when it is obtained as concatenation and/or composition of multiple piecewise linear
+components. If f is obtained by concatenating two piecewise linear functions f1 and f2, we can easily
+obtain [Lf
+k, U f
+k ] = [Lf1
+k1, U f1
+k1 ] ∩ [Lf2
+k2, U f2
+k2 ]. However, if f is obtained as a composition of two piecewise
+linear functions f1 and f2, the calculation of the interval is given by the following lemma.
+Lemma 2. Consider the composition of two piecewise linear functions, that is, f(X(z)) = (f2 ◦
+11
+
+f1)(X(z)). Given a real value of z, the interval [Lf2
+k , U f2
+k ] in the input domain of f2 can be computed
+as
+Lf2
+k2 =
+max
+j:(∆f2
+k2γf1)j<0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+,
+U f2
+k2 =
+min
+j:(∆f2
+k2γf1)j>0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+,
+where βf1 + γf1z is the output of f1 (i.e., the input of f2). Moreover, ∆f2
+k2 and δf2
+k2 are obtained by
+verifying the value of βf1 + γf1z. Then, the interval of the composite function is obtained as follows:
+[Lf
+k, U f
+k ] = [Lf1
+k1, U f1
+k1 ] ∩ [Lf2
+k2, U f2
+k2 ]
+The proof is provided in Appendix A.3.
+Here, the variables βfk and γfk can be recursively
+computed through layers as
+βfk+1 = Ψfk
+k βfk + ψfk
+k
+and
+γfk+1 = Ψfk
+k γfk.
+Lemma 2 indicates that the intervals in which X(z) decreases can be forwardly propagated through
+these layers. This means that the lower bound LAi
+k
+and upper bound U Ai
+k
+of the current piece in the
+piecewise linear function in Eq. (15) can be automatically computed by forward propagation of the
+intervals of the relevant piecewise linear components.
+6
+Experiment
+We only highlight the main results. More details (methods for comparison, network structure, etc.)
+can be found in the Appendix A.4.
+Experimental setup.
+We compared our proposed method with the naive method, over-conditioning
+(OC) method, and Bonferroni correction. To investigate the false positive rate (FPR) we consid-
+erd, 1000 null images X = (X1, ..., Xn) and 1000 reference images Xref = (Xref
+1 , ..., xref
+n ), where
+s = sref = 0 and ε, εref ∼ N(0, In), for each n ∈ {64, 256, 1024, 4096}. To investigate the true positive
+rate (TPR), we set n = 256 and generated 1,000 images, in which si = signal for any i ∈ S where
+S is the “true” salient region whose location is randomly determined. si = 0 for any i ̸∈ S and
+ε ∼ N(0, In). We set ∆ ∈ {1, 2, 3, 4}. Reference images were generated in the same way as in the case
+of FPR. In all experiments, we set the threshold for selecting the salient region τ = 0 in the mean
+null test and τ = 5 in the global null test . We set the significance level α = 0.05. We used CAM as
+the saliency method in all experiments.
+Numerical results.
+The results of FPR control are presented in Fig. 2. The proposed method, OC,
+and Bonferroni successfully controlled the FPR in both the mean and global null test cases, whereas
+12
+
+64
+256
+1024
+4096
+n
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive Rate(FPR)
+Proposed
+OC
+Bonferroni
+Naive
+(a) Mean null test
+64
+256
+1024
+4096
+n
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+False Positive Rate(FPR)
+Proposed
+OC
+Bonferroni
+Naive
+(b) Global null test
+Figure 2: False Positive Rate (FPR) comparison.
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+True Positive Rate(TPR)
+Proposed
+OC
+Bonferroni
+(a) Mean null test
+1
+2
+3
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+True Positive Rate(TPR)
+Proposed
+OC
+Bonferroni
+(b) Global null test
+Figure 3: True Positive Rate (FPR) comparison.
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 4: Mean null test for image without tumor (pnaive = 0.00, pselective = 0.78).
+the others could not. Because naive methods failed to control the FPR, we no longer considered their
+TPR. The results of the TPR comparison are shown in Fig. 3. The proposed method has the highest
+TPR in all cases. The Bonferroni method has the lowest TPR because it is conservative owing to
+considering the number of all possible hypotheses. The OC method also has a low TPR than the
+proposed method because it considers several extra conditions, which causes the loss of TPR.
+Real data experiments.
+We examined the brain image dataset extracted from the dataset used
+in Buda et al. [2019], which included 939 and 941 images with and without tumors, respectively. The
+results of the mean null test are presented in Figs. 4 and 5. The results of the global null test are
+presented in Figs. 6 and 7. The naive p-value remains small even when the image has no tumor region,
+which indicates that naive p-values cannot be used to quantify the reliability of DNN-based salient
+regions. The proposed method successfully identified false positive and true positive detections.
+7
+Conclusion
+In this study, we proposed a novel method to conduct statistical inference on the significance of
+DNN-driven salient regions based on the concept of conditional SI. We provided a novel algorithm for
+efficiently and flexibly conducting conditional SI for salient regions. We conducted experiments on
+13
+
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 5: Mean null test for image with a tumor (pnaive = 0.00, pselective = 1.92 × 10−4).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 6: Global null test for image without tumor (pnaive = 0.03, pselective = 0.46)
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+Figure 7: Global null test for image with a tumor (pnaive = 0.00, pselective = 1.51 × 10−3).
+both synthetic and real-world datasets to demonstrate the performance of the proposed method.
+Acknowledgements
+This work was partially supported by MEXT KAKENHI (20H00601), JST CREST (JPMJCR21D3),
+JST Moonshot R&D (JPMJMS2033-05), JST AIP Acceleration Research (JPMJCR21U2), NEDO
+(JPNP18002, JPNP20006), and RIKEN Center for Advanced Intelligence Project.
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+A
+Appendix
+A.1
+Proof of Lemma 1
+In the mean null test, according to the second condition in (10), we have
+ΩX,Xref = ΩXobs,Xref
+obs
+⇔
+�
+I2n − ηMXη⊤
+MX
+η⊤
+MXηMX
+� � X
+Xref
+�
+= ΩXobs,Xref
+obs
+⇔
+� X
+Xref
+�
+= ΩXobs,Xref
+obs +
+ηMX
+∥ηMX∥2 η⊤
+MX
+� X
+Xref
+�
+.
+By defining a = qXobs,Xref
+obs, b =
+ηMX
+∥ηMX ∥2 , z = η⊤
+MX
+� X
+Xref
+�
+, we obtain the result in Lemma 1.
+In the global null test, according to the second condition in (10),
+UX,Xref = UXobs,Xref
+obs
+⇔ P ⊥
+MX
+� X
+Xref
+�
+= UXobs,Xref
+obs
+⇔ (I2n − PMX)
+� X
+Xref
+�
+= UXobs,Xref
+obs
+⇔
+� X
+Xref
+�
+= UXobs,Xref
+obs + VXobs,Xref
+obsσ−1
+����PMX
+� X
+Xref
+����� .
+By defining a = UXobs,Xref
+obs, b = VXobs,Xref
+obs, z = σ−1 ���PMX
+� X
+Xref
+���� , we obtain the result in Lemma
+1.
+A.2
+Examples of piecewise linear functions
+Examples of piecewise linear components in a trained CNN with X ∈ R2 are provided as follows:
+18
+
+ReLU: Consider f is ReLU function. Then, K(f) = 4, ψk = (0 0)⊤ for any k ∈ [4],
+Ψf
+1 =
+�
+�0
+0
+0
+0
+�
+� , Pf
+1 =
+�
+�
+�X : X1 < 0,
+X2 < 0
+�
+�
+� ,
+Ψf
+2 =
+�
+�0
+0
+0
+1
+�
+� , Pf
+2 =
+�
+�
+�X : X1 < 0,
+X2 ≥ 0
+�
+�
+� ,
+Ψf
+3 =
+�
+�1
+0
+0
+0
+�
+� , Pf
+3 =
+�
+�
+�X : X1 ≥ 0,
+X2 < 0
+�
+�
+� ,
+Ψf
+4 =
+�
+�1
+0
+0
+1
+�
+� , Pf
+4 =
+�
+�
+�X : X1 ≥ 0,
+X2 ≥ 0
+�
+�
+� .
+This can be similarly extended to the case of Leaky ReLU.
+Max-pooling: Consider f(X) = max{X1, X2}. Then, it is represented as a piecewise linear function
+with K(f) = 2, ψk = (0) for any k ∈ [2],
+Ψf
+1 =
+�
+1
+0
+�
+, Pf
+1 = {X : X1 ≥ X2} ,
+Ψf
+2 =
+�
+0
+1
+�
+, Pf
+2 = {X : X1 < X2} .
+Convolution and matrix-vector multiplication: In a neural network, the multiplication results between
+the weight matrix and the output of the previous layer and its summation with the bias vector is a
+linear function. In a CNN, the convolution operation is obviously a linear function.
+Upsampling: Consider f is the upsampling operation on X ∈ R2, then it can be represented as a
+piecewise linear function with K(f) = 1, ψ1 = (0 0 0 0)⊤,
+Ψf
+1 =
+�
+�1
+1
+0
+0
+0
+0
+1
+1
+�
+�
+⊤
+,
+Pf
+1 = R2.
+Sigmoid and hyperbolic tangent: If there is any specific demand to use non-piecewise linear activation
+functions, we can apply a piecewise-linear approximation approach to these functions.
+A.3
+Proof of Lemma 2
+At f1, given a a real value z, the input is βf0 + γf0z = a1:n + b1:nz. By checking the value of this
+input, we can easily obtain the polytope
+{∆f1
+k1(βf0 + γf0z) ≤ δf1
+k1},
+k1 ∈ [K(f1)],
+that βf0 + γf0z belongs to. Based on the obtained polytope, we can calculate the interval [Lf1
+k1, U f1
+k1 ],
+Lf1
+k1 =
+max
+j:(∆f1
+k1γf0)j<0
+(δf1
+k1)j − (∆f1
+k1βf0)j
+(∆f1
+k1γf0)j
+and
+U f1
+k1 =
+min
+j:(∆f1
+k1γf0)j>0
+(δf1
+k1)j − (∆f1
+k1βf0)j
+(∆f1
+k1γf0)j
+.
+Moreover, based on the obtained polytope, we can easily obtain Ψf1
+k1 and ψf1
+k1, k1 ∈ [K(f1)]. Therefore,
+the output of the first layer at z can be defined as
+f1(z) = Ψf1
+k1(βf0 + γf0z) + ψf1
+k1
+= βf1 + γf1z,
+19
+
+where βf1 = Ψf1
+k1βf0 + ψf1
+k1 and γf1 = Ψf1
+k1γf0. Next, we input βf1, γf1 to f2.
+At the 2nd layer, similarly, the input is βf1 + γf1z. By checking the value of this input, we can
+easily obtain the polytope
+{∆f2
+k2(βf1 + γf1z) ≤ δf2
+k2},
+k2 ∈ [K(f2)],
+that βf1 + γf1z belongs to. Based on the obtained polytope, we can calculate the interval [Lf2
+k2, U f2
+k2 ],
+Lf2
+k2 =
+max
+j:(∆f2
+k2γf1)j<0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+and
+U f2
+k2 =
+min
+j:(∆f2
+k2γf1)j>0
+(δf2
+k2)j − (∆f2
+k2βf1)j
+(∆f2
+k2γf1)j
+.
+Moreover, based on the obtained polytope, we can easily obtain Ψf2
+k2 and ψf2
+k2, k2 ∈ [K(f2)]. Therefore,
+the output of the first layer at z can be defined as
+f2(z) = Ψf2
+k2(βf1 + γf1z) + ψf2
+k2
+= βf2 + γf2z,
+where βf2 = Ψf2
+k2βf1 + ψf2
+k2 and γf2 = Ψf2
+k2γf1.
+A.4
+Experimental details.
+Methods for comparison.
+We compared our proposed method with the following approaches:
+• Naive: the classical z-test is used to calculate the naive p-value.
+• Bonferroni: the number of all possible hypotheses are considered to account for the selection
+bias. The p-value is computed by pbonferroni = min(1, pnaive ∗ 2n)
+• Over-conditioning (OC): additionally conditioning on the observed activeness and inactiveness
+of all the nodes. The limitation of this method is its low statistical power due to over-conditioning.
+Network structure.
+In all the experiments, we used the network structure shown in Fig. 8.
+Experimental setting on brain image dataset.
+We examine the brain image dataset extracted
+from the dataset used in Buda et al. [2019], which includes 941 images without tumors (C1) and 939
+images with tumors (C2). We selected 50 images from C1 as reference images. We used 841 images
+from C1 and 889 images from C2 for DNN training. The remaining images from C1 and C2 are used
+for demonstrating the advantages of the proposed selective p-value.
+More results on brain image dataset.
+Additional results are shown in Figs. 11, 12, 9 and 10
+20
+
+Conv
+MaxPooling
+GAP
+UpSampling
+Image
+Saliency Map
+Prediction
+FC
+Weight
+( 𝑛, 𝑛, 1)
+( 𝑛, 𝑛, 4)
+( 𝑛/2, 𝑛/2,4)
+(4)
+(1)
+( 𝑛, 𝑛, 4)
+( 𝑛/2, 𝑛/2,4)
+CAM
+( 𝑛, 𝑛, 4)
+( 𝑛, 𝑛, 4)
+Figure 8: Network structure.
+21
+
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 0.01, pselective = 0.47
+Figure 9: Inference on salient regions for images without tumor (mean null test).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 0.00, pselective = 2.82 × 10−4
+Figure 10: Inference on salient regions for images where there exists a tumor (mean null test).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 3.00 × 10−4, pselective = 0.29
+Figure 11: Inference on salient regions for images without tumor (global null test).
+Input Image
+Saliency Map
+Salient Region
+Reference Image
+Reference Region
+(a) pnaive = 0.00, pselective = 2.66 × 10−20
+Figure 12: Inference on salient regions for images where there exists a tumor (global null test).
+22
+

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@@ -0,0 +1,3078 @@
+arXiv:2301.12007v1  [math.OC]  27 Jan 2023
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC
+OPTIMIZATION PROBLEMS
+POUYA SAMPOURMAHANI∗, MOHAMMADHOSSEIN MOHAMMADISIAHROUDI, TAM ´AS TERLAKY
+Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA, 18015
+Abstract. Second-order conic optimization (SOCO) can be considered as a special case of semidefinite optimiza-
+tion (SDO). In the literature it has been advised that a SOCO problem can be embedded in an SDO problem using
+the arrow-head matrix transformation. However, a primal-dual solution pair cannot be mapped simultaneously
+using the arrow-head transformation as we might lose complementarity and duality in some cases. To address this
+issue, we investigate the relationship between SOCO problems, and their SDO counterpart. Through derivation of
+standard semidefinite representations of SOCO problems, we introduce admissible mappings. We show that the
+proposed mappings preserve both feasibility and optimality. Further, we discuss how the optimal partition of a
+SOCO problem maps to the optimal partition of its SDO counterpart.
+Keywords. Second-order conic Optimization; Semidefinite Optimization; Semidefinite Representation; Mapping;
+Optimal Partition.
+2020 Mathematics Subject Classification. 90C25, 90C22, 90C99.
+1. INTRODUCTION
+In the hierarchy of convex optimization problems, second-order conic optimization (SOCO)
+problems can be seen as a special case of semidefinite optimization (SDO) problems. SOCO
+problems minimize a linear function over the intersection of an affine space with the Cartesian
+product of second-order cones, also known as Lorentz cones. An SDO problem consists of
+minimizing a linear objective function over the intersection of the cone of positive semidefinite
+matrices with an affine space. SDO encompasses other subclasses of conic optimization prob-
+lems namely linear optimization (LO), and SOCO, in the hierarchy. This means that each one
+can be represented as a special case of SDO [1].
+In this paper, we focus on the relationship of SOCO and SDO. We investigate their relation-
+ship in order to gain theoretical insight and realize how these problems get mapped to each
+other. Only a few papers [9, 11] were devoted to study this relationship from a theoretical point
+of view. Sim and Zhao [9], in particular, studied the relationship between a SOCO problem
+and its counterpart SDO problem. They provided a mapping based on the direct correspon-
+dence between the dual problems of SOCO and SDO. Their SDO representation is defined on
+the product of some cones of positive semidefinite matrices, which is a special case of standard
+∗Corresponding author.
+E-mail address: [email protected] (P. Sampourmahani), [email protected] (M. Mohammadisiahroudi),
+[email protected] (T. Terlaky).
+Received xx, x, xxxx; Accepted xx, x, xxxx.
+©2023 Communications in Optimization Theory
+1
+
+2
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+SDO and needs further analysis. In this paper, we extend their analysis by considering the actual
+standard case which returns an SDO representation through a large positive semidefinite cone.
+Furthermore, we propose a framework that allows full description of the point-to-set map
+from SOCO to its SDO counterpart. Then, we analyze how the optimal partition of a SOCO
+problem is mapped to that of SDO, and vice versa. This is important in understanding the
+relationship between these two problems as we are mapping between an index-based partition
+and a subspace-based partition.
+Throughout this paper, the following notation is used. The Lorentz cone of dimension ni
+is denoted by L ni, and Rn denotes the n-dimensional Euclidean space. Superscripts are used
+to represent cone-related information, and subscripts are used for matrix and vector entries.
+For a given matrix A, Ai j represents the (i, j)-th entry, while Ai denotes the i-th matrix. The
+notation (.;.;...;.) denotes the concatenation of the column vectors. The set of all p×q matrices
+with real entries is denoted by Rp×q. For a symmetric matrix X, X ⪰ 0 (X ≻ 0) means X is
+positive semidefinite (positive definite). Furthermore, the trace operator is denoted by Tr(.).
+The remaining notations will be introduced at appropriate places.
+This paper is structured as follows. Section 2 reviews the preliminaries required for this pa-
+per. Section 3 studies the relationship between SOCO and SDO relying on the correspondence
+of dual problems. Section 4 takes the other direction and proposes mappings focusing on corre-
+spondence of primal problems as the starting point. Section 5 analyzes how the optimal partition
+of SOCO maps to that of it’s SDO counterpart. Section 6 concludes the paper, summarizing our
+results.
+2. PRELIMINARIES
+Let L ni denote the Lorentz cone of dimension ni, and L n = L n1 ×L n2 ×...×L nr, where
+n = ∑r
+i=1 ni. Then, the primal and dual SOCO problems are defined as follows
+z∗
+PSOCO := min (c1)Tx1 +...+(cr)Txr
+s.t. A1x1 +...+Arxr = b,
+xi ∈ L ni,
+for i = 1,...,r,
+(PSOCO)
+z∗
+DSOCO := max bTy
+s.t. (Ai)Ty+si = ci,
+for i = 1,...,r,
+si ∈ L ni,
+for i = 1,...,r,
+(DSOCO)
+where ci ∈ Rni, Ai ∈ Rm×ni, b ∈ Rm. We define the feasible sets of the primal-dual problems as
+follows,
+FPSOCO = {(x1;x2;...;xr) ∈ L n : A1x1 +...+Arxr = b},
+FDSOCO = {(y;s1;s2;...;sr) ∈ Rm ×L n : (Ai)Ty+si = ci for i = 1,...,r},
+and the sets of optimal solutions as
+P∗
+SOCO = {x∗ = (x1;x2;...;xr) ∈ FPSOCO : cTx∗ = z∗
+PSOCO},
+D∗
+SOCO = {(y∗,s∗) = (y;s1;s2;...;sr) ∈ FDSOCO : bTy∗ = z∗
+DSOCO},
+respectively. An optimal solution of SOCO, if there exists any, is denoted by (x∗;y∗;s∗).
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+3
+Let Arw(·) denote the arrow-head (Lorentz) transformation [8, 10], with the structure of
+Arw(xi) :=
+� xi
+1
+(xi
+2:ni)T
+xi
+2:ni
+xi
+1Ini−1
+�
+,
+where (xi
+2:ni)T denotes the vector (xi
+2,...,xi
+ni). Then, the Jordan product is defined as
+xi ◦si = Arw(xi)si = Arw(si)xi =
+�
+(xi)Tsi
+xi
+1si
+2:ni +si
+1xi
+2:ni
+�
+,
+i = 1,...,r.
+Any feasible solutions satisfying x◦s = 0 is called complementary. Here, we have
+x◦s := (x1 ◦s1,x2 ◦s2,...,xr ◦sr).
+Feasible solution are complementary if and only if they are optimal with zero duality gap.
+Definition 2.1. An optimal solution (x∗;y∗;s∗) ∈ P∗
+SOCO × D∗
+SOCO is called maximally com-
+plementary if x∗ ∈ ri(P∗
+SOCO) and (y∗;s∗) ∈ ri(D∗
+SOCO). Further, (x∗;y∗;s∗) is called strictly
+complementary if x∗ +s∗ ∈ int(L n).
+Next, we define the primal and dual SDO problems.
+z∗
+PSDO := min Tr(CX)
+s.t. Tr(AiX) = bi
+for all i = 1,...,m,
+X ⪰ 0,
+(PSDO)
+z∗
+PSDO := max bTy
+s.t.
+m
+∑
+i=1
+yiAi +S = C,
+S ⪰ 0,
+(DSDO)
+where X,S,C, and Ai for i = 1,...,m are n×n symmetric matrices, and b,y ∈ Rm. We define
+the feasible sets of SDO problems as
+FPSDO = {X ∈ Sn : Tr(AiX) = bi,i = 1,...,m,X ⪰ 0},
+FDSDO = {(y,S) ∈ Rm ×Sn :
+m
+∑
+i=1
+yiAi +S = C,S ⪰ 0},
+where Sn denotes the set of n ×n symmetric matrices. The sets of optimal solutions for a pair
+of SDO problems are
+P∗
+SDO = {X∗ ∈ FPSDO : Tr(CX∗) = z∗
+PSDO},
+D∗
+SDO = {(y∗,S∗) ∈ FDSDO : bTy∗ = z∗
+DSDO}.
+A feasible and an optimal solution of SDO are denoted as (X,y,S), and (X∗,y∗,S∗), respectively.
+Any feasible solution (X,y,S) satisfying XS = 0 is called complementary. Similar to SOCO, a
+feasible solution is optimal and yields zero duality gap if and only if it is complementary.
+Definition 2.2. A primal-dual optimal solution (X∗,y∗,S∗) ∈ P∗
+SDO × D∗
+SDO is called maxi-
+mally complementary if X∗ ∈ ri(P∗
+SDO) and (y∗,S∗) ∈ ri(D∗
+SDO). A maximally complementary
+optimal solution (X∗,y∗,S∗) is called strictly complementary if X∗ +S∗ ≻ 0.
+
+4
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Next, we define the optimal partitions of SOCO and SDO. The notion of the optimal partition
+of LO can be extended to SOCO [8]. Even though a SOCO problem can be embedded in SDO,
+the optimal partition in SOCO may be more nuanced when it is defined and analyzed directly
+in the SOCO setting. In SOCO, the index set {1,...,r} of the second-order cones is partitioned
+into four subsets
+¯
+B,
+¯
+N , ¯
+R, and
+¯
+T , where
+¯
+T is further partitioned to
+¯
+T := ( ¯T1, ¯T2, ¯T3) as
+follows,
+¯
+B := { i | xi
+1 > ||xi
+2:ni||2, for some x ∈ P∗
+SOCO},
+¯
+N := { i | si
+1 > ||si
+2:ni||2, for some s ∈ D∗
+SOCO},
+¯
+R := { i | xi
+1 = ||xi
+2:ni||2 > 0,si
+1 = ||si
+2:ni||2 > 0, for some (x;y;s) ∈ P∗
+SOCO ×D∗
+SOCO},
+¯
+T1 := { i | xi = si = 0, for all (x;y;s) ∈ P∗
+SOCO ×D∗
+SOCO},
+¯
+T2 := { i | si = 0, for all (y;s) ∈ D∗
+SOCO, and xi
+1 = ||xi
+2:ni||2 > 0, for some x ∈ P∗
+SOCO},
+¯
+T3 := { i | xi = 0, for all x ∈ P∗
+SOCO, and si
+1 = ||si
+2:ni||2 > 0, for some (y;s) ∈ D∗
+SOCO}.
+It should be highlighted that, due to the convexity of the optimal set,
+¯
+B,
+¯
+N , ¯
+R, and
+¯
+T are
+mutually disjoint and their union is the index set {1,...,r}. Therefore, it follows from the
+complementarity condition that for all (˜x∗; ˜y∗; ˜s∗) ∈ P∗
+SOCO ×D∗
+SOCO, ˜xi = 0 for all i ∈
+¯
+N , and
+˜si = 0 for all i ∈ ¯
+B [8].
+For SDO, let B := R(X∗) and N := R(S∗), where (X∗,y∗,S∗) is a maximally comple-
+mentary optimal solution, meaning that we have R(X) ⊆ B and R(S) ⊆ N for all (X,y,S) ∈
+P∗
+SDO × D∗
+SDO. By the complementarity condition, the subspaces B and N are orthogonal.
+Moreover, let subspace T , be the orthogonal complement to B+N . The partition (B,N ,T )
+of Rn is called the optimal partition of an SDO problem. We can represent X∗ and S∗ using
+a common eigenvector basis, Q∗, as X∗ = Q∗Λ(X∗)(Q∗)T, and S∗ = Q∗Λ(S∗)(Q∗)T, where
+Λ(X∗) and Λ(S∗) corresponds to the diagonal matrices containing the eigenvalues of X∗ and S∗,
+respectively. Thus, we have R(X∗) = R(Q∗Λ(X∗)), and R(S∗) = R(Q∗Λ(S∗)). In particular,
+the columns of Q∗ corresponding to the positive eigenvalues of X∗ and S∗ can be chosen as an
+orthonormal basis for B and N , respectively [8].
+Current literature [1, 2, 3, 4, 5, 7, 9, 8, 10] suggests that a SOCO problem can be embedded
+in an SDO problem using the arrow-head matrix transformation,
+Arw(xi) :=
+� xi
+1
+(xi
+2:ni)T
+xi
+2:ni
+xi
+1Ini−1
+�
+⪰ 0 ⇔ xi ∈ L ni.
+(2.1)
+However, this transformation cannot be used to map both primal and dual solutions at the same
+time. Upon using the arrow-head representation of vectors xi and si simultaneously, we might
+lose duality and complementarity. The following example illustrates that we may lose comple-
+mentarity.
+Example 2.3. Let (x;y;s) be an optimal solution of SOCO and assume that there exist at least
+one index i ∈ R. For all i ∈ R, we can represent a solution as
+xi = ζ i
+
+
+1
+xi
+2:ni
+||xi
+2:ni||2
+
+,
+si = ξ i
+
+
+1
+si
+2:ni
+||si
+2:ni||2
+
+,
+xi
+2:ni
+||xi
+2:ni||2
+= −
+si
+2:ni
+||si
+2:ni||2
+,
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+5
+where ζ i = xi
+1 ≥ 0, ξ i = si
+1 ≥ 0, and for at least one (x∗;y∗;s∗) we have both ζ i,ξ i > 0. Without
+loss of generality, and for the sake of simplicity, assume that ζ i = ξ i = 1. Moreover, let
+uj =
+xi
+j
+||xi
+2:ni||2
+= −
+si
+j
+||si
+2:ni||2
+,
+j = 2,...,ni,
+and u = (u2;...;uni). Then, using the arrow-head matrix transformation, we have
+Xi = Arw(xi) =
+�1
+uT
+u
+Ini−1
+�
+,
+Si = Arw(si) =
+� 1
+−uT
+−u
+Ini−1
+�
+.
+While xi ◦si = 0, this transformation does not preserve complementarity as we have
+XiSi =
+�1
+uT
+u
+Ini−1
+�� 1
+−uT
+−u
+Ini−1
+�
+=
+�
+0
+[0]1×(n−1)
+[0](n−1)×1
+Ini−1 −uuT
+�
+̸= 0.
+Example 2.3 shows that the arrow-head matrix transformation is not sufficient to represent a
+primal-dual pair of SOCO problems as an SDO problem. Thus, it seems worth exploring the
+actual relationship between an instance of SOCO and it’s SDO counterpart.
+To address this issue, Sim and Zhao [9] started from a SOCO dual problem and exploited the
+arrowhead representation (2.1) of the dual SOCO problem, to obtain the SDO dual as follows,
+max bTy
+s.t.
+m
+∑
+i=1
+yiArw(aj
+(i))+S j = Arw(cj) for all j = 1,...,r,
+S j ⪰ 0 for all j = 1,...,r,
+(DSZ)
+where aj
+(i) denotes ith row of the matrix A corresponding to Lorentz cone j. Observe that S j as a
+linear combination of arrow-head matrices is an arrow-head matrix, too. Using this dual model,
+we get the SDO primal problem as
+min
+r
+∑
+j=1
+Tr(Arw(cj)X j)
+s.t.
+r
+∑
+j=1
+Tr(Arw(aj
+(i))X j) = bi for all i = 1,...,m,
+X j ⪰ 0 for all j = 1,...,r.
+(PSZ)
+They showed that X j = Arw(xj) is not a feasible solution for the SDO primal problem (PSZ).
+In fact, primal feasible solutions of (PSZ) are fully dense, and do not have an arrow-head
+structure. To fix this issue, they proposed the mapping
+MR(xj) =
+� 1
+4θ j
+1
+2(xj
+2:n)T
+1
+2xj
+2:n
+x j
+1−∥x j
+2:n∥
+2(n−1) I + x j
+2:n(x j
+2:n)T
+θ j
+�
+,
+(2.2)
+where θ j = xj
+1 +∥x j
+2:n∥+
+�
+(xj
+1 +∥xj
+2:n∥)2 −4∥xj
+2:n∥2.
+In our study, a key concept is the notion of admissible map which is defined next.
+
+6
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Definition 2.4. A mapping M is called admissible if it preserves feasibility and objective func-
+tion value, i.e.
+(x,y,s) ∈ FPSOCO ×FDSOCO ⇒ M (x,y,s) ∈ FPSDO ×FDSDO,
+(X,y,S) ∈ FPSDO ×FDSDO ⇒ M −1(X,y,S) ∈ FPSOCO ×FDSOCO,
+cTx = Tr(CX), bTy = bTy.
+The mapping of Sim and Zhao [9] is admissible, and they proved that it maps a solution from
+the boundary (interior) of the Lorentz cone to a solution on the boundary (interior) of the cone
+of semidefinite matrices. In this paper, we seek to extend their approach and explore mappings
+that satisfy the definition of admissible mapping. Although the mapping of [9] is a point to
+point map, the image of (x,y,s) might be a point or a set. In addition, the SDO representation
+of SOCO of [9], (PSZ) and (DSZ), is defined using the product of multiple cones of positive
+semidefinite matrices, but we use a more general approach to get an SDO representation in
+standard form. The major goal of this paper is clarifying more the relationship between SOCO
+and the related SDO by developing different mappings and exploring the relationship between
+the optimal partitions of these problems.
+Without loss of generality, in Sections 3 and 4, we first present the results in case of a single
+second-order cone, and then we generalize the results to the multiple cone case. To this end, we
+consider the following primal and dual problems,
+z∗
+P1
+SOCO = min
+�
+cTx : Ax = b, x ∈ L n�
+,
+(P1
+SOCO)
+z∗
+D1
+SOCO = max
+�
+bTy : ATy+s = c, (y,s) ∈ Rm ×L n�
+,
+(D1
+SOCO)
+with feasible sets FP1
+SOCO = {x ∈ L n : Ax = b} and FD1
+SOCO = {(y,s) ∈ Rm×L n : ATy+s = c},
+and optimal solution sets P1
+SOCO
+∗ = {x ∈ FP1
+SOCO : cTx = z���
+P1
+SOCO} and D1
+SOCO
+∗ = {(y,s) ∈
+FD1
+SOCO : bTy = z∗
+D1
+SOCO}, respectively.
+3. FROM SOCO TO SDO: STARTING FROM THE DUAL SIDE
+One can derive the SDO counterpart of a SOCO problem starting with either the primal or
+dual SOCO problem. In this section, similar to [9], we initiate the derivation from the dual
+side of SOCO. Thus, as mentioned earlier, we preserve the arrow-head structure of the matrix S
+corresponding to dual solution s.
+3.1. Derivation and Solution Mapping. We utilize the arrow-head transformation to the vec-
+tor c and the rows of matrix A,
+⃗C = Arw(c),
+⃗Ai = Arw(a(i)),
+i = 1,2,...,m.
+Since in the SOCO dual s = c−ATy, by applying the arrow-head structure to A and c, we have
+that S = ⃗C −∑m
+i=1 yi⃗Ai has the arrow-head structure, as it is a linear combination of arrow-head
+matrices. Therefore, the SDO counterpart of the SOCO dual problem (D1
+SOCO) is as follows,
+z∗
+DD
+SDO := max
+�
+bTy :
+m
+∑
+i=1
+yi⃗Ai +S = ⃗C, S ⪰ 0
+�
+,
+(DD
+SDO)
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+7
+which has the following dual,
+z∗
+PD
+SDO := min
+�
+Tr(⃗CX) : Tr(⃗AiX) = bi,
+i = 1,...,m,
+X ⪰ 0
+�
+,
+(PD
+SDO)
+as it’s SDO primal problem. For the SDO problems (PD
+SDO) and (DD
+SDO), let
+FDD
+SDO = {(y,S) ∈ Rm ×Sn :
+m
+∑
+i=1
+yi⃗Ai +S = ⃗C,S ⪰ 0},
+FPD
+SDO = {X ∈ Sn : Tr(⃗AiX) = bi,i = 1,...,m,X ⪰ 0}
+represent the feasible sets, and
+DD
+SDO
+∗ = {(y,S) ∈ FDD
+SDO : bTy = z∗
+DD
+SDO},
+PD
+SDO
+∗ = {X ∈ FPD
+SDO : Tr(⃗CX) = z∗
+PD
+SDO},
+represent the optimal solution sets, respectively. The following theorem provides a point to set
+admissible mapping, see Definition 2.4 for r = 1, based on the (D1
+SOCO) and (P1
+SOCO), and their
+representations (DD
+SDO), (PD
+SDO).
+Theorem 3.1. Consider the SOCO problem pairs (P1
+SOCO) and (D1
+SOCO) with (x,y,s) ∈ FP1
+SOCO ×
+FD1
+SOCO, and the SDO problem pairs (PD
+SDO) and (DD
+SDO) with (X,y,S) ∈ FPD
+SDO × FDD
+SDO.
+Then, mapping (X,y,S) = M (x,y,s) with
+S = Arw(s),
+y = y,
+X =
+
+
+X11
+X12
+...
+X1n
+X12
+X22
+...
+X2n
+...
+...
+...
+...
+X1n
+X2n
+...
+Xnn
+
+ ⪰ 0 with
+
+
+∑n
+i=1Xii
+X12
+...
+X1n
+
+ =
+
+
+x1
+x2
+2...
+xn
+2
+
+,
+is a point-to-set admissible mapping. In addition, the inverse mapping denoted by (x,y,s) =
+M −1(X,y,S), with
+s = Arw−1(S),
+y = y,
+x =
+�
+∑n
+i=1 Xii,2X12,...,2X1n
+�T ,
+is a point-to-point admissible mapping.
+Proof. The proof of this theorem is presented in Appendix A.
+□
+The following corollaries restate that the provided mapping preserves the objective function
+value.
+Corollary 3.2. We have z∗
+P1
+SOCO = z∗
+PD
+SDO, and z∗
+D1
+SOCO = z∗
+DD
+SDO.
+Corollary 3.3. A feasible solution (x,y,s) ∈ FP1
+SOCO ×FD1
+SOCO is optimal for a pair of SOCO
+problems (P1
+SOCO), and (D1
+SOCO) with optimal value (z∗
+P,z∗
+D) if and only if the mapped solution
+(X,y,S) is optimal for the SDO problems (PD
+SDO), and (DD
+SDO) with optimal value (z∗
+P,z∗
+D) .
+Corollary 3.4. A feasible solution (x,y,s) ∈ FP1
+SOCO ×FD1
+SOCO is optimal for a pair of SOCO
+problems (P1
+SOCO), and (D1
+SOCO) with zero duality gap if and only if the mapped solution
+(X,y,S) is optimal for the SDO problems (PD
+SDO), and (DD
+SDO) with zero duality gap, i.e.,
+x◦s = 0 ⇐⇒ Tr(XS) = 0
+
+8
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Note that these results are valid regardless of duality (strong / weak with gap), status of the
+SOCO problems. Furthermore, observe that one can propose different admissible mappings
+that satisfies the conditions presented in Theorem 3.1. In Section 3.2, we propose a rank-one
+mapping, which is the simplest option. In Section 3.3, we show that when x ∈ int(L n), full
+rank mappings can also be obtained, which map a solution in the interior of SOCO to a solution
+in the interior of the SDO cone.
+3.2. Rank-one Mapping. In this section, we construct a rank-one matrix X for vector x that
+satisfies the conditions in Theorem 3.1. Thus, we introduce the vector β ∈ Rn, and define the
+rank-one matrix.
+X = ββ T,
+with
+n
+∑
+i=1
+β 2
+i = x1,
+β1β j = xj
+2 for all j = 2,...,n.
+We need to solve this n-variable-n-equation system. The solution of this system is β = 0 if
+x = 0, and if x ̸= 0 then
+β =
+1
+�
+2(x1 +δ)
+(x1 +δ,x2,...,xn)T,
+where δ =
+�
+(x1)2 −||x2:n||2. It is easy to see that if we are on the boundary of the second-
+order cone, then δ = 0. Otherwise, we have δ ̸= 0. Using this vector, we can construct a
+suitable matrix X.
+Theorem 3.5. Consider the rank-one mapping with
+X =
+�
+[0]n×n
+if x = (0,0,...,0),
+DMR1(x)
+otherwise.
+(3.1)
+where
+DMR1(x) = ββ T =
+
+
+x1+δ
+2
+x2
+2
+...
+xn
+2
+x2
+2
+x2
+2
+2[x1+δ]
+...
+x2xn
+2[x1+δ]
+...
+...
+...
+...
+xn
+2
+x2xn
+2[x1+δ]
+...
+x2n
+2[x1+δ]
+
+
+.
+Then, (3.1) together with (y,S) = (y,Arw(s)) is a point-to-point admissible mapping.
+Proof. The proof is straightforward as it is enough to show that matrix DMR1(x) satisfies the
+conditions in Theorem 3.1.
+□
+3.3. Higher Rank Mapping. In this section, we show that when a SOCO solution is in the
+interior of the cone, i.e. x ∈ int(L n), then we can use a full rank mapping, i.e.
+DMRn(x) =
+n
+∑
+i=1
+β i(β i)T.
+Theorem 3.6. There exist mappings DMR where rank(DMR(¯x)) = n for x ∈ int(L n).
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+9
+Proof. The proof of this theorem is presented in Appendix B.
+□
+One can easily modify Algorithm 1 (presented in proof of Theorem 3.6) to map a solution
+x ∈ int(L n) to a matrix DMRk(x) with rank 1 ≤ k ≤ n. This means that the primal feasible set
+of a SOCO can be mapped to different subsets of the SDO primal feasible region, e.g., rank-one
+mapping maps the primal feasible set of a SOCO to a one-dimensional face of the SDO primal
+feasible set.
+Recall that Theorem 1 of [9] proposes the mapping
+MR(x) =
+� 1
+4θ
+1
+2xT
+2:n
+1
+2x2:n
+x1−∥x2:n∥
+2(n−1) I + x2:nxT
+2:n
+θ
+�
+,
+where θ = x1 +∥x2:n∥+
+�
+(x1 +∥x2:n∥)2 −4∥x2:n∥2. They showed that this map is admissible.
+Moreover, it has full rank when x1 > ∥x2:n∥, i.e. x ∈ int(L n), and it is a rank-one matrix when
+x1 = ∥x2:n∥. This map also proves Theorem 3.6, while our proof follows a different approach.
+In our approach, we can generate different mappings and explore the feasible set of SDO by
+changing parameter ε in Algorithm 1. However, to prove that the set of all maps with different
+rank produced by our approach can build the whole feasible set of the SDO representation is an
+ongoing research.
+Given the mapping of Sim and Zhao [9], consider the case in which the solution is on the
+boundary of the second-order cone, i.e. x1 = ∥x2:n∥. Then, θ = 2x1, and
+MR1(x) =
+� 1
+2x1
+1
+2xT
+2:n
+1
+2x2:n
+x2:nxT
+2:n
+2x1
+�
+.
+We can see that this is exactly identical to the rank one mapping we presented in Theorem 3.5.
+We can write it as
+MR1(x) = ν1(ν1)T,
+where
+ν1 =
+1
+√
+θ
+�1
+2θ,x2,...,xn
+�T
+.
+In order to construct MR(x) as the sum of rank-one matrices, we define
+ν j =
+�
+x1 −∥x2:n∥
+2(n−1)
+ej
+for j = 2,...,n,
+where ej is a unit vector with 1 in element j. By this setting, we have
+MR(x) =
+n
+∑
+j=1
+ν j(ν j)T.
+If the solution x of the SOCO primal problem is on the boundary of the cone, i.e. x1 = ∥x2:n∥,
+then we get
+ν1 =
+1
+√2x1
+(x1,x2,...,xn)T ,
+ν j = 0,
+for j = 2,...,n.
+
+10
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+This results in the rank one mapping MR1(x). On the other hand, if the solution is in the interior
+of the cone, i.e. x1 > ∥x2:n∥, then ν j ̸= 0 for j = 2,...,n, and we can construct the full rank
+mapping MR(x). Note that one cannot take a combination of first k vectors ν j to construct a
+rank-k mapping as it would not be admissible since it violates the conditions given in Theorem
+3.1, i.e. sum of diagonals will not be equal to x1. To overcome this issue and construct a rank-k
+mapping, let N ⊆ {2,...,n} with |N | = k. One can take the following definition of ν j,
+ν j =
+�
+x1 −∥x2:n∥
+2(k −1)
+ej
+for j ∈ N , and ν j = 0 for j /∈ N .
+The resulting matrix MRk(x) has rank k, and one can easily see that it satisfies the conditions
+of Theorem 3.1. Considering k = n, this choice of ν j results in identical mapping to MR(x).
+Next theorem shows that we can have mappings with different ranks when we are mapping a
+solution from interior of the Lorentz cone.
+Theorem 3.7. Let ρ(x) = max{rank(M (x)) : M is admissible map}. We have
+• ρ(x) = n if x ∈ int(L n).
+• ρ(x) = 1 if x ∈ ∂(L n).
+Proof. Proof of this theorem is similar to Theorem 1 of [9].
+□
+In the next section, we generalize our result for the case there are multiple Lorentz cones.
+3.4. Generalization to Multiple SOCs. In this section, we extend our rank-one mapping to
+the case with multiple second-order cones. We can use similar conditions as in Theorem 3.1 to
+extend our mapping to the case of multiple second-order cones. Although, instead of the “Arw”
+operator, we need to introduce a new operator called “DArw” which constructs a block diagonal
+matrix with arrow-head matrices of input vectors. First, we transform the objective coefficient
+vectors and coefficient matrices into proper block-diagonal structure as
+˜C = DArw(c1,c2,...,cr) =
+
+
+⃗C1
+⃗C2
+...
+⃗Cr
+
+,
+where ⃗Ci is the arrow-head matrix corresponding to vector ci for all i = 1,...,r. Moreover, we
+define a similar block diagonal matrix
+˜Aj = DArw(A1
+j,A2
+j,...,Ar
+j) =
+
+
+⃗A1
+j
+⃗A2
+j
+...
+⃗Ar
+j
+
+
+for all j = 1,...,m,
+where Ai
+j corresponds to the row j of matrix Ai for all i = 1,...,r. Now, the SDO representation
+of the SOCO problem can be derived as (PSDO) and (DSDO). Compared to the SDO represen-
+tation of [9], (PSZ) and (DSZ), our representation is a standard SDO over one cone of positive
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+11
+semidefinite matrices of dimension n×n. Given the introduced notations ˜C and ˜A, we have
+z∗
+P ˜D
+SDO := min Tr( ˜C ˜X)
+s.t. Tr( ˜Ai ˜X) = bi
+for all i = 1,...,m,
+˜X ⪰ 0,
+(P ˜D
+SDO)
+and the dual problem is
+z∗
+D ˜D
+SDO := max bTy
+s.t.
+m
+∑
+i=1
+yi ˜Ai + ˜S = ˜C,
+˜S ⪰ 0.
+(D ˜D
+SDO)
+Analogously, we can define the sets of feasible and optimal solutions for the problems (P ˜D
+SDO)
+and (D ˜D
+SDO).
+We can present the following theorem to specify admissible mappings for the general form.
+Theorem 3.8. Consider the SOCO problem pairs (PSOCO) and (DSOCO) with
+(x1;x2;...;xr;y;s1;s2;...;sr) ∈ FPSOCO ×FDSOCO,
+and SDO problem pairs (P ˜D
+SDO) and (D ˜D
+SDO) with ( ˜X, ˜y, ˜S) ∈ FP ˜D
+SDO × FD ˜D
+SDO. Then, the fol-
+lowing mapping, denoted by ( ˜X, ˜y, ˜S) = M (x1;x2;...;xr;y;s1;s2;...;sr), with
+˜S = DArw(s1;...;sr),
+˜y = y,
+˜X ⪰ 0
+s.t.
+
+
+∑ui+ni
+j=ui ˜Xj j
+˜Xui,ui+1
+...
+˜Xui,ui+ni
+
+ =
+
+
+xi
+1
+xi
+2
+2...
+xin
+2
+
+
+, for all i = 1,...,r
+coupled with the inverse mapping denoted by (x1;x2;...;xr;y;s1;s2;...;sr) = M −1( ˜X, ˜y, ˜S) with
+(s1;...;sr) = DArw−1( ˜S),
+y = ˜y,
+(x1;...;xr) =
+
+
+∑ui+ni
+j=ui ˜Xj j
+2 ˜Xui,ui+1
+...
+2 ˜Xui,ui+ni
+
+,
+where ui = 1+∑i−1
+k=0 ni and n0 = 0, is an admissible mapping.
+Proof. The proof of this theorem is analogous to that of Theorem 3.1.
+□
+Remark 3.9. Note that a solution of (P ˜D
+SDO) is not necessarily block-diagonal.
+
+12
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Using the conditions in Theorem 3.7, we can write
+˜S = DArw(s1,s2,...,sr) =
+
+
+Arw(s1)
+Arw(s2)
+...
+Arw(sr)
+
+,
+and,
+˜y = y.
+We can define matrix ˜X using our rank one mapping with vector ˜β. Vector ˜β consists of n
+elements partitioned into r sub-vectors each corresponds to the cones in the problem. Thus, we
+have
+˜X = ˜β ˜β T,
+where
+˜β = ((β 1)T,(β 2)T,...,(β r)T)T.
+One can calculate vector ˜β in the closed form as follows,
+β i =
+1
+�
+2(xi
+1 +δ i)
+�
+xi
+1 +δ i,xi
+2,...,xi
+ni
+�
+,
+where δ i =
+�
+(xi
+1)2 −||xi
+2:n||2. We can also have a separate rank-one mapping for each Lorentz
+cone as ˜X = ∑r
+i=1(β)i(β)T
+i , where
+(β j)i =
+
+
+
+1
+√
+2(xi
+1+δ i)
+�
+xi
+1 +δ i,xi
+2,...,xi
+ni
+�
+if j = i,
+0
+otherwise.
+In this case, ˜X is a block diagonal matrix with rank equal to r. If we choose (β j)i ∈ Rnj for
+j ∈ {1,...,r}/{i}, then ˜X = ∑r
+i=1(β)i(β)T
+i will be a positive semidefinite matrix, which is not
+necessarily block diagonal. Since all the input matrices are block diagonal, it is straightforward
+to check that the mapping will remain admissible. For any Lorentz cone i with xi
+1 > ∥xi
+2:n∥, the
+corresponding part of the matrix ˜X can have any rank from 1 to ni using higher rank mappings
+proposed in the previous section. For instance, if the solution x is in the interior of all Lorentz
+cones, i.e. xi
+1 > ∥xi
+2:n∥ for i = 1,...,r, then we can find ˜X = ∑r
+i=1∑nr
+k=1(β)k
+i (β)k
+i
+T whose rank is
+n×r. In this case, it is straightforward to produce mapping with arbitrary rank form 1 to n×r
+by using Algorithm 1 for filling diagonal blocks and fill other parts with arbitrary numbers.
+4. FROM SOCO TO SDO: STARTING FROM THE PRIMAL SIDE
+In this section, we analyze the case when we start to reformulate the standard primal SOCO
+as an equivalent SDO problem, which requires more complex reformulation and was left un-
+touched by Sim and Zhao [9]. The purpose is to investigate the SOCO-SDO relationship starting
+from the primal side and answer the following questions.
+(1) What happens if we force matrix X to be arrow-head?
+(2) Does this derivation results in similar mappings between the SOCO problems and their
+SDO counterparts?
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+13
+4.1. Derivation and Solution Mapping. Recall SOCO problems (P1
+SOCO) and (D1
+SOCO) and
+their corresponding feasible and optimal solution sets from Section 2. Recall that we have
+x ∈ L n if and only if X = Arw(x) ⪰ 0, thus starting from (P1
+SOCO) requires a different choice
+of ⃗C and ⃗Ai in order to properly represent the primal constraint of SOCO in an equivalent SDO
+reformulation. Thus, we define
+⃗C = Arw(1
+nc1, 1
+2c2,..., 1
+ncn),
+⃗Ai = Arw(1
+nai1, 1
+2ai2,..., 1
+2ain),
+i = 1,2,...,m.
+which leads to preserving aT
+(i)x = Tr(⃗AiX) = bi, i = 1,...,m, where X = Arw(x). Using the just
+introduced arrow-head representations of c, a(i), and x, we write the following SDO problem.
+min
+�
+Tr(⃗CX) : Tr(⃗AiX) = bi,
+i = 1,...,m,
+X ⪰ 0
+�
+.
+However, in this SDO problem X is a semidefinite matrix without arrow-head structure. Thus, in
+order to represent (P1
+SOCO) using this SDO model, we need to enforce the arrow-head structure
+on matrix X. In other words, we need to translate the arrow-head structure requirement into
+linear constraints. Thus, we introduce symmetric matrices ˇAhl for 2 ≤ h < l ≤ n with (ˇAhl)hl =
+(ˇAhl)lh = 1 for 2 ≤ h < l ≤ n, and all other entries are zero. Moreover, we introduce matrix ˆAk,
+with (ˆAk)11 = 1,(ˆAk)kk = −1 for k = 2,...,n, and rest of the entries are zero. Therefore, we write
+the following model
+min Tr(⃗CX)
+s.t. Tr(⃗AiX) = bi,
+i = 1,...,m
+Tr(ˇAhlX) = 0,
+h = 2,...,n−1, h < l ≤ n
+Tr(ˆAkX) = 0,
+k = 2,...,n
+X ⪰ 0,
+(PP
+SDO)
+which is an accurate SDO representation of (P1
+SOCO) as the arrow-head structure of the matrix
+X is enforced by the linear constraints. Let z∗
+PP
+SDO denote the optimal objective function value
+of (PP
+SDO), and
+FPP
+SDO = {X ∈ Sn :Tr(⃗AiX) = bi,i = 1,...,m,
+Tr(ˇAhlX) = 0,h = 2,...,n, h < l,
+Tr(ˆAkX) = 0,k = 2,...,n,
+X ⪰ 0},
+and
+PP
+SDO
+∗ = {X ∈ FPP
+SDO : Tr(⃗CX) = z∗
+PP
+SDO},
+
+14
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+be the feasible and optimal solution sets of problem (PP
+SDO), respectively. Next, we present the
+dual model of the SDO problem (PP
+SDO) as
+max bTv
+s.t.
+m
+∑
+i=1
+vi⃗Ai + ∑
+h̸=1,h<l
+whl ˇAhl +
+n
+∑
+k=2
+ukˆAk +S =⃗C,
+S ⪰ 0,
+(DP
+SDO)
+where whl denotes the dual variable corresponding to the matrix dedicated for setting entries
+(h,l) and (l,h) in X equal to zero, and uk corresponds to the linear constraints that are setting
+elements on the diagonal of each block equal to each other for that specific block.
+Similarly, let z∗
+DP
+SDO denote the optimal objective function value for the dual model (DP
+SDO), and
+FDP
+SDO =
+�
+(v,w,u,S) ∈ Rm ×R
+(n−1)(n−2)
+2
+×R(n−1) ×Sn :
+m
+∑
+i=1
+vi⃗Ai + ∑
+h̸=1,h<l
+whl ˇAhl +
+n
+∑
+k=2
+uk ˆAk +S =⃗C, S ⪰ 0
+�
+and
+DP
+SDO
+∗ = {(v,w,u,S) ∈ FDP
+SDO : bTv = z∗
+DP
+SDO},
+be the feasible and optimal solution sets of problem (DP
+SDO), respectively.
+Let’s analyze the structure of S. First, recall problem (D1
+SOCO), where we have
+s j = cj −
+m
+∑
+i=1
+yiai j,
+for all j = 1,...,n.
+Considering that, we seek to preserve the objective function value throughout the mapping, we
+have v = y. Therefore, using the non-zero structure of the matrices ˇAhl and ˆAk, we have
+S =
+
+
+1
+n(s1)−∑n
+k=2 uk
+1
+2(s2)
+...
+...
+1
+2(sn)
+1
+2(s2)
+1
+n(s1)+u2
+−w23
+...
+−w2n
+...
+−w23
+1
+n(s1)+u3
+...
+...
+...
+...
+...
+...
+−w(n−1)n
+1
+2(sn)
+−w2n
+...
+−w(n−1)n
+1
+n(s1)+un
+
+
+.
+(4.1)
+Furthermore, we have free variables w and u which can take values such that S is positive
+semidefinite. Now, let y = (v;w;u), and b = [b;0 (n−1)(n−2)
+2
+×1;0n−1×1]. Then, we can present the
+following theorem which presents a point to set admissible mapping, see Definition 2.4, with
+r = 1, based on (P1
+SOCO) and (D1
+SOCO), and their representations (PP
+SDO) and (DP
+SDO).
+Theorem 4.1. Consider the SOCO problem pairs (P1
+SOCO) and (D1
+SOCO) with (x,y,s) ∈ FP1
+SOCO ×
+FD1
+SOCO and SDO problem pairs (PP
+SDO) and (DP
+SDO) with (X,v,w,u,S) ∈ FPP
+SDO × FDP
+SDO.
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+15
+Then, the mapping (X,v,w,u,S) = M (x,y,s) defined as
+X = Arw(x),
+v = y,
+S =
+
+
+s1
+n −∑n
+k=2 uk
+s2
+2
+...
+...
+sn
+2
+s2
+2
+s1
+n +u2
+−w23
+...
+−w2n
+...
+−w23
+s1
+n +u3
+...
+...
+...
+...
+...
+...
+−w(n−1)n
+sn
+2
+−w2n
+...
+−w(n−1)n
+s1
+n +un
+
+
+,
+where w ∈ R
+(n−1)(n−2)
+2
+and u ∈ R(n−1) taking any values such that S ⪰ 0 is a point-to-set admis-
+sible mapping. The inverse mapping is denoted by (x,y,s) = M −1(X,v,w,u,S), with
+x = Arw−1(X),
+y = v,
+s =
+
+
+c1 −∑m
+i=1viai1
+...
+cn −∑m
+i=1viain
+
+.
+Proof. The proof of this theorem is presented in Appendix C.
+□
+Remark 4.2. The proposed mapping in Theorem 4.1 is not necessarily a point to point mapping
+regarding dual variables of (D1
+SOCO) and (DP
+SDO).
+Analogous to Corollaries 3.2-3.4, the proposed mapping in Theorem 4.1 preserves optimality
+and complementarity.
+4.2. Rank-one Mapping. Using the intuition from the dual side’s section, we seek to represent
+matrix S using a rank one matrix. Hence, we propose
+S = η(η)T,
+with
+n
+∑
+i=1
+(ηi)2 = s1,
+η1ηj = s j
+2 for all j = 2,...,n.
+We need to solve an n-variable-n-equation system. The solution of this system is
+η =
+1
+�
+2(s1 +δ ′)
+�
+s1 +δ ′,s2,...,sn
+�T,
+where δ ′ =
+�
+s1 −||s2:n||2. Using this vector, we can construct matrix S. Thus, we can compute
+uk for k = 2,...,n, and whl for h ̸= 1,h < l as follows,
+uk = (ηk)2 − 1
+ns1 =
+s2
+k
+2(s1 +δ ′) − 1
+ns1,
+k = 2,...,n,
+whl = ηhηl =
+shsl
+2(s1 +δ ′),
+h ̸= 1,h < l.
+
+16
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Theorem 4.3. Consider the rank-one mapping
+S =
+�
+[0]n×n
+if s = (0,0,...,0),
+PMR1
+otherwise.
+(4.2)
+where
+PMR1 = η(η)T =
+
+
+s1+δ ′
+2
+s2
+2
+...
+sn
+2
+s2
+2
+s2
+2
+2[s1+δ ′]
+...
+s2sn
+2[s1+δ ′]
+...
+...
+...
+...
+sn
+2
+s2sn
+2[s1+δ ′]
+...
+s2n
+2[s1+δ ′]
+
+
+.
+Then, (4.2) together with (X,y) = (Arw(x),y) is a point-to-point admissible mapping.
+Proof. The proof is similar to that of Theorem 3.5.
+□
+4.3. Higher Rank Mapping. The derivation of full rank mapping is similar to the derivation
+discussed in Section 3.3, as we show that when a SOCO solution s ∈ int(L n), then we can use
+a full rank mapping, i.e.
+PMR =
+n
+∑
+i=1
+ηi(ηi)T.
+The proof of existence of a full rank mapping is similar to the proof presented in Appendix B.
+Theorem 4.4. There exist mappings PMR where rank(PMR(¯s)) = n for s ∈ int(L n).
+Proof. The proof of this theorem is similar to the proof presented in Appendix B. The difference
+is that we need to set π1 = s in the proof.
+□
+The reason why the procedure to show the existence of full rank mapping is similar for both
+directions is that although the mappings seem to be different, in fact both satisfy analogous
+conditions about the first row and column and the trace of the mapping matrix. From this point
+of view, they are very similar. In fact, the update vector in the procedure is meant to be satisfying
+those conditions. Thus, we can follow the procedure in Appendix B for both mappings.
+Similar to Theorem 1 of [9], we can develop the following mapping for the primal side.
+MR(s) =
+� 1
+4θ′
+1
+2sT
+2:n
+1
+2s2:n
+s1−∥s2:n∥
+2(n−1) I + s2:nsT
+2:n
+θ ′
+�
+,
+where θ′ = s1 +∥s2:n∥ +
+�
+(s1 +∥s2:n∥)2 −4∥s2:n∥2. Here, we can perform the same analysis
+we did in Section 3.3, too. Considering the case in which the solution is on the boundary of the
+second-order cone, i.e. s1 = ∥s2:n∥. Then, θ′ = 2s1, and
+MR1(s) =
+� 1
+2s1
+1
+2sT
+2:n
+1
+2s2:n
+s2:nsT
+2:n
+2s1
+�
+.
+We can see that this is exactly identical to the rank one mapping we presented in Theorem 4.3.
+We can also write it as
+MR1(s) = γ1(γ1)T
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+17
+where
+γ1 =
+1
+√
+θ′
+�1
+2θ′,s2,...,sn
+�T
+.
+To obtain MR(s) as the sum of rank-one matrices, we define
+γ j =
+�
+s1 −∥s2:n∥
+2(n−1)
+ej
+forj = 2,...,n,
+where ej is a unit vector with 1 in element j. By this setting, we have
+MR(s) =
+n
+∑
+j=1
+γ j(γ j)T.
+If the solution s of the SOCO dual problem is on the boundary of the cone, i.e. s1 = ∥s2:n∥, then
+we get
+γ1 =
+1
+√2s1
+(s1,s2,...,sn)T ,
+γ j = 0,
+for j = 2,...,n.
+This leads to the rank one mapping MR1(s). On the other hand, if the solution s ∈ int(L n),
+i.e. s1 > ∥s2:n∥, then γ j ̸= 0 for j = 2,...,n, and we obtain the full rank mapping MR(s). To
+construct an admissible rank-k mapping, here one cannot take a combination of first k vectors
+γ j as it violates the conditions given in Theorem 4.1, i.e. the sum of the diagonal elements will
+not be equal to s1. Recall the definition of N from Section 3.3. The correct choice of γ j to
+construct a rank-k mapping is
+γ j =
+�
+s1 −∥s2:n∥
+2(k −1)
+ej
+for j ∈ N , and γ j = 0 for j /∈ N .
+The resulting matrix MR(s) has rank k, and one can easily see that it satisfies the conditions of
+Theorem 4.1. Analogous to the dual side, we have the following theorem.
+Theorem 4.5. Let ρ(s) = max{rank(M (s)) : M is admissible map}. We have
+• ρ(s) = n if s ∈ int(L n).
+• ρ(s) = 1 if s ∈ ∂(L n).
+Proof. Proof of this theorem is similar to Theorem 1 of [9].
+□
+4.4. Generalization to Multiple SOCs. Similar to the discussion in Subsection 3.4, here we
+adopt the mapping presented in Theorem 3.8 to the primal side. Here we need to define the
+following notations. Let,
+˜C =
+
+
+⃗C1
+⃗C2
+...
+⃗Cr
+
+,
+where
+⃗Ck = Arw(1
+nck
+1, 1
+2ck
+2,..., 1
+2ck
+n)
+and
+˜Aj = DArw(⃗A1
+j,⃗A2
+j,...,⃗Ar
+j)
+for all j = 1,...,m,
+
+18
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+where
+⃗Ak
+j = Arw(1
+nak
+i1, 1
+2ak
+i2,..., 1
+2ak
+in)
+for all k = 1,...,r.
+Similar to the structure of ˜C and ˜Aj, matrix ˜X needs to have a block-diagonal structure. Thus,
+we need to enforce this structure on it. To this end, we define an operator. Let Cf(i) = j, be an
+operator which returns the corresponding cone j to input row i of the block-diagonal matrices.
+Let set I consists of all entries that need to be zero due to being either off-arrow within the
+blocks, or the off-block-diagonal structure.
+I =
+�
+(h,l) ∈ [1,
+r
+∑
+i=1
+ni]2
+�����
+�
+∑
+Cf(h)
+i=0 ni < l,
+∑
+Cf(h)−1
+i=0
+ni +1 < h < ∑
+Cf(h)
+i=0 ni,
+and
+h < l ≤ ∑
+Cf(h)
+i=0 ni.
+�
+,
+K =
+�
+k ∈ [1,
+r
+∑
+i=1
+ni]
+�����
+Cf(k)−1
+∑
+i=0
+ni +1 < k ≤
+Cf(k)
+∑
+i=0
+ni
+�
+,
+where n0 = 0. Note that here we introduce matrices ˜ˇAhl and ˜ˆAk which are generalizations of
+��Ahl and ˆAk, for the multiple cone case, respectively. In detail, ˜ˇAhl enforces all entries off-block-
+diagonal and off-arrow within each block to be zero. Moreover, ˜ˆAk guarantees that in each
+block, diagonal entries are equal by setting entry (k,k) equal to −1 and entry (∑
+Cf(k)−1
+t=1
+nt +
+1,∑
+Cf(k)−1
+t=1
+nt +1) equal to +1. By this notation, we have
+min Tr(˜C ˜X)
+s.t. Tr(˜Aj ˜X) = bj,
+j = 1,...,m
+Tr(˜ˇAhl ˜X) = 0,
+(h,l) ∈ I
+Tr(˜ˆAk ˜X) = 0,
+k ∈ K
+˜X ⪰ 0.
+(P ˜P
+SDO)
+Then, we dualize and get
+max bTy
+s.t.
+m
+∑
+j=1
+yj ˜Aj + ∑
+h,l∈I
+vhl ˜ˇAhl + ∑
+k∈K
+zk˜ˆAk + ˜S = ˜C,
+˜S ⪰ 0.
+(D ˜P
+SDO)
+In similar fashion to other introduced models, we can define the sets of feasible and optimal
+solutions corresponding to models (P ˜P
+SDO) and (D ˜P
+SDO).
+Next, we can present the following theorem.
+Theorem 4.6. Consider the SOCO problem pairs (PSOCO) and (DSOCO) with
+(x1;x2;...;xr;y;s1;s2;...;sr) ∈ FPSOCO ×FDSOCO,
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+19
+and SDO problem pairs (P ˜P
+SDO) and (D ˜P
+SDO) with ( ˜X, ˜y, ˜S) ∈ FP ˜P
+SDO ×FD ˜P
+SDO. Then, the map-
+ping ( ˜X, ˜y,v,z, ˜S) = M (x,y,s) defined as
+˜X = DArw(X1,X2,...,Xr),
+˜y = y,
+˜S =
+
+
+S1
+S2
+...
+Sr
+
+ ⪰ 0 s.t.
+
+
+∑ui+ni
+j=ui ˜Sj j
+˜Sui,ui+1
+...
+˜Sui,ui+ni
+
+ =
+
+
+si
+1
+si
+2
+2...
+sin
+2
+
+
+with vectors v and z taking values such that ˜S ⪰ 0. Furthermore, the inverse mapping denoted
+by (x,y,s) = M −1( ˜X, ˜y,v,z, ˜S), with
+(x1;...;xr) with xi =
+
+
+˜xui,ui
+˜xui+1,ui
+...
+˜xui+ni,ui
+
+,
+y = ˜y,
+(s1;...;sr) with si =
+
+
+∑ui+ni
+j=ui ˜Sj j
+2 ˜Sui,ui+1
+...
+2 ˜Sui,ui+ni
+
+,
+where ui = 1+∑i−1
+k=0 ni and n0 = 0, is an admissible mapping.
+Proof. The proof of this theorem is analogous to that of Theorem 3.1.
+□
+We observe that although in the general case the mapping is similar to that of the dual side,
+but the details of the mapping are different. This mapping is different from the other one since it
+requires more work to enforce the arrow-head structure on matrix ˜X. An analogous analysis on
+different rank mappings can be done for the generalized multiple cone case, with the difference
+that here we have the arrow-head structure for matrix ˜X, and we can compute matrix ˜S with
+different ranks. We skip that similar analysis here for brevity.
+Now, that we have the generalized standard mapping for both sides, we can proceed with
+studying the mapping of the optimal partitions on both sides in the next section.
+5. MAPPING THE OPTIMAL PARTITION
+As shown in the previous sections, the proposed mappings represent a solution of SOCO
+depending on where it is located in the cone. For a solution on the boundary of the second-order
+cone, all admissible maps provide a rank-one positive semidefinite matrix, while a solution in
+the interior of the Lorentz cone can be mapped to semidefinite matrices with different ranks.
+This correspondence can be analyzed in order to see how the optimal partition of SOCO is
+mapped to that of the derived SDO counterparts. For mapping the optimal partition, maximally
+complementary solutions are of particular interest. First, we define a helpful notation, and then
+the following theorem discusses the preservation of maximal complementarity.
+Definition 5.1 (Proper Map). Mapping M is a proper map if M is admissible and rank(X) =
+ρ(x) for all x ∈ L n, or rank(S) = ρ(s) for all s ∈ L n
+Based on this definition, a rank-one mapping is not proper but map MR of [9] is proper. In
+the subsequent theorem, we show that a proper mapping preserves maximal complementarity
+using eigenvalues of the mapped solution. We used the mapping approach of Section 3 which
+
+20
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+starts from dual side. Let λ X
+i denotes the ith eigenvalue of a matrix X. Then, the eigenvalues
+of an arrow-head matrix X = Arw(x) are λ X
+1 = x1 − ∥x2:n∥, λ X
+2 = ··· = λ X
+n−1 = x1, and λ X
+n =
+x1 +∥x2:n∥, see e.g. [1].
+Theorem 5.2. For a maximally complementary solution (x;y;s) ∈ P∗
+SOCO×D∗
+SOCO, the mapped
+solution ( ˜X, ˜y, ˜S) = M (x;y;s) ∈ P ˜D∗
+SDO ×D ˜D∗
+SDO is maximally complementary if M is a proper
+map.
+Proof. Given a maximally complementary solution (x;y;s) ∈ P∗
+SOCO ×D∗
+SOCO, one can analyze
+the result of mapping based how the cones are partitioned. The following analysis holds for the
+dual side direction. In this direction, we have matrix ˜S as a block diagonal of arrow-head
+matrices. Since M is proper map, the eigenvalues of matrix ˜X can be partitioned based on
+Lorentz cones. If vector xi is on the non-zero boundary of Lorentz cone, the all corresponding
+eigenvalues are non-zero except the first one. If the vector xi is in the interior of Lorentz cone,
+the corresponding eigenvalues are positive. For the point of the second-order cone, all the
+corresponding eigenvalues are zero. Thus, we have
+if i ∈ ¯
+B =
+�
+xi
+1 > ∥xi
+2:ni∥
+→ full rank matrix Xi with λ Xi
+1 ,λ Xi
+2 ,...,λ Xi
+ni > 0,
+si = 0
+→ zero Si matrix with λ Si
+1 = ... = λ Si
+ni = 0,
+if i ∈
+¯
+N =
+�
+xi = 0
+→ zero matrix Xi with λ Xi
+1 = ... = λ Xi
+ni = 0,
+si
+1 > ∥si
+2:ni∥
+→ arrow-head matrix Si with λ Si
+1 ,λ Si
+2 ,...,λ Si
+ni > 0,
+if i ∈ ¯
+R =
+�
+xi
+1 = ∥xi
+2:ni∥ > 0
+→ rank-one matrix Xi with λ Xi
+1 > 0,λ Xi
+2 = ... = λ Xi
+ni = 0
+si
+1 = ∥si
+2:ni∥ > 0
+→ arrow-head Si matrix with λ Si
+1 = 0,λ Si
+2 ,...,λ Si
+ni > 0,
+if i ∈ ¯
+T1 =
+�
+xi = 0
+→ zero matrix Xi with λ Xi
+1 = ... = λ Xi
+ni = 0,
+si = 0
+→ zero matrix Si with λ Si
+1 = ... = λ Si
+ni = 0,
+if i ∈ ¯
+T2 =
+�
+xi
+1 = ∥xi
+2:ni∥ > 0
+→ rank-one matrix Xi with λ Xi
+1 > 0,λ Xi
+2 = ... = λ Xi
+ni = 0
+si = 0
+→ zero matrix Si with λ Si
+1 = ... = λ Si
+ni = 0,
+if i ∈ ¯
+T3 =
+�
+xi = 0
+→ zero matrix Xi with λ Xi
+1 = ... = λ Xi
+ni = 0,
+si
+1 = ∥si
+2:ni∥ > 0
+→ arrow-head matrix Si with λ Si
+1 = 0,λ Si
+2 ,...,λ Si
+ni > 0,
+Now, using that M is a proper, M maps a given maximally complementary optimal SOCO so-
+lution to ( ˜X, ˜y, ˜S) ∈ P ˜D∗
+SDO ×D ˜D∗
+SDO. Here, we show that this solution is maximally complemen-
+tary for the SDO counterpart problem. The proof goes by contradiction. Let us assume to the
+contrary that ( ˜X, ˜y, ˜S) is not a maximally complementary solution. Let ( ˆX, ˆy, ˆS) ∈ P ˜D∗
+SDO×D ˜D∗
+SDO
+be a maximally complementary solution. Since both ( ˜X, ˜y, ˜S) and ( ˆX, ˆy, ˆS) are optimal, then
+ˆS ˜X = 0. Accordingly, if λ ˜Si
+j > 0 then λ ˆSi
+j > 0. Now it is enough to show that there are no i, j
+such that λ ˜Si
+j = 0 and λ ˆSi
+j > 0. If there exist such index, by doing inverse mapping, the mapped
+solution in SOCO will have larger partition set either for
+¯
+N , ¯
+R, or
+¯
+T3 which contradicts with
+assumption that solution (x;y;s) being a maximally complementary solution of SOCO. With
+similar reasoning we can show that there are no i, j such that λ ˜Xi
+j
+= 0 and λ ˆXi
+j
+> 0. Thus, we
+can conclude that a proper map preserves maximal complementarity.
+□
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+21
+One can prove similar theorem as stated as follows for the mapping of Section 4 starting from
+primal side.
+Theorem 5.3. For a maximally complementary solution (x;y;s) ∈ P∗
+SOCO×D∗
+SOCO, the mapped
+solution ( ˜X, ˜y, ˜S) = M (x;y;s) ∈ P ˜P∗
+SDO ×D ˜P∗
+SDO is maximally complementary if M is a proper
+map.
+Proof. The proof is analogous to the proof of Theorem 5.2.
+□
+5.1. Mapping the Optimal Partition. Based on the analysis described in the proof of The-
+orem 5.2, one can develop the mapping for the optimal partitions. Based on the sign of the
+eigenvalues, if positive, their corresponding eigenvectors generate the subspaces B and N ,
+and if zero, to subspace T of the optimal partition of SDO. First we consider mapping starting
+from the dual side as discussed in Section 3.4 and Section 4.4. The summary of optimal parti-
+tion mapping on the dual side is presented in Table 1, where ei
+j represents the jth eigenvector
+corresponding to λ j for the semidefinite cone i.
+B
+N
+T
+if i ∈ ¯
+B
+{ei
+1,...,ei
+ni}
+/0
+/0
+if i ∈
+¯
+N
+/0
+{ei
+1,...,ei
+ni}
+/0
+if i ∈ ¯
+R
+{ei
+1}
+{ei
+2,...,ei
+ni}
+/0
+if i ∈ ¯T1
+/0
+/0
+{ei
+1,...,ei
+ni}
+if i ∈ ¯T2
+{ei
+1}
+/0
+{ei
+2,...,ei
+ni}
+if i ∈ ¯T3
+/0
+{ei
+2,...,eini}
+{ei
+1}
+TABLE 1. The Dual Side Optimal Partition Mapping
+A similar analysis can be conducted for the primal side derivation, where matrix X has an
+arrow-head structure and we represent matrix S using zero, rank-one or full rank matrices, based
+on the position of vector s in the second-order cone. The optimal partition mapping is presented
+in Table 2.
+
+22
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+B
+N
+T
+if i ∈ ¯
+B
+{ei
+1,...,ei
+ni}
+/0
+/0
+if i ∈
+¯
+N
+/0
+{ei
+1,...,ei
+ni}
+/0
+if i ∈ ¯
+R
+{ei
+2,...,ei
+ni}
+{ei
+1}
+/0
+if i ∈ ¯T1
+/0
+/0
+{ei
+1,...,ei
+ni}
+if i ∈ ¯T2 {ei
+2,...,ei
+ni}
+/0
+{ei
+1}
+if i ∈ ¯T3
+/0
+{ei
+1}
+{ei
+2,...,ei
+ni}
+TABLE 2. The Primal Side Optimal Partition Mapping
+We can see that the result of mapping the optimal partition for the two sides are different.
+We observe identical outcome for partitions ¯
+B,
+¯
+N ,
+¯
+T1, which was predictable as they include
+the simple case where at least one of the variables x and s is zero. However, notable difference
+arises for the ¯
+R, ¯T2, and ¯T3 partitions depending on whether the dual or the primal variable’s
+representation is forced to be arrow-head.
+Consider partition ¯
+R, where both x and s are on the non-zero boundary of the second-order
+cone. Adopting the dual (primal) side approach, variable s (x) will have arrow-head represen-
+tation, and the other has a rank one representation. We have seen in Example 2.3, this partition
+is where complementarity will not be preserved if we apply arrow-head representation on both
+primal and dual sides. While our derivations guarantee feasibility and duality, we can see that
+complementarity is also preserved as the arrow-head matrix has only one zero eigenvalue, while
+the rank-one matrix has only one none-zero. For partitions
+¯
+T2 and
+¯
+T3, the difference in tables
+simply comes from which variable is non-zero and its corresponding arrow-head representation
+eigenvalues.
+From the geometric point of view, we can see that for partitions ¯
+B and
+¯
+N , where the respec-
+tive solution is in the interior of the second-order cone, the mapped solution is in the interior of
+the semidefinite cone. For partition ¯
+R, we observe that the solution pair, which both are on the
+non-zero boundary of the second-order cone is mapped to a face of the semidefinite cone. For
+partition
+¯
+T1, the zero solution pair is mapped to the origin. Finally, for partitions
+¯
+T2, and
+¯
+T3,
+the solutions are on the boundary of the second-order cone in the primal and dual, respectively.
+We can see that the result of the mapping is a face of the semidefinite cone, for both cases,
+depending on if the primal or dual variable is on the boundary on the second-order cone.
+6. CONCLUSION
+In this paper, we study the relationship between a SOCO problem and its SDO representa-
+tion. Knowing about the fact that SOCO can be considered as a special case of SDO, we extend
+the literature by investigating both the primal and the dual side SDO representations of a SOCO
+problem. We demonstrate that using arrow-head matrix transformation on the primal or dual
+SOCO problem, does not result in an arrow-head matrix variable on its dual. In fact, nothing
+forces the dual variable to be arrow-head. We usually end up with dense matrices which can be
+represented by either rank-one or full rank mappings, based on the position of SOCO solutions
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+23
+in the second-order cone. We propose low-rank to full rank mappings which are admissible,
+meaning that they preserve feasibility and objective function value. First of all, these mappings
+are not unique. One can come up with different mappings that satisfies these conditions. In ad-
+dition, the dense structure of mapped solutions gives us an intuition about why solving SOCO
+as a SOCO problem is more efficient than solving as an SDO problem. In the SDO representa-
+tion, we have to deal with dense matrices when solving the SDO counterpart. Furthermore, we
+investigated the relationship between the optimal partitions of these problems. The optimal par-
+tition of SOCO is an index-based partition, while that of SDO is a subspace-based partition. We
+discussed how these partitions map to each other based on the eigenvalue analysis of mapped
+solutions.
+The theoretical study of the relationship between SOCO and its SDO counterparts remains
+with unsolved questions. One interesting question to look into is to study how degeneracy and
+singularity degree properties are affected throughout mappings.
+
+24
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+APPENDIX A. PROOF OF THEOREM 3.1
+Proof. To prove that the presented mapping in Theorem 3.1 is admissible, we need to show that
+it complies with the definition of admissible mapping, i.e. preservation of feasibility and the
+objective function value.
+Step 1. First, we show that if (y,s) ∈ FD1
+SOCO, then (y,S) = (y,Arw(s)) ∈ FDD
+SDO. Since S =
+Arw(s) and s ∈ L n, we know that S is positive semidefinite [1]. The only thing remains to
+prove is that ∑n
+i=1 yi⃗Ai +S = ⃗C. Since (y,s) is feasible, i.e., ATy+s = c, we have
+n
+∑
+i=1
+yi⃗Ai +S =
+
+
+∑n
+i=1yiai1 +s1
+∑n
+i=1yiai2 +s2
+...
+∑n
+i=1yiain +sn
+∑n
+i=1yiai2 +s2
+∑n
+i=1yiai1 +s1
+...
+...
+∑n
+i=1yiain +sn
+∑n
+i=1yiai1 +s1
+
+
+=
+
+
+c1
+c2
+...
+cn
+c2
+c1
+...
+...
+cn
+c1
+
+ = ⃗C.
+Step 2. We show that if x = (x1,...,xn) ∈ FP1
+SOCO, then matrix X as defined in Theorem 3.1,
+belongs to FPD
+SDO. By construction, X is positive semidefinite and we need to just show that
+Tr(⃗AiX) = bi. Based on the construction of X and feasibility of x, i.e., Ax = b, we have
+Tr(⃗AiX) = ai1(
+n
+∑
+i=1
+Xii)+2ai2X12 +...+2ainX1n
+= ai1x1 +ai2x2 +...+ainxn = bi.
+Step 3. It is obvious that bTy = bTy. Similar to step 2, we have
+Tr(⃗CX) = c1(
+n
+∑
+i=1
+Xii)+2c2X12 +...+2cnX1n
+= c1x1 +c2x2 +...+cnxn = cTx.
+Step 4. In this step, we show that if X ∈ FPD
+SDO, then x ∈ FP1
+SOCO. Suppose that X ∈ FPD
+SDO,
+then it is positive semidefinite. We need to show that x ∈ L n. To do so, we utilize that in a
+positive semidefinite matrix every principal submatrix, in particular every 2-by-2 submatrix is
+positive semidefinite [6]. Thus,
+|Xi j| ≤
+�
+XiiXj j
+for all i = 1,...,n and for all j = 1,...,n
+Thus, we have
+X2
+12 +X2
+13 +...+X2
+1n ≤ (X11X22)+(X11X33)+...+(X11Xnn)
+(A.1)
+≤ X11(X22 +X33 +...+Xnn),
+(A.2)
+�
+X2
+12 +X2
+13 +...+X2
+1n ≤
+�
+X11(X22 +X33 +...+Xnn)
+(A.3)
+≤ X11 +X22 +X33 +...+Xnn
+2
+,
+(A.4)
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+25
+where (A.4) is derived using the arithmetic-geometric mean inequality. The expression can be
+rewritten as
+n
+∑
+i=1
+Xii ≥ 2
+�
+X2
+12 +X2
+13 +...+X2
+1n
+=
+�
+(2X12)2 +(2X13)2 +...+(2X1n)2,
+or simply,
+n
+∑
+i=1
+Xii ≥
+�
+(2X12)2 +(2X13)2 +...+(2X1n)2,
+x1 ≥
+�
+(x2)2 +...+(xn)2 = ||x2:n||2,
+which shows that x ∈ L n. Now, we need to prove that Ax = b. Thus, we have
+Aix = ai1x1 +ai2x2 +...+ainxn
+for all i = 1,...,m,
+= ai1(
+n
+∑
+i=1
+Xii)+ai2(2X12)+...+ain(2X1n)
+= Tr(⃗AiX) = bi.
+Step 5. Finally, we need to show that if (y,S) ∈ FDD
+SDO, then (y,s) ∈ FD1
+SOCO. As (y,S) =
+(y,Arw(s)), according to [1], we know that S = Arw(s) ⪰ 0, implies s ∈ L n. Then, the only
+property that remains to prove is that ∑m
+i=1 yiAi + s = c. By feasibility of (y,S), we know that
+∑n
+i=1yi⃗Ai +S = ⃗C, i.e.,
+
+
+∑n
+i=1 yiai1 +s1
+∑n
+i=1 yiai2 +s2
+...
+∑n
+i=1 yiain +sn
+∑n
+i=1 yiai2 +s2
+∑n
+i=1 yiai1 +s1
+...
+...
+∑n
+i=1 yiain +sn
+∑n
+i=1 yiai1 +s1
+
+ =
+
+
+c1
+c2
+...
+cn
+c2
+c1
+...
+...
+cn
+c1
+
+,
+where each element of the matrix is a constraint in (D1
+SOCO) which means ∑m
+i=1 yiAi +s = c.
+Considering all of the previous steps, we conclude that presented mapping in Theorem 3.1 is
+an admissible mapping.
+□
+
+26
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+APPENDIX B. PROOF OF THEOREM 3.6
+Proof. A rank-n mapping of ¯x can be constructed by Algorithm 1.
+Algorithm 1 Full Rank Mapping
+π1 = ¯x,
+Choose sufficiently small ε
+for k = 1 : n−1 do
+τk ← πk
+2
+�
+τk
+k+1 = τk
+k+1 +ε
+if πk
+k+1 ≥ 0
+τk
+k+1 = τk
+k+1 −ε
+if πk
+k+1 < 0
+Calculate
+β k =
+1
+2
+�
+τk
+1+ ˆδ k
+2
+(τk
+1 + ˆδ k,τk
+2,...,τk
+n)T, where ˆδ k =
+�
+(τk
+1)2 −||τk
+2:n||2.
+πk+1 = πk −τk
+end for
+Calculate
+β n =
+1
+2
+�
+πn
+1+ ˆδ n
+2
+(πn
+1 + ˆδ n,πn
+2,...,πn
+n)T, where ˆδ n =
+�
+(πn
+1)2 −||πk
+2:n||2.
+If ∥πk
+2:n∥ < πk
+1 for all k, then we have
+0 < ||πk
+2:n||
+2
+≤ ||τk
+2:n|| < τk
+1 ≤ πk
+1
+2 < πk
+1,
+
+
+
+πk
+j
+2 ≤ τk
+j
+if π j ≥ 0
+πk
+j
+2 ≥ τk
+j
+if π j < 0
+,
+j = 2,...,n.
+(B.1)
+We need to show that in all loops ∥πk
+2:n∥ < πk
+1. For k = 1 this holds since ∥x2:n∥ < x1. Using
+induction, we need to prove ∥πk+1
+2:n ∥ < πk+1
+1
+assuming ∥πk
+2:n∥ < πk
+1. We have
+πk+1
+1
+−∥πk+1
+2:n ∥ = (πk
+1 −τk
+1)−∥(πk
+j −τk
+j)2:n∥ ≥ τk
+1 −∥τk
+2:n∥ > 0.
+Thus, πk
+1 > ∥πk
+2:n∥ for all k. The proposed MR is also admissible since
+n
+∑
+j=1
+MRj j =
+n
+∑
+k=1
+n
+∑
+j=1
+(β k
+j )2 = x1
+MR1j =
+n
+∑
+k=1
+β k
+j β k
+1 = x j
+2 .
+We need to prove that the generated β k vectors are linearly independent. We have
+β k =
+1
+2
+�
+τk
+1+ ˆδ k
+2
+(τk
+1 + ˆδ,τk
+2,...,τk
+n)T, where ˆδ k =
+�
+(τk
+1)2 −||τk
+2:n||2.
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+27
+Let us define the matrix
+B =
+
+
+τ1
+1 + ˆδ 1
+τ2
+1 + ˆδ 2
+...
+τn
+1 + ˆδ n
+τ1
+2
+τ2
+2
+...
+τn
+2
+...
+...
+...
+...
+τ1
+n
+τ2
+n
+...
+τn
+n
+
+
+We still need to prove that B has full rank. Now, w.l.o.g. we may assume that x ≥ 0. One can
+easily extend the following proof to general x. Then we have
+B =
+
+
+τ1
+1 + ˆδ 1
+τ2
+1 + ˆδ 2
+...
+τn
+1 + ˆδ n
+τ1
+2
+τ2
+2
+...
+τn
+2
+...
+...
+...
+...
+τ1
+n
+τ2
+n
+...
+τn
+n
+
+
+=
+
+
+π1
+1
+2 + ˆδ 1
+π2
+1
+2 + ˆδ 2
+...
+πn
+1 + ˆδ n
+π1
+2
+2 +ε
+π2
+2
+2
+...
+πn
+2
+π1
+3
+2
+π2
+3
+2 +ε
+...
+πn
+3
+...
+...
+...
+...
+π1n
+2
+π2n
+2
+...
+πn
+n
+
+
+=
+
+
+x1
+2 + ˆδ 1
+x1
+4 + ˆδ 2
+...
+x1
+(2)n + ˆδ n
+x2
+2 +ε
+x2
+4 − ε
+2
+...
+x1
+(2)n −
+ε
+(2)n−1
+x3
+2
+x3
+4 +ε
+...
+x1
+(2)n −
+ε
+(2)n−2
+...
+...
+...
+...
+xn
+2
+xn
+4
+...
+x1
+(2)n − ε
+(2)
+
+
+=
+
+
+x1
+2
+x1
+4
+...
+x1
+(2)n
+x2
+2
+x2
+4
+...
+x1
+(2)n
+x3
+2
+x3
+4
+...
+x1
+(2)n
+...
+...
+...
+...
+xn
+2
+xn
+4
+...
+x1
+(2)n
+
+
++
+
+
+ˆδ 1
+ˆδ 2
+...
+ˆδ n−1
+ˆδ n
+ε
+−ε
+2
+...
+−
+ε
+(2)n−2
+−
+ε
+(2)n−1
+0
+ε
+...
+−
+ε
+(2)n−3
+−
+ε
+(2)n−2
+...
+...
+...
+...
+...
+0
+0
+...
+ε
+− ε
+(2)
+
+
+,
+where
+
+
+ˆδ 1
+ˆδ 2
+...
+ˆδ n
+
+ =
+
+
+�
+(x1
+2 )2 −(x2
+2 +ε)2 −(x3
+2 )2 −···−(xn
+2 )2
+�
+(x1
+4 )2 −(x2
+4 − ε
+2)2 −(x3
+4 +ε)2 −···−(xn
+4 )2
+...
+�
+( x1
+(2)n)2 −( x1
+(2)n −
+ε
+(2)n−1)2 −(x3
+4 −
+ε
+(2)n−1)2 −···−( xn
+(2)n − ε
+2)2
+
+
+.
+
+28
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+By doing row eliminations on both matrices to sparsify the second matrix, we get
+
+
+x1
+2
+2x1
+4
+...
+n x1
+(2)n
+x2
+2
+2x2
+4
+...
+n x1
+(2)n
+x3
+2
+2x3
+4
+...
+n x1
+(2)n
+...
+...
+...
+...
+xn
+2
+2xn
+4
+...
+n x1
+(2)n
+
+
++
+
+
+0
+0
+...
+0
+¯δ
+ε
+0
+...
+0
+0
+0
+ε
+...
+0
+0
+...
+...
+...
+...
+...
+0
+0
+...
+ε
+0
+
+
+where
+¯δ =
+ˆδ 1
+(2)n−1 +
+ˆδ 2
+(2)n−2 +···+
+ˆδ n−1
+2
++ ˆδ n.
+As we can see ¯δ, as sum of strictly positive numbers, is strictly positive, and the matrix has
+full rank. By choosing small ε, we can show that B has full rank. The last piece is to find
+appropriate value for ε. We need to choose ε so that
+(x1
+2 )2 −(x2
+2 +ε)2 −(x3
+2 )2 −···−(xn
+2 )2 > 0
+(x1
+4 )2 −(x2
+4 − ε
+2)2 −(x3
+4 +ε)2 −···−(xn
+4 )2 > 0
+...
+( x1
+(2)n)2 −( x1
+(2)n −
+ε
+(2)n−1)2 −(x3
+4 −
+ε
+(2)n−1)2 −···−( xn
+(2)n − ε
+2)2 > 0.
+Let ρ = (x1)2 −∑n
+i=2(xi)2 > 0. Then, we have
+4ε2 +4x2ε −ρ < 0
+20ε2 +(8x3 −4x2)ε −ρ < 0
+...
+Since the second derivatives of all quadratic forms in these inequalities are positive, they are
+convex. Since coefficients of ε2 and ρ are strictly positive, they have two distinct roots. Thus,
+any epsilon in the intersection of all these intervals satisfies the required conditions. The in-
+tersection of them is non-empty since we have a valid choice ε = x1−∥x2:n∥
+2(n−1)
+as discussed in
+Section 3.3.
+□
+APPENDIX C. PROOF OF THEOREM 4.1
+Proof. Proof is similar to that of Theorem 3.1, showing that under the proposed setting (x,y,s)
+and (X,y,S) = M (x,y,s) satisfy the primal and dual constraints and preserves the objective
+function value. To this end, we need to take the following steps.
+Step 1. First, we show that if x ∈ FP1
+SOCO, then X = Arw(x) ∈ FPP
+SDO. First, we know that
+since x ∈ L n, then X = Arw(x) ⪰ 0. Next, we need to show that X satisfies the SDO primal
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+29
+constraint. Thus,
+Tr(⃗AiX) = 1
+nai1(
+n
+∑
+i=1
+Xii)+ 2
+2ai2X12 +...+ 2
+2ainX1n
+= ai1x1 +ai2x2 +...+ainxn
+= a(i)x = bi,
+for all i = 1,...,m.
+The second equality represents the product of ith row of matrix A and vector x. Hence, we can
+conclude that X satisfies the SDO primal constraint.
+Step 2. Next, we show that if (y,s) ∈ FD1
+SOCO, then (y,S) ∈ FDP
+SDO. By definition and using u
+and w in construction of S, we have S ⪰ 0. Next, we need to show that S satisfies the constraint
+of (DP
+SDO). Thus, since (y,s) is feasible for (D1
+SOCO), i.e. ∑m
+i=1 yiai j +sj = cj for all j = 1,...,n,
+we can write
+m
+∑
+i=1
+yi⃗Ai + ∑
+h̸=1,h<l
+whl ˜Ahl +
+n
+∑
+k=2
+uk ˆAk +S =
+
+
+1
+n ∑m
+i=1 yiai1
+1
+2 ∑m
+i=1 yiai2
+...
+1
+2 ∑m
+i=1 yiain
+1
+2 ∑m
+i=1 yiai2
+1
+n ∑m
+i=1 yiai1
+...
+...
+1
+2 ∑m
+i=1 yiain
+1
+n ∑m
+i=1 yiai1
+
+
++
+
+
+0
+0
+...
+...
+0
+0
+0
+w23
+...
+w2n
+...
+w23
+...
+...
+...
+...
+...
+...
+...
+w(n−1)n
+0
+w2n
+...
+w(n−1)n
+0
+
+
++
+
+
+∑n
+k=2 uk
+0
+...
+...
+...
+0
+0
+−u2
+0
+...
+...
+0
+...
+0
+...
+...
+...
+...
+...
+...
+−ui
+...
+...
+...
+...
+...
+...
+0
+0
+0
+...
+...
+0
+−un
+
+
++
+
+
+s1
+n −∑n
+k=2 uk
+s2
+2
+...
+...
+sn
+2
+s2
+2
+s1
+n +u2
+−w23
+...
+−w2n
+...
+−w23
+s1
+n +u3
+...
+...
+...
+...
+...
+...
+−w(n−1)n
+sn
+2
+−w2n
+...
+−w(n−1)n
+s1
+n +un
+
+
+=
+
+
+1
+nc1
+1
+2c2
+...
+1
+2cn
+1
+2c2
+1
+nc1
+...
+...
+1
+2cn
+1
+nc1
+
+ = ⃗C.
+
+30
+SAMPOURMAHANI, MOHAMMADISIAHROUDI, TERLAKY
+Step 3. It is obvious that bTy = bTy. Similar to step 1, we have
+Tr(⃗CX) = 1
+nc1(
+n
+∑
+i=1
+Xii)+ 2
+2c2X12 +...+ 2
+2cnX1n
+= c1x1 +c2x2 +...+cnxn
+= cTx.
+Step 4. In this step, we need to show that if X ∈ FPP
+SDO, then x ∈ FP1
+SOCO. To this end, we
+know that as X = Arw(x) is positive semidefinite, then x ∈ L n. Next, we need to show that
+Ax = b. Thus, we have
+Aix = ai1x1 +ai2x2 +...+ainxn
+= 1
+nai1(nx1)+2(1
+2ai2)x2 +...+2(1
+2ain)xn
+= Tr(⃗AiX) = bi,
+for all i = 1,...,m.
+Step 5. Finally, we need to show that if (y,S) ∈ FDP
+SDO, then (y,s) ∈ FD1
+SOCO. Suppose that
+S ∈ FDP
+SDO, then it is positive semidefinite. We need to show that s ∈ L n. To do so, recall that
+in a positive semidefinite matrix every principal submatrix, in particular every 2-by-2 submatrix
+is positive semidefinite [6]. Thus,
+|Si j| ≤
+�
+SiiSj j
+for all i = 1,...,n and for all j = 1,...,n
+Thus, we have
+S2
+12 +S2
+13 +...+S2
+1n ≤ (S11S22)+(S11S33)+...+(S11Snn)
+(C.1)
+≤ S11(S22 +S33 +...+Snn),
+(C.2)
+�
+S2
+12 +S2
+13 +...+S2
+1n ≤
+�
+S11(S22 +S33 +...+Snn)
+(C.3)
+≤ S11 +S22 +S33 +...+Snn
+2
+,
+(C.4)
+where (C.4) is derived using the arithmetic-geometric mean inequality. The expression can be
+rewritten as
+n
+∑
+i=1
+Sii ≥ 2
+�
+S2
+12 +S2
+13 +...+S2
+1n
+=
+�
+(2S12)2 +(2S13)2 +...+(2S1n)2,
+or simply,
+s1 ≥
+�
+(s2)2 +...+(sn)2 = ||s2:n||2,
+which according to definition of (4.1) shows that s ∈ L n. Next, we need to show that vector
+s =
+
+
+c1 −∑m
+i=1yiai1
+...
+cn −∑m
+i=1yiain,
+
+
+satisfies the SOCO dual constraint. This is obvious since each entry of this vector is equal to
+the corresponding entry of vector s in the definition of (D1
+SOCO).
+□
+
+ON SEMIDEFINITE REPRESENTATIONS OF SECOND-ORDER CONIC OPTIMIZATION PROBLEMS
+31
+Acknowledgements
+This research is supported by the National Science Foundation (NSF) under Grant No. 2128527.
+REFERENCES
+[1] Farid Alizadeh and Donald Goldfarb. Second-order cone programming. Mathematical Programming,
+95(1):3–51, 2003.
+[2] Roberto Andreani, Gabriel Haeser, Leonardo M Mito, C H´ector Ram´ırez, and Thiago P Silveira. Global
+convergence of algorithms under constant rank conditions for nonlinear second-order cone programming.
+Journal of Optimization Theory and Applications, 195(1):42–78, 2022.
+[3] Aharon Ben-Tal and Arkadi Nemirovski. Lectures on Modern Convex Optimization: Analysis, Algorithms,
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+[4] Etienne De Klerk. Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applica-
+tions, volume 65. Springer Science & Business Media, 2006.
+[5] Daniel Dueri, Jing Zhang, and Behc¸et Ac¸ikmese. Automated custom code generation for embedded, real-
+time second order cone programming. In Proceedings of the 19th IFAC World Congress. Ed. E. Boje, and X.
+Xia, volume 47, pages 1605–1612. Elsevier, 2014.
+[6] Roger A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 2012.
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+

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+Platoon Leader Selection, User Association and
+Resource Allocation on a C-V2X based highway: A
+Reinforcement Learning Approach
+Mohammad Farzanullah and Tho Le-Ngoc
+Department of Electrical & Computer Engineering, McGill University, Montr´eal, QC, Canada
+Abstract—We consider the problem of dynamic platoon leader
+selection, user association, channel assignment, and power allo-
+cation on a cellular vehicle-to-everything (C-V2X) based high-
+way, where multiple vehicle-to-vehicle (V2V) and vehicle-to-
+infrastructure (V2I) links share the frequency resources. There
+are multiple roadside units (RSUs) on a highway, and vehicles can
+form platoons, which has been identified as an advanced use case
+to increase road efficiency. The traditional optimization methods,
+requiring global channel information at a central controller, are
+not viable for high-mobility vehicular networks. To deal with this
+challenge, we propose a distributed multi-agent reinforcement
+learning (MARL) for resource allocation (RA). Each platoon
+leader, acting as an agent, can collaborate with other agents for
+joint sub-band selection and power allocation for its V2V links,
+and joint user association and power control for its V2I links.
+Moreover, each platoon can dynamically select the vehicle most
+suitable to be the platoon leader. We aim to maximize the V2V
+and V2I packet delivery probability in the desired latency using
+the deep Q-learning algorithm. Simulation results indicate that
+our proposed MARL outperforms the centralized hill-climbing
+algorithm, and platoon leader selection helps to improve both
+V2V and V2I performance.
+Index Terms—New radio, cellular vehicle-to-everything, rein-
+forcement learning, resource allocation
+I. INTRODUCTION
+Cellular vehicle-to-everything (C-V2X) is a vehicular stan-
+dard that enables communication between vehicles and other
+entities on the road, such as pedestrians and infrastructure, to
+increase road safety and efficiency [1]. The C-V2X system
+consists of communication between different entities, such
+as vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I),
+vehicle-to-network (V2N), and vehicle-to-pedestrian (V2P)
+communications. The C-V2X is envisioned to support high
+throughput, ultra-reliable, and low latency communications.
+3GPP has provided the 5G new radio (NR) standards
+for vehicular communications concerning architecture in TS
+23.287 [2], advanced use cases in TS 22.886 [3], and eval-
+uation methodology in TR 37.885 [4]. One component of
+infrastructure will be the roadside units (RSUs). RSU is a
+stationary wireless C-V2X device that can exchange messages
+with vehicles and other C-V2X entities. It uses the PC5
+side-link interface to communicate with the vehicles and
+transmit information about road signs and traffic lights [3].
+It can also receive information from the vehicles to make
+a dynamic map of the surroundings and share it with other
+vehicles/pedestrians. Furthermore, we consider the use-case of
+platooning, where multiple vehicles form a train-like structure
+and travel closely together in a line. The platoon leader (PL)
+organizes communications between vehicles. Vehicle platoon-
+ing has been identified as an advanced use case in [3] and has
+gained significant interest since it reduces fuel consumption
+and traffic congestion. The RSU and platoon will need to
+exchange a maximum of 1200 bytes in 500 ms for real-
+time traffic updates and 600 bytes in 10 ms for conditional
+automated driving [3]. We grouped these two requirements
+for an aggregate of 624 bytes in 10 ms. Moreover, the PL
+and members need to exchange 50-1200 bytes in 10 ms for
+cooperative driving and up to 2000 bytes in 10 ms for collision
+avoidance [3]. We aggregate these service requirements and
+keep the exchange of 1200-2800 bytes in 10 ms for our
+simulations. An intelligent resource allocation (RA) design is
+necessary for these stringent requirements.
+The 5G new radio (NR) C-V2X supports two RA modes
+for sidelink PC5 communications: mode 1, the under-coverage
+mode, and mode 2, the out-of-coverage mode [5]. In mode 1,
+the gNB allocates the communication resources to vehicles.
+Meanwhile, in mode 2, the vehicles autonomously select
+the resources. For mode 2, the current RA technique used
+in standards is the sensing-based Semi-Persistent Scheduling
+(SPS) algorithm, which periodically selects random resources.
+However, the probability of resource selection collision can be
+high, and many works have considered either improving the
+SPS algorithm or alternate techniques to increase reliability.
+[6] proposes a novel sensing-based SPS algorithm in an urban
+scenario to reduce the collision probability. The authors in
+[7] considered a highway scenario and suggested a stochastic
+reservation scheme for aperiodic traffic. For the platooning
+use case, [8] shows that the SPS algorithm can not achieve
+the required performance.
+Due to the fast channel variations in vehicular networks,
+centralized optimization schemes that require global channel
+state information (CSI) will no longer be feasible. The high
+CSI overhead and the corresponding increase in latency make
+such methods impractical. To deal with this issue, distributed
+RA algorithms have been suggested in the literature, e.g., [9],
+[10]. Furthermore, traditional optimization techniques have
+limitations, requiring complete information about the environ-
+ment and needing to be retrained for rapidly varying envi-
+ronments [11]. Recently, distributed multi-agent reinforcement
+learning (MARL) has been proposed as an alternative approach
+to resolving such issues. The authors in [12] used Deep Q
+Networks (DQN) for joint channel assignment and power
+allocation to maximize the V2V delivery probability and V2N
+arXiv:2301.03145v1  [cs.MA]  9 Jan 2023
+
+sum-rate in an urban setting, where the V2V links share the
+time-frequency resources with V2N links. Inspired by these
+results, [13] used double DQN for a platoon-based scenario
+for the same objectives. [14] uses the actor-critic method for
+mode selection, subchannel selection, and power control in an
+urban platoon scenario to increase the transmission probability
+of cooperative awareness messages.
+This paper considers a highway C-V2X system consisting
+of multiple platoons. We consider the periodic payload deliv-
+ery from PLs to RSUs, termed V2I links. Furthermore, we
+consider the periodic transmission of messages from PLs to
+members, termed V2V links. We assume a limited spectrum
+is available for the V2V and V2I transmission, and they share
+the frequency resources for efficient spectrum usage. Given
+this system, this paper formulates a dynamic PL selection,
+user association, channel assignment, and power level control
+to maximize the packet delivery probability for both V2V and
+V2I links. Reliability is defined as the successful transmission
+of the packet within a time constraint T [12], [13].
+We utilize MARL in a distributed manner. The RL works
+on a trial-and-error strategy, and each agent slowly improves
+the action taken based on the feedback from the vehicular
+environment. We use the Deep Q-learning algorithm, which
+DeepMind developed for Atari video games [15]. Deep Q-
+Learning has been used for joint channel assignment and
+power allocation in C-V2X systems [12], [13]. However,
+we also use Deep Q-learning for user association and PL
+selection. As per our knowledge, dynamic PL selection has
+not been investigated in the literature. In our work, there
+are multiple collaborative agents for PL selection, V2V joint
+channel assignment and power allocation, and V2I joint user
+association and power allocation. The objective is to increase
+reliability for both V2V and V2I links. Simulation results
+indicate that the proposed MARL algorithm can outperform
+other benchmarks, such as the hill-climbing algorithm, which
+requires global CSI at the central controller. Moreover, the
+dynamic PL selection offers a gain in V2V and V2I reliability.
+II. SYSTEM MODEL AND PROBLEM FORMULATION
+Road Side Unit (RSU)
+V2V link
+V2I link
+Platoon Leader
+Platoon Member
+Fig. 1.
+Illustrative C-V2X based highway, where multiple platoon leaders
+are transmitting to RSUs using the V2I links, and each platoon leader is
+transmitting to its platoon members using the V2V links.
+As illustrated in Fig. 1, we consider a highway-based C-
+V2X System, outlined in [4]. The highway consists of 3
+lanes on both sides. The roadside units (RSUs) are placed in
+the middle of the highway, with 100 m between them. RSU
+is a stationary communicating device capable of exchanging
+messages with other V2X devices. We consider there are K
+RSUs on the highway. Furthermore, we consider there are
+M platoons, with O vehicles in each platoon. The PL is
+required to share the real-time data with the RSUs so that
+the RSUs can form a dynamic map of the surrounding traffic.
+We refer to the message exchange from PLs to RSUs as
+V2I communication. Moreover, the PLs need to periodically
+transmit the cooperative awareness messages and the traffic
+data received from RSUs to the platoon members. In this
+paper, we refer to the communication from the PL to platoon
+members as V2V communications. Each platoon is denoted as
+m, and the platoon leader is denoted as m′. In our simulations,
+the PL selection is dynamic, and all vehicles in a platoon are
+candidates for becoming the PL. The vehicles in a platoon are
+in the same lane, each with a single PL at a given time. It is
+considered that the vehicles are separated by a fixed distance
+of d meters and are traveling with a velocity of v m/s. It is
+considered that the vehicles and RSUs use a single antenna to
+transmit/receive the signal.
+We consider that a fixed and limited number of sub-bands
+are available for both V2V and V2I links, denoted as N.
+Each sub-band has a bandwidth of W. The PL needs to
+transmit a payload of size BV 2I to the RSUs, and a payload
+of BV 2V to the platoon members, within a time constraint
+of T. We assume that both V2V and V2I links use the
+same spectrum for efficient spectrum utilization. However, all
+the V2V and V2I links can interfere, making an intelligent
+design for interference management necessary. The set of
+RSUs, platoons, and sub-bands are denoted as K, M, and N,
+respectively. Meanwhile, the set of members in each platoon
+is denoted as Vm, m = 1, . . . , O. In the paper, Lab refers to
+the large-scale fading power from transmitter a to receiver b.
+The small-scale fading power from a to b in the sub-band n
+is given by gab[n]. The ρab[n] is used as an indicator function
+set to 1 if the link ab reuses the sub-band n and 0 otherwise.
+The V2I links consist of communication from the PL to the
+RSU. Each PL m′ is required to transmit a fixed payload of
+BV 2I to the RSUs within a time constraint of T. We assume
+each RSU has a sub-band preassigned to it. Each RSU k will
+experience interference from other V2V and V2I links using
+the same sub-band n. Thus, the typical signal-to-interference
+(SINR) for a V2I link m′k can be written as:
+SINRm′k[n] = Pm′kLm′kgm′k[n]
+Ik + σ2
+k
+(1)
+Here, the interference at the RSU receiver is denoted by Ik
+Ik =
+�
+k∈K
+�
+a̸=m
+ρa′k[n]Pa′kLa′kga′k[n]+
+M
+�
+m=1
+�
+o∈Vm
+ρm′o[n]Pm′oLm′ogm′o[n]
+(2)
+In (1), the σ2
+k is the noise power at the RSU. For simplicity,
+we assume that the noise power at all RSUs is equal.
+
+Given the SINR, the achievable rate for the V2I link from
+m can be written as:
+Rm = W log2(1 + SINRm′k[n])
+(3)
+Meanwhile, the V2V link consists of communications be-
+tween the PL and the members. Each PL m′ is required to
+transmit a fixed payload of size BV 2V to each of its members o
+in the time constraint T. The platoon member o will experience
+interference from the other V2V and V2I links using the same
+sub-band n. The SINR for a platoon member o in platoon m
+can be written as:
+SINRmo[n] = Pm′oLm′ogm′o[n]
+Io + σ2o
+(4)
+Here, the interference at member o is denoted by Io
+Io =
+�
+k∈K
+�
+m∈M
+ρm′k[n]Pm′kLm′o,kgm′o,k[n]+
+M
+�
+a=1,a̸=m
+ρa′x[n]Pa′xLa′,moga′,mo[n]
+(5)
+where x denotes the platoon members in platoon a.
+Given the SINR, the achievable rate for the platoon member
+o in platoon m can be written as:
+Rmo =
+W
+O − 1 log2(1 + SINRmo[n])
+(6)
+where we divide the bandwidth equally among the platoon
+members.
+A. Problem Formulation
+We consider a multi-objective optimization problem, where
+we simultaneously maximize the payload delivery proba-
+bility for the V2V and V2I links. For the V2I link, the
+objective for each PL is to transmit the payload BV 2I
+to the RSUs within a time limit of T. This is given by
+P(∆T
+�T
+t=1
+�N
+n=1 ρmk[n, t] ≥ BV 2I), ∀m ∈ M, where
+∆T is the channel coherence time. Meanwhile, for the V2V
+links, the objective is to maximize the delivery of pay-
+load BV 2V
+within a time limit of T. This is given by
+P(∆T
+�T
+t=1
+�N
+n=1 ρmo[n, t] ≥ BV 2V ), ∀m ∈ M, o ∈ Vm.
+Due to the spectrum sharing between the V2V and V2I
+links, we need to optimize two competing objectives of simul-
+taneously maximizing the V2V and the V2I payload delivery
+probability. To achieve this, we use MARL for multiple
+objectives:
+1) Platoon Leader selection: For each platoon m, the PL
+will be selected dynamically and periodically. The pla-
+toon will decide which vehicle is the most suitable for
+being the leader so that both objectives can be met.
+2) Joint User Association and power allocation for V2I
+links: Each PL m′ will need to decide which RSU k
+it needs to be served by, along with its transmit power
+level Pm′k
+3) Joint channel assignment and power allocation for V2V
+links: Each PL m′ needs to decide the channel n and
+transmit power Pm′o to transmit to its platoon members.
+III. OPTIMAL ACTION SELECTION USING DEEP
+REINFORCEMENT LEARNING
+A. Reinforcement Learning and Deep Q Learning
+Reinforcement Learning (RL) is a discipline of Machine
+Learning (ML) where an agent can make a sequence of
+decisions by interacting with an environment. Based on the
+reward received by taking action, the agent learns to become
+intelligent. The agent aims to take actions that maximize
+the long-term cumulative reward. Markov Decision Process
+(MDP) is used to model an RL problem. According to the
+Markov property, the current state captures all relevant infor-
+mation from history. At each time-step t, the agent observes
+the environment through the state st and takes an action at.
+The agent receives a reward rt and transitions into a new
+state st+1. In RL, the goal of the agents is to maximize the
+cumulative reward Gt it receives in the long run, given by
+Gt = �∞
+k=0 γkrk+t where γ ∈ [0, 1] represents the discount
+factor which reduces the present value of the future rewards.
+The action-value function Qπ(s, a) is defined as the ex-
+pected return by taking an action a in state s by following
+a policy π: Qπ(s, a) = Eπ[Gt|St = s, At = a] where the
+expectation is taken over all possible transitions following the
+distribution π. The goal of RL is to find the optimal policy
+π∗ that maximizes the Q-function over all the policies.
+Q-learning is an off-policy RL algorithm that learns the
+value of an action a in a state s. It repetitively updates the
+action-value function for each state-action pair (s, a) until
+they converge to the optimal action-value function Q∗(s, a).
+The update equation is given by: Q(st, at) ← Q(st, at) +
+α[rt+1 + γ maxa′ Q(st+1, a′) − Q(st, at)] where α represents
+the learning rate. If the Q-function is estimated accurately,
+the optimal policy π∗ at a given state s would be to select
+the action a∗ that yields the highest value. The Deep Q-
+Learning [16] uses a Deep Neural Network (DNN) as a
+function approximator to learn the Q-function. The state space
+is input to the DNN, and it learns to predict the Q-value for
+each output action. The state-action space is explored with a
+soft policy such as ϵ-greedy, where the agent takes random
+action at a given state st at time t with a probability of
+ϵ. Otherwise, greedy action a∗ = arg maxa∈A Q∗(st, a) is
+selected. The tuple < st, at, rt, st+1 > is stored in the replay
+memory at each time instance. At each time-step, a mini-
+batch is sampled from the replay memory to update the DNN
+parameters θ, and the gradient descent algorithms are used to
+update the parameters θ.
+B. Multi-Agent Reinforcement Learning for optimization
+In this section, we formulate the multi-agent RL algorithm
+for optimization. There will be three different types of agents,
+all based within the vehicles in the platoon. The first agent
+type will be the PL selection, which will dynamically decide
+the PL every 100 ms. The second type of agent will be the
+
+V2V platoon, which needs to determine the joint channel
+assignment and power allocation. Meanwhile, the third type
+of agent will be the V2I agent, which will need to optimize
+joint user association and power allocation. All the agents will
+interact with the environment and learn to take optimal actions
+by trial and error. Furthermore, we use a common reward for
+all the agents to ensure collaboration. Moreover, each agent
+has a separate DQN and only uses its own experience to train
+the DNN.
+We develop two phases for the MARL problem: training
+and testing. During the training phase, each agent can access
+the common reward to train the DQN. Meanwhile, during the
+testing phase, each agent uses the trained DQN to select the
+action optimally.
+1) State and Action Space for Platoon Leader Selection:
+For the platoon leader selection agent, the state space Zpl(t)
+consist of the measurements at time-step t. The state space
+consists of the following measurements: i) The large-scale fad-
+ing information between all members within a platoon m, i.e.,
+{Lab}(a,b)∈Vm; ii) The large-scale fading information between
+all vehicles in platoon m to all RSUs, i.e., {Lok}k∈K,o∈Vm.
+Meanwhile, the action space consists of the PL selection. The
+action is updated every 100 ms.
+2) State and Action Space for V2V agent:
+The state
+space of the V2V agent, denoted by Zv2v(t), consists of
+the measurements from the last time-step t and consists of
+the following groups: i) Direct channel measurements from
+the PL m′ to the members, i.e., {Lm′ogm′o[n]}o∈Vm ii)
+The interfering channels from other PLs sharing the same
+sub-band with the V2V agent m, which occupies the sub-
+band n, i.e., {ρa′x[n]Pa′xLa′,moga′mo}a∈M,a̸=m,x∈Va iii) The
+interfering channels from the V2I links to the RSU, i.e.,
+{ρm′k[n]Pm′kLm′o,kgm′ok}o∈Vm,k∈K iv) The remaining pay-
+load and time limitation after the current time-step.
+Meanwhile, the action space consists of the combination
+of sub-band selection and power allocation. The sub-band
+consists of N disjoint sub-bands, and the power levels are
+broken down into multiple discrete levels in the range [0, Pd],
+where Pd denotes the maximum power.
+3) State and Action Space for V2I agent: The state space
+of the V2I agent, denoted by Zv2i(t), for the PL m′, consists
+of the measurements from the last time-step t and consists of
+the following groups: i) Direct channel measurements from
+the PL m′ to all the RSUs, i.e., {Lm′kgm′k}k∈K ii) The
+remaining payload and time remaining after current time-step.
+iii) The training iteration number e and the agent’s probability
+of random action selection ϵ.
+The action space consists of the RSU selected and the power
+level. We assume that each RSU uses a fixed sub-band. There
+are K RSUs to select from and transmit at a power divided
+into multiple discrete levels in the range [0, Pd].
+4) Reward function design: We use a common reward
+for all the agents in our proposed MARL design to ensure
+collaboration. We have a multi-objective problem, which is
+to maximize the payload delivery probability for the PL to
+RSU V2I links, and maximize the payload delivery probability
+for the PL to platoon members V2V links, within the time
+constraint T. The V2V and V2I agents need to select actions
+to minimize interference between each other. To achieve this
+purpose, we define the reward at time-step t, denoted as rt,
+as:
+rt = wc
+M
+�
+m=1
+�
+o∈Vm
+Umo(t) + wd
+M
+�
+m=1
+Vm(t)
+(7)
+where Umo(t) is the contribution towards the reward of the
+V2V link m′o and Vm(t) is the contribution of the V2I link
+from PL m′ to RSU. Furthermore, wc, wd ∈ [0, 1] are weights
+to balance the two objectives.
+Umo(t) is the achievable rate of the PL to platoon member
+link mo, defined as:
+Umo(t) =
+�
+Rmo(t),
+if Bmo(t) ≥ 0,
+U,
+otherwise.
+(8)
+where Bmo(t) is the remaining payload for the V2V link mo at
+time-step t. Furthermore, if the payload intended for link mo
+has been delivered, the agent is given an award U, which needs
+to be greater than the maximum rate achievable, to indicate
+to the agent the successful transmission of the payload. U is
+a hyperparameter that needs to be adjusted empirically [12].
+Similarly, Vm(t) is the achievable rate of the PL to RSU
+link, defined as:
+Vm(t) =
+�
+Rm(t),
+if Bm(t) ≥ 0,
+V,
+otherwise.
+(9)
+where Bm(t) is the payload that PL m′ needs to transmit to
+the RSUs. V is a hyperparameter, which needs to be greater
+than the maximum achievable rate of the V2I link.
+C. Training Algorithm and Testing Strategy
+We devise the problem as an episodic setting, where each
+episode corresponds to the time limit T for the V2V and V2I
+links to complete their transmission. Each episode consists
+of multiple time-steps t. The vehicle location and large-scale
+fading are updated every 100 ms [4]. Meanwhile, the small-
+scale fading is updated at each time-step t, changing the state
+space for the V2V and V2I agents and prompting the agents to
+adjust their actions. Each agent stops its transmission once its
+payload has been delivered. The training is centralized, where
+each agent has access to the common reward rt. Deep Q-
+Learning is used to train the agent. The algorithm is outlined
+in Algorithm 1.
+During the testing phase, each agent observes the state. The
+state is input to the trained DQN, which is used to select the
+optimal action. The testing is implemented in a distributed
+manner, where each agent takes action based on their local
+state observation only.
+IV. ILLUSTRATIVE RESULTS
+This section presents the simulation results to illustrate the
+performance of our algorithm. We consider a highway setting
+as described in TR 37.885 [4], with the carrier frequency
+of 6 GHz. The technical report provides all details, such
+
+Algorithm 1 Training Algorithm
+1: Initiate the environment and generate the V2I and V2V links
+2: Initiate the DQN with random parameters θ
+3: for each episode i do
+4:
+if i%20 = 0 then
+5:
+Update the vehicle locations and large-scale fading
+6:
+for each PL selection agent n1 do
+7:
+Observe the state st1 for PL selection and take action
+at1
+8:
+end for
+9:
+end if
+10:
+for each time-step t do
+11:
+Update small-scale channel fading
+12:
+for each V2V agent n2 do
+13:
+Observe the state st2
+14:
+end for
+15:
+for each V2I agent n3 do
+16:
+Observe the state st3
+17:
+end for
+18:
+All agents take actions simultaneously according to the
+ϵ−greedy policy and receive the common reward rt
+19:
+for each agent {n1, n2, n3} do
+20:
+Observe the next state st+1
+21:
+Store et = [st, at, rt, st+1] in the replay memory
+22:
+end for
+23:
+end for
+24:
+for each agent {n1, n2, n3} do
+25:
+Uniformly sample mini-batch data D from replay memory
+26:
+Train the deep Q-networks using the mini-batch data.
+27:
+end for
+28: end for
+as evaluation scenarios, vehicle drop and mobility modeling,
+RSU deployment, and channel models for V2V and V2I links.
+The small-scale fading is modeled as Rayleigh fading. We
+consider a highway with a length of 1 km, with 3 lanes for
+traffic on both sides. The RSUs are placed in the middle of
+the highway, with a distance of 100 m between them. We use
+option A in UE drop options in Section 6.1.2 of TR 37.885
+[4]. The type 3 vehicles (bus/tracks) are used, with a length of
+13 m and 2 m distance between each in a platoon. All vehicles
+travel with a velocity of 140 km/h. Each V2V platoon consists
+of 3 vehicles. Moreover, the antenna on vehicles is placed in
+the middle of each vehicle. As per TS 22.185 [3], the V2V
+and V2I links need to complete their transmission in 10 ms.
+However, we set T as 5 ms, assuming the other 5 ms will
+be used for communication in other directions, i.e., platoon
+member to platoon leader. The main simulation parameters
+are listed in Table I.
+The DQN was implemented in Python using the Tensorflow
+package. The DNN for all 3 types of agents consisted of 3
+hidden layers. The DNN of PL selection agents had 71, 35,
+and 17 neurons, the DNN of V2V agents had 100, 50, and
+24 neurons; and the DNN of V2I agents had 166, 83, and
+40 neurons in their hidden layers, respectively. The rectified
+linear unit (ReLU) was us as the activation function for all
+3 types of agents. RMSProp was used for optimization for
+all agents, and learning rates of 0.0001, 0.0001, and 0.001
+were used for PL selection agents, V2V agents, and V2I
+agents, respectively. The wc and wd in (7) were set as 0.3
+and 0.7, respectively. Meanwhile, the hyperparameters U in
+TABLE I
+SIMULATION PARAMETERS
+Carrier frequency
+6 GHz
+Bandwidth of each sub-band
+1 MHz
+Number of sub-bands N
+2
+Number of RSUs K
+11
+Number of platoons M
+[4,6]
+Number of vehicles in each platoon O
+3
+Vehicle velocity v
+140 km/h
+Tx power for V2V links
+[23, 15, 5, -100] dBm
+Tx power for V2I links
+[23, -100] dBm
+Vehicle Antenna gain
+3 dBi
+Vehicle receiver noise figure
+9 dB
+Noise PSD
+-169 dBm/Hz
+Time constraint T
+5 ms
+Platoon Leader update interval
+100 ms
+V2V payload BV 2V
+[1200,.....,2800] bytes
+V2I payload BV 2I
+624 bytes
+(8) and V in (9) were selected as 25 and 15, respectively. The
+training phase consisted of 2000 episodes, and the testing was
+performed for 100 episodes. The ϵ-greedy policy was used
+during the training, and the value of ϵ was reduced linearly
+from 1 to 0.02 for 1600 episodes. The training was performed
+setting BV 2V as 2400 bytes and was varied between 1200-
+2800 bytes during testing. Meanwhile, BV 2I was set to 624
+bytes during the training and testing phases.
+We developed three benchmarks for comparison: 1) Hill-
+climbing algorithm [17]: Hill-climbing algorithm is a local
+search optimization method, guaranteed to reach a local opti-
+mum. It is a centralized iterative algorithm, which starts with a
+random solution, and then iteratively keeps improving it until
+it reaches an optimum. The algorithm is used as an upper
+benchmark in our paper. 2) Greedy Algorithm: Each agent
+uses the best link available to transmit at maximum power. 3)
+RL algorithm without PL selection: We run the RL algorithm,
+fixing the leading vehicle in each platoon as the platoon leader.
+This is to show the effectiveness of PL selection agents.
+0
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+1800
+2000
+Training episodes
+2
+3
+4
+5
+6
+7
+8
+Cummulative reward per episode
+Fig. 2. Cumulative reward per episode for M = 4
+Fig. 2 shows the cumulative reward for each episode for
+the case of 4 platoons. The reward increases during training,
+indicating that the agents can collaborate.
+
+6
+7
+8
+9
+10
+11
+12
+13
+14
+V2V payload size BV2V ( 
+ 200 bytes)
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Average V2V payload success rate
+Proposed RL algorithm
+Hill climbing algorithm
+RL algorithm without PL selection
+Greedy Algorithm
+4 platoons
+6 platoons
+(a) Average V2V payload delivery probability
+6
+7
+8
+9
+10
+11
+12
+13
+14
+V2V payload size BV2V ( 
+ 200 bytes)
+0.8
+0.82
+0.84
+0.86
+0.88
+0.9
+0.92
+0.94
+0.96
+0.98
+1
+Average V2I payload success rate
+Proposed RL algorithm
+Hill climbing algorithm
+RL algorithm without PL selection
+Greedy Algorithm
+4 platoons
+6 platoons
+(b) Average V2I payload delivery probability
+Fig. 3. V2V and V2I performance for M = 4 and M = 6.
+Fig. 3 shows the reliability of V2V and V2I links as we
+increase BV 2V for the cases of 4 and 6 platoons. Fig. 3a
+shows that the V2V performance decreases as we increase the
+packet size. This is because a larger payload size requires
+a longer time to transmit. For 4 platoons, we achieve a
+reliability of 1 for up to 2200 bytes, outperforming all the other
+benchmarks. Fig. 3b shows that for 4 platoons, the payload
+success rate for V2I links is 0.9975 for all cases. However,
+the hill-climbing and greedy algorithm performance decrease,
+indicating more significant interference as we increase V2V
+payload size. When we increase the number of platoons to 6,
+the performance gap between the proposed algorithm and the
+hill-climbing algorithm increases, which shows the superiority
+of our algorithm for a higher number of agents. Furthermore,
+it can be seen that dynamic platoon leader selection improves
+performance for both V2V and V2I links.
+V. CONCLUSION
+In this work, we proposed a distributed multi-agent rein-
+forcement learning algorithm to optimize the performance of
+the V2V and the V2I links. The V2V and V2I links used
+the same spectrum, making an intelligent resource allocation
+design necessary to manage interference. Each platoon leader
+had an agent for joint channel assignment and power allocation
+for V2V links, and another agent for joint user association
+and power allocation for V2I links. Further, another agent was
+able to select the platoon leader, to maximize the reliability
+of both V2V and V2I links. Based on RL, the agents could
+collaborate to take optimal actions. The proposed approach
+is decentralized, and the agents were able to make decisions
+based on their local state observations only. Simulation results
+indicate that the proposed algorithm could perform well for
+variable V2V packet size and different numbers of platoons,
+outperforming the centralized hill-climbing algorithm. More-
+over, the PL selection improved reliability for both V2V and
+V2I links.
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+

LtE1T4oBgHgl3EQfZAR1/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf,len=480
+page_content='Platoon Leader Selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' User Association and  Resource Allocation on a C-V2X based highway: A  Reinforcement Learning Approach  Mohammad Farzanullah and Tho Le-Ngoc  Department of Electrical & Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' McGill University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Montr´eal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' QC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Canada  Abstract—We consider the problem of dynamic platoon leader  selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' user association,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' channel assignment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' and power allo-  cation on a cellular vehicle-to-everything (C-V2X) based high-  way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' where multiple vehicle-to-vehicle (V2V) and vehicle-to-  infrastructure (V2I) links share the frequency resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' There  are multiple roadside units (RSUs) on a highway, and vehicles can  form platoons, which has been identified as an advanced use case  to increase road efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The traditional optimization methods,  requiring global channel information at a central controller, are  not viable for high-mobility vehicular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To deal with this  challenge, we propose a distributed multi-agent reinforcement  learning (MARL) for resource allocation (RA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each platoon  leader, acting as an agent, can collaborate with other agents for  joint sub-band selection and power allocation for its V2V links,  and joint user association and power control for its V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Moreover, each platoon can dynamically select the vehicle most  suitable to be the platoon leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We aim to maximize the V2V  and V2I packet delivery probability in the desired latency using  the deep Q-learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Simulation results indicate that  our proposed MARL outperforms the centralized hill-climbing  algorithm, and platoon leader selection helps to improve both  V2V and V2I performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Index Terms—New radio, cellular vehicle-to-everything, rein-  forcement learning, resource allocation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' INTRODUCTION  Cellular vehicle-to-everything (C-V2X) is a vehicular stan-  dard that enables communication between vehicles and other  entities on the road, such as pedestrians and infrastructure, to  increase road safety and efficiency [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The C-V2X system  consists of communication between different entities, such  as vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I),  vehicle-to-network (V2N), and vehicle-to-pedestrian (V2P)  communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The C-V2X is envisioned to support high  throughput, ultra-reliable, and low latency communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  3GPP has provided the 5G new radio (NR) standards  for vehicular communications concerning architecture in TS  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='287 [2], advanced use cases in TS 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='886 [3], and eval-  uation methodology in TR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='885 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' One component of  infrastructure will be the roadside units (RSUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' RSU is a  stationary wireless C-V2X device that can exchange messages  with vehicles and other C-V2X entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It uses the PC5  side-link interface to communicate with the vehicles and  transmit information about road signs and traffic lights [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  It can also receive information from the vehicles to make  a dynamic map of the surroundings and share it with other  vehicles/pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we consider the use-case of  platooning, where multiple vehicles form a train-like structure  and travel closely together in a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The platoon leader (PL)  organizes communications between vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Vehicle platoon-  ing has been identified as an advanced use case in [3] and has  gained significant interest since it reduces fuel consumption  and traffic congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The RSU and platoon will need to  exchange a maximum of 1200 bytes in 500 ms for real-  time traffic updates and 600 bytes in 10 ms for conditional  automated driving [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We grouped these two requirements  for an aggregate of 624 bytes in 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the PL  and members need to exchange 50-1200 bytes in 10 ms for  cooperative driving and up to 2000 bytes in 10 ms for collision  avoidance [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We aggregate these service requirements and  keep the exchange of 1200-2800 bytes in 10 ms for our  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' An intelligent resource allocation (RA) design is  necessary for these stringent requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The 5G new radio (NR) C-V2X supports two RA modes  for sidelink PC5 communications: mode 1, the under-coverage  mode, and mode 2, the out-of-coverage mode [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In mode 1,  the gNB allocates the communication resources to vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Meanwhile, in mode 2, the vehicles autonomously select  the resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For mode 2, the current RA technique used  in standards is the sensing-based Semi-Persistent Scheduling  (SPS) algorithm, which periodically selects random resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  However, the probability of resource selection collision can be  high, and many works have considered either improving the  SPS algorithm or alternate techniques to increase reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  [6] proposes a novel sensing-based SPS algorithm in an urban  scenario to reduce the collision probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The authors in  [7] considered a highway scenario and suggested a stochastic  reservation scheme for aperiodic traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For the platooning  use case, [8] shows that the SPS algorithm can not achieve  the required performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Due to the fast channel variations in vehicular networks,  centralized optimization schemes that require global channel  state information (CSI) will no longer be feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The high  CSI overhead and the corresponding increase in latency make  such methods impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To deal with this issue, distributed  RA algorithms have been suggested in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', [9],  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, traditional optimization techniques have  limitations, requiring complete information about the environ-  ment and needing to be retrained for rapidly varying envi-  ronments [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Recently, distributed multi-agent reinforcement  learning (MARL) has been proposed as an alternative approach  to resolving such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The authors in [12] used Deep Q  Networks (DQN) for joint channel assignment and power  allocation to maximize the V2V delivery probability and V2N  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='03145v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='MA]  9 Jan 2023  sum-rate in an urban setting, where the V2V links share the  time-frequency resources with V2N links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Inspired by these  results, [13] used double DQN for a platoon-based scenario  for the same objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' [14] uses the actor-critic method for  mode selection, subchannel selection, and power control in an  urban platoon scenario to increase the transmission probability  of cooperative awareness messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  This paper considers a highway C-V2X system consisting  of multiple platoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We consider the periodic payload deliv-  ery from PLs to RSUs, termed V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we  consider the periodic transmission of messages from PLs to  members, termed V2V links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume a limited spectrum  is available for the V2V and V2I transmission, and they share  the frequency resources for efficient spectrum usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Given  this system, this paper formulates a dynamic PL selection,  user association, channel assignment, and power level control  to maximize the packet delivery probability for both V2V and  V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Reliability is defined as the successful transmission  of the packet within a time constraint T [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We utilize MARL in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The RL works  on a trial-and-error strategy, and each agent slowly improves  the action taken based on the feedback from the vehicular  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We use the Deep Q-learning algorithm, which  DeepMind developed for Atari video games [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Deep Q-  Learning has been used for joint channel assignment and  power allocation in C-V2X systems [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' However,  we also use Deep Q-learning for user association and PL  selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' As per our knowledge, dynamic PL selection has  not been investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In our work, there  are multiple collaborative agents for PL selection, V2V joint  channel assignment and power allocation, and V2I joint user  association and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The objective is to increase  reliability for both V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Simulation results  indicate that the proposed MARL algorithm can outperform  other benchmarks, such as the hill-climbing algorithm, which  requires global CSI at the central controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the  dynamic PL selection offers a gain in V2V and V2I reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' SYSTEM MODEL AND PROBLEM FORMULATION  Road Side Unit (RSU)  V2V link  V2I link  Platoon Leader  Platoon Member  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Illustrative C-V2X based highway, where multiple platoon leaders  are transmitting to RSUs using the V2I links, and each platoon leader is  transmitting to its platoon members using the V2V links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 1, we consider a highway-based C-  V2X System, outlined in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The highway consists of 3  lanes on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The roadside units (RSUs) are placed in  the middle of the highway, with 100 m between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' RSU  is a stationary communicating device capable of exchanging  messages with other V2X devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We consider there are K  RSUs on the highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we consider there are  M platoons, with O vehicles in each platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The PL is  required to share the real-time data with the RSUs so that  the RSUs can form a dynamic map of the surrounding traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We refer to the message exchange from PLs to RSUs as  V2I communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the PLs need to periodically  transmit the cooperative awareness messages and the traffic  data received from RSUs to the platoon members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In this  paper, we refer to the communication from the PL to platoon  members as V2V communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each platoon is denoted as  m, and the platoon leader is denoted as m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In our simulations,  the PL selection is dynamic, and all vehicles in a platoon are  candidates for becoming the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The vehicles in a platoon are  in the same lane, each with a single PL at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It is  considered that the vehicles are separated by a fixed distance  of d meters and are traveling with a velocity of v m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It is  considered that the vehicles and RSUs use a single antenna to  transmit/receive the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We consider that a fixed and limited number of sub-bands  are available for both V2V and V2I links, denoted as N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Each sub-band has a bandwidth of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The PL needs to  transmit a payload of size BV 2I to the RSUs, and a payload  of BV 2V to the platoon members, within a time constraint  of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume that both V2V and V2I links use the  same spectrum for efficient spectrum utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' However, all  the V2V and V2I links can interfere, making an intelligent  design for interference management necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The set of  RSUs, platoons, and sub-bands are denoted as K, M, and N,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, the set of members in each platoon  is denoted as Vm, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' , O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In the paper, Lab refers to  the large-scale fading power from transmitter a to receiver b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The small-scale fading power from a to b in the sub-band n  is given by gab[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The ρab[n] is used as an indicator function  set to 1 if the link ab reuses the sub-band n and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The V2I links consist of communication from the PL to the  RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each PL m′ is required to transmit a fixed payload of  BV 2I to the RSUs within a time constraint of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume  each RSU has a sub-band preassigned to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each RSU k will  experience interference from other V2V and V2I links using  the same sub-band n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Thus, the typical signal-to-interference  (SINR) for a V2I link m′k can be written as:  SINRm′k[n] = Pm′kLm′kgm′k[n]  Ik + σ2  k  (1)  Here, the interference at the RSU receiver is denoted by Ik  Ik =  �  k∈K  �  a̸=m  ρa′k[n]Pa′kLa′kga′k[n]+  M  �  m=1  �  o∈Vm  ρm′o[n]Pm′oLm′ogm′o[n]  (2)  In (1), the σ2  k is the noise power at the RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For simplicity,  we assume that the noise power at all RSUs is equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Given the SINR, the achievable rate for the V2I link from  m can be written as:  Rm = W log2(1 + SINRm′k[n])  (3)  Meanwhile, the V2V link consists of communications be-  tween the PL and the members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each PL m′ is required to  transmit a fixed payload of size BV 2V to each of its members o  in the time constraint T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The platoon member o will experience  interference from the other V2V and V2I links using the same  sub-band n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The SINR for a platoon member o in platoon m  can be written as:  SINRmo[n] = Pm′oLm′ogm′o[n]  Io + σ2o  (4)  Here, the interference at member o is denoted by Io  Io =  �  k∈K  �  m∈M  ρm′k[n]Pm′kLm′o,kgm′o,k[n]+  M  �  a=1,a̸=m  ρa′x[n]Pa′xLa′,moga′,mo[n]  (5)  where x denotes the platoon members in platoon a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Given the SINR, the achievable rate for the platoon member  o in platoon m can be written as:  Rmo =  W  O − 1 log2(1 + SINRmo[n])  (6)  where we divide the bandwidth equally among the platoon  members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Problem Formulation  We consider a multi-objective optimization problem, where  we simultaneously maximize the payload delivery proba-  bility for the V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For the V2I link, the  objective for each PL is to transmit the payload BV 2I  to the RSUs within a time limit of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' This is given by  P(∆T  �T  t=1  �N  n=1 ρmk[n, t] ≥ BV 2I), ∀m ∈ M, where  ∆T is the channel coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, for the V2V  links, the objective is to maximize the delivery of pay-  load BV 2V  within a time limit of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' This is given by  P(∆T  �T  t=1  �N  n=1 ρmo[n, t] ≥ BV 2V ), ∀m ∈ M, o ∈ Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Due to the spectrum sharing between the V2V and V2I  links, we need to optimize two competing objectives of simul-  taneously maximizing the V2V and the V2I payload delivery  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To achieve this, we use MARL for multiple  objectives:  1) Platoon Leader selection: For each platoon m, the PL  will be selected dynamically and periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The pla-  toon will decide which vehicle is the most suitable for  being the leader so that both objectives can be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  2) Joint User Association and power allocation for V2I  links: Each PL m′ will need to decide which RSU k  it needs to be served by, along with its transmit power  level Pm′k  3) Joint channel assignment and power allocation for V2V  links: Each PL m′ needs to decide the channel n and  transmit power Pm′o to transmit to its platoon members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' OPTIMAL ACTION SELECTION USING DEEP  REINFORCEMENT LEARNING  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Reinforcement Learning and Deep Q Learning  Reinforcement Learning (RL) is a discipline of Machine  Learning (ML) where an agent can make a sequence of  decisions by interacting with an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Based on the  reward received by taking action, the agent learns to become  intelligent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The agent aims to take actions that maximize  the long-term cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Markov Decision Process  (MDP) is used to model an RL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' According to the  Markov property, the current state captures all relevant infor-  mation from history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' At each time-step t, the agent observes  the environment through the state st and takes an action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The agent receives a reward rt and transitions into a new  state st+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' In RL, the goal of the agents is to maximize the  cumulative reward Gt it receives in the long run, given by  Gt = �∞  k=0 γkrk+t where γ ∈ [0, 1] represents the discount  factor which reduces the present value of the future rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The action-value function Qπ(s, a) is defined as the ex-  pected return by taking an action a in state s by following  a policy π: Qπ(s, a) = Eπ[Gt|St = s, At = a] where the  expectation is taken over all possible transitions following the  distribution π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The goal of RL is to find the optimal policy  π∗ that maximizes the Q-function over all the policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Q-learning is an off-policy RL algorithm that learns the  value of an action a in a state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It repetitively updates the  action-value function for each state-action pair (s, a) until  they converge to the optimal action-value function Q∗(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The update equation is given by: Q(st, at) ← Q(st, at) +  α[rt+1 + γ maxa′ Q(st+1, a′) − Q(st, at)] where α represents  the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' If the Q-function is estimated accurately,  the optimal policy π∗ at a given state s would be to select  the action a∗ that yields the highest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The Deep Q-  Learning [16] uses a Deep Neural Network (DNN) as a  function approximator to learn the Q-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The state space  is input to the DNN, and it learns to predict the Q-value for  each output action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The state-action space is explored with a  soft policy such as ϵ-greedy, where the agent takes random  action at a given state st at time t with a probability of  ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Otherwise, greedy action a∗ = arg maxa∈A Q∗(st, a) is  selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The tuple < st, at, rt, st+1 > is stored in the replay  memory at each time instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' At each time-step, a mini-  batch is sampled from the replay memory to update the DNN  parameters θ, and the gradient descent algorithms are used to  update the parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Multi-Agent Reinforcement Learning for optimization  In this section, we formulate the multi-agent RL algorithm  for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' There will be three different types of agents,  all based within the vehicles in the platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The first agent  type will be the PL selection, which will dynamically decide  the PL every 100 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The second type of agent will be the  V2V platoon, which needs to determine the joint channel  assignment and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, the third type  of agent will be the V2I agent, which will need to optimize  joint user association and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' All the agents will  interact with the environment and learn to take optimal actions  by trial and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, we use a common reward for  all the agents to ensure collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, each agent  has a separate DQN and only uses its own experience to train  the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We develop two phases for the MARL problem: training  and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' During the training phase, each agent can access  the common reward to train the DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, during the  testing phase, each agent uses the trained DQN to select the  action optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  1) State and Action Space for Platoon Leader Selection:  For the platoon leader selection agent, the state space Zpl(t)  consist of the measurements at time-step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The state space  consists of the following measurements: i) The large-scale fad-  ing information between all members within a platoon m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=',  {Lab}(a,b)∈Vm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' ii) The large-scale fading information between  all vehicles in platoon m to all RSUs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {Lok}k∈K,o∈Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Meanwhile, the action space consists of the PL selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The  action is updated every 100 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  2) State and Action Space for V2V agent:  The state  space of the V2V agent, denoted by Zv2v(t), consists of  the measurements from the last time-step t and consists of  the following groups: i) Direct channel measurements from  the PL m′ to the members, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {Lm′ogm′o[n]}o∈Vm ii)  The interfering channels from other PLs sharing the same  sub-band with the V2V agent m, which occupies the sub-  band n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {ρa′x[n]Pa′xLa′,moga′mo}a∈M,a̸=m,x∈Va iii) The  interfering channels from the V2I links to the RSU, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=',  {ρm′k[n]Pm′kLm′o,kgm′ok}o∈Vm,k∈K iv) The remaining pay-  load and time limitation after the current time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Meanwhile, the action space consists of the combination  of sub-band selection and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The sub-band  consists of N disjoint sub-bands, and the power levels are  broken down into multiple discrete levels in the range [0, Pd],  where Pd denotes the maximum power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  3) State and Action Space for V2I agent: The state space  of the V2I agent, denoted by Zv2i(t), for the PL m′, consists  of the measurements from the last time-step t and consists of  the following groups: i) Direct channel measurements from  the PL m′ to all the RSUs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', {Lm′kgm′k}k∈K ii) The  remaining payload and time remaining after current time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  iii) The training iteration number e and the agent’s probability  of random action selection ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The action space consists of the RSU selected and the power  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We assume that each RSU uses a fixed sub-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' There  are K RSUs to select from and transmit at a power divided  into multiple discrete levels in the range [0, Pd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  4) Reward function design: We use a common reward  for all the agents in our proposed MARL design to ensure  collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We have a multi-objective problem, which is  to maximize the payload delivery probability for the PL to  RSU V2I links, and maximize the payload delivery probability  for the PL to platoon members V2V links, within the time  constraint T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The V2V and V2I agents need to select actions  to minimize interference between each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' To achieve this  purpose, we define the reward at time-step t, denoted as rt,  as:  rt = wc  M  �  m=1  �  o∈Vm  Umo(t) + wd  M  �  m=1  Vm(t)  (7)  where Umo(t) is the contribution towards the reward of the  V2V link m′o and Vm(t) is the contribution of the V2I link  from PL m′ to RSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, wc, wd ∈ [0, 1] are weights  to balance the two objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Umo(t) is the achievable rate of the PL to platoon member  link mo, defined as:  Umo(t) =  �  Rmo(t),  if Bmo(t) ≥ 0,  U,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  (8)  where Bmo(t) is the remaining payload for the V2V link mo at  time-step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore, if the payload intended for link mo  has been delivered, the agent is given an award U, which needs  to be greater than the maximum rate achievable, to indicate  to the agent the successful transmission of the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' U is  a hyperparameter that needs to be adjusted empirically [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Similarly, Vm(t) is the achievable rate of the PL to RSU  link, defined as:  Vm(t) =  �  Rm(t),  if Bm(t) ≥ 0,  V,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  (9)  where Bm(t) is the payload that PL m′ needs to transmit to  the RSUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' V is a hyperparameter, which needs to be greater  than the maximum achievable rate of the V2I link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Training Algorithm and Testing Strategy  We devise the problem as an episodic setting, where each  episode corresponds to the time limit T for the V2V and V2I  links to complete their transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each episode consists  of multiple time-steps t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The vehicle location and large-scale  fading are updated every 100 ms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, the small-  scale fading is updated at each time-step t, changing the state  space for the V2V and V2I agents and prompting the agents to  adjust their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each agent stops its transmission once its  payload has been delivered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The training is centralized, where  each agent has access to the common reward rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Deep Q-  Learning is used to train the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The algorithm is outlined  in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  During the testing phase, each agent observes the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The  state is input to the trained DQN, which is used to select the  optimal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The testing is implemented in a distributed  manner, where each agent takes action based on their local  state observation only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' ILLUSTRATIVE RESULTS  This section presents the simulation results to illustrate the  performance of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We consider a highway setting  as described in TR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='885 [4], with the carrier frequency  of 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The technical report provides all details,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' such  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Algorithm 1 Training Algorithm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='1: Initiate the environment and generate the V2I and V2V links  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='2: Initiate the DQN with random parameters θ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='3: for each episode i do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='4:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='if i%20 = 0 then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='5:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Update the vehicle locations and large-scale fading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='6:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each PL selection agent n1 do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='7:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Observe the state st1 for PL selection and take action  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='at1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='8:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='10:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each time-step t do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='11:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Update small-scale channel fading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='12:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each V2V agent n2 do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='13:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Observe the state st2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='14:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='15:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each V2I agent n3 do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='16:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='Observe the state st3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='17:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='18:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='All agents take actions simultaneously according to the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='ϵ−greedy policy and receive the common reward rt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='19:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='for each agent {n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n3} do  20:  Observe the next state st+1  21:  Store et = [st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' st+1] in the replay memory  22:  end for  23:  end for  24:  for each agent {n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' n3} do  25:  Uniformly sample mini-batch data D from replay memory  26:  Train the deep Q-networks using the mini-batch data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  27:  end for  28: end for  as evaluation scenarios, vehicle drop and mobility modeling,  RSU deployment, and channel models for V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The small-scale fading is modeled as Rayleigh fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We  consider a highway with a length of 1 km, with 3 lanes for  traffic on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The RSUs are placed in the middle of  the highway, with a distance of 100 m between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' We use  option A in UE drop options in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='2 of TR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='885  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The type 3 vehicles (bus/tracks) are used, with a length of  13 m and 2 m distance between each in a platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' All vehicles  travel with a velocity of 140 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each V2V platoon consists  of 3 vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Moreover, the antenna on vehicles is placed in  the middle of each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' As per TS 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='185 [3], the V2V  and V2I links need to complete their transmission in 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  However, we set T as 5 ms, assuming the other 5 ms will  be used for communication in other directions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=', platoon  member to platoon leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The main simulation parameters  are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  The DQN was implemented in Python using the Tensorflow  package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The DNN for all 3 types of agents consisted of 3  hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The DNN of PL selection agents had 71, 35,  and 17 neurons, the DNN of V2V agents had 100, 50, and  24 neurons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' and the DNN of V2I agents had 166, 83, and  40 neurons in their hidden layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The rectified  linear unit (ReLU) was us as the activation function for all  3 types of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' RMSProp was used for optimization for  all agents, and learning rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='0001, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='001  were used for PL selection agents, V2V agents, and V2I  agents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The wc and wd in (7) were set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='3  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' the hyperparameters U in  TABLE I  SIMULATION PARAMETERS  Carrier frequency  6 GHz  Bandwidth of each sub-band  1 MHz  Number of sub-bands N  2  Number of RSUs K  11  Number of platoons M  [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='6]  Number of vehicles in each platoon O  3  Vehicle velocity v  140 km/h  Tx power for V2V links  [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' -100] dBm  Tx power for V2I links  [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' -100] dBm  Vehicle Antenna gain  3 dBi  Vehicle receiver noise figure  9 dB  Noise PSD  169 dBm/Hz  Time constraint T  5 ms  Platoon Leader update interval  100 ms  V2V payload BV 2V  [1200,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=',2800] bytes  V2I payload BV 2I  624 bytes  (8) and V in (9) were selected as 25 and 15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The  training phase consisted of 2000 episodes, and the testing was  performed for 100 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The ϵ-greedy policy was used  during the training, and the value of ϵ was reduced linearly  from 1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='02 for 1600 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The training was performed  setting BV 2V as 2400 bytes and was varied between 1200-  2800 bytes during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Meanwhile, BV 2I was set to 624  bytes during the training and testing phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  We developed three benchmarks for comparison: 1) Hill-  climbing algorithm [17]: Hill-climbing algorithm is a local  search optimization method, guaranteed to reach a local opti-  mum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' It is a centralized iterative algorithm, which starts with a  random solution, and then iteratively keeps improving it until  it reaches an optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The algorithm is used as an upper  benchmark in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 2) Greedy Algorithm: Each agent  uses the best link available to transmit at maximum power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3)  RL algorithm without PL selection: We run the RL algorithm,  fixing the leading vehicle in each platoon as the platoon leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  This is to show the effectiveness of PL selection agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  0  200  400  600  800  1000  1200  1400  1600  1800  2000  Training episodes  2  3  4  5  6  7  8  Cummulative reward per episode  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Cumulative reward per episode for M = 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 2 shows the cumulative reward for each episode for  the case of 4 platoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The reward increases during training,  indicating that the agents can collaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  6  7  8  9  10  11  12  13  14  V2V payload size BV2V (  200 bytes)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='95  1  Average V2V payload success rate  Proposed RL algorithm  Hill climbing algorithm  RL algorithm without PL selection  Greedy Algorithm  4 platoons  6 platoons  (a) Average V2V payload delivery probability  6  7  8  9  10  11  12  13  14  V2V payload size BV2V (  200 bytes)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='98  1  Average V2I payload success rate  Proposed RL algorithm  Hill climbing algorithm  RL algorithm without PL selection  Greedy Algorithm  4 platoons  6 platoons  (b) Average V2I payload delivery probability  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' V2V and V2I performance for M = 4 and M = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3 shows the reliability of V2V and V2I links as we  increase BV 2V for the cases of 4 and 6 platoons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3a  shows that the V2V performance decreases as we increase the  packet size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' This is because a larger payload size requires  a longer time to transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' For 4 platoons, we achieve a  reliability of 1 for up to 2200 bytes, outperforming all the other  benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 3b shows that for 4 platoons, the payload  success rate for V2I links is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='9975 for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' However,  the hill-climbing and greedy algorithm performance decrease,  indicating more significant interference as we increase V2V  payload size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' When we increase the number of platoons to 6,  the performance gap between the proposed algorithm and the  hill-climbing algorithm increases, which shows the superiority  of our algorithm for a higher number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Furthermore,  it can be seen that dynamic platoon leader selection improves  performance for both V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' CONCLUSION  In this work, we proposed a distributed multi-agent rein-  forcement learning algorithm to optimize the performance of  the V2V and the V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The V2V and V2I links used  the same spectrum, making an intelligent resource allocation  design necessary to manage interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Each platoon leader  had an agent for joint channel assignment and power allocation  for V2V links, and another agent for joint user association  and power allocation for V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Further, another agent was  able to select the platoon leader, to maximize the reliability  of both V2V and V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Based on RL, the agents could  collaborate to take optimal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' The proposed approach  is decentralized, and the agents were able to make decisions  based on their local state observations only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Simulation results  indicate that the proposed algorithm could perform well for  variable V2V packet size and different numbers of platoons,  outperforming the centralized hill-climbing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' More-  over, the PL selection improved reliability for both V2V and  V2I links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  REFERENCES  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Abou-Zeid, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Pervez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Adinoyi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Aljlayl, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Yanikomeroglu,  “Cellular V2X transmission for connected and autonomous vehicles  standardization, applications, and enabling technologies,” IEEE Con-  sumer Electronics Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 91–98, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='  [2] “3rd Generation Partnership Project;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Technical Specification Group  Services and System Aspects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' Architecture enhancements for 5G System  (5GS) to support Vehicle-to-Everything (V2X) services (Release 17) ,”  TS 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='287 V17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content='0, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
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+page_content=' Technical Specification Group  Services and System Aspects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfZAR1/content/2301.03145v1.pdf'}
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@@ -0,0 +1,2165 @@
+Draft version February 1, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Planetesimal Initial Mass Functions following Diffusion Regulated Gravitational Collapse
+Konstantin Gerbig
+1 and Rixin Li (李日新)
+2
+1Department of Astronomy, Yale University, 52 Hillhouse Ave, New Haven, CT 06511, USA
+2Center for Astrophysics and Planetary Science, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
+ABSTRACT
+The initial mass function (IMF) of planetesimals is of key importance for understanding the
+initial stages of planet formation, yet theoretical predictions so far have been insufficient in
+explaining the variety of IMFs found in simulations. Here, we connect diffusion-tidal-shear limited
+planetesimal formation within the framework of a Toomre-like instability in the particle mid-plane of
+a protoplanetary disk to an analytic prediction for the planetesimal IMF. The shape of the IMF is
+set by the stability parameter Qp, which in turn depends on the particle Stokes number, the Toomre
+Q value of the gas, the local dust concentration and the local diffusivity. We compare our prediction
+to high-resolution numerical simulations of the streaming instability and planetesimal formation via
+gravitational collapse. We find that our IMF prediction agrees with numerical results, and is consistent
+with both the ‘planetesimals are born big’ paradigm and the power law description commonly found
+in simulations.
+1. INTRODUCTION
+Planetesimals
+are
+the
+initial
+building
+blocks
+of
+planets and their Initial Mass Function (IMF) is
+consequently of great interest.
+The planetesimal
+formation
+process
+in
+protoplanetary
+disks,
+which
+ultimately dictates the planetesimal IMF, is connected
+to
+an
+ensemble
+of
+challenges,
+one
+of
+the
+most
+prominent of which is the so-called meter barrier (see
+the review by Chiang & Youdin 2010).
+Particles
+exceeding sizes of order meters are expected to be
+limited
+in
+their
+capacity
+to
+grow
+via
+continued
+coagulation due to both rapid radial drift inwards and
+increased relative velocities that result in preferentially
+destructive collisions (Birnstiel et al. 2012).
+One
+well-received solution to this growth barrier is to
+rapidly form planetesimals via gravitational collapse of
+over-dense particle clouds. Initial ideas focused on the
+gravitational instability of the entire particle mid-plane
+(Safronov 1969; Goldreich & Ward 1973).
+However,
+Kelvin-Helmholtz stirring prevents razor-thin particle
+settling and thus renders such a global gravitational
+instability challenging (Weidenschilling 1980; Sekiya
+1998; Youdin & Shu 2002; Johansen et al. 2006a). On
+the other hand, local patches have been shown to
+Corresponding author: Konstantin Gerbig
+[email protected]
+collapse if local particle concentrations are sufficiently
+high (Johansen et al. 2006b). In particular, instabilities
+energized by the relative dust-gas streaming velocity
+(Youdin & Goodman 2005; Squire & Hopkins 2018)
+can concentrate particles to densities sufficient for
+gravitational collapse to trigger and planetesimals to
+form (e.g., Johansen et al. 2015; Gerbig et al. 2020). In
+the past, the IMF of planetesimals has been obtained
+by performing numerical simulations of this setup and
+then counting produced planetesimals, a process that
+resulted in power-laws of various kinds (Simon et al.
+2016; Sch¨afer et al. 2017; Li et al. 2019).
+Recently, the formation of planetesimals has been
+connected to the diffusion of particles as well as their
+stability to stellar tidal forces (Klahr & Schreiber 2016;
+Gerbig et al. 2020; Klahr & Schreiber 2020, 2021).
+This framework envisions particles to be subject to a
+diffusive flux away from the concentration maximum,
+much how pressurized gas clouds resist collapse in star
+formation. In addition, any particle cloud on the verge
+of gravitational collapse must be stable to stellar tidal
+gravity. These effects limit planetesimal formation on
+small scales and large scales respectively, thus together
+prescribe a characteristic scale on which planetesimal
+formation is expected to occur.
+This in turn, can be
+translated to a characteristic planetesimal mass, which
+Klahr & Schreiber (2020) hypothesized to be the center
+of a Gaussian-shaped IMF, in agreement with IMFs of
+arXiv:2301.13297v1  [astro-ph.EP]  30 Jan 2023
+
+ID2
+Gerbig and Li
+primordial asteroids (Delbo et al. 2019), yet seemingly
+implying a mismatch to numerically obtained IMFs.
+In this paper,
+we connect these two paradigms
+by deriving planetesimal IMFs within the framework
+of diffusion-tidal-shear limited planetesimal formation.
+Thereby, we argue that the probability density function
+of a given scale to collapse scales with the scale’s
+Toomre-like growth rate.
+We also directly test our
+prediction by conducting numerical simulations using
+proven setups in ATHENA (Stone et al. 2008), that
+produce the streaming instability (Youdin & Goodman
+2005) and planetesimal formation. In the process, we
+for the first time conduct diffusion measurements in
+large-scale stratified streaming instability simulations,
+as well as develop a method for obtaining local particle
+concentrations that are appropriate for characterizing
+the onset of planetesimal formation.
+The paper is structured as follows.
+In Sect. 2
+we review the Toomre-like instability for planetesimal
+formation.
+In Sect. 3, we derive planetesimal IMFs,
+which we compare to numerical simulations in Sect. 4.
+We discuss our results, namely applicability, caveats and
+implications, in Sect. 5.
+2. TOOMRE-LIKE INSTABILITY
+2.1. Stability parameter
+The stability of self-gravitating particles subject to
+a diffusive flux induced by coupling to turbulent gas
+velocities has been studied in numerous occasions
+(Youdin 2011; Gerbig et al. 2020; Klahr & Schreiber
+2020, 2021), and can be assessed using a Toomre-like
+value Qp, such as
+Qp ≡
+�
+δr
+St
+1
+Z
+csΩ
+πGΣg
+=
+�
+δr
+St
+Q
+Z
+(1)
+where Z = Σp/Σg is the (local) dust concentration,
+Q = csΩ/(πGΣg) is the standard Toomre value (Toomre
+1964), St = tsΩ is the dimensionless stopping time,
+and δr
+= Dp,r/(csH) is the dimensionless (radial)
+diffusion coefficient for particles.
+Ω is the orbital
+frequency, cs the sound-speed of the gas, H = cs/Ω the
+disk pressure scale-height, Σp and Σg particle and gas
+surface densities respectively, and G is the gravitational
+constant.
+In the Epstein regime, the stopping time
+relates to particle size a, and dust material density ρ•
+via ts = ρ•a/(
+√
+2πΣg). We consider dust and pebbles
+where ts is such that St remains below unity (e.g.,
+Birnstiel et al. 2012).
+This corresponds to well or at
+least marginally coupled particles.
+If Qp < 1, the system is unstable and expected to
+collapse and form planetesimals.
+We review the
+corresponding
+instability
+analysis
+in
+the
+following
+section.
+2.2. Dynamical equations and dispersion relation
+Following (Klahr & Schreiber 2021), we start with the
+a set of dynamical equations for dust particles in the
+shearing sheet approximation and under the assumption
+of a razor-thin disk. We adopt the coordinates r, φ, z
+for radial, azimuthal and vertical directions respectively,
+and consider a patch at distance R from a solar-mass
+star. The linearized set of equation is given by
+1
+Σp
+∂Σ′
+p
+∂t + ∂v′
+r
+∂r = 0,
+(2)
+∂v′
+r
+∂r − 2Ωv′
+φ = − 1
+Σp
+Dp,r
+ts
+∂Σ′
+p
+∂r − ∂Φ′
+∂r − v′
+r
+ts
+,
+(3)
+∂v′
+φ
+∂t + Ωv′
+r
+2
+= 0,
+(4)
+where the prime denotes perturbed quantities.
+They
+describe mass continuity, and conservation of radial and
+azimuthal momenta.
+Notably, we ignore an explicit
+azimuthal drag term as in e.g., Youdin (2011), an
+assumption that is justified at dust-to-gas ratios of
+order unity.
+Instead, coupling to the gas is assumed
+to be wholly described by gas pressure counteracting
+radial contraction and the diffusive flux in the radial
+momentum equation, which can be understood as an
+effective pressure flux induced by turbulent gas motions.
+As a consequence, there is also no mass diffusion term
+in the continuity equation.
+We continue by introducing axisymmetric WKB waves
+scaling with Σ′
+p ∝ exp(−i(kr − ωt)). Φ′ = −2πGΣ′/|k|
+is the potential for a perturbed disk assuming ρp(k, z) =
+(kΣp/2) exp(−|k|z). The resulting dispersion relation is
+given by (Eq. (B22) in Klahr & Schreiber 2021),
+ω2
+0 = δr
+Stc2
+sk2 − 2πΣpG|k| + Ω2,
+(5)
+and can be expressed in terms of the stability parameter
+Qp
+ω2
+0
+Ω2 = δr
+St(kH)2 − 2
+Qp
+�
+δr
+St|k|H + 1.
+(6)
+Here, ω2
+0 is defined as ω2
+0 = ω(ω − i/ts), and represents
+the complex frequency without the drag term in Eq. (3).
+Note that the herein used solution to the Poisson
+equation does not take into account a softening term
+caused by the particle layer thickness (see Eq. (12) in
+Youdin 2011).
+
+Planetesimal IMFs under diffusion regulated collapse
+3
+10−2
+10−1
+100
+101
+k[kfgm]
+−6
+−4
+−2
+0
+2
+ω2
+0[Ω−2]
+Qp = 0.4
+Qp = 0.5
+Qp = 0.6
+Qp = 0.8
+10−1
+100
+Qp
+10−3
+10−2
+10−1
+100
+λ[H]
+λfgm
+lc
+2π
+�
+δ/StH
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(γΩ)
+Figure 1. Dispersion relation ω2
+0 in Eq. (5) (left panel) and growth rates γ in Eq. (11) (right panel) for different values of
+Qp. Parameters are δ = 10−5 and St = 0.4. The gray shaded region in the left panel indicates stability against axisymmetric
+perturbations. As Qp decreases, more modes become unstable, and the fastest growing mode (dashed lines in right panel) shifts
+to smaller scales. For reference, we plot the scale 2π
+�
+δ/StH (dotted line right panel).
+10−4
+10−2
+100
+mp[MCeres]
+0.0000
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+p(mp)
+Qp = 0.15
+Qp = 0.4
+Qp = 0.8
+10−4
+10−2
+100
+mp[MCeres]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+P(mp)
+δr = δz = 10−6
+δr = δz = 10−5
+10−4
+10−2
+100
+mp[MCeres]
+10−2
+10−1
+100
+N(> mp)
+Figure 2. Probability density functions (left panel), cumulative probability density functions (center panel) and initial mass
+functions (right panel) for the growth rates in Fig. 2 where St = 0.4. Diffusion is isotropic and set to δ = 10−6 (solid lines)
+and δ = 10−5 (dashed lines). Different colors correspond to different values of Qp, all of which chosen to be unstable. We use
+Eq. (16) for converting unstable scale to seed mass, which assumes collapse at Hill density, and requires knowledge of the aspect
+ratio which we set to h = 0.05. We assume the system to produce N = 1000 planetesimals total to calculate the normalized
+IMF N(< mp).
+2.3. Fastest growing mode and growth rates.
+The
+fastest
+growing
+mode
+is
+found
+by
+solving
+∂ω2
+0/∂k = 0, which yields
+kfgm =
+�
+St
+δr
+1
+QpH .
+(7)
+The complex frequency of the fastest growing mode is
+simply
+ω2
+0,fgm
+Ω2
+= 1 − 1
+Q2p
+,
+(8)
+which highlights that only for Qp < 1 exist k for which
+ω2
+0 < 0 and the instability can grow, thus justifying the
+definition of Qp as a stability parameter. We can also
+calculate largest and smallest unstable scales λ = 2π/k
+by solving ω2
+0 = 0, resulting in
+λmin/max
+λfgm
+= 1
+Qp
+�
+1
+Qp
+±
+�
+1
+Q2p
+− 1
+�
+,
+(9)
+with λfgm = 2π/kfgm. The range of unstable scales is
+λmax − λmin = 2λfgm/Qp
+�
+1/Q2p − 1, which increases
+for decreasing Qp.
+Note, that the fastest growing mode relates to the
+critical cloud radius lc in Klahr & Schreiber (2020) and
+Gerbig et al. (2020) via
+λfgm = 2π
+�
+δr
+StQpH = 6πQplc.
+(10)
+
+4
+Gerbig and Li
+The growth rate γ(k) can be found solving γ = iω
+or equivalently γ(γ + 1/ts) = −ω2
+0 (Klahr & Schreiber
+2021):
+γ(k)
+Ω
+= − 1
+2St +
+�
+1
+4St2 − ω2
+0(k)
+Ω2
+.
+(11)
+Hence, the fastest growing mode kfgm grows with
+γ(k = kfgm)
+Ω
+= − 1
+2St +
+�
+1
+4St2 + 1
+Q2p
+− 1
+(12)
+Dispersion relation and growth rates are shown for a set
+of unstable Qp in Fig. 1. In the right panel, we also
+plot λfgm, lc as well as the scale 2π
+�
+δr/StH which is of
+order the radial extent of small-scale particle structures
+(Gerbig et al. 2020).
+3. PLANETESIMAL INITIAL MASS FUNCTIONS
+3.1. Planetesimal masses at Hill density
+In order for a region in the particle disk to be
+stable against stellar tidal gravity, its mass must be be
+contained within its Hill-radius.
+In other words, the
+region’s density must be at least Hill density
+ρH = 9Ω2
+4πG.
+(13)
+Assuming a given scale k = 2π/λ is unstable under the
+previously discussed dispersion relation, at Hill density
+ρH, the mass available to the produced planetesimal1
+can be estimated with
+mp = π
+4
+� λ
+2π
+�2
+Σp(ρp = ρH)
+(14)
+We assumed that the seed mass has access to a region of
+size λ/2π. The surface density if the mid-plane has Hill-
+density can be estimated with Σp =
+√
+2πHpρp, where
+the particle scale height relates to the vertical diffusion
+coefficient via (Youdin & Lithwick 2007)
+Hp =
+�
+δz
+StH.
+(15)
+Together, the mass associated with an unstable scale λ
+is of order,
+mp = 9
+64
+�
+2
+π3 h3
+�
+δz
+St
+� λ
+H
+�2
+M⊙,
+(16)
+1 The mass mp is only equal to the initial planetesimal mass if
+the entire particle cloud collapses to material density of the
+planetesimal, and thus should be understood as an approximate
+mass scale.
+As such, in Klahr & Schreiber (2020, 2021), this
+mass scale is called ‘equivalent mass’, and associated with some
+conversion efficiency that quantifies the fraction that ultimately
+ends up in the formed planetesimal.
+and equivalently, the unstable scale λ that is expected
+to produce a planetesimal of mass of order mp is
+λ
+H =
+�
+64
+9
+�
+π3
+2
+1
+h3
+�
+St
+δz
+mp
+M⊙
+�1/2
+(17)
+Here h = H/R is the disk aspect ratio, typically of order
+0.03 < h < 0.1. The mass associated with the fastest
+growing mode is given by
+mp,fgm = 9
+8
+�π
+2
+�
+δz
+St
+δr
+Sth3Q2
+pM⊙
+(18)
+which scales as
+mp,fgm ≈ 0.37MCeres ·
+� δz
+10−5
+� 1
+2 � δr
+10−5
+�
+·
+�0.1
+St
+� 3
+2 � h
+0.05
+�3 �Qp
+1
+�2
+(19)
+Under isotropic diffusion, this equals the characteristic
+mass in Klahr & Schreiber (2020) if Qp ≈ 0.28.
+Note, that the Toomre instability a priori does not
+require Hill-density to operate. Indeed, as pointed out
+by Klahr & Schreiber (2021), if the vertical density
+structure is not set by vertical diffusion, but instead
+stringently follows ρp(k, z) = (kΣp/2) exp(−|k|z), then
+the fastest growing linear instability would be achieved
+at a mid-plane density of ρp(k = kfgm, z = 0) = 2ρH/9.
+By presupposing Hill density, our approach detaches
+from this assumption of mode-dependent stratification,
+and in the process excludes Toomre unstable clouds that
+fail to withstand tidal gravity and are thus of little
+physical importance for the IMF.
+3.2. IMF from Toomre growth rate
+Given this context of Toomre-like instability and
+Hill density, we proceed by providing predictions for
+the IMF. Our ansatz is to take the mode-depended
+instability growth rates as a probability density function
+p(λ) for unstable scales, and then convert unstable
+modes to planetesimal masses via the recipes discussed
+in Sect. 3.1. This yields a probability density function
+p(mp) such that the probability of a seed mass within
+[mp, mp + dmp] is p(mp)dmp. This first order approach
+agrees with intuition in that fastest growing modes
+should most preferentially collapse,
+slowly growing
+modes only sometimes, stable modes never.
+We write the probability density function p(λ) for
+scale λ to collapse as
+p(λ)dλ ∝ max[ˆγ−1γ(λ), 0]
+(20)
+
+Planetesimal IMFs under diffusion regulated collapse
+5
+ˆγ−1
+is
+a
+normalization
+constant
+given
+by
+ˆγ =
+� λmax
+λmin γ(λ)dλ. Since λ ∝ m1/2
+p
+via Eq. (17), we can
+map p(λ)dλ onto the probability density function for
+seed planetesimal masses p(mp)dmp.
+The
+probability
+density
+function
+p(mp),
+the
+cumulative probability function P(m′
+p ≤ mp), and the
+resulting IMF are shown in Fig. 2, for St = 0.4 and
+different values of δ and Qp.
+The smaller Qp, the
+flatter the IMF, as more scales become unstable. The
+steepest IMF is achieved for Qp → 1.
+Indeed, for
+Qp = 1, the probability density function becomes a
+Dirac delta function, i.e., p(mp) = δ(mp,H − mp).
+Increasing diffusivity shifts the IMF to larger masses,
+provided Qp remains constant, which would require a
+corresponding decrease in Q/Z.
+3.3. Comparison to statistical approaches
+Our ansatz is distinct from past means of deriving
+planetesimal IMFs. Cuzzi et al. (2008, 2010); Hartlep
+& Cuzzi (2020) approach the problem statistically,
+and consider turbulent clustering of particles.
+In
+particular, the assumed scale invariance of the turbulent
+spectrum implies the statistically appearance of regions
+of highly enhanced particle density.
+The argument is
+that only sufficiently dense clumps can withstand ram
+pressure disruption and thus contract to planetesimals
+in a process called primary accretion, thus limiting
+the formation of planetesimals on the low-mass end.
+Our mechanism likewise prohibits the formation of
+arbitrarily small planetesimals, yet the physical intuition
+differs in that (1) the system is, in fact, gravitationally
+unstable under the Toomre-like instability discussed in
+Sect. 2 resulting in (2) ram pressure being negligible
+compared to diffusion (also see Klahr & Schreiber
+(2020)), and (3), that the smallest planetesimals are
+those resulting from the smallest scale that can be
+gravitationally unstable under the dispersion relation in
+Eq. (5).
+Another statistical ansatz was taken by Hopkins &
+Christiansen
+(2013),
+where
+turbulent
+density
+fluctuations can render local regions of the (gas) disk
+unstable to gravitational collapse.
+The initial mass
+function of collapsing clumps therein depends on the
+critical density for self-gravitating clumps and as well
+as the properties of the ambient turbulence.
+In our
+ansatz, the gas remains stable throughout. Moreover,
+we take all clumps to collapse exactly at Hill density in
+Eq. (13). Different masses are the result of collapse of
+differently sized regions, i.e. on different scales λ.
+4. NUMERICAL TESTS
+We perform high-resolution numerical tests using
+ATHENA (Stone et al. 2008; Bai & Stone 2010a) to
+−0.1
+0.0
+0.1
+y/H
+t = 60.00 Ω−1
+−0.1
+0.0
+0.1
+x/H
+−0.05
+0.00
+0.05
+z/H
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+log(Σp/Σg)
+−2
+0
+2
+log(max(ρp)y/ρ0)
+Figure 3. Map of the particle surface density at after 60
+Ω−1, which is when self-gravity is turned on. Top panel is
+vertically integrated, wheares bottom panel is azimuthally
+integrated. The white dashed lines mark the border of the
+physical simulation domain and the ghost cells. Since, one
+of the two filaments at 60 Ω−1 is located right at the radial
+simulation boundary, we chose to include the ghost cells in
+this figure, which results in that filament being depicted
+twice.
+directly test our predictions for the IMF. The procedure
+is similar to that in e.g. Li et al. (2019); Gerbig
+et al. (2020), in that we let a small patch around
+a protoplanetary disk mid-plane evolve into some
+turbulent state.
+We then turn on self-gravity with
+different values for Qp by varying the self-gravity
+parameter ˆG in code units. The measurement of Qp,
+as well as the calculation of the predicted IMF requires
+measurement of radial and vertical diffusion, prior to
+turning on-self gravity. Since the streaming instability
+will concentrate particles into filaments, we must also
+determine the local particle concentration.
+4.1. Numerical Setup
+We employ ATHENA (Stone et al. 2008) to solve the
+hydrodynamic equations on an Eulerian grid including
+Lagrangian super-particles (Bai & Stone 2010a). Our
+numerical setup is similar to simulations of streaming
+instability regulated planetesimal formation such as
+Johansen et al. (2007); Simon et al. (2016); Li et al.
+(2019); Gerbig et al. (2020), in that we use the local
+
+6
+Gerbig and Li
+−0.1
+0.0
+0.1
+x/H
+−0.1
+0.0
+0.1
+y/H
+Q = 16, t = 60 Ω−1
+−0.1
+0.0
+0.1
+x/H
+Q = 8, t = 60 Ω−1
+−0.1
+0.0
+0.1
+x/H
+Q = 4, t = 60 Ω−1
+−3.5
+−3.0
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+log(Σp/Σg)
+101
+103
+NColumns
+all columns
+Q = 4
+Q = 8
+Q = 16
+Z
+Figure 4. Maps of the columns that, at 60 Ω−1, contain at least one cell at or above Hill-density for Q = 16, Q = 8, and Q = 4
+(top panels, from left to right), and corresponding column density histograms (bottom panel). The bottom panel also indicates
+the chosen values for Z in dashed lines, i.e. Z = 0.137, Z = 0.100, and Z = 0.0075 for Q = 16, Q = 8, and Q = 4 respectively.
+The full surface density map of this snapshot is shown in Fig. 3.
+shearing box approximation (Goldreich & Lynden-Bell
+1965) with coordinates (x, y, z).
+We consider a non-
+magnetized gas with an isothermal equation of state.
+Gas is initialized in hydrostatic equilibrium. Particles
+are forced into a super-Keplerian rotation by an external
+pressure gradient, which we parameterize using
+Π = ηvK
+cs
+,
+(21)
+where
+η
+=
+−(1/2)h2d ln ρ/(d ln r).
+This
+is
+mathematically equivalent to real disks where particles
+orbit Keplerian, and gas experiences sub-Keplerian
+forcing (see Bai & Stone 2010a for details on the
+pressure gradient implementation in ATHENA). Note,
+that Π relates to the pressure gradient parameter used
+in Schreiber & Klahr (2018); Gerbig et al. (2020)
+simply via Π = β/2.
+Particles are initialized with a
+narrower Gaussian, however at an scale height of ηr/2,
+which is the characteristic scale of Kelvin-Helmholtz
+instability in protoplanetary disks (Gerbig et al. 2020).
+As
+such,
+particles
+are
+not
+expected
+to
+undergo
+significant settling or lofting due to vertical stellar
+gravity or Kelvin-Helmholtz stirring respectively.
+The pressure gradient and the resulting relative
+velocity between particle and gas flow energize the linear
+streaming instability (Youdin & Goodman 2005; Squire
+& Hopkins 2018), which then saturates non-linearly and
+in the process concentrates particles into high density
+regions (Johansen & Youdin 2007). The degree of which
+the streaming instability operates, and as a consequence
+the strength of particle diffusion and concentration prior
+to gravitational collapse and planetesimal formation, is
+largely set by two quantities. First, the Stokes number
+St, and second, the ratio of metallicity and pressure
+gradient Z0/Π (Sekiya & Onishi 2018).
+The former
+quantifies the coupling of particles to the gas flow, in
+particular the gas turbulence. The latter traces the ratio
+of dust abundance relative to gas and dust layer scale-
+height, and thus maps onto the mid-plane dust-to-gas
+ratio. Note, that the metallicity Z0 is the global (as in
+simulation domain averaged) particle concentration and
+thus not necessarily equal to the local enhancement Z we
+introduced in Eq. (1). We elaborate on this important
+difference in Sect. 4.2.
+In this work, we choose St = 0.4, Z0 = 0.02 and
+Π = 0.05. This setup is specifically designed to provide
+favorable conditions for the streaming instability and,
+provided Qp < 1, planetesimal formation, in order to
+produce a large number of planetesimals which allows
+for a more statistically robust determination of the
+numerical IMF. Our simulations use a computational
+
+Planetesimal IMFs under diffusion regulated collapse
+7
+domain of 0.2H × 0.2H × 0.15H, with a resolution of
+2560/H (i.e., ∆x ≈ 3.9 · 10−4H) and Npar = 226 ≈
+6.71·107 particles. The vertical extent is slightly reduced
+from a cube to mitigate computational costs and is
+still tall enough because the particle layer remains thin
+all the time.
+Moreover, in the vertical direction, we
+adopt outflow boundary conditions that are known to
+reduce boundary artifacts, especially in shorter boxes
+(Li et al. 2018). In the radial and azimuthal directions,
+the standard shearing-periodic boundary conditions are
+imposed.
+Following the precedent set by many past works of
+streaming instability, we express the results of our scale-
+free simulations in the dimensionless unit system of
+dynamical timescale Ω−1, H, and ρ0. The simulation
+is run for t = 60Ω−1 without self-gravity. This allows
+for sufficient amount of time in the non-linear phase of
+the streaming instability to measure diffusion.
+Figure.
+3 shows the vertically
+(top
+panel)
+and
+azimuthally
+(bottom
+panel)
+integrated
+particle
+densities at 60Ω−1, i.e.
+just before self-gravity is
+turned on.
+The streaming instability in a stratified
+disk collects particles into two azimuthally elongated
+filaments,
+which are enhanced in particle density
+relative to the prescribed average of Z0 = 0.02.
+Following the snapshot at 60 Ω−1 depicted in Fig. 3
+we turn on self-gravity,
+the strength of which is
+parameterized by the self-gravity parameter
+˜G
+≡
+4πGρ0/Ω2, which relates to Q via Q = 4/(
+√
+2π ˜G). Self-
+gravity is required for the concept of Hill-density to be
+meaningful. Specifically, the Hill-density depends on the
+self-gravity parameter via
+ρH
+ρ0
+= 9
+˜G
+= 9
+�π
+8 Q
+(22)
+We conduct three self-gravity runs with Q = 16, Q = 8
+and Q = 4 which correspond to Hill densities of ρH/ρ0 ∼
+90, ρH/ρ0 ∼ 45 and ρH/ρ0 ∼ 23 respectively. Note that,
+we can also now associate a Hill-radius rH with a given
+mass m, i.e.,
+rH = R
+� m
+M⊙
+� 1
+3
+=
+�4πρ0m
+˜G
+� 1
+3
+=
+�4π
+9 ρHm
+� 1
+3
+. (23)
+4.2. Local particle concentration
+The stability parameter Qp depends on the local
+particle concentration Z = Σp/Σg. This is importantly
+not equal to the initial, global particle concentration
+Z0, which in our case is set to Z0 = 0.02. The reason
+for this lies within predominanetly radial concentration
+of particle surface density within two filaments as
+evident in Fig. 3, where the typical particle column
+density exceeds Z = 0.02 by between one and two
+orders of magnitude.
+This challenge in applying the
+diffusion-limited collapse criterion to proven shearing
+box simulation simulations of the streaming instability
+was also recognized in Gerbig et al. (2020), where
+the radial extent of streaming instability filaments was
+related to a radial enhancement. While this argument
+was sufficient to demonstrate the general applicability
+of a diffusion-limited collapse criterion to simulations,
+it lacks the precision to reliably predict the Qp as it
+by construction cannot account for additional azimuthal
+enhancement within filaments.
+Because of this, in this work, we choose a different
+pathway and utilize the requirement of Hill-density
+for planetesimal formation to measure the average
+particle concentration of the cells that are expected to
+participate in planetesimal formation. More precisely, in
+order for a given vertical column to be counted towards
+the metallicity, it must contain at least one cell at or
+above Hill density.
+Figure. 4 shows maps of the columns that satisfy this
+requirement for all three Q-values. Note, we are here
+depicting the same snapshot at 60Ω−1 as in Fig. 3,
+i.e.
+before self-gravity has impacted the simulation.
+We are merely evaluating which columns satisfy our
+requirement for a given Hill-density.
+A comparison
+to Fig. 3 confirms that this scheme constrains the
+metallicity measurement to the over-dense filaments
+only. In the bottom panel of Fig. 4, we show histograms
+of the surface density of both the entire simulation
+domain in grey, as well only those columns that contain
+at least one cell above Hill density. The dashed lines
+indicate the average particle concentration for the three
+runs given, and correspond to Z = 0.137, Z = 0.100,
+and Z = 0.075 for Q = 16, Q = 8, and Q = 4
+respectively.
+4.3. Radial diffusion
+We measure radial particle diffusion via (Youdin &
+Lithwick 2007; Johansen & Youdin 2007; Schreiber &
+Klahr 2018; Baehr et al. 2022)
+δr
+csH = Dp,r = 1
+2
+∂⟨|xi(t) − xi(t0)|2⟩i
+∂t
+,
+(24)
+The idea is,
+that as the turbulent state evolves
+particles, the underlying diffusion will widen the particle
+distribution, and as such allow for a calculation of
+a value for particle diffusion.
+If the particles were
+normally distributed, this would correspond to the
+time evolution of the distribution’s variance.
+Note,
+that since we are investigating the streaming instability
+in stratified disks, this method cannot be used to
+
+8
+Gerbig and Li
+−0.8
+−0.6
+−0.4
+−0.2
+0.0
+∆x/H
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Nbin/Npar
+t0 = 18Ω−1
+∆t =6Ω−1
+∆t =15Ω−1
+∆t =24Ω−1
+∆t =42Ω−1
+−0.3
+−0.2
+−0.1
+0.0
+∆x/H
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+Nbin/Npar
+t0 = 50.25Ω−1
+∆t =3.75Ω−1
+∆t =5.0Ω−1
+∆t =7.5Ω−1
+∆t =9.75Ω−1
+Figure 5.
+Evolution of an initial particle distribution
+at different times ∆t for t0
+= 18Ω−1 (top panel) and
+t0 = 50.25Ω−1 (bottom panel).
+Due to the two over-
+dense filaments the particle distribution develops bi-modality
+— an effect not seen in unstratified streaming instability
+simulations (e.g., Johansen & Youdin 2007)
+determine vertical diffusivity which we will measure in
+the subsequent section.
+Figure. 5 shows the evolution of the initial particle
+distribution from 18Ω−1 to 60Ω−1, when we turn on
+self-gravity.
+Our particle distribution depicts much
+stronger non-Gaussianity than Johansen & Youdin
+(2007) Fig. 17,
+since our simulation is stratified.
+This leads the streaming instability to produce as
+azimuthally-extended filaments, where particles drift
+slower due to the enhanced local dust-to-gas ratio. This
+can lead to multiple peaks in the particle distribution
+— in our specific case there are two peaks as there are
+two dominant filaments.
+If we take all particles into account when evaluating
+Eq. (24), we will determine a global value for the
+radial diffusivity.
+However, the strength of diffusion
+is expected to vary locally and correlate with particle
+density (see e.g., Schreiber & Klahr 2018).
+As such,
+we sort all particles by local density and measure
+the evolution of the distribution up to 60Ω−1 for
+each density bin, and then calculate the corresponding
+diffusion coefficients via Eq. (24).
+Fig. 6 shows that
+denser regions are less diffusive than less dense regions.
+More specifically, up to a dust-to-gas ratio of order 101,
+the diffusion is approximately constant at δr = 1.3·10−4.
+This is a value consistent with that found in Li & Youdin
+(2021).
+For more dense regions,
+the diffusion decreases
+significantly down to δ ∼ 10−6 for ρp/ρ0 > 4 · 102. This
+turbophoretic behavior of the particles (Caporaloni
+et al. 1975; Belan et al. 2014) makes it challenging to
+pinpoint an exact value of δ that is appropriate for the
+determination
+of
+Qp
+and
+the
+subsequent
+IMF.
+Additionally,
+calculation
+of
+δ
+requires
+temporal
+averaging. In Fig. 6 we show multiple averaging times
+in order to highlight the trend of longer averaging
+times yielding larger diffusivity.
+In this work, we choose diffusivities by evaluating
+δ(ρp) at the particle density weighted average
+˜ρp(Q) =
+�
+i,ρp,i>ρH(Q) Niρp,i
+�
+i,ρp,i>ρH(Q)
+.
+(25)
+Note that we only count those particles in Hill-stable
+regions.
+The resulting densities are indicated in the
+right panel of Fig. 6. We evaluate the radial diffusion
+coefficient δr (see Eq. 24) with a second-order one-side
+derivative at t = 60/Ω (by supplying edge order=2
+to numpy.gradient).
+This results in diffusivities of
+δr = 2.4 · 10−6, δr = 2.0 · 10−6, and δr = 2.6 · 10−6
+for Q = 16, Q = 8, and Q = 4 respectively.
+We
+acknowledge
+that
+due
+to
+the
+steepness
+of
+δx(ρH/ρ0) at high dust-to-gas ratios, as well as the
+dependence on averaging time ∆tave, there remains a
+relatively large ambiguity about the most appropriate
+values for δr.
+Indeed, we find any diffusivity within
+1 · 10−6 ≲ δx ≲ 1 · 10−5 justifiable, and note that this
+full range is consistent with diffusivities obtained by
+Schreiber & Klahr (2018) at similar dust-to-gas ratios
+(although with smaller particles).
+4.4. Vertical diffusion
+While vertical diffusion is not required to calculate
+the stability parameter Qp which arises from a purely
+2D
+consideration,
+it
+is
+required
+to
+calculate
+planetesimal masses by setting the surface density
+necessary to achieve tidal-stability.
+We first measure
+the particle scale-height in each vertical column of grid
+cells by calculating the standard deviation of the
+
+Planetesimal IMFs under diffusion regulated collapse
+9
+52
+54
+56
+58
+60
+t [Ω−1]
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+⟨|xi(t) − xi(t0 = 50Ω−1)|2⟩i [H2]
+10−1
+100
+101
+102
+103
+ρp/ρ0
+10−6
+10−5
+10−4
+δx
+t = 60Ω−1
+∆tavg = 1Ω−1
+∆tavg = 2Ω−1
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Number of particles
+×106
+ρH(Q = 16)/ρ0
+ρH(Q = 8)/ρ0
+ρH(Q = 4)/ρ0
+˜ρp(Q)/ρ0
+Figure 6. Radial diffusion depends on particle density. Left panel: Evolution of the variance of the particle distribution for
+different densities. Colors correspond to densities. Right panel: diffusion vs particle density. Colors correspond to density
+and map onto the left panel. Different line styles indicate different averaging times when calculating the gradient in Eq. (24).
+We overlay a histogram of the density distribution of the particles, where vertical solid lines indicate Hill density for chosen
+Q-values. Vertical dashed-dotted lines correspond to the weighted average density ˜ρp(Q), which pinpoint the diffusivities to
+δr = 2.4 · 10−6, δr = 2.0 · 10−6, and δr = 2.6 · 10−6 for Q = 16, Q = 8, and Q = 4 respectively
+−0.1
+0.0
+0.1
+x/H
+−0.1
+0.0
+0.1
+y/H
+−0.1
+0.0
+0.1
+x/H
+−0.1
+0.0
+0.1
+y/H
+10−3
+10−2
+10−1
+Σp/Σg
+10−6
+10−5
+δz
+Q = 16
+Q = 8
+Q = 4
+0.002
+0.004
+0.006
+0.008
+0.010
+Hp/H
+−6.5
+−6.0
+−5.5
+−5.0
+−4.5
+log(δz)
+Figure 7. Map of particle scale height Hp(left panel) and vertical diffusion coefficient δz(center panel) at 60 Ω−1. The right
+panel shows the mean vertical diffusion ℏδz vs the local particle concentration (compare to top panel of Fig. 3). The shaded
+region characterizes ℏδz ± σδ, where σδ is the standard deviation of the vertical diffusivities at a given Σp/Σ. The vertical
+lines indicate the the determined values for Z (see Sect. 4.2, Fig. 4), which set the chosen values for vertical diffusivities to
+δz = 2.3 · 10−6, δz = 3.1 · 10−6, and δz = 3.3 · 10−6 for Q = 16, Q = 8, and Q = 4 respectively.
+vertical particle positions, and then convert to vertical
+diffusivity δz using Eq. (15). Similar procedures were
+also done by e.g., Bai & Stone (2010b); Yang et al.
+(2018); Li & Youdin (2021) when analyzing the vertical
+diffusivity within a stratified particle layer subject to
+the streaming instability. The resulting maps are seen
+in the left and center panel of Fig. 7. The right panel
+of Fig. 7 depicts the mean vertical diffusivity plotted vs
+local particle concentration, where the gray shaded
+region indicates one standard deviation from the mean.
+While there is significant spread of diffusivity for a
+given particle concentration, the vertical diffusivity
+behaves similar to the radial diffusivity in that it
+decreases in more concentrated regions. While in the
+plateau region, the diffusion anisotropy is of order
+δr/δz ∼ 10 which is consistent with e.g., Fig. 6 in Li &
+
+10
+Gerbig and Li
+Table 1. Simulation parameters for the three self-gravity
+runs.
+For all simulations, St = 0.4, Z0 = 0.02, Π =
+0.05, and self-gravity is turned on after 60Ω−1. Calculated
+masses assume h = 0.05 and collapse at Hill-density. Our
+simulations have a radial and azimuthal domain size of 0.2H,
+and the smallest resolved scale is ∆x = 3.9 × 10−4H. The
+first two rows are prescribed values. Rows three, four and
+five correspond to quantities measured at 60Ω−1 as described
+in Sects. 4.2, 4.3 and 4.4. Subsequent rows are properties
+calculated within the diffusion-limited collapse framework
+outlined in Sects. 2 and 3.
+Q
+16
+8
+4
+ρH/ρ0
+90.2
+45.1
+22.6
+Z
+0.137
+0.100
+0.075
+δr
+2.42 · 10−6
+2.04 · 10−6
+2.60 · 10−6
+δz
+2.35 · 10−6
+3.07 · 10−6
+3.28 · 10−6
+Qp
+0.287
+0.180
+0.136
+λmin[H]
+2.26 · 10−3
+1.29 · 10−3
+1.09 · 10−3
+λfgm[H]
+4.42 · 10−3
+2.56 · 10−3
+2.18 · 10−3
+λmax[H]
+1.05 · 10−1
+1.56 · 10−1
+2.34 · 10−1
+mp,min[MCeres]
+1.16 · 10−4
+4.32 · 10−5
+3.22 · 10−5
+mp,fgm[MCeres]
+4.44 · 10−4
+1.70 · 10−4
+1.27 · 10−4
+mp,max[MCeres]
+0.253
+0.630
+1.47
+Youdin (2021), in denser regions the diffusion is of
+order isotropic.
+Indeed, we evaluate δz(Σp/Σ) at the
+particle concentrations identified in Sect. 4.2, which
+yields δz
+=
+2.3 · 10−6,
+δz
+=
+3.1 · 10−6,
+and
+δz = 3.3 · 10−6 for Q = 16, Q = 8, and Q = 4
+respectively.
+This
+relatively
+low,
+and
+isotropic
+diffusion
+is
+consistent with high particle concentrations damping
+diffusion from gas, regardless of direction (Ida et al.
+2021).
+4.5. Planetesimals and mass scalings
+Table 1 summarizes the simulation parameters, the
+measured
+quantities
+δr,
+δz
+and
+local
+particle
+concentration Z, as well as the resulting predicted
+properties, namely Qp as well as characteristic size and
+mass scales. All three simulations have Qp < 1 and are
+thus expected to be gravitationally unstable.
+This is
+unsurprising as the simulation parameters were chosen
+in
+line
+with
+past
+setups
+proven
+to
+produce
+planetesimals (e.g., Li et al. 2019).
+Our smallest
+resolved scale is ∆x = 3.9 · 10−4H, which is about an
+order
+of
+magnitude
+smaller
+than
+λmin
+for
+our
+simulations.
+We thus expect to capture the small
+scales
+of
+diffusion
+regulated
+collapse
+in
+these
+simulations.
+On the other hand, the domain size is
+0.2H, which is of order λmax.
+Moreover, the largest
+scale is further limited by the maximum extent of
+over-dense regions, which radially and vertically is of
+order ηr
+=
+ΠH (Gerbig et al. 2020),
+which for
+Π = 0.05 is just 5 · 10−2H, and thus almost an order of
+magnitude less than λmax. Hence, we do not expect to
+the simulation to collapse on the largest unstable scales
+and produce planetesimals on the very high mass end.
+Fig. 8 shows snapshots of all three runs at 62Ω−1
+which is 2.0Ω−1 after self-gravity is turned on.
+In
+addition
+to
+the
+practical
+benefit
+of
+reducing
+computational costs,
+we chose this short time for
+self-gravity to act, in order to capture a relatively
+pristine mass distribution that is devoid of mergers.
+We used the clump finder algorithm PLAN (Li 2019; Li
+et al. 2019) to identify gravitationally bound clumps2,
+which are produced in all three runs. Indeed, all runs
+collapse rapidly enough for diffusion measurements to
+be unfeasible post 60 Ω−1.
+Thus, we cannot confirm
+whether
+or
+not
+diffusion
+increases
+in
+this
+gravo-turbulent state as it does in Klahr & Schreiber
+(2021).
+Note, that the requirement of Hill-density imposes a
+physical mass unit onto the analytic prediciton, and only
+implicitly depends on distance to the star (see Klahr &
+Schreiber 2020, for a discussion on the radial dependence
+of the diffusion-limited collapse criterion). On the other
+hand, the numerical results, i.e. the densities of bound
+clumps in Fig. 8 are fully scale-free in gas surface density
+ρ0. However, the two can be connected using the self-
+gravity parameter. More specifically, the mass unit of
+the simulation M0 can be expressed as
+M0 = ρ0H3 = h3 ˜GM⊙
+4π =
+1
+4
+√
+2π3
+h3
+Q M⊙
+(26)
+= 5.21 · 102
+� Q
+16
+�−1 � h
+0.05
+�3
+MCeres,
+(27)
+which introduces the same cubic dependence on aspect
+ratio as the analytic predictions in e.g., Eq. (16). The
+aspect ratio relates to the pressure gradient parameter
+via
+h =
+�
+−2η d ln ρ
+d ln r
+�1/2
+= −2Πd ln ρ
+d ln r .
+(28)
+2 Throughout
+this
+paper,
+we
+use
+‘planetesimals’
+and
+‘gravitationally
+bound
+clumps’
+relatively
+interchangeably.
+The latter is more precise, as our simulations do not resolve the
+scale of planetesimals. Likewise, the Toomre-like paradigm only
+predicts the mass of a Hill-stable region subject to gravitational
+collapse.
+The actual planetesimal mass requires knowledge of
+contraction efficiency and subsequent accretion of additional
+pebbles.
+
+Planetesimal IMFs under diffusion regulated collapse
+11
+−0.10
+−0.05
+0.00
+0.05
+0.10
+x [H]
+−0.10
+−0.05
+0.00
+0.05
+0.10
+y [H]
+Q = 16, Qp = 0.29
+−0.10
+−0.05
+0.00
+0.05
+0.10
+x [H]
+Q = 8, Qp = 0.18
+−0.10
+−0.05
+0.00
+0.05
+0.10
+x [H]
+Q = 4, Qp = 0.14
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(Σp/Σg)
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(Σp/Σg)
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+log(Σp/Σg)
+Figure 8. Snapshots of the three self-gravity simulations at 62Ω−1, which is 2Ω−1 after self-gravity is turned on. All three
+simulations formed gravitationally bound clumps indiciated by red circles.
+Throughout this work, we choose h = 0.05, which
+together with our simulation parameter of Π = 0.05
+implicitly assumes d ln ρ/d ln r = −1/2. These choices
+are relatively generic for disk models (compare to e.g.,
+Dullemond et al. 2007; Gerbig et al. 2019; Gerbig &
+Laughlin 2022).
+The smallest mass above which a planetesimal is well
+resolved is the mass associated with a Hill-radius that
+equals the cell-size, i.e., mrH=∆x.
+This limitation is
+caused by the accuracy of the self-gravity solver set
+by the finite grid scale (Simon et al. 2016).
+PLAN
+automatically discards clumps, if any, less massive than
+mrH=∆x.
+Using Eq. (23) and the mass scaling in
+Eq. (26), this mass limit is independent of Q, i.e.
+mrH=∆x = 9
+4π ρH∆x3 = 9
+4π
+�ρH
+ρ0
+� �∆x
+H
+�3
+M0
+= 81
+64h3
+�∆x
+H
+�3
+M⊙ = 1.97 · 10−5MCeres
+(29)
+for ∆x = 3.9 · 10−4H. As shown in Table 1, mrH=∆x
+is smaller than mp,min, and λmin is larger than 2∆x
+for all Q, suggesting we sufficiently resolve the smallest
+expected planetesimals.
+4.6. IMFs and K-S test
+Figure 9 shows the IMFs obtained from the snapshot
+shown in Fig. 8 in solid lines.
+We compare these
+numerically obtained IMFs, to the predicted IMF from
+the Toomre-like instability paradigm, evaluated for the
+properties shown in Tab. 1.
+Hereby, we draw N
+planetesimals from the theoretical PDF, where N is
+the number of bound clumps found in the respective
+simulation. We conduct this random draw 1000 times
+in order to obtain an average theoretical IMF (dashed
+curves), together with one and three σ intervals (dark
+and light shaded regions respectively). In addition, we
+draw a dotted vertical line, corresponding to the mass
+associated with the fastest growing mode of the IMF
+prediction.
+In order to assess the quality of the analytical
+predictions,
+we
+perform
+two
+sample
+Kolmogorov-Smirnov (K-S) tests, which quantify the
+likelihood p of two samples (in our case the numerical
+and
+analytically
+obtained
+planetesimal
+masses)
+originating from the same underlying distribution. The
+null hypothesis, that is the two samples indeed coming
+from the same distribution, can be rejected if p < 0.05.
+For the growth-rate based PDF, the K-S test yields
+p-values of 0.501, 0.395, and 0.0629 for Q = 16, Q = 8,
+and Q = 4 respectively.
+The numerical IMFs are
+therefore consistent with the analytical IMFs obtained
+from the Toomre-like analysis.
+5. DISCUSSION
+We developed a framework of analytically obtaining
+planetesimal IMFs, based on the Toomre-like instability
+in the particle layer in protoplanetary disks. In Sect. 4
+we presented a simulation of the streaming instability,
+measured diffusivity and particle concentration for three
+self-gravity runs, and then compared resulting Qp-values
+and corresponding analytical IMFs to the numerically
+obtained IMFs. We find that the IMFs are consistent
+with each other, as seen Fig. 9.
+Specifically, our
+prediction can explain a number of key properties of the
+planetesimal IMF, which we will outline in the following.
+Our IMF prediction explains the somewhat counter-
+intuitive property of the least massive clumps being
+present in the simulation with the most mass, i.e.
+Q = 4. Since this simulation has the smallest Qp with
+Qp = 0.14, the fastest growing mode shifts to smaller
+scales (see Fig. 1) and thus the most likely planetesimal
+becomes less massive.
+
+12
+Gerbig and Li
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+mp[MCeres]
+10−2
+10−1
+100
+N(> mp)
+p = 5.01e-01
+Q = 16, Z = 0.137
+δz = 2.3e-06, δr = 2.4e-06
+Qp = 0.29, N = 90
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+mp[MCeres]
+p = 3.95e-01
+Q = 8, Z = 0.1
+δz = 3.1e-06, δr = 2.0e-06
+Qp = 0.18, N = 102
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+mp[MCeres]
+p = 6.29e-02
+Q = 4, Z = 0.075
+δz = 3.3e-06, δr = 2.6e-06
+Qp = 0.14, N = 84
+Figure 9. Initial mass functions N(> mp)for the three self-gravity runs. The numerically obtained IMFs are shown in solid
+lines and are based on the snapshots seen in Fig. 8. The IMF resulting from the growth rate-based PDF are shown in dashed
+lines. The dark and light shaded regions indicate one and three standard deviations from the distribution of IMFs that originates
+from only drawing N ∼ 100 planetesimals from the distribution. The vertical dotted lines indicate the masses associated with
+the fastest growing mode mp,fgm. Lastly, the p-value from the K-S-Test is shown in the top right corner of each panel.
+Next, we observe that the IMF becomes flatter if Q
+and therefore Qp decreases, which is due to the increase
+of the range of unstable scales.
+Notably, all three
+numerical IMFs fall short of the analytical prediction
+on the high mass end. As alluded to in Sect. 4.5, this
+is expected as these planetesimals would have to form
+from scales that exceed the maximum scale of particle
+over-densities ηr (see Gerbig et al. 2020). Indeed, the
+idea that the most massive planetesimal is not set by
+the largest Toomre-scale λ but by ηr, is consistent
+with our simulations where the most massive clump
+has ∼ 0.1MCeres in all three simulations, and also with
+Fig. 10 in Simon et al. (2016), where the most massive
+planetesimal shortly after self-gravity is turned on is
+relatively independent of ˜G (or equivalently Q).
+Lastly, the fact that λfgm is much closer to λmin
+when Qp is small, provides an explanation for why only
+high-resolution simulations can sufficiently capture the
+turnover in the mass frequency distribution at the low-
+mass end of planetesimal IMFs (Li et al. 2019).
+5.1. Caveats
+In this section, we outline a number of caveats that are
+to be kept in mind when relating the herein presented
+theory and analysis to real disks.
+First, unstable scales λ were converted to planetesimal
+masses by assuming the contraction of a circular
+sheet of radius λ/(2π) at Hill-density.
+While such a
+conversion agrees with timescale arguments presented
+in e.g. Gerbig et al. (2020); Klahr & Schreiber (2020),
+it certainly is an order of magnitude calculation.
+In
+particular, as pointed out in Polak & Klahr (2022), a
+sphere at Hill-density is not fully tidally stable.
+The
+Hill-sphere is derived by considering the tidal forces on
+either end of the Roche lobe in the restricted three-
+body potential. This has the Hill-sphere extend beyond
+the tidally-stable Roche lobe, and implies Hill-density is
+insufficient for tidal stability by an order of unity factor.
+Moreover, it is not the case that all unstable scales are
+at Hill-density exactly. In fact, as evidenced by Fig. 6,
+many particles are located within regions which exceed
+Hill-density by up to an order of magnitude.
+Additionally,
+our
+model
+assumed
+planetesimal
+formation to be perfectly efficient.
+Fig. 8 however
+demonstrates that there are still particles in the
+filaments that are not bound to any clump. Moreover,
+previous
+works
+showed
+that
+higher
+resolution
+simulations may extend the mass distribution to lower
+masses (Li et al. 2019). Another possibility that is not
+considered in our model is a single clump forming
+binary planetesimals upon contraction to material
+density (Nesvorn´y et al. 2019, 2021).
+Further, our
+model does not take into account the possibility of
+“clumps
+within
+clumps”
+which
+would
+result
+in
+double-counting of planetesimals (compare to Hopkins
+&
+Christiansen
+2013,
+who
+consider
+this
+when
+investigating the statistical instability of the gas disk).
+We note,
+that while multiplicity and inefficient
+contraction would shift the IMF towards smaller masses,
+collapse beyond Hill-density and double counting would
+shift the IMF towards larger masses. We expect that
+this allows the numerically obtained IMFs to remain
+consistent with our analytical predictions.
+Other simplifying assumptions include gas density
+being considered constant.
+For stable, yet small Q
+values, the gas develops a mid-plane cusp with a non-
+
+Planetesimal IMFs under diffusion regulated collapse
+13
+neglible vertical density gradient (see e.g., Armitage
+2015, for a review). We also assumed a mono-disperse
+dust population with a constant Stokes number. This
+choice is likely reasonable to first order as in real disks,
+most solid mass is contained in the largest grains (e.g.,
+Birnstiel et al. 2011; McNally et al. 2021), and the
+largest grains dominate dust-gas interactions in the mid-
+plane and participate most vigorously in planetesimal
+formation (Yang & Zhu 2021).
+Still, a multi-species
+dust fluid can alter the turbulent behavior of the system
+itself (see e.g., Krapp et al. 2019, who pointed out
+the damping effect multi-species dust can have on the
+streaming instability).
+Therefore, the assumption of
+single-species dust is to be kept in mind when applying
+our results to real disks.
+Lastly, we acknowledge that our numerical setup does
+not perfectly replicate the assumptions the Toomre-like
+instability is based on. More specifically, the Toomre
+analysis,
+assumes a thin,
+two-dimensional disk of
+constant density which is then linearly perturbed in form
+of axisymmetric waves.
+This picture evidently differs
+from the turbulent state, i.e.
+non-linear streaming
+instability in vertically-stratified disks, our simulations
+are in when self-gravity is turned on.
+We attempted
+to mitigate this discrepancy by evaluating diffusivities
+and concentration locally, rather than globally, which,
+as discussed in Sect. 4.6, yielded consistent IMFs.
+5.2. The shape of the initial mass function in real disks
+The diffusion-tidal-shear collapse paradigm suggests
+that the shape of planetesimal IMF depends on the
+gravitational,
+Toomre-like
+instability
+and
+on
+the
+stability parameter Qp
+only.
+While independent
+information on diffusivities δr and δz, as well as Stokes
+number St, and disk aspect ratio h are required to
+match the IMF to the appropriate mass range, they do
+not affect the steepness of the IMF (for a fixed Qp). As
+such, as a disk region collects more massive particles, it
+experiences an increase in average grain size, cools or
+becomes less turbulent, its Qp-value decreases, thus
+flattening the planetesimal IMF. Within this context,
+the role of the streaming instability (or other processes
+affecting particle dynamics) for the planetesimal IMF
+in this work, is to set the initial condition at the onset
+of gravitational collapse. The key question within our
+framework therefore is to determine what value for Qp
+takes on in real disks.
+Our simulations, much like many in previous works
+(e.g., Simon et al. 2016; Sch¨afer et al. 2017; Li et al. 2019;
+Gerbig et al. 2020) show collapse as soon as self-gravity
+is turned on. The turbulent state before self-gravity is
+turned on is therefore not physical, and thus the Qp-
+values and IMFs we calculate not necessarily reflective
+of real disks.
+Indeed, as discussed in Gerbig et al.
+(2020), if a system softly evolves towards instability —
+for example by collecting more particle mass — its Qp-
+value likewise softly evolve from Qp > 1 to Qp < 1.
+If Qp ∼ 1 is sufficient for the system to fully collapse
+and form planetesimals as assumed in Klahr & Schreiber
+(2020, 2021), then the resulting IMF is expected to be
+rather steep (Polak & Klahr 2022). On the other hand,
+the system may be unable to fully contract at Qp ∼ 1
+if it is below Hill-density.
+To a similar effect, as the
+contraction time, given by (Gerbig et al. 2020),
+tc =
+Ω
+4πGρpSt =
+�π
+8
+�
+δz
+δr
+Qp
+St Ω−1,
+(30)
+only falls below the dynamical timescale Ω−1 once Qp
+is substantially less than unity, it is plausible that the
+system can collect relevant amounts of mass during
+contraction.
+Both situations would lead to smaller
+values for Qp and thus flatter IMFs like the ones in our
+simulations in Fig. 9.
+One observational constraint may be provided by
+the Asteroids size distribution,
+which,
+at least in
+part, constitutes pristine planetesimals from the Solar
+System’s formation era (see Klahr et al. 2022, for a
+review). Taking, the collisional evolution into account,
+it is possible to reconstruct model of the primordial size
+distribution of Asteroids (see e.g., Delbo et al. 2019).
+Within the here presented framework, the “knee” at
+∼ 100 km (e.g., Gladman et al. 2001) typically found
+in the resulting mass functions can be interpreted as
+an indication of a marginally unstable origin system
+with Qp close to unity — that is assuming the asteroids
+in question indeed formed via gravitational collapse of
+locally over-dense regions, which very well may not have
+been the case.
+Cold Classical Kuiper Belt Objects
+(KBOs) are believed to be even more primordial with
+little to no collisional evolution (Morbidelli & Nesvorn´y
+2020).
+Kavelaars et al. (2021) found a exponentially
+tapered power law as the best fit for the Cold Classical
+KBO mass distribution, and connected the lack of
+large planetesimals (> 400 km in size) to streaming
+instability regulated planetesimal formation. Our work
+is consistent with this interpretation, as Qp imposes a
+limit on the maximum planetesimal mass.
+One often invoked pathway of forming planetesimals
+and circumventing the meter barrier, that is worth
+discussing in the context of our work, is the existence
+of long-lived pressure bumps associated with disk sub-
+structures. While such structures are robustly confirmed
+observationally (e.g., Andrews et al. 2018), their role
+in producing the first generation of planetesimals is
+
+14
+Gerbig and Li
+elusive,
+as already existing planets seem to most
+promisingly explain the sub-structures in the first
+place (Teague et al. 2021).
+Either way, a pressure
+bump is characterized by very small pressure gradients,
+which diminishes the relative velocity between dust
+and gas.
+If Π = 0, then particles can trivially settle
+razor-thin and go gravitationally unstable (as seen in
+e.g., Abod et al. 2019).
+In this state, both vertical
+and radial diffusivities are expected to trend towards
+zero, implying Qp
+→
+0 as well, which our work
+associates with the planetesimal IMF shifting towards
+very small masses. Unless hierarchical mergers between
+small planetesimal immediately after formation is very
+efficient, such a bottom-heavy mass function seems
+unlikely. We therefore suspect that either the majority
+of planetesimal formation does not occur in Π = 0
+regions; or, that other mechanisms, possibly gravito-
+turbulence (see e.g., Riols et al. 2017, for the gas disk),
+provide a lower bound on diffusivities, thus also setting
+a lower limit on Qp.
+To conclude, our analysis that connects the PDF of
+formed
+planetesimals
+to
+the
+growth
+rates
+of
+the
+Toomre-like instability of a particle layer subject to a
+diffusive flux, unifies both the flat, power-law shaped,
+IMFs previously obtained numerically (e.g., Simon
+et al. 2016; Li et al. 2019), and the ‘Asteroids are born
+big’ (Morbidelli et al. 2009) paradigm that arises when
+investigating marginally unstable systems (Klahr &
+Schreiber 2020).
+The analytically obtained IMFs are
+consistent with our numerical setups. Further work is
+required
+to
+test
+the
+predictions
+for
+different
+numerically setups, and in particular, to assess how the
+value of Qp at the on-set of planetesimal formation,
+and thus the steepness of the resulting IMF, depend on
+disk properties and radii.
+Such an analysis informs
+initial conditions for planet formation models (e.g.,
+Emsenhuber
+et
+al.
+2021),
+and
+conversely,
+if
+the
+formation
+of
+an
+observed
+exoplanetary
+population
+presupposes a specific planetesimal IMF(e.g., Batygin
+& Morbidelli 2023, for rocky Super-Earths), provides
+constraints
+on
+the
+disk
+state
+during
+the
+era
+of
+planetesimal formation.
+6. ACKNOWLEDGEMENTS
+This work and KG and RL benefited from the
+2022 Exoplanet Summer Program in the Other Worlds
+Laboratory (OWL) at the University of California,
+Santa Cruz, a program funded by the Heising-Simons
+Foundation. The authors thank Greg Laughlin, Hubert
+Klahr, Andrew Youdin, Ruth Murray-Clay and Malena
+Rice for insightful comments and discussions.
+Software:
+Athena (Stone et al. 2008), PLAN (Li
+2019), NumPy (Harris et al. 2020), Matplotlib (Hunter
+2007), CMasher (van der Velden 2020)
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+“A Handbook of Integer Sequences” Fifty Years Later
+N. J. A. Sloane,
+The OEIS Foundation Inc.,
+11 So. Adelaide Ave., Highland Park, NJ 08904, USA
+Email: [email protected]
+January 8, 2023
+Abstract
+Until 1973 there was no database of integer sequences. Someone coming across
+the sequence 1,2,4,9,21,51,127,... would have had no way of discovering that
+it had been studied since 1870 (today these are called the Motzkin numbers,
+and form entry A001006 in the database). Everything changed in 1973 with the
+publication of A Handbook of Integer Sequences, which listed 2372 entries. This
+report describes the fifty-year evolution of the database from the Handbook to its
+present form as The On-Line Encyclopedia of Integer Sequences (or OEIS), which
+contains 360,000 entries, receives a million visits a day, and has been cited 10,000
+times, often with a comment saying “discovered thanks to the OEIS”.
+1
+Introduction
+Number sequences arise in all branches of science: for example, 1,1,2,4,9,20,48,115,...
+gives the number of rooted trees with n nodes (A000081,1 see also Fig. 1), and in daily
+life: how many pieces can you cut a pancake into with n knife-cuts? (The pieces need
+not all be the same size.) That one is easy: 1,2,4,7,11,16,..., n(n+1)/2+1 (A000124).
+But what is the answer for cutting up an (ideal) bagel, or torus? That is a lot harder:
+with a sharp knife you might get a few terms, perhaps 1,2,6,13,..., but probably not
+enough to guess the formula, which is n(n2 + 3n + 8)/6 for n > 0. For that you would
+need to to consult the database: go to https://oeis.org and enter “cutting bagel”,
+or go directly to A003600.
+My fascination with these sequences began in 1964 when I was a graduate student
+at Cornell University in Ithaca, NY, studying neural networks. I had encountered a
+sequence of numbers, 1,8,78,944,13800,..., and I badly needed a formula for the n-th
+term, in order to determine the rate of growth of the terms (this would indicate how
+long the activity in this very simple neural network would persist). I will say more
+about this sequence in Section 2.1.
+I noticed that although several books in the Cornell library contained sequences
+somewhat similar to mine, as far as I could tell this particular sequence was not men-
+tioned. I expected to have to analyze many related sequences, so in order to keep track
+of the sequences in these books, I started recording them on 3” × 5” file cards.
+1Six-digit numbers prefixed by A refer to entries in the current version of the Handbook, The On-Line
+Encyclopedia of Integer Sequences [11].
+1
+arXiv:2301.03149v1  [math.NT]  9 Jan 2023
+
+Figure 1: Left: one of 48 unlabeled rooted trees with 7 nodes (the root node is at the
+bottom); center: four cuts of a pancake can produce 11 pieces; right: three cuts of a
+bagel can produce 13 pieces.
+The collection grew rapidly as I searched though more books, and once the word got
+out, people started sending me sequences. Richard Guy was an enthusiastic supporter
+right from the start.
+In 1973 I formalized the collection as A Handbook of Integer
+Sequences, which was published by Academic Press (Fig. 2). It contained 2372 entries.
+Figure 2: Front cover of the Handbook. The embossed figures show side views of the two
+ways of folding a strip of three (blank) stamps, and the five ways of folding a strip of
+four stamps. The full sequence begins 1,1,2,5,14,38,120,353,1148,3527,..., A001011.
+No formula is known.
+Once the book appeared, the flood of correspondence increased, and it took twenty
+years to prepare the next version.
+Simon Plouffe helped a great deal, and in 1995
+Academic Press published our sequel, The Encyclopedia of Integer Sequences, with 5487
+entries. From this point on the collection grew even more rapidly. I waited a year,
+until it had doubled in size, and then put it on the Internet, calling it The On-Line
+Encyclopedia of Integer Sequences.
+In the rest of this article I will first say more about the evolution of the database:
+the Handbook (§2.1), the 1995 Encyclopedia (§2.2), the On-Line Encyclopedia (§2.3),
+and the OEIS Foundation (§2.4). The next sections describe the database itself: what
+2
+
+AHANDBOOKOF
+INTEGER SEQUENCES
+N.JLA.SLOANEsequences are—or are not—included (§3.1), how the database is used (§3.2), the layout
+of a typical entry (§3.3), the arrangement of the entries (§3.4), and a Fact Sheet (§3.5).
+The final sections describe some especially interesting sequences: Recam´an’s sequence
+(§4.1), Iteration of number-theoretic functions (§4.2), Gijswijt’s sequence (§4.3), Lex-
+icographically Earliest Sequences (§4.4), The Stepping Stones problem (§4.5), Stained
+glass windows (§4.6), and other sequences I would have liked to include (§4.7).
+Several open questions are mentioned to which I would very much like to know the
+answers.
+Notation.
+a(n) will denote the n-th term of the sequence being discussed, and
+A001006(n) (for example) would denote the n-th term of A001006. σ(n) is the sum of
+the divisors of n (A000203).
+2
+Evolution of the database
+2.1
+The Handbook of Integer Sequences
+Once the collection had grown to a few hundred entries, I entered them on punched
+cards,2 which made it easier to check and sort them. The Handbook was type-set directly
+from the punched cards. There were a few errors in the book, but almost all of them
+were caused by errors in the original publications. Accuracy was a primary concern in
+that book, as it is today in the OEIS.
+The book was an instant success. It was, I believe, the world’s first dictionary of
+integer sequences (and my original title said Dictionary rather than Handbook). Many
+people said “What a great idea”, and wondered why no one had done it before. Martin
+Gardner recommended it in the Scientific American of July 1974. Lynn A. Steen, writing
+in the American Mathematical Monthly said “Incomparable, eccentric, yet very useful.
+Contains thousands of ‘well-defined and interesting’ infinite integer sequences together
+with references for each ... If you ever wondered what comes after 1,2,4,8,17,35,71,...,
+this is the place to look it up”’.
+Harvey J. Hindin, writing from New York City, exuberantly concluded a letter to
+me by saying: “There’s the Old Testament, the New Testament, and the Handbook of
+Integer Sequences.”
+I never did find the sequence that started it all in the literature, but I learned P´olya’s
+theory of counting, and with John Riordan’s help found the answer, which appears in
+[13] and A000435.
+2These were never called “punch cards” (sic). To anyone who worked with them in the 1960s, “punch
+cards” sounds like “grill cheese” (sic) for “grilled cheese”, or “barb wire” (sic) for “barbed wire”, both
+of which I have recently seen in print.
+3
+
+2.2
+The Encyclopedia of Integer Sequences
+Following the publication of the Handbook, a large amount of correspondence ensued,
+with suggestions for further sequences and updates to the entries. By the early 1990’s
+over a cubic meter of new material had accumulated.
+A Canadian mathematician,
+Simon Plouffe, offered to help in preparing a revised edition of the book, and in 1995
+The Encyclopedia of Integer Sequence, by me and Simon Plouffe, was published by
+Academic Press. It contained 5487 sequences, occupying 587 pages. By now punched
+cards were obsolete, and the entries were stored on magnetic tape.
+2.3
+The On-Line Encyclopedia of Integer Sequences
+Again, once the book appeared, many further sequences and updates were submitted
+from people all over the world. I waited a year, until the size of the collection had
+doubled, to 10000 entries, and then in 1996 I launched The On-Line Encyclopedia of
+Integer Sequences (now usually called simply the OEIS) on the Internet. From 1996
+until October 26, 2009, it was part of my homepage on the AT&T Labs website.
+Incidentally, in 2004 the database was mentioned by the Internet website slashdot
+(“News for Nerds. Stuff that Matters”), and this brought so much traffic to my Bell
+Labs homepage that it briefly crashed the whole Bell Labs website. My boss was quite
+proud of this, since it was a rare accomplishment for the Mathematics and Statistics
+Research Center.
+2.4
+The OEIS Foundation
+In 2009, in order to ensure the long-term future of the database, I set up a non-profit
+foundation, The OEIS Foundation Inc., a 501(c)(3) Public Charity, whose purpose is
+to own, maintain and raise funds to support The On-Line Encyclopedia of Integer Se-
+quences or OEIS.
+On October 26, 2009, I transferred the intellectual property of The On-Line Ency-
+clopedia of Integer Sequences to the Foundation. A new OEIS with multiple editors was
+launched on November 11, 2010.
+Since then it has been possible for anyone in the world to propose a new sequence
+or an update to an existing sequence. To do this, users must first register, and then
+submissions are reviewed by the editors before they become a permanent part of the
+OEIS. Technically the OEIS is now a “moderated wiki”.
+I started writing this article on November 11, 2022, noting that this marked twelve
+years of successful operation of the online OEIS, and also that the database is in its
+59th year of existence.
+4
+
+3
+The database today
+3.1
+What sequences are included?
+From the very beginning the goal of the database has been to include all “interest-
+ing” sequences of integers.
+This is a vague definition, but some further examples
+will make it clearer.
+The database includes a huge number of familiar and unfa-
+miliar sequences from mathematics (the prime numbers 2,3,5,7,11,13,..., A000040;
+60,168,360,504,660,1092,..., the orders of noncyclic simple groups, A001034), com-
+puter science (0,1,3,5,8,11,14,..., Number of comparisons needed for merge sort,
+A001855), physics (see “self-avoiding walks on lattices”, Ising model, etc., e.g. A002921),
+chemistry (the enumeration of chemical compounds was one of the motivations behind
+P´olya’s theory of counting, see e.g. A000602), and not least, from puzzles and I.Q.
+tests (1,8,11,69,99,96,111,..., the “strobogrammatic” numbers, guess!, or see A000787;
+4,14,23,34,42,50,59,..., the numbered stops on the New York City A train subway,
+A011554. That entry has links to a map and the train schedule).
+Sequences that have arisen in the course of someone’s work—especially if published—
+have always been welcomed. On the other hand, sequences that have been proposed
+simply because they were missing from the database are less likely to be accepted.
+There are a few hard and fast rules. The sequence must be well-defined and the
+terms must not be time-dependent—if the next term is only known to be either 14 or
+15, for instance, then the sequence must end with the last term that is known for certain.
+The sequence may not have any missing terms or gaps. In the case of Mersenne primes,
+for instance (A000043) it is common for later primes to be known before all intermediate
+numbers having been tested. The later primes get mentioned in comments, but they
+are not as part of the main sequence until their position has been confirmed.
+Very short sequences and sequences that are subsequences of many other sequences
+are not accepted. A sequence for which the only known terms are 2,3,5,7 would not
+be accepted since it is matched by a large number of existing sequences. The definition
+may not involve an arbitrary but large parameter (primes ending in 1 are fine, A030430,
+but not primes ending in 2023).3
+Most entries give an ordered list of integers. But triangles of numbers are included
+by reading them row-by-row: Pascal’s triangle becomes 1, 1,1, 1,2,1, 1,3,3,1,...,
+A007318.
+Doubly-infinite square arrays are included by reading them by antidiago-
+nals: the standard multiplication table for positive integers becomes 1, 2,2, 3,4,3,
+4,6,6,4,..., A003991.
+Sequences of fractions are included as a linked pair giving the numerators and denom-
+inators separately (the Bernoulli numbers are A027641/A027642). Important individual
+real numbers are included by giving their decimal or continued fraction expansions (for
+π see A000796 and A001203). A relatively small number of sequences of nonintegral
+3The OEIS Wiki, which has a great deal of useful information about the database, has a section
+listing additional examples of “what not to submit”.
+5
+
+real numbers are included by rounding them to the nearest integer, or by taking floors
+or ceilings (the imaginary parts of the zeros of Riemann’s zeta function give A002410).
+Two less obvious sources for sequences are binomial coefficient identities and number-
+theoretic inequalities. The values of either side of the identity
+n
+∑
+k=0
+(2n
+k )
+2
+= 1
+2(4n
+2n) − 1
+2(2n
+n )
+2
+[7, (3.68)] give A036910. From the inequality σ(n) < n√n for n > 2, [10, Sect. III.1.1.b],
+we get the integer sequence ⌊n√n⌋ − σ(n), A055682. The point is that if you want to
+know if this inequality is known, you look up the difference sequence, and find A055682
+and a reference to the proof. Many more sequences of these two types should be added
+to the database.
+3.2
+How the database is used
+The main applications of the database are in identifying sequences or in finding out the
+current status of a known sequence. Barry Cipra has called it a mathematical analogue
+of a “fingerprint file”. You encounter a number sequence, and wish to know if anyone
+has ever come across it before.
+If your sequence is in the database, the reply will give a definition, the first 50 or
+so terms, and, when available, formulas, references, computer code for producing the
+sequence, links to any relevant web sites, and so on.
+Figures 3 and 4 show what happens if you submit 1,2,5,14,42,132,429, the first few
+Catalan numbers, one of the most famous sequences of all.
+Figure 3: The result of submitting 1,2,5,14,42,132,429 to the database. This figure
+shows the banner at the top of the reply. There are 26 matches, ranked in order of
+importance, the top match being the one we want, the Catalan numbers. A shortened
+version of the top match is shown in the next figure.
+6
+
+TheOEisissupportedbythemanygenerousdonorstotheOEisFoundation
+013627
+THE ON-LINE ENCYCLOPEDIA
+20
+OF INTEGER SEOUENCES
+?
+10221121
+founded in 1964 by N. J. A. Sloane
+1.2.5.14,42.132,429
+Search
+Hints
+(GreetingsfromTheOn-LineEncyclopediaofIntegerSequences!)
+Search:seq:1.2.5.14.42.132.429
+Displaying1-10of26resultsfound.
+page 1 2 3
+Sort:relevanceIreferencesInumberImodifiedIcreated
+Format:longIshortIdataA000108
+Catalan numbers: C(n) = binomial(2n,n)/(n + 1) = (2n)!/(n!(n + 1)!).
+(Formerly M1459 N0577)
+DATA
+1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...
+COMMENTS
+These were formerly sometimes called Segner numbers.
+A very large number of combinatorial interpretations are known -
+see references, esp. R. P. Stanley, Catalan Numbers, Camb., 2015.
+This is probably the longest entry in the OEIS, and rightly so.
+The solution to Schr¨oder’s first problem: number of ways to insert n pairs
+of parentheses in a word of n + 1 letters. E.g., for n = 2 there are 2 ways:
+((ab)c) or (a(bc)); for n=3 there are 5 ways: ((ab)(cd)), (((ab)c)d), ...
+...
+REFERENCES
+The large number of references and links demonstrates the ubiquity
+of the Catalan numbers.
+R. Alter, Some remarks and results on Catalan numbers, pp. 109-132
+in Proc. Louisiana Conf. Combinatorics, Graph Theory and
+Computer Science. Vol. 2, edited R. C. Mullin et al., 1971.
+M. Bona, ed., Handbook of Enumerative Combinatorics, CRC Press, 2015
+L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 53.
+J. H. Conway & R. K. Guy, The Book of Numbers, Springer, 1995, 96-106.
+...
+LINKS
+Robert G. Wilson v, Table of n, a(n) for n = 0..1000
+...
+F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Disc. Math.,
+Vol. 204, No. 1-3 (1999), 73-112.
+...
+W. G. Brown, Historical Note on a Recurrent Combinatorial Problem,
+Amer. Math. Monthly,. 72, No. 9 (1965), 973-977.
+...
+FORMULA
+Recurrence: a(n) = 2 ∗ (2 ∗ n − 1) ∗ a(n − 1)/(n + 1) with a(0) = 1.
+...
+MAPLE
+A000108 := n-¿binomial(2*n, n)/(n+1);
+...
+MATHEMATICA
+A000108[n ] := (2 n)!/n!/(n+1)!
+...
+PARI
+a(n)=binomial(2*n, n)/(n+1)
+...
+KEYWORD
+core,nonn,easy,nice
+AUTHOR
+N. J. A. Sloane
+Figure 4: The entry for the Catalan number A000108. The full entry has over 750 lines,
+which have been edited here to show samples of the different fields.
+I could have chosen a simpler example, like the Fibonacci numbers, but I have a
+particular reason for choosing the Catalan numbers. When the OEIS was new, people
+would sometimes say to me that they had a sequence they were trying to understand,
+and would I show them how to use the database.
+At least twice when I used the
+Catalan sequence as an illustration, they said, why, that is my sequence, how on earth
+did you know? It was no mind-reading trick, the Catalan numbers are certainly the
+most common sequence that people don’t know about. This entry is the longest—and
+7
+
+one of the most important—in the whole database.
+If we do not find your sequence in the database, we will send you a message inviting
+you to submit it (if you consider it is of general interest), so that the next person who
+comes across it will be helped, and your name will go on record as the person who
+submitted it.
+The second main use of the database is to find out the latest information about a
+particular sequence.
+Of course we cannot hope to keep all 360000 entries up-to-date. But when a new
+paper is published that mentions the OEIS, Google will tell us, and we then add links to
+that paper from any sequence that it mentions. People have told us that this is one of
+the main ways they use the OEIS. After all, even a specialist in (say) permutation groups
+cannot keep track of all the papers published worldwide in that area. And if a paper in
+a physics journal happens to mention a number-theoretic sequence, for example, that is
+unlikely to be noticed by mathematicians.
+There are also many other ways in which the database has proved useful.
+For example, it is an excellent source of problems to work on. The database is con-
+stantly being updated. Every day we get thirty to fifty submissions of new sequences,
+and an equal number of comments on existing entries (new formulas, references, ad-
+ditional terms, etc.).
+The new sequences are often sent in by non-mathematicians,
+and are a great source of problems. You can see the current submissions at https:
+//oeis.org/draft. Often enough you will see a sequence that is so interesting you
+want to drop everything and work on it. And remember that we are always in need of
+more volunteer editors. In fact anyone who has registered with the OEIS can suggest
+edits, you do not even need to be an official editor. We have been the source of many
+international collaborations.
+There is also an educational side: several people have told us that they were led into
+mathematics through working as an editor. Here is a typical story.
+Subject: Reminiscence from a young mathematician
+I wanted to relay a bit of nostalgia and my heartfelt thanks. Back in the late 1990s, I was a high
+school student in Oregon. While I was interested in mathematics, I had no significant mathematically
+creative outlet until I discovered the OEIS in the course of trying to invent some puzzles for myself. I
+remember becoming a quite active contributor through the early 2000s, and eventually at one point, an
+editor. My experience with the OEIS, and the eventual intervention of one of my high school teachers,
+catalyzed my interest in studying mathematics, which I eventually did at [...] College. I went on to a
+Ph.D. in algebraic geometry at the University of [...], and am currently at [...].
+I wanted to thank you for seriously engaging with an 18-year old kid, even though I likely submitted
+my fair share of mathematically immature sequences. I doubt I would have become a mathematician
+without the OEIS!
+A less-obvious use of the database is to quickly tell you how hard a problem is. I use
+it myself in this way all the time. Is the sequence “Catalan” or “Collatz”? If a sequence
+comes up in your own work, or when reviewing someone else’s work, it is useful to know
+right away if this is a well-understood sequence, like the Catalan numbers, or if it is one
+of the notoriously intractable problems like the Collatz or 3x + 1 problem (A006577).
+8
+
+Finally, the OEIS is a welcome escape when you feel the world is falling apart. Take
+a look at Scott Shannon’s drawings of stained glass windows in A331452; or Jonathan
+Wild’s delicate illustrations of the ways to draw four circles (in A250001); or ´Eric An-
+gelini’s “1995” puzzle (A131744) or any of his “lexicographically earliest sequences”
+(A121053, A307720, and many more); or find better solutions to the Stepping Stones
+Problem (§4.5,A337663). You can find brand new problems at any hour of the day or
+night by looking at the stack of recent submissions: but beware, you may see a problem
+there that will keep you awake for days. Or search in the database for phrases like “It
+appears that ...”, or “Conjecture: ...”, or “It would be nice to know more!”
+3.3
+Layout of a typical entry
+This is a good place to mention some of the features of an OEIS entry. Most of the
+fields (see Figs. 3 and 4) are self-explanatory. At the top it tells you how many matches
+were found to your query (26 in the example). These are ranked in order of importance.
+The DATA section shows the start of the sequence, usually enough terms to fill a few
+lines on the screen (typically 300 to 500 decimal digits). Often one wants more terms
+than are shown, and the first link in the entry will point to a plain text file with perhaps
+10000 or 20000 terms. That file will have a name like b001006.txt, and is called the
+“b-file” for the sequence. Some entries also have much larger tables, giving a million or
+more terms.
+If you click the “graph” button near the top of the reply, you will be shown two plots
+of the sequence, and if you click the “listen” button, you could listen to the sequence
+played on an instrument of your choice. The default instrument is the grand piano, and
+the terms of the sequence would be mapped to the 80 keys by reducing the numbers
+mod 80 and adding 1.
+I conclude this section with a philosophical comment.
+When you are seriously trying to analyze a sequence, and are prepared to spend
+any amount of time needed (searching for a formula or recurrence, for instance), you
+need all the help you can get, which is why we provide the b-files and other data files,
+and why we give computer programs in so many languages. This is also the reason we
+give as many references and links as possible for a sequence. Even if the reference is to
+an ancient or obscure journal, or one that has been accused as being “predatory”, we
+still give the reference, especially for sequences that are not well-understood. The same
+thing holds for formulas, comments, and cross-references to other sequences. When you
+are desperate, you will accept help from anywhere. And do not forget “Superseeker”!
+3.4
+Arrangement of the entries
+The entries in the database are (virtually) arranged in two different ways, the first
+essentially chronological, the second lexicographic.
+9
+
+The first is by their absolute identification number, or A-number.4 Once the collec-
+tion reached a few hundred entries, I sorted them into lexicographic order and numbered
+them A1, A2, A3, .... A1 gives the number of symmetry groups of order n, A2 is the
+famous Kolakoski sequence, and so on. This numbering is still used today, only A1 has
+become A000001, A2 is A000002, ..., and as each new submission comes in it gets a
+number from the stack. Current sequences are being issued numbers around A360000.
+Rejected A-numbers are recycled, so there are no gaps in the order. We reached 100000
+entries in 2004, and 250000 in 2015. The present growth rate is about 12000 new entries
+each year.
+The second arrangement is a kind of lexicographic ordering.
+First I describe an
+idealized, theoretical, lexicographic order. Sequences of nonnegative numbers can be
+arranged in lexicographic (or dictionary) order.
+For example, sequences beginning
+1,2,4,... come before 1,2,5,..., 1,2,4,3,..., 1,3,..., etc., but after 1,2,3,....
+Also
+1,2,4,... comes after the two-term sequence 1,2 (because blanks precede numbers).
+More formally, we compare the two sequences term-by-term, and in the first position
+where they differ whichever is smaller (or blank) is the lexicographically earlier sequence.
+For sequences with negative terms, we ignore the signs and sort according to the
+absolute values.
+Here is the actual ordering used in the OEIS. The sequences are arranged (virtually)
+into a version of lexicographic order, according to the following rules. First, delete all
+minus signs. Then find the first term that is greater than 1, and discard all the terms
+before it. What’s left determines its position in the lexicographic order. For example,
+to place −1,0,1,1,2,1,17,3,2,1,... in the ordering, we would ignore the terms before
+the underlined 2, and consider the sequence as beginning 2,1,17,3,2,1,....
+Sequences that contain only 0s, 1s and −1s are sorted into lexicographic order by
+absolute value and appear at the beginning of the ordering. The first sequence in the
+database is therefore the zero sequence A000004.
+In this way every sequence has a unique position in the ordering. The sequences
+have been sorted in this way since the 1960s. For the first ten years the punched card
+entries were physically sorted into this order.
+When you look at an OEIS entry, A005132 say (the subject of Section 4.1), towards
+the bottom you will see two lines like5
+Sequence in context: A277558 A350578 A335299 * A064388 A064387 A064389
+Adjacent sequences: A005129 A005130 A005131 * A005133 A005134 A005135
+which tell you the three entries immediately before and after that entry in the lexico-
+graphic ordering, and the three entries before and after it in the A-numbering. The
+asterisks represent the sequence you are looking at. The first group can be useful if you
+are uncertain about a term in your sequence, the second in case you want to look at
+other sequences submitted around that time.
+Today the sequences are actually stored internally in an SQLite database. However,
+4The sequences in the 1973 and 1995 books were numbered N0001, ..., and M0001, ..., respectively.
+5If you don’t see these, click on the A-number at the top of the entry.
+10
+
+the punched card format has been so useful that when you view a sequence, as in Fig. 4,
+it is still presented to you in something very like the old punched card format.
+3.5
+Summary: “A Handbook of Integer Sequences” today
+– Now The On-Line Encyclopedia of Integer Sequences or OEIS: https://oeis.org
+– Accurate information about 360000 sequences.
+– Definition, formulas, references, links, programs. View as list, table, graph, music!
+– Traffic: 1 million hits/day.
+– 30 new entries, 50 updates every day.
+– Often called one of best math sites on the Web. Fingerprint file for mathematics.
+– Street creds: 10000 citations.
+– A moderated Wiki, owned by OEIS Foundation, a 501(c)(3) public charity.
+– Uses: to see if your sequence is new, to find references, formulas, programs.
+– Catalan or Collatz? (Very easy or very hard?)
+– Source of fascinating research problems;6 low-hanging fruit from recent submis-
+sions.
+– Accessible (free, friendly).
+– Fun (1,2,4,6,3,9,12,8,10,5,15,...?). Interesting, educational. Escape.
+– Addictive (better than video games).
+– Has led many people into mathematics.
+– One of the most successful international collaborations, a modest contribution
+towards world peace.
+– Need editors.
+4
+Some favorite sequences
+I’m sometimes asked what my favorite sequence is. This is a difficult question. I’m
+tempted to reply by saying: If you were the keeper of the only zoo in the world, how
+would you answer that question? (Because that is roughly the situation I’m in.) Would
+you pick one of the exotic animals, a giraffe, a kangaroo, or a blue whale? Or one of the
+essential animals, like a horse, a cow, or a duck? If the question came from a visiting
+alien, of course, there is only one possible answer: a human being.
+6Look for “Conjecture”, “It appears that”, “It would be nice to”, ...
+11
+
+For sequences, the essential ones are the primes, the powers of 2, the Catalan num-
+bers, or (especially if the question came from an alien with no fingers or toes), the
+counting sequence 0, 1, 2, 3, 4, ... (A001477).
+But here I’ll mention a few that are fairly exotic. The Recam´an and Gijswijt se-
+quences have simple recursive definitions, yet are astonishingly hard to understand.
+4.1
+Recam´an’s sequence (A005132)
+This remarkable sequence has resisted analysis for over 30 years, even though we have
+computed an astronomical number of terms. It was contributed to the database by
+Bernardo Recam´an Santos in 1991.
+The definition is deceptively simple. The first term is 0. We now add or subtract 1,
+then we add or subtract 2, then add or subtract 3, and so on. The rule is that we always
+first try to subtract, but we can only subtract if that leaves a nonnegative number that
+is not yet in the sequence. Otherwise we must add.
+Here is how the sequence starts. We have the initial 0. We can’t subtract 1, because
+that would give a negative number, so we add 1 to 0. So the second term is 1. We can’t
+subtract 2 from 1, so we add it, getting the third term 1+2 = 3. Again we can’t subtract
+3, for that would give 0, which has already appeared, so we add 3, getting the fourth
+tern 3 + 3 = 6.
+Now we must add or subtract 4, and this time we can subtract, because 6−4 = 2, and
+2 is nonnegative and a number that hasn’t yet appeared. So at this point the sequence
+is 0,1,3,6,2. Then it continues 7(= 2 + 5), 13(= 7 + 6),20(= 13 + 7),12(= 20 − 8), and so
+on. The first 16 terms are
+0,1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,...
+When adding rather than subtracting, repeated terms are permitted (42 is repeated
+at the 24th term).
+Edmund Harriss has found an elegant was to draw the sequence as a spiral on the
+number line. Start at 0, and when we subtract n, draw a semicircle of diameter n to
+the left from the last point, or to the right if we are adding n. Draw the semicircles
+alternately below and above the horizontal axis so as to produce a smooth spiral.
+The main question about this sequence is: Does every positive number appear?
+What makes this sequence so interesting is that certain numbers (for reasons we do not
+understand) are extremely reluctant to appear. 4 does not appear until 131 steps, and
+19 takes 95734 steps.
+A group of us at AT&T Bell Labs worked on this sequence in 2001, and developed a
+way to greatly speed up the computation. Allan Wilks used it to compute the first 1015
+terms, and found that 2406 (which had been missing for a long time) finally appeared
+at step 394178473633984.
+At this point the smallest missing number was 852655 = 5⋅31⋅5501. Benjamin Chaffin
+has continued this work, and in 2018 reached 10230 terms. However, 852655 was still
+12
+
+Figure 5: Harriss’s drawing of the first 64 terms of Recam´an’s sequence. (The tiny
+initial semicircle, at the extreme left, is below the axis. It has diameter 1 and joins the
+points 0 and 1. It continues as a semicircle of diameter 2, above the axis, joining the
+points 1 and 3.)
+missing, and there has been no progress since then.
+Thirty years ago I thought that every number would eventually appear.
+Now I
+am not so sure. My current belief is that there are two possibilities: either there are
+infinitely many numbers that never appear, and 852655 just happens to be the smallest
+of them, but has no other special property. A similar phenomenon appears to occur
+when iterating various number-theoretic functions—see the next section.
+Or, every
+number will eventually appear (just as presumably every one of Shakespeare’s plays will
+eventually appear in the expansion of π in base 60), although we may never be able
+to extend the sequence far enough to hit 852655. For the latest information about this
+sequence (or any other sequence mentioned in this article), consult the OEIS.
+Open question: Does 852655 appear in A005132?
+4.2
+Iteration of number-theoretic functions
+Many mysterious sequences arise from the iteration of number-theoretic functions. A
+classic problem concerns the iteration of the function f(n) = σ(n) − n, the sum of the
+“aliquot parts” of n (see Guy [8, §B6]). For an initial value of n, what happens to the
+trajectory n,f(n),f(f(n)),...? All n < 276 terminate by entering a cycle (such n are
+called “perfect”, “amicable”, or “sociable” numbers), or reaching a prime, then 1, then
+0. But it appears likely that n = 276 and perhaps infinitely many even numbers, will
+never terminate. The trajectory of 276 is A008892, and the b-file gives the first 2140
+13
+
+terms, term 2140 being a 213-digit number. The continuation of the trajectory has
+stalled at term 2051, where a 202-digit number is waiting to be factored. A098007 gives
+the number of distinct terms in the trajectory of n, or −1 if the trajectory is unbounded.
+The value of A098007(276) is unknown.
+If indeed 276 does go to infinity, it is natural to ask, how did 276 know it was destined
+to be the first immortal number under the map f? The answer may be that there are
+infinitely many immortal numbers, and 276 just happens to be the first. It got lucky,
+that’s all! Just as 852655 got lucky in Recam´an’s problem.
+A similar question, also discussed by Guy [8, §B41], which has received much less
+attention, concerns the map g(n) = (σ(n) + φ(n))/2, where φ(n) is the Euler totient
+function A000010. The trajectory may end at 1, a prime, or a fraction, or it may increase
+monotonically to infinity. Sequence A292108 gives the number of steps in the trajectory,
+or −1 if the trajectory is infinite. All numbers n < 270 have finite trajectories, but it
+appears that 270 goes increases forever. The trajectory of 270 is A291789. Andrew
+Booker has given a heuristic argument showing that almost all numbers go to infinity.
+What makes 270 the first immortal number under g? Again I suspect it just got lucky!
+Open questions: Does the trajectory of 276 under f increase forever? What about
+the trajectory of 270 under g?
+4.3
+Gijswijt’s sequence (A090822)
+For this sequence it will be helpful to remember that chemists do not write H − H − O,
+they write H2O, they do not write AlAlAlSOOOOSOOOO, they write Al3(SO4)2.
+We will apply a similar compression to sequences of numbers, except that we indicate
+repetition by superscripts rather than by subscripts.
+For this problem, when we look at a sequence of numbers, we want to write it in the
+form XY Y ...Y , or XY k, where X and Y are themselves sequences of numbers, X can
+be missing, and the exponent k is as large as possible.
+For example, we can write 1,2,2,2,2 as XY k, where X = 1, Y = 2, and k = 4. The
+highest k we can achieve for a sequence is called its “curling number”. So 1,2,2,2,2
+has curling number 4. Think of an animal with its head looking to the left, with a very
+curly tail. X represents the head and body of the animal, and Y k represents the curls
+in its tail.
+Consider the sequence 3,2,4,4,2,4,4,2,4,4. We could take X = 3,2,4,4,2,4,4,2 and
+Y = 4, getting XY 2, with k = 2, or we could take X = 3, Y = 2,4,4, getting XY 3, with
+k = 3, which is larger. So this sequence has curling number 3.
+Remember that X may be missing. So the sequence with a single term 99, say, can
+be written as Y 1 where Y is the number 99, and it has curling number 1. (The notion
+of curling number is independent of the base in which the numbers are written.)
+We are now ready to define Dion Gijswijt’s absolutely brilliant sequence, which he
+sent to the OEIS in 2004.
+14
+
+The rule for finding the next term is simple: it is the curling number of the sequence
+so far. And you start with 1. That’s the sequence!
+So let’s construct it. We start with 1, and the curling number of 1 is 1. So now we
+have 1,1. This has curling number 2, so now we have 1,1,2. At each step we recompute
+the curling number, and make that the next term.
+Here are the first few generations.
+1
+1 1
+1 1 2
+1 1 2 1
+1 1 2 1 1
+1 1 2 1 1 2
+1 1 2 1 1 2 2 (we took Y = 1 1 2)
+1 1 2 1 1 2 2 2
+1 1 2 1 1 2 2 2 3
+and we have found the first 3, at the 9th term. After a while, a 4 appears at term 220.
+But Gijswijt was unable to find a 5, and left that question open when he submitted
+the sequence. Some Bell Labs colleagues computed many millions of terms, but no 5
+appeared.
+Finally, over the course of a long weekend, Fokko van der Bult (a fellow student of
+Gijswijt’s in Amsterdam) and I independently showed that there is a 5. In fact there
+are infinitely many 5’s, but the first one does not appear until about term 101023. The
+universe would be cold long before any computer search would find it.
+In the paper we wrote about the sequence [4], we also conjectured that the first time
+a number N > 4 appears is at about term
+2 ↑ (2 ↑ (3 ↑ (4 ↑ (5 ↑ ... ↑ (N − 1))))),
+where the up-arrows (↑) indicate exponentiation. This is a tower of exponents of height
+N − 1.
+A very recent manuscript by a student of Gijswijt’s, Levi van de Pol, still under
+review, has extended our work, and may have proved the above conjecture.
+I cannot resist adding a further comment about curling numbers, which if true shows
+that the Gijswijt sequence is in a sense universal.
+The Curling Number Conjecture asserts that if any finite starting sequence is ex-
+tended by the rule that the next term is the curling number of the sequence so far, then
+eventually the curling number will be 1.
+If true, this implies that if the starting sequence contains no 1s, then the sequence
+eventually becomes Gijswijt’s sequence [5, Th. 23]. In fact I conjecture that this is true
+for any starting sequence.
+Open question: Is the Curling Number Conjecture true?
+15
+
+4.4
+Lexicographically Earliest Sequences
+Although there is no space to discuss them in detail, let me just mention that there
+are many fascinating and difficult sequences in the OEIS whose definition has the form
+“Lexicographically Earliest Sequence of distinct positive numbers with the property that
+...”, where now we are using lexicographic in its pure sense, as defined in Section 3.4.
+A favorite example is the EKG (or ECG) sequence A064413, whose definition is the
+lexicographically earlier infinite sequence of distinct positive numbers with the property
+that each term after the first has a nontrivial common factor with the previous term [9].
+Other L.E.S. examples are the Yellowstone permutation A098550 [2], the Enots Wolley
+sequence A336957 (the name suggests the definition), and the Binary Two-Up sequence
+A354169 [6].
+Open question: Show that the terms of the Enots Wolley sequence are precisely 1,
+2, and all numbers with at least two distinct prime factors.
+4.5
+The Stepping Stones Problem (A337663)
+This lovely problem was invented in 2020 by two undergraduates, Thomas Ladouceur
+and Jeremy Rebenstock. You have an infinite chessboard, and a handful of brown stones,
+which are worth one point each. You also have an infinite number of white stones, of
+values 2, 3, 4,..., one of each value. Suppose you have n brown stones. You start by
+placing them anywhere on the board. Now you place the white stones, trying to place
+as many as you can. The rules are that you can only place a white stone labeled k on
+a square if the values of the stones on the eight squares around it add up to k. And
+you must place the white stones in order, first 2, then 3, and so on. You stop when
+you cannot place the next higher-numbered white stone. The goal is to maximize the
+highest value that you place. Call this a(n).
+9
+5
+10
+11
+4
+1
+12
+8
+3
+2
+16∗
+6
+1
+15
+13
+7
+14
+Figure 6: A solution to the Stepping Stones problem for two starting stones. The high
+point a(2) = 16 here is indicated by an asterisk, as it is in the next three tables.
+Say we start with n = 2 brown stones. There are infinitely many squares where they
+can be placed, but it turns out that the best thing is to place them so they are separated
+diagonally by a single blank square, as in Fig. 6. Now we start trying to place the white
+16
+
+stones. The 2 stone has to go between the two brown (or 1) stones, and then the 3 goes
+on a square adjacent to the 1 and the 2. There is now a choice for where the 4 goes, but
+the choice shown in Fig. 6 is the best. (After we have placed the 4, the neighbors of the
+3 no longer add up to 3, but that is OK. It is only when we place the 3 that its neighbors
+must add to 3.) Continuing in this way, we eventually reach 16. There is nowhere to
+place the 17, so we stop. Ladouceur and Rebenstock showed, using a computer and
+considering all possible arrangements, that 16 is the highest value that can be attained
+with two starting stones. So a(2) = 16.
+This is clearly a hard problem, since the number of possibilities grows rapidly with
+the number of brown stones. Only six terms of this sequence are known: a(1) through
+a(6) are 1,16,28,38,49,60. A solution for n = 4 found by Arnauld Chevallier is shown in
+Fig. 7. There are lower bounds for larger values of n which may turn out to be optimal.
+For n = 7,...,10 the current best constructions give 71,80,90,99. See A337663 for the
+latest information.
+35
+18
+36
+23
+21
+32
+17
+1
+14
+9
+12
+20
+34
+16
+15
+5
+4
+8
+26
+27
+31
+10
+1
+3
+19
+25
+1
+28
+11
+2
+6
+33
+29
+24
+13
+22
+1
+7
+37
+30
+38∗
+Figure 7: A solution to the Stepping Stones problem for four starting stones.
+We don’t know how fast a(n) grows. There have been a series of upper and lower
+bounds, initiated by Robert Gerbicz and Andrew Howroyd. The simple linear construc-
+tion shown in Fig. 8 shows that a(n) ≥ 6(n − 1) for n ≥ 3.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+1
+1
+1
+10
+18∗
+17
+16
+15
+14
+13
+12
+11
+Figure 8: Every additional 1 on the middle row increases the number of white stones
+by 6, showing that a(n) ≥ 6(n − 1) for n ≥ 3.
+17
+
+By combining the constructions of Figs. 6 and 8, Menno Verhoeven obtained a(n) ≥
+6n + 3 for n ≥ 3 (Table 9).
+25
+24
+1
+26
+23
+27
+22
+28
+21
+1
+29
+20
+30
+19
+31
+9
+5
+10
+11
+18
+1
+32
+4
+1
+17
+33∗
+12
+8
+3
+2
+16
+6
+1
+15
+13
+7
+14
+Figure 9: Combining the constructions of of Figs. 6 and 8 gives a(n) ≥ 6n + 3 for n ≥ 3.
+The case n = 5 is shown. For other values of n, adjust the height of the “chimney” on
+the right.
+The best lower bound for large n is due to Robert Gerbicz, who has shown by a
+remarkable extension of the construction in Figs. 8 and 9 that limn→∞ a(n)/n > 6. (A
+preliminary version of his bound gives a(n) > 6.0128n−5621 for all n, although the exact
+values of the constants have not been confirmed.) In his construction the “chimney” on
+the right of Fig. 9 gets expanded into a whole trellis.
+One might think that with a sufficiently clever arrangement, perhaps extending the
+construction in Fig. 8 so that the path wraps around itself in a spiral, one could achieve
+large numbers with only a few starting stones. But a simple counting argument due
+to Robert Gerbicz shows this is impossible. The current best upper bound is due to
+Jonathan F. Waldmann, who has shown that a(n) < 79n + C for some constant C. See
+A337663 for the latest information, including proofs of of the results mentioned here.
+Open question: Improve the upper and lower bounds on a(n).
+4.6
+Stained glass windows
+In 1998 Poonen and Rubinstein [12] famously determined the numbers of vertices and
+cells in the planar graph formed from a regular n-gon by joining every pair of vertices
+by a chord. The answers are in A006561 and A007678. Lars Blomberg, Scott Shannon,
+and I have studied versions of this question when the regular n-gon is replaced by other
+18
+
+Figure 10: A 4×2 grid of squares with every pair of boundary points joined by a chord.
+The graph has 213 vertices and 296 cells.
+The cells are color-coded to distinguish
+triangles (red), quadrilaterals (yellow), and pentagons (blue).
+polygons, for instance by a square in which n equally-spaced points are placed along each
+side and each pair of boundary points is joined by a chord. We also studied rectangles,
+triangles, etc. In most cases we were unable to find formulas for the numbers of vertices
+or cells, but we collected a lot of data, and the graphs, when colored, often resemble
+stained glass windows (see [3] and the illustrations in A331452 and other sequences
+cross-referenced there).7 So we consoled ourselves with the motto: if we can’t solve it,
+make art!
+The most promising case to analyze seemed to be the n × 2 grid (although we did
+not succeed even there).
+Open question: How many vertices and cells are there in the graph for the n×2 grid,
+7There is no fee for downloading images in the OEIS, but if you use any of them, please credit the
+source!
+19
+
+as illustrated for n = 4 in Fig. 10? Sequences A331763 and A331766 give the first 100
+terms, yet even with all that data we have not found a formula.
+The case of an n × n grid seems even harder.
+Figure 11 shows the 6 × 6 graph.
+Sequences A331449 and A255011 give the numbers of vertices and cells for n ≤ 42.
+A334699 enumerates the cells by number of sides.
+Figure 11: A 6×6 grid with every pair of boundary points joined by a chord. There are
+4825 vertices and 6264 cells.
+In the summer of 2022 Scott Shannon and I considered several other families of planar
+graphs. I cannot resist showing one of Shannon’s graphs, a 16 × 16 grid, illustrating the
+16th term of A355798 (Fig. 12). There are 61408 cells. Although Shannon has calculated
+40 terms of this sequence, again no formula is known.
+20
+
+Figure 12: Scott Shannon’s “Magic Carpet” graph, illustrating A355798(16).
+4.7
+If I had more space
+If I had had more space I would also have discussed some very interesting sequences
+arising from:
+– Dissecting a square to get a regular n-gon (A110312).
+– Gerrymandering (A341578, A348453, and many others).
+– In how many ways can circles overlap? (A250001).
+– The Inventory sequence A342585.
+– Kaprekar’s junction numbers (A006064, [1]).
+– The kissing number problem (A001116, A257479).
+– The neural network problem that started it all (A000435).
+– Squares in the plane (A051602).
+And, maybe, meta-sequences such as A051070 (a(n) is the nth term of An) and
+A107357 (the nth term is 1 + the nth term of An).
+21
+
+A final comment: there are many videos on the Internet of talks I have given about
+sequences. There are over twenty videos that Brady Haran and I have made that have
+appeared on the Youtube Numberphile channel (and have been viewed over eight million
+times). See for example “Terrific Toothpick Patterns”.
+5
+Acknowledgments
+I would like to thank some good friends who have helped me and the OEIS over the
+years: David L. Applegate, William Cheswick, Russ Cox, Susanna S. Cuyler, Harvey
+P. Dale, Ronald L. Graham, Richard K. Guy, Marc LeBrun, John Riordan, and Doron
+Zeilberger.
+There are many active volunteer editors, and it is impossible to thank them all.
+But I would like to give particular thanks to J¨org Arndt, Michael S. Branicky, Michael
+De Vlieger, Amiram Eldar, Charles R. Greathouse IV, Maximilian F. Hasler, Alois P.
+Heinz, Andrew Howroyd, Sean A. Irvine, Michel Marcus, Richard J. Mathar, Peter
+Munn, Hugo Pfoertner, Kevin Ryde, Jon E. Schoenfield, R´emy Sigrist, and Chaiwah
+Wu.
+I also thank the members of the Board of Trustees of the OEIS Foundation, past
+and present, for all their help, both to me personally and to the OEIS.
+Figure credits: Figure 1(c): Clifford A. Pickover. Figure 5: Edmund Harriss. Fig-
+ures 6, 7, 8, and 9 are based on communications from Thomas Ladouceur and Jeremy
+Rebenstock, Arnauld Chevallier, Skylark Xentha Murphy-Davies, and Menno Verho-
+even, respectively. Figures 10 and 11: Lars Blomberg and Scott Shannon. Figure 12:
+Scott R. Shannon. Other figures: the author.
+References
+[1] M. A. Alekseyev and N. J. A. Sloane, On Kaprekar’s Junction Numbers, J. Com-
+binat. and Number Theory, 2023, in press.
+[2] D. L. Applegate, H. Havermann, B. Selcoe, V. Shevelev, N. J. A. Sloane, and
+R. Zumkeller, The Yellowstone permutation, J. Integer Seqs., 18 (2015), #15.6.7.
+[3] L. Blomberg, S. R. Shannon, and N. J. A. Sloane, Graphical enumeration and
+stained glass windows, 1: Rectangular grids, Integers, Ron Graham Memorial Vol-
+ume 21A (2021), #A5.
+[4] F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane, and Allan
+Wilks, A slow-growing sequence defined by an unusual recurrence, J. Integer Seqs.,
+10 (2007), #07.1.2.
+[5] B. Chaffin, J. P. Linderman, N. J. A. Sloane and A. R. Wilks, On curling numbers
+of integer sequences, J. Integer Seqs., 16 (2013), #13.4.3.
+22
+
+[6] M. De Vlieger, T. Scheuerle, R. Sigrist, N. J. A. Sloane, and W. Trump, The binary
+Two-Up sequence, arXiv:2209.04108, Sep. 11 2022.
+[7] H. W. Gould, Combinatorial Identities, Morgantown, WV, 1972.
+[8] R. K. Guy, Unsolved Problems in Number Theory, 3rd. ed., Springer, 2010.
+[9] J. C. Lagarias, E. M. Rains, and N. J. A. Sloane, The EKG sequence, Experimental
+Math., 11 (2002), 437–446.
+[10] D. S. Mitrinovi´c, J. S´andor and B. Crstici, Handbook of Number Theory, Kluwer,
+Dordrecht, 1996.
+[11] The OEIS Foundation Inc. (2023), The On-Line Encyclopedia of Integer Sequences,
+https://oeis.org.
+[12] B. Poonen and M. Rubinstein, The number of intersection points made by the
+diagonals of a regular polygon, SIAM J. Discrete Mathematics, 11.1 (1998) 135–
+156.
+[13] J. Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J.
+Austral. Math. Soc., 10 (1969), 278–282.
+2020 Mathematics Subject Classification: 05-00, 11-00, 11Bxx, 48-00, 68-00
+23
+

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+Experimental characterization of an ultra-broadband dual-mode 
+symmetric Y–junction based on metamaterial waveguides 
+Raquel Fernández de Caboa,*, Jaime Vilasa,b, Pavel Chebenc, Aitor V. Velascoa, David González-
+Andraded 
+a Instituto de Óptica Daza de Valdés, Consejo Superior de Investigaciones Científicas (CSIC), 121 Serrano, Madrid 28006, Spain 
+b Alcyon Photonics S.L., 11 Génova, Madrid 28004, Spain 
+c National Research Council Canada, 1200 Montreal Road, Bldg. M50, Ottawa K1A 0R6, Canada 
+d Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France 
+* Corresponding author: [email protected] 
+ARTICLE INFO 
+ 
+Keywords:  
+Silicon photonics 
+Subwavelength grating 
+metamaterial 
+Power splitter  
+Y-junction 
+Ultra-broadband 
+Fabrication tolerant 
+ 
+ 
+ 
+ 
+ 
+ 
+ABSTRACT
+ 
+Silicon photonic integrated circuits routinely require 3-dB optical power dividers with 
+minimal losses, small footprints, ultra-wide bandwidths, and relaxed manufacturing 
+tolerances to distribute light across the chip and as a key building block to form more 
+complex devices. Symmetric Y–junctions stand out among other power splitting 
+devices owing to their wavelength-independent response and a straightforward design. 
+Yet, the limited resolution of current fabrication methods results in a minimum feature 
+size (MFS) at the tip between the two Y–junction arms that leads to significant losses 
+for the fundamental mode. Here we propose to circumvent this limitation by leveraging 
+subwavelength metamaterials in a new type of ultra-broadband and fabrication-
+tolerant Y–junction. An exhaustive experimental study over a 260 nm bandwidth 
+(1420–1680 nm) shows excess loss below 0.3 dB for the fundamental transverse-
+electric mode (TE0) for a high-resolution lithographic process (MFS ~ 50 nm) and less 
+than 0.5 dB for a fabrication resolution of 100 nm. Subwavelength Y–junctions with 
+deterministically induced errors of ±10 nm further demonstrated robust fabrication 
+tolerances. Moreover, the splitter exhibits excess loss lower than 1 dB for the first-
+order transverse-electric mode (TE1) within a 100 nm bandwidth (1475–1575 nm), 
+using high-resolution lithography.  
+1. Introduction 
+Photonic integrated circuits (PICs) built on the silicon-
+on-insulator (SOI) platform benefit from high modal 
+confinement, small footprints, energy efficiency and 
+large-scale production, whilst driving-down costs thanks 
+to the compatibility with complementary metal-oxide-
+semiconductor (CMOS) fabrication processes [1–3]. 
+These compelling advantages substantially extend the 
+scope of photonic integration beyond telecom and 
+datacom to emerging applications with a far-reaching 
+impact. These include 5G mobile communications [4], 
+the Internet of Things [5], quantum photonics [6], light 
+detection and ranging [7], spectrometry [8,9] and 
+biochemical sensing [10], also enabling lab-on-a-chip 
+solutions [11]. 
+The complexity leap of the aforementioned 
+applications requires an increasing number of on-chip 
+components that take advantage of either multimode or 
+broadband operations. Specifically, 3-dB optical power 
+splitters are key components extensively used in light 
+distribution or as building blocks for more intricate 
+arrangements, including optical switches, multiplexers 
+or integrated Mach-Zehnder interferometers [12,13]. 
+Sequentially concatenated 3-dB power splitters are often 
+utilized to implement 1×N dividers [14,15], requiring 
+compact and low-loss designs. For datacom applications 
+such as mode-division multiplexing [16] or multitarget 
+sensing [17], power splitters with broad bandwidths are 
+required. Different power division architectures have 
+been reported based, among others, on symmetric Y–
+junctions [18,19], multimode interference (MMI) 
+couplers [20–22], inverse tapers [23], adiabatic tapers 
+[24], directional and adiabatic couplers [25–27], slot 
+waveguides [28] and photonic crystal structures [29–31]. 
+MMI devices offer good fabrication tolerances and 
+compact footprints, and their operational bandwidth can 
+be optimized through geometry design [20,21] or 
+partially shallowly etched regions [22]. Inverse tapers 
+[23] provide efficient mode evolution but typically 
+present narrow bandwidths. Adiabatic tapers [24] exhibit 
+wider bandwidths, but the performance of transverse-
+electric (TE) polarized light has larger fluctuations across 
+the operation bandwidth due to the higher sensitivity to 
+sidewall roughness. Conversely, the bandwidth of 
+conventional directional couplers is restricted due to the 
+strong wavelength-dependence of the evanescent 
+coupling. Asymmetric [25] and bent [26] directional 
+
+couplers have been demonstrated with reduced 
+wavelength dependence, but it is not sufficient for ultra-
+broadband optical systems. Additional power division 
+architectures with low losses over extensive operational 
+bandwidth also include adiabatic couplers [27], slot 
+waveguides [28] photonic crystal power splitters [29–
+31], inverse design methods [32,33], pixelated meta-
+structures [34], and deep neural networks [35].  
+Amongst these power splitters, symmetric Y–
+junctions are one of the most common alternatives given 
+their 
+polarization- 
+and 
+wavelength-independent 
+response, and their straightforward design. Symmetric 
+Y–junctions consist of a stem waveguide ramifying into 
+two arms of the same width. However, these structures 
+typically present significant loss for fundamental modes 
+at the junction, especially as the tilt angle between the 
+two branching arms increases [36]. Two basic 
+mechanisms are responsible for this loss: the wavefront 
+mismatch due to the abrupt tilt angle, and the 
+transformation of the mode profile in the interface 
+between the stem and the arms [37]. In order to reduce 
+the effect of the tilt angle, s-bend shaped waveguides can 
+be employed for the arms [38], whereas the junction 
+region can be tapered to ensure an approximately 
+adiabatic mode evolution. Despite these design 
+optimizations, minimum feature size (MFS) limitations 
+of fabrication technologies lead to an imperfect tip 
+between the two Y–junction arms. This MFS constraint 
+causes considerable losses on the fundamental mode as 
+the maximum of its power profile coincides with the 
+junction tip [39]. Conversely, this same effect is 
+negligible for the first-order mode, which presents a 
+minimum of its power profile at the waveguide center. 
+Several approaches to optimized Y–junction designs 
+have been proposed to circumvent the effect of the MFS 
+at the tip, including tapered and slotted waveguides 
+[19,27,28], core size optimization [40] or particle swarm 
+optimization algorithms [18]. Nevertheless, ultra-
+broadband, low-loss and fabrication-tolerant solutions 
+are still sought after. 
+Subwavelength gratings (SWG) provide a powerful 
+tool for improving the performance of a wide range of 
+photonic devices [41–45]. SWGs are based on periodic 
+grating waveguides with a period (Λ) significantly 
+smaller than the operating wavelength (λ), hence 
+behaving as a homogenous metamaterial (i.e. a medium 
+with tailored optical properties). This behavior inhibits 
+diffraction and enables refractive index and dispersion 
+engineering [44]. Subwavelength engineering has 
+become a strong design method for the realization of 
+integrated silicon photonics components, including  
+fiber-chip couplers [41], reconfigurable filters [46] and 
+gradient-index lenses [47]. SWG metamaterials have 
+also been successfully applied to several power splitters 
+such as asymmetric directional couplers [48], three-guide 
+directional couplers [49], inverse tapers [45,50], slot 
+adiabatic waveguides [51], MMI devices [52–54], 
+waveguide crossings [55] and Y–junctions [56]. 
+Recently, we proposed an architecture for a high-
+performance and fabrication-tolerant SWG Y–junction 
+[57]. However, this previous study only covered 
+simulation results and preliminary measurements for 
+TE0. 
+In this paper, we present a comprehensive 
+experimental study of a 
+dual-mode Y–junction 
+engineered with subwavelength metamaterials for 
+deconfinement of the fundamental mode near the 
+junction tip and mitigating losses. Specifically, we 
+greatly expand our original study [57] by including 
+accurate TE0 mode measurements through cascaded 
+splitters and TE1 mode measurements through auxiliary 
+mode multiplexers, for both SWG and conventional Y-
+junctions. We also measure biased devices in order to 
+study the fabrication tolerances of our device considering 
+both 50 nm and 100 nm MFSs. Our power splitter yields 
+a measured fundamental transverse-electric mode (TE0) 
+excess loss (EL) of less than 0.3 dB considering a high-
+resolution fabrication process with MFS of 50 nm and 
+below 0.5 dB for a resolution scenario with MFS of 100 
+nm within the 1420–1680 nm wavelength range. 
+Moreover, the splitter exhibits excess loss lower than 1 
+dB for the first-order transverse-electric mode (TE1) 
+within a 100 nm bandwidth (1475–1575 nm), in the high-
+resolution process. 
+2. Device design 
+Our proposed SWG Y–junction, presented in Fig. 1(a), 
+operates according to similar principles as a conventional 
+symmetric Y–junction, illustrated in Fig. 2(a). The 
+structure of the SWG and conventional Y–junctions 
+comprise two single-mode output arms and a multimode 
+input stem waveguide that supports the first two TE 
+modes. When TE0 is injected into the stem waveguide, 
+its power is equally divided into two in-phase TE0 modes, 
+one in each output arm. When the stem waveguide is 
+excited with TE1, two TE0 modes of equal power but with 
+a relative phase difference of π are generated in the two 
+output arms [39]. Therefore, notice that the Y-junction 
+operates simultaneously as a splitter and a mode 
+converter for TE1 mode. In the conventional symmetric 
+Y–junction, MFS limitations at the junction tip results in 
+significant TE0 loss penalty. By applying SWG 
+engineering, modal confinement near the junction tip is 
+reduced and the TE0 mode transition at the stem-arm 
+interface is smoothed, significantly reducing losses. 
+Both splitters were designed for a 220-nm-thick Si 
+core. Our SWG Y–junction includes an input strip 
+waveguide of width Ws = 1.2 m, optimized to avoid a 
+weak confinement of the Bloch–Floquet TE1 mode that 
+would lead to high TE1 excess losses (ELTE1) due to 
+substrate leakage or mode radiation. Using an adiabatic 
+input taper of length Lti = 10 m, the strip waveguide is 
+progressively adapted to a subwavelength stem 
+waveguide of the same width and a length Lc = 13 m. 
+The SWG stem splits into two SWG single-mode arms 
+of width W = 500 nm and length Lb = 12.3 m. Each 
+SWG arm is shaped as an s-bend to avoid abrupt tilt 
+angles at the junction tip. After the SWG s-bends, each 
+arm waveguide transforms into a strip output waveguide 
+by means of an output taper of length Lto = 6 m. The 
+
+final separation between arms (Ha) is 1.5 m. An initial 
+arm offset (Hoff) is included to account for two alternative 
+MFS scenarios. We considered an MFS of 50 nm for 
+high-resolution fabrication processes and an MFS of 100 
+nm.  
+In both SWG arms and stem, Λ was set to 220 nm to 
+prevent Bragg reflection while maintaining SWG feature 
+sizes above the target MFS. In order to minimize mode 
+mismatch at the junction, we set different duty cycles in 
+the stem (DCs =as/Λ) and the arms (DCa=aa/Λ), where 
+as and aa are the lengths of the SWG silicon segments in 
+the stem and arms, respectively. For DCs = 0.5, the best 
+EL balance between TE0 and TE1 modes was achieved 
+for DCa = 0.6 when considering an MFS of 50 nm and 
+for DCa = 0.55 with MFS of 100 nm. The design 
+procedure followed is described in further detail in [57]. 
+Notice that fabrication tolerances and the effect of 
+temperature variations was also studied, showing 
+negligible performance degradation for ±10 K. Figures 
+1(b) and 1(c) show scanning electron microscope (SEM) 
+images of the fabricated SWG Y–junctions for MFS 50 
+nm and 100 nm, respectively.  
+In order to compare the performance of our SWG 
+device with a conventional splitter, we designed 
+conventional Y–junctions with different MFSs (Fig. 
+2(a)). These devices also comprise an input multimode 
+stem waveguide of width W0 = 1 m, which is adapted to 
+the width at the junction (Wt = 2W + Hoff) by means of a 
+taper of length Lt = 4 m. The two single-mode s-shaped 
+output arms maintain the same arm width (W), length (Lb) 
+and final separation (Ha) as in the SWG splitter. The 
+initial offset between the arms (Hoff) was also included to 
+consider the MFSs of 50 nm and 100 nm. Additionally, 
+we also included a third study-case with Hoff = 0 nm, 
+which ideally would result in a perfect tip. SEM images 
+of the fabricated conventional Y–junctions for Hoff of 0 
+nm, 50 nm and 100 nm can be seen in Figs. 2(b), 2(c) and 
+2(d), respectively. It can be observed that conventional 
+Y–junctions with Hoff = 0 nm and Hoff = 50 nm, as 
+fabricated, present comparable tip dimensions. This 
+leads to almost analogous performance for both devices, 
+as discussed in more detail in the following sections. 
+ 
+Fig. 2. Conventional Y–junction schematic (a), and SEM images of the tip for three resolutions:  ideal 0 nm (b), and realistic 50 nm (c) and 100 nm (d). 
+  
+ 
+Fig. 1. Subwavelength Y–junction schematic (a) and SEM images for devices with an MFS of 50 nm (b) and 100 nm (c). 
+ 
+
+TEi
+TE
++
+W
+(b)
+(c)
+(d)
+462nm
+468nm
+455nm
+947nm
+956nm
+42nm
+993nm
+nm
+200nm
+200nm
+200nmTEi
+TE0
+(b)
+C
+114nm
+134nm
+117nm
+127nm
+493nm
+486nm
+100nm
+200nm
+1157nm
+200nm
+L1163nm3. Fabrication and experimental characterization 
+The device was fabricated using electron-beam 
+lithography in a commercial foundry [58]. The SOI wafer 
+has a silicon layer thickness of 220 nm and a 2-µm-thick 
+buried oxide (BOX). The mask pattern was defined by 
+exposing the resist to a 100 keV electron-beam 
+lithography system, followed by an anisotropic reactive 
+ion etching process that transfers the pattern to the Si 
+layer. A SiO2 cladding with a thickness of 2.2 µm was 
+deposited by chemical vapor deposition. Finally, a deep 
+etch process was applied to smooth the chip facets, 
+allowing efficient fiber-chip edge coupling by using 
+high-efficiency broadband SWG edge couplers [41]. 
+Experimental characterization was carried out using 
+two tunable lasers to sweep the wavelength from 1420 
+nm to 1680 nm, coupled to a three-paddle fiber 
+polarization controller, a linear polarizer and a half-wave 
+plate. TE polarized light was coupled into the chip using 
+a lensed polarization-maintaining optical fiber. Light at 
+the chip output was collected by a 40× microscope 
+objective, directed to a Glan-Thompson polarizer, and 
+focused onto a germanium photodetector. 
+3.1 Fundamental transverse-electric mode (TE0) 
+In [57], 3D finite-difference time-domain simulations 
+were performed for the SWG Y-junction, which 
+provided excess loss for the TE0 (ELTE0) mode below 0.3 
+dB in a 350 nm bandwidth for the worst-case MFS 
+scenario of 100 nm. Given the challenge of measuring 
+losses of this order of magnitude in a stand-alone 
+configuration, we implemented cascaded structures with 
+multiple stages. That is, 1 to 4 stages of concatenated Y–
+junctions: 1×2, 1×4, 1×8 and 1×16 structures. SEM 
+image of a 1×16 structure and the different stages are 
+shown in Fig. 3(a), with a close-up view of the SWG Y–
+junction in Fig. 3(b). The 50 nm MFS is within the limit 
+of the fabrication resolution offered by the foundry and, 
+as it can be seen in the SEMs, the device was fabricated 
+correctly. Reference waveguides with the same length 
+and number of bends as the cascaded structures were also 
+included to determine Y–junction excess loss. 
+The measured excess loss for different cascaded Y–
+junctions is presented in Fig. 4. The ELTE0 of the SWG 
+splitter designed for MFS = 50 nm (𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50) is plotted in 
+Fig. 4(a) and for MFS = 100 nm (𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100) in Fig. 4(c). 
+The ELTE0 of the conventional Y–junction is shown in 
+Fig. 4(b) for MFS = 50 nm (𝐸𝐿𝑇𝐸0
+𝐶𝑜𝑛𝑣,50) and for MFS = 100 
+nm (𝐸𝐿𝑇𝐸0
+𝐶𝑜𝑛𝑣,100) in Fig. 4(d). Conventional Y–junctions 
+with Hoff = 0 nm were also measured but have not been 
+included in Fig. 4 for clarity, since they are very similar 
+to results with Hoff = 50 nm. This similarity is caused by 
+the MFS limitation of the fabrication process and verifies 
+our initial assumption on e-beam fabrication MFS. That 
+is, even when the nominal design includes a perfect tip 
+(Hoff = 0), experimental MFS limitations will induce an 
+imperfect tip, in this case, similar to the design of Hoff = 
+50 nm. Notwithstanding, results for this Hoff are included 
+in Figs. 5 and 8 for direct comparison. For both MFSs of 
+50 nm and 100 nm, the SWG splitter has a substantially 
+reduced ELTE0 compared to the conventional Y–junction.  
+Figure 4 shows the overall excess loss increment as 
+more cascaded stages are included. To obtain the ELTE0 
+relative to a single Y-junction, we performed a linear 
+regression with the measured losses in each cascaded 
+stage. Figure 5 shows the resulting ELTE0 per splitter, that 
+is, the average loss per cascaded splitter calculated 
+through the slope of the linear regression. The SWG Y–
+junction (solid line) exhibits a flat response in the entire 
+measured bandwidth from 1420 nm to 1680 nm, with an 
+𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50 lower than 0.3 dB and 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 below 0.5 dB in 
+the full 260 nm spectrum. By contrast, the conventional 
+Y–junction 
+(dotted 
+line) 
+shows 
+a 
+performance 
+degradation towards shorter wavelengths, resulting in a 
+loss penalty, especially for an MFS of 100 nm (see Fig. 
+5(b)). Considering the high-resolution fabrication (MFS 
+ 
+Fig. 4. Excess loss of the 1×2, 1×4, 1×8 and 1×16 cascaded structures, 
+for SWG Y–junction with 50 nm MFS (a), conventional Y–junction 
+with 50 nm MFS (b), SWG Y–junction with 100 nm MFS (c), and 
+conventional Y–junction with 100 nm MFS (d). 
+ 
+Fig. 3. SEM images of the SWG Y–junction in cascaded configuration 
+1×16 (a) and inset of the device (b). 
+
+SWG.MFS = 50
+over
+3
+1x2
+1x2
+2.5
+1x4
+2.5
+1x4
+1x8
+1x8
+2
+1x16
+2
+1x16
+1.5
+(aB)
+1.5
+n
+1
+EL
+1
+0.5
+0.5
+0
+0
+-0.5
+0.5
+1420
+1485
+1550
+1615
+1680
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+Wavelength (nm)
+6
+6
+1x2
+1x2
+5
+1x4
+1x4
+1x8
+1x8
+1x16
+1x16
+4
+(dB)
+dB)
+m
+73
+2
+0
+0
+1420
+1485
+1550
+1615
+1680
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+Wavelength(nm)(a)
+20μm
+4cascaded stages
+3cascaded stages
+2 cascaded stages
+istage
+(b)
+3μm= 50 nm), our device consistently outperforms the 
+conventional Y–junction, with 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50 < 0.23 dB in a 
+215 nm bandwidth (1420 nm – 1635 nm) as presented in 
+Fig. 5(a). As previously explained, it can be seen that 
+excess losses for conventional Y-junctions with an Hoff = 
+0 nm and Hoff = 50 nm are very similar, only differing in 
+a mean deviation of less than 0.06 dB. As the MFS 
+increases to 100 nm, the negative impact on the 
+performance of the conventional Y–junction is more 
+pronounced, with an 𝐸𝐿𝑇𝐸0
+𝐶𝑜𝑛𝑣,100 above 0.57 dB in the 1420 
+nm –1680 nm window. The SWG device, on the other 
+hand, yields improved performance over the entire 
+measured bandwidth with 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 below 0.46 dB.   
+Devices with deterministically induced dimension 
+variations (Δδ) of ±10 nm were incorporated in the mask 
+layout to measure the robustness of the SWG Y–junction 
+to fabrication errors. Figure 6(a) shows 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50, 
+demonstrating that performance is preserved despite the 
+presence of geometric variations, and even exhibiting a 
+slight improvement for over-etching deviations (i.e., Δδ 
+= -10 nm, SEM shown in Fig. 6(c)). In contrast, for the 
+MFS of 100 nm (Fig. 6(b)), 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 is slightly improved 
+for Δδ = +10 nm (SEM shown in Fig. 6(f)). For Δδ = -10 
+nm (SEM shown in Fig. 6(d)), 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100performance 
+degrades towards longer wavelengths. The largest 
+fabrication bias was observed in waveguide width, 
+narrowing the designed stem waveguide (Ws = 1200 ± 10 
+nm) by approximately 40 nm.  
+3.2 First-order transverse-electric mode (TE1) 
+In order to characterize the TE1 mode division, a mode 
+multiplexer [59] was included in combination with the 
+SWG Y–junction, as schematically shown in Fig. 7. 
+When TE0 is injected through the upper input port of the 
+mode multiplexer, the TE0 mode is generated at the 
+output multimode waveguide. When the lower input port 
+is excited with TE0, mode evolution results in TE1 at the 
+mode multiplexer output. Two mode multiplexers in 
+back-to-back were used as reference to extract TE1 mode 
+excess loss of the Y–junctions.  
+Figure 8(a) shows ELTE1 measurements for MFS = 50 
+nm in both SWG (𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50) and conventional (𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,50) 
+Y–junctions, as well as for the conventional Y–junction 
+with Hoff = 0 nm. Likewise, ELTE1 for MFS = 100 nm in 
+ 
+Fig. 6. Tolerances to fabrication errors of Δδ = ±10 nm for the SWG 
+Y–junction with MFS of 50 nm (a) and 100 nm (b). Excess loss per 
+splitter was measured through linear regression of cascaded stages. 
+SEM images of devices with Δδ = -10 nm for an MFS of 50 nm (c) 
+and 100 nm (d). SEM images of devices with Δδ = +10 nm for an MFS 
+of 50 nm (e) and 100 nm (f). 
+ 
+Fig. 7. Schematic of the structure employed for the characterization of TE1 mode (a) and SEM images of the mode multiplexer (b). 
+ 
+Fig. 5. EL per splitter measured through linear regression of the 
+response of four cascaded stages for SWG (solid line) and conventional 
+(dotted line) Y–junctions, for MFS of 50 nm (a) and 100 nm (b). Results 
+obtained for the conventional splitter with ideal resolution (MFS = 0 
+nm) are also shown in panel (a). 
+
+b)1.5
+1.5
+Nominal
+20=-10nm
+Nominal
+A=-10nm
+1
+40=+10nm
+1
+A0=+10nm
+【dB]
+0.5
+0.5
+0
+-0.5
+-0.5
+1420
+1485
+1550
+1615
+1680
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+Wavelength(nm)
+(C)
+103nm
+125nm
+483nm
+(d)
+105nm
+116nm
+-479nm
+56nm
+f107nm
+L1146nm
+200nm
+-1146nm
+200nm
+(e)
+123nm
+147nm
+507nm
+(f).124nm
+136m
+-497nm
+37nm
+f88nm
+L1168nm
+200nm
+1166nm
+200nmTEi
+TE
+(b)
+2μm
+15.16μm
+5.04μm
+0.50μm
+0.15μm7
+0.74μm
+0.15μmr
+0.25μm
+0.52μmConv MFS = O nm
+..
+Conv MFS = 50 nm
+SWG MFS =50 nm
+1
+(ap)
+0.5
+0
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)
+ConvMFS= 100 nm
+SWG MFS = 100 nm
+(ap)
+0.5
+0
+1420
+1485
+1550
+1615
+1680
+Wavelength(nm)SWG (𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,100) and conventional (𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,100) Y–
+junctions are depicted in Fig. 8(b). As previously 
+mentioned, 𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,50 and 𝐸𝐿𝑇𝐸1
+𝐶𝑜𝑛𝑣,100 are negligible due to 
+TE1 mode odd symmetry, with a power minimum at the 
+center of the multimode stem. Our device exhibits an 
+𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50 below 1 dB over a 100 nm bandwidth ranging 
+from 1475 to 1575 nm. Within a 170 nm bandwidth 
+(1420 – 1590 nm), 𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50 and 𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,100 only increase by 
+0.5 dB. Compared to the conventional Y–junction, the 
+degradation of the TE1 mode for our SWG Y–junction 
+arises from the selection of the stem waveguide width of 
+WS = 1200 nm, as a compromise between ELTE0 and 
+ELTE1. Note that increasing the width of the SWG stem 
+waveguide results in a stronger modal confinement that 
+prevents TE1 mode radiation, but increasingly penalizes 
+TE0 due to the resulting mode profile at the junction tip 
+[57]. Nevertheless, the performance of the proposed 
+device compares very favorably to state-of-the-art 
+higher-order mode power splitters (see Table 1).  
+SWG Y–junctions with Δδ = ±10 nm were also 
+included in combination with the mode multiplexer 
+structures to study fabrication tolerances for TE0 and TE1 
+mode division mux/demux and are shown in Figs. 9(a) 
+and (b). Figure 9(a) shows that 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,50 is lower than 0.5 
+dB for both under-etching and over-etching errors in the 
+full 1420 nm – 1680 nm bandwidth. Figure 9(b) shows 
+that 𝐸𝐿𝑇𝐸0
+𝑆𝑊𝐺,100 remain <1 dB for Δδ = -10 nm and <0.7 dB 
+for Δδ = +10 nm, for the same bandwidth. These results 
+corroborate the robustness of the SWG Y–junction to 
+fabrication errors for TE0 mode, shown in section 3.1 
+(see Figs. 6(a) and (b)). The performance for TE1 mode 
+division is more sensitive to fabrication errors in the 
+width of the SWG stem owing to the weaker confinement 
+of the Bloch-Floquet TE1 mode compared to the Bloch-
+Floquet TE0 mode. While under-etching results in 
+𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,50 < 2.1 dB and 𝐸𝐿𝑇𝐸1
+𝑆𝑊𝐺,100 < 1.7 dB for the full 
+bandwidth, over-etching errors are negligible at shorter 
+wavelengths and increase towards longer wavelengths. 
+4. Conclusions 
+The detailed experimental study conducted in this work 
+demonstrates the broadband performance and relaxed 
+fabrication tolerances of SWG-based Y–junctions. Two 
+resolutions were investigated to account for different 
+fabrication processes, namely MFS of 50 nm and 100 
+nm. Accurate measurements in cascaded splitters 
+demonstrate excess loss for the fundamental TE mode 
+lower than 0.3 dB for MFS = 50 nm and below 0.5 dB 
+for MFS = 100 nm, in a 260 nm bandwidth (1420 nm – 
+1680 nm). Characterization of the first-order TE mode 
+was performed in combination with a mode multiplexer, 
+showing excess loss lower than 1 dB over a 100 nm 
+ 
+Fig. 8. ELTE1 measurements for SWG (solid) and conventional (dotted) 
+Y–junctions for MFS of 50 nm (a) and 100 nm (b). Conventional Y–
+junction (dashed) with MFS = 0 nm is also shown in panel (a). 
+ 
+Fig. 9. Tolerances to fabrication errors of the SWG Y–junction for 
+TE0 (dashed line) and TE1 (solid line) polarizations, for MFS of 50 nm 
+(a) and 100 nm (b). 
+Table 1. Experimental performance comparison of state-of-the-art multimode power dividers. 
+Ref. 
+Design method 
+EL (dB) 
+Bandwidth (nm) 
+MFS (nm) 
+Length (µm) 
+Functionality 
+[33] 
+Inverse 
+design 
+subwavelength 
+axisymmetric 
+1.5 
+60 
+30 
+2.88 
+2-mode splitter 
+[34] 
+Pixelated meta-structure 
+1.5 
+40 
+30 
+4.5 
+3-mode splitter 
+[60] 
+MMI coupler 
+0.76 
+60 
+1000 
+86.5 
+2-mode splitter 
+[61] 
+Tapered directional coupler 
+0.7 
+30 
+200 
+25 
+2-mode splitter 
+[62] 
+Densely-packed waveguide array 
+1 
+28 
+110 
+46 
+2-mode splitter 
+This work 
+SWG Y–junction 
+1 
+100 
+50 
+41.3 
+2-mode splitter & converter 
+This work 
+SWG Y–junction 
+1.5 
+170 
+100 
+41.3 
+2-mode splitter & converter 
+ 
+
+a
+2.5
+-Conv MFS = O nm
+...
+Conv MFS = 50 nm
+2
+SWG MFS = 50 nm
+B
+1.5
+1
+0.5
+0
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength (nm)
+b)
+2.5
+Conv MFS = 100 nm
+SWG MFS = 100 nm
+2
+1.5
+1
+0.5
+0
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength(n-TE
+2.5
+Nominal
+48=-10nm
+2
+14§=+10nm
+(dB)
+1.5
+0.5
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength (nm)
+TE
+TEi
+2.5
+Nominal
+145=-10nm
+40=+10nm
+B
+d
+0.5
+0
+-0.5
+1420
+1470
+1520
+1570
+1620
+Wavelength(nm)bandwidth (1475 nm – 1575 nm) for the 50 nm MFS. 
+SWG Y–junctions with deterministically induced errors 
+of -10 nm and +10 nm were also measured to analyze 
+resilience 
+to 
+over- 
+and 
+under-etching 
+errors. 
+Experimental results demonstrate robust fabrication 
+tolerances for the fundamental TE mode. Our SWG-
+engineered metamaterial Y–junction opens up promising 
+prospects for improving performance of diverse silicon 
+photonic integrated circuits where power splitters are 
+ubiquitous component, such as on-chip high bandwidth 
+communication systems and broadband spectroscopic 
+systems. 
+Credit authorship contribution statement 
+Raquel Fernández de Cabo: Software, Formal 
+Analysis, Validation, Data curation, Writing – original 
+draft. Jaime Vilas: Experimental validation, Writing – 
+review & editing. Pavel Cheben: Conceptualization, 
+Writing – review & editing. Aitor V. Velasco: 
+Conceptualization, Resources, Funding acquisition, 
+Supervision, Writing – review & editing. David 
+González-Andrade: Methodology, Supervision, Project 
+administration, Writing – review & editing. 
+Declaration of Competing Interest 
+The authors declare that they have no known competing 
+financial interests or personal relationships that could 
+have appeared to influence the work reported in this 
+paper.  
+Acknowledgements 
+This work has been funded by the Spanish Ministry of 
+Science 
+and 
+Innovation 
+(RTI2018-097957-B-C33, 
+RED2018-102768-T, 
+PID2020-115353RA-I00); 
+the 
+Spanish State Research Agency and the European Social 
+Fund 
+Plus 
+under 
+grant 
+PRE2021-096954; 
+the 
+Community of Madrid – FEDER funds (S2018/NMT-
+4326); the Horizon 2020 research and innovation 
+program under Marie Sklodowska-Curie grant No. 
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+arXiv:2301.13315v1  [physics.atom-ph]  30 Jan 2023
+Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+1
+Probing Gravity for one Minute
+with an Optical-Lattice Atom Interferometer
+C.D. Panda, M. Tao, J. Eggelhof, M. Ceja, A. Reynoso, V. Xu, and H. M¨uller
+University of California, Berkeley, CA 94720, USA
+We have realized an atom interferometer that probes gravitational potentials
+by holding, rather than dropping, atoms.
+Up to one minute of coherence
+times are realized by suspending the spatially separated atomic wave packets
+in an optical lattice that is mode-filtered by an optical cavity. This trapped
+configuration suppresses phase variance due to vibrations by four to five orders
+of magnitude, overcoming the dominant noise source in atom-interferometric
+gravimeters.
+Recent progress in characterizing and reducing interferometer
+decoherence led to major increases in coherence and precision, paving the way
+to measurements of dark-energy candidates and probes of the quantum nature
+of gravity through measuring the gravity of source masses with record precision
+and spatial resolution.
+Light-pulse atom interferometry1 has been used to measure the fine struc-
+ture constant,2 the gravitational constant,3 to test the equivalence princi-
+ple,4 to look for laboratory signatures of dark energy,5 or to search for CPT
+or Lorentz violation.6,7 Atom interferometers are among the most accurate
+and sensitive tools for measuring gravity, in and out of the laboratory.8
+The achieved precision is limited by two main factors: coherence time and
+sensitivity to vibrations. In atom interferometers performed with atomic
+fountains, coherence is limited by the available free-fall time to below three
+seconds (for ∼ 10 m drop towers).
+Our interferometer suspends atoms against Earth’s gravity in an optical
+lattice that is formed by the mode of an optical cavity (Fig. 1). We observe
+up to 1 minute of coherence time (Fig. 2), which is by far the longest-lasting
+realization of the fundamental notion that a massive quantum particle can
+be in a superposition of being located in two different places at once and
+still three times longer than we have realized with the same experiment
+in a previous publication.9 Achieving this in a conventional free-fall atom
+interferometer would require a drop tower 4.5 km tall.
+
+Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+2
+Δz
+Fig. 1.
+Lattice interferometer atom trajectories.
+Ultracold Cesium atoms are
+launched against gravity.
+A pair of π/2 pulses creates a spatial superposition state
+where two atomic wavepackets are separated by a distance ∆z. After a time tA, atoms
+reach the apex of their free-fall trajectory and are loaded in a far-detuned optical lattice
+formed by the mode of an optical cavity (horizontal stripes) and remain held for a time
+τ. The two wavepackets are recombined and their interference is measured.
+Atoms held at fixed locations can measure extremely weak fields, such
+as the gravity due to small milligram source masses, for many seconds. This
+is orders of magnitude longer than possible with atomic fountains and rep-
+resents a major sensitivity boost for fundamental-physics experiments, such
+as tests of fifth forces,10 searches for dark energy in the laboratory,5 pro-
+posed measurements of the gravitational Aharonov–Bohm effect,11 or tests
+on whether gravity can mediate entanglement between quantum mechan-
+ical systems.12 The optical-lattice interferometer is also significantly less
+susceptible to noise due to acousto-mechanical vibrations than its fountain
+counterpart. Holding the atoms in an optical lattice for 60 seconds provides
+104–105 times suppression of environmental vibrations in the 1–1000 Hz vi-
+bration band, making operation near the standard quantum limit possible.
+Bringing optical-lattice technology, common to atomic-clock metrol-
+ogy,13 to atom interferometry is nontrivial, as spatial-superposition states
+are particularly susceptible to external forces. The interaction of the atoms
+with the optical lattice causes shifts of the atomic partial wavepacket phases
+that are large compared to the measured signals. This common-mode phase
+
+B
+元/2
+元/2
+元/2
+元/2
++hk.
+X
+ZProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+3
+Fig. 2.
+One-minute interference fringes measured with the lattice interferometer.
+is intrinsically subtracted by the interferometer, but any imperfections in
+the optical potential can lead to phase shifts and decoherence. Precise con-
+trol of the optical field is therefore necessary, which we achieve at least
+partially by using an optical cavity as a mode-filter for the optical lattice.
+We empirically observe a robust scaling in the decoherence rate of
+our interferometer, where the contrast follows an exponential decay, C =
+C0 exp(−τ/τC) with C0 the starting contrast. τC is inversely proportional
+to the wavepacket separation ∆z and trap depth U: 1/τC = U∆z/c, where
+c = 110 ± 30 µm Er s is a measured constant and Er is the recoil energy
+of the Cs D2 transition (Fig. 3a).
+The decoherence results from resid-
+ual transverse motion of the atoms in the one-dimensional optical lattice,
+where atoms sample tiny differences in the optical potential holding the top
+and bottom partial wavepackets. We confirm this by observing increased
+contrast when only selecting cold atoms by loading atoms in the narrower
+higher-order Laguerre–Gaussian modes of the cavity (Fig. 3b). Planned
+reductions in the atom sample temperature and control of lattice imperfec-
+tions promise to increase coherence times past the one-minute mark.
+We have described an optical-lattice atom interferometer with coher-
+ence times reaching one minute, realizing the longest spatial superposi-
+tion states to date, more than an order of magnitude longer than atomic-
+fountain interferometers.
+This geometry comes along with intrinsic ad-
+vantages: vibration-noise rejection and micron-scale spatial resolution of
+the measured signals. Cognizance and reduction of decoherence and atom-
+source upgrades promise to push this technology to record precision and
+ultra-long coherence. This opens a new era in atom interferometry, with
+
+Time offset (μus)
+-20-10.0 10.20
+-20-1001020-20-1001020
+0.4
+T
+0.2
+AA
+0.0
+As
+-0.2
+-0.4
+0.2109
+1.0509
+5.1109
+15.0509
+30.0309
+60.0609
+Interferometer hold time (seconds)Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022
+4
+Contrast decay constant c (
+�
+m Er s)
+Transverse mode
+�
+�
+�
+�
+■
+■
+■
+■
+◆
+◆
+◆
+◆
+▲
+▲
+▲
+▲
+▼
+▼
+▼
+▼
+0
+2
+4
+6
+8
+10
+12
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+U (Er)
+1/τC (s)
+Δz (μm)
+▼ 1.8
+▲ 4.2
+●
+6.5
+■
+9.4
+◆ 11.3
+a
+b
+Fig. 3.
+Interferometer coherence. a.) Measured linear scaling of 1/τC with trap
+depth U and separation, ∆z.
+b.)
+The magnitude of the contrast decay constant
+c increases when the optical-lattice radius is lowered through the use of higher-order
+Laguerre–Gaussian modes of the cavity.
+applications in gravimetry, searches for fifth forces, a measurement of the
+gravitational Aharonov–Bohm effect in the absence of forces, or new tests
+of CPT and Lorentz violation.
+References
+1. A.D. Cronin, J. Schmiedmayer, and D.E. Pritchard, Rev. Mod. Phys. 81,
+1051 (2009).
+2. L. Morel, Z. Yao, P. Clade, and S. Guellati-Kh´elifa, Nature 588, 61 (2020).
+3. G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G.M. Tino, Nature
+510, 518 (2014).
+4. P. Asenbaum, C. Overstreet, M. Kim, J. Curti, and M. A. Kasevich, Phys.
+Rev. Lett. 125, 191101 (2020).
+5. M. Jaffe et al., Nature Phys. 13, 938 (2017).
+6. K. Chung, S. Chiow, S. Herrmann, S. Chu, and H. M”uller, Phys. Rev. D
+80, 016002 (2009).
+7. Q.G. Bailey and V.A. Kosteleck´y, Phys. Rev. D 74, 045001 (2006).
+8. G.M. Tino, Quantum Sci. Technol. 6, 024014 (2021).
+9. V. Xu, M. Jaffe, C.D. Panda, S.L. Kristensen, L.W. Clark, and H. M¨uller,
+Science 366, 745 (2019).
+10. P. Wolf, P. Lemonde, A. Lambrecht, S. Bize, A. Landragin, and A. Clairon,
+Phys. Rev. A 75, 063608 (2007).
+11. M.A. Hohensee, B. Estey, P. Hamilton, A. Zeilinger, and H. M¨uller, Phys.
+Rev. Lett. 108, 230404 (2012).
+12. D. Carney, H. M¨uller, and J.M. Taylor, PRX Quantum 2, 030330 (2021).
+13. G.E. Marti, R.B. Hutson, A. Goban, S.L. Campbell, N. Poli, and J. Ye, Phys.
+Rev. Lett. 120, 103201 (2018).
+
+500
+400
+300
+I
+200
+I
+100
+LG 00
+LG 10
+LG 20
+0

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='13315v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='atom-ph]  30 Jan 2023  Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  1  Probing Gravity for one Minute  with an Optical-Lattice Atom Interferometer  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Panda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Tao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Eggelhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Ceja, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Reynoso, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Xu, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' M¨uller  University of California, Berkeley, CA 94720, USA  We have realized an atom interferometer that probes gravitational potentials  by holding, rather than dropping, atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Up to one minute of coherence  times are realized by suspending the spatially separated atomic wave packets  in an optical lattice that is mode-filtered by an optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This trapped  configuration suppresses phase variance due to vibrations by four to five orders  of magnitude, overcoming the dominant noise source in atom-interferometric  gravimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Recent progress in characterizing and reducing interferometer  decoherence led to major increases in coherence and precision, paving the way  to measurements of dark-energy candidates and probes of the quantum nature  of gravity through measuring the gravity of source masses with record precision  and spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Light-pulse atom interferometry1 has been used to measure the fine struc-  ture constant,2 the gravitational constant,3 to test the equivalence princi-  ple,4 to look for laboratory signatures of dark energy,5 or to search for CPT  or Lorentz violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='6,7 Atom interferometers are among the most accurate  and sensitive tools for measuring gravity, in and out of the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='8  The achieved precision is limited by two main factors: coherence time and  sensitivity to vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' In atom interferometers performed with atomic  fountains, coherence is limited by the available free-fall time to below three  seconds (for ∼ 10 m drop towers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Our interferometer suspends atoms against Earth’s gravity in an optical  lattice that is formed by the mode of an optical cavity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' We observe  up to 1 minute of coherence time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 2), which is by far the longest-lasting  realization of the fundamental notion that a massive quantum particle can  be in a superposition of being located in two different places at once and  still three times longer than we have realized with the same experiment  in a previous publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='9 Achieving this in a conventional free-fall atom  interferometer would require a drop tower 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='5 km tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  2  Δz  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Lattice interferometer atom trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Ultracold Cesium atoms are  launched against gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  A pair of π/2 pulses creates a spatial superposition state  where two atomic wavepackets are separated by a distance ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' After a time tA, atoms  reach the apex of their free-fall trajectory and are loaded in a far-detuned optical lattice  formed by the mode of an optical cavity (horizontal stripes) and remain held for a time  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' The two wavepackets are recombined and their interference is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Atoms held at fixed locations can measure extremely weak fields, such  as the gravity due to small milligram source masses, for many seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This  is orders of magnitude longer than possible with atomic fountains and rep-  resents a major sensitivity boost for fundamental-physics experiments, such  as tests of fifth forces,10 searches for dark energy in the laboratory,5 pro-  posed measurements of the gravitational Aharonov–Bohm effect,11 or tests  on whether gravity can mediate entanglement between quantum mechan-  ical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='12 The optical-lattice interferometer is also significantly less  susceptible to noise due to acousto-mechanical vibrations than its fountain  counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Holding the atoms in an optical lattice for 60 seconds provides  104–105 times suppression of environmental vibrations in the 1–1000 Hz vi-  bration band, making operation near the standard quantum limit possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Bringing optical-lattice technology, common to atomic-clock metrol-  ogy,13 to atom interferometry is nontrivial, as spatial-superposition states  are particularly susceptible to external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' The interaction of the atoms  with the optical lattice causes shifts of the atomic partial wavepacket phases  that are large compared to the measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This common-mode phase  B  元/2  元/2  元/2  元/2  +hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  X  ZProceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  One-minute interference fringes measured with the lattice interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  is intrinsically subtracted by the interferometer, but any imperfections in  the optical potential can lead to phase shifts and decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Precise con-  trol of the optical field is therefore necessary, which we achieve at least  partially by using an optical cavity as a mode-filter for the optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  We empirically observe a robust scaling in the decoherence rate of  our interferometer, where the contrast follows an exponential decay, C =  C0 exp(−τ/τC) with C0 the starting contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' τC is inversely proportional  to the wavepacket separation ∆z and trap depth U: 1/τC = U∆z/c, where  c = 110 ± 30 µm Er s is a measured constant and Er is the recoil energy  of the Cs D2 transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  The decoherence results from resid-  ual transverse motion of the atoms in the one-dimensional optical lattice,  where atoms sample tiny differences in the optical potential holding the top  and bottom partial wavepackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' We confirm this by observing increased  contrast when only selecting cold atoms by loading atoms in the narrower  higher-order Laguerre–Gaussian modes of the cavity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Planned  reductions in the atom sample temperature and control of lattice imperfec-  tions promise to increase coherence times past the one-minute mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  We have described an optical-lattice atom interferometer with coher-  ence times reaching one minute, realizing the longest spatial superposi-  tion states to date, more than an order of magnitude longer than atomic-  fountain interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  This geometry comes along with intrinsic ad-  vantages: vibration-noise rejection and micron-scale spatial resolution of  the measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' Cognizance and reduction of decoherence and atom-  source upgrades promise to push this technology to record precision and  ultra-long coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' This opens a new era in atom interferometry, with  Time offset (μus)  20-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
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+page_content='0609  Interferometer hold time (seconds)Proceedings of the Ninth Meeting on CPT and Lorentz Symmetry (CPT’22), Indiana University, Bloomington, May 17–26, 2022  4  Contrast decay constant c (  �  m Er s)  Transverse mode  �  �  �  �  ■  ■  ■  ■  ◆  ◆  ◆  ◆  ▲  ▲  ▲  ▲  ▼  ▼  ▼  ▼  0  2  4  6  8  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
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+page_content='3  a  b  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  Interferometer coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=') Measured linear scaling of 1/τC with trap  depth U and separation, ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content=')  The magnitude of the contrast decay constant  c increases when the optical-lattice radius is lowered through the use of higher-order  Laguerre–Gaussian modes of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
+page_content='  applications in gravimetry, searches for fifth forces, a measurement of the  gravitational Aharonov–Bohm effect in the absence of forces, or new tests  of CPT and Lorentz violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFQT4oBgHgl3EQfZDbf/content/2301.13315v1.pdf'}
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+arXiv:2301.04722v1  [math.PR]  11 Jan 2023
+RATE OF CONVERGENCE IN MULTIPLE SLE USING
+RANDOM MATRIX THEORY
+ANDREW CAMPBELL, KYLE LUH, AND VLAD MARGARINT
+Abstract. We provide an order of convergence for a version of the Carath´eodory
+convergence for the multiple SLE model with a Dyson Brownian motion driver
+towards its hydrodynamic limit, for β = 1 and β = 2. The result is obtained
+by combining techniques from the field of Schramm-Loewner Evolutions with
+modern techniques from random matrices. Our approach shows how one can
+apply modern tools used in the proof of universality in random matrix theory,
+in the field of Schramm-Loewner Evolutions.
+1. Introduction and main results
+Schramm-Loewner Evolution (SLE) and random matrix theory (RMT) are two
+active and well-studied fields of research within modern probability theory [2,38].
+The SLE was introduced by Oded Schramm in 2000 in his study of scaling limits
+of various discrete processes [49]. RMT appeared earlier in the statistical work of
+Wishart [61] and the pioneering physics of Wigner [60]. Both SLE and RMT have
+been thriving areas of mathematical research since their advent.
+When studying SLE theory, one introduces the notion of compact hulls, which
+are compact sets with simply connected complements in the upper half-plane. If
+Kt is a growing set of hulls parameterized by t ∈ [0, T ] and the growth is local in
+some sense, then it is known that gt := gKt obeys the Loewner differential equation
+∂tgt(z) =
+2
+gt(z) − Wt
+where Wt is referred to as the driving function and captures the local growth of Kt.
+SLE are the random curves corresponding to the gt when the driving function is a
+constant multiple of Brownian motion, that we denote by √κBt, for κ ≥ 0. With
+probability one, gt is continuous up to the boundary and the limit
+γ(t) = lim
+y→0 g−1
+t
+(√κBt + iy)
+exists and is continuous in time, by the Rohde-Schramm Theorem [46]. The curve
+γ(t) is called the SLE trace. Also, it can be shown that with probability one, gt is
+a continuous family of conformal maps from Ht to H, where Ht is the unbounded
+component of the complement in H of γ(t), for t ∈ [0, T ] [46]. Moreover, the nature
+of the curve changes as κ increases from simple a.s. when κ ∈ [0, 4], to having
+double points a.s. for κ ∈ (4, 8) and space-filling a.s., for κ ≥ 8. For different
+parameters κ, the SLE models the scaling limits of an astoundingly diverse set of
+discrete models. For instance, it was proved in [39] that the scaling limit of the loop
+erased random walk (with the loops erased in a chronological order) converges in the
+K. Luh was supported in part by the Ralph E. Powe Junior Faculty Enhancement Award.
+1
+
+2
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+scaling limit to SLEκ with κ = 2. Moreover, other two dimensional discrete models
+from Statistical Mechanics including the Ising model cluster boundaries, Gaussian
+free field interfaces, percolation on the triangular lattice at critical probability, and
+Uniform spanning trees were proved to converge in the scaling limit to SLE for
+values of κ = 3, κ = 4, κ = 6 and κ = 8 respectively in the series of works [52], [50],
+[51] and [39].
+One can consider more generally the Loewner equation driven by a time-dependent
+real-valued measure µt
+∂
+∂tgt(z) =
+�
+R
+µt(dx)
+gt(z) − x,
+g0(z) = z.
+When the driving measure µt is a Dirac-delta mass at location √κBt, we recover
+the previous SLE maps. In the case µt = �N
+i=1 ωi(t)δUi(t), for some non-intersecting
+continuous functions Ui(t) ∈ R (called driving functions), and weights ωi(t) ∈ R+,
+we obtain the multi-slit Loewner equation with driving functions Ui(t), i = 1, · · · , n.
+In this work, we consider the case ωi(t) = 1/N, for all t ∈ [0, T ].
+For a real parameter β > 0, Dyson Brownian motion (DBM) is defined by the
+following system of N equations
+(1)
+dλ(i)
+t
+=
+2
+√Nβ dB(i)
+t
++ 2
+N
+�
+j̸=i
+dt
+λ(j)
+t
+− λ(i)
+t
+,
+for i = 1, 2, ..., N.
+Due to its connections with other fields, an important Loewner equation is the
+multiple SLE with DBM as a driver. The multiple SLE maps that are obtained
+when the driving measure is an empirical measure on N DBM particles are denoted
+in this paper by gN
+t (z). This model was introduced by Cardy in [10], and studied
+further by Lawler and Healey in [30], in connection with the quantum Calogero-
+Sutherland model and Conformal Field Theory.
+More works on the connection
+between Multiple SLE and CFT can be found in [40] and [48]. In the case of N = 2
+curves, perturbations of this model in the parameter β have been studied in [11].
+We note that the parameters β in the DBM model and κ in SLE theory are related
+via β = 8/κ.
+We refer to the multiple SLE model with Dyson Brownian motion as a driver
+as the simultaneously growing multiple SLE model. There is also a version of the
+multiple SLE that has non-simultaneous growth that has received a lot of attention
+in the previous years. There have been several results on the multiple SLE model
+in both the upper half-plane and the unit disk versions [5, 12, 13, 15, 31, 32, 35, 37,
+40,45,47,58,62,63].
+In [14], the authors consider the N → ∞ limit of multiple SLE driven by DBM. In
+particular they show that the empirical measure of the initial positions converges to
+a probability measure µ0, then gN
+t converges in distribution with respect to locally
+uniform convergence to g∞
+t
+solving
+(2)
+∂
+∂tg∞
+t (z) = M ∞
+t (z),
+g0(z) = z,
+Where M ∞
+t
+is a solution to the complex Burgers equation
+(3)
+� ∂M∞
+t (z)
+∂t
+= −2M ∞
+t (z) ∂M∞
+t (z)
+∂z
+, t > 0,
+M ∞
+0 (z) =
+�
+R
+2
+z−xdµ0(x).
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+3
+Their result serves as the multiple SLE analog of Wigner’s famous semicircle law
+in random matrix theory.
+We consider this model and we obtain more refined
+information by providing an order of convergence of this model in a weaker ver-
+sion of the Carath´eodory type convergence. We aim in future works to study the
+full Carath´eodory convergence by strengthening the estimates as we approach the
+multiple SLE hull.
+In this work, we combine elements of the proof of Local Laws in random matrix
+theory, such as resolvent techniques, with elements of the SLE theory. In other
+words, we apply modern techniques from random matrix theory to the analysis of
+SLE. Local Laws are a very important research direction in random matrix theory
+in the last years (see [9], [16], [18], [17], [19], [20], [21], [22], [25], [24], [26], [23],
+[42], [44], [54], [53], [57], [56], [55] for a non-exhaustive list). They are one of the
+fundamental ingredients in proving the universality of Wigner ensembles in random
+matrix theory (see [27]).
+Given the outstanding developments in the proof of the universality in RMT
+using the analysis of the DBM, the interaction between multiple SLE and random
+matrices will provide many avenues to explore. The approach in the current work
+represents one of the possible directions of exploration between these two major
+fields of Probability theory. In a different direction, which we aim to explore in the
+future, one can study the geometry of the multiple SLE curves using the analysis
+of the Dyson Brownian motion drivers, as well as good approximation schemes of
+the model (see, for example, [28], [59], [33] in the one SLE curve case). Yet another
+possibility is to study the continuity of the multiple SLE model in the parameter
+β, motivated by the great interest and progress throughout the years in this yet
+unresolved conjecture in the one SLE curve case (see [6], [7], [29], [34]). In addition,
+the fact that the multiple SLE curves grow from the positions of the drivers along
+with some knowledge about the structure of the drivers gives the possibility of
+defining new observables in order to study the convergence of discrete models to
+the multiple SLE. Examples of such observables include the statistics of the kth
+smallest distance between drivers, for k ≥ 1, (see [8]) or the probability of having
+no drivers in a symmetric region about the origin (see [41] for β = 2).
+Although our result can be obtained for general bounded initial conditions, we
+state it in the case in which all the Dyson Brownian motion particles start from
+the origin. We prefer this choice for the simplicity of the notation and exposition.
+Theorem 1.1. Let β = 1 or β = 2, and let us consider Dyson Brownian motion
+beginning at the origin. Let KT be the multiple SLE hull at time T > 0. Then, for
+any ε > 0, for the multiple SLE maps for N curves, we have that
+sup
+t∈[0,T ], z∈G
+|gN
+t (z) − g∞
+t (z)| = O
+�
+1
+N 1/3−ε
+�
+,
+with overwhelming probability1, for a given G ⊂ H \ KT .
+Remark 1.2. It is well-known that for the special values of the parameters β = 1,
+β = 2 and β = 4, the Dyson Brownian motion particles statistics can be understood
+using matrices as these values correspond to the well-studied models of the Gauss-
+ian Orthogonal Ensemble, Gaussian Unitary Ensemble (GUE), and the Gaussian
+1An event E holds with overwhelming probability if, for every p > 0, P(E) ≥ 1 − Op(n−p); see
+Definition 3.1 for details.
+
+4
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Symplectic Ensemble (GSE) respectively. An n × n real symmetric matrix A is
+drawn from the Gaussian Orthogonal Ensemble (GOE) if the upper-triangular en-
+tries Aij, 1 ≤ i ≤ j ≤ n are independent Guassian random variables, where Aij has
+mean zero and variance 1+δij
+n
+and δij is the Kronecker delta. The GUE and GSE
+ensembles are defined similarly with complex and quaternic Gaussian off-diagonal
+entries. We study the cases β = 1 and β = 2 respectively as they correspond to
+the critical parameters κ = 8 and κ = 4 in SLE theory. We expect that a similar
+analysis will hold for the case β = 4 that corresponds to the value κ = 2.
+We note that the N −(1/3−ǫ) order of convergence to the hydrodynamic limit
+of multiple SLE is obtained via an estimate in [44] which is, to the best of our
+knowledge, the best stability estimate in this setting available in the literature.
+Theorem 1.1 relies on the following technical result.
+Theorem 1.3. Let β = 1 or β = 2, and let us consider Dyson Brownian motion
+started from the origin
+�
+λ(1)
+t , . . . , λ(N)
+t
+�
+and M N
+t
+: C+ → C− defined by
+M N
+t (z) = 1
+N
+N
+�
+j=1
+2
+z − λ(j)
+t
+.
+Let M ∞
+t
+: C+ → C− be the solution to the complex Burgers equation
+(4)
+� ∂M∞
+t (z)
+∂t
+= −2M ∞
+t (z) ∂M∞
+t (z)
+∂z
+, t > 0,
+M ∞
+0 (z) = 2
+z.
+Then for any compact set G ⊂ C+, ε > 0, and fixed t ∈ [0, T ]
+(5)
+sup
+z∈G
+��M N
+t (z) − M ∞
+t (z)
+�� = OG,ε
+�
+t
+N
+1
+3 −ε
+�
+,
+with overwhelming probability.
+The remainder of the paper is organized into several sections. In the second
+section, we present probabilistic estimates involving the multiple SLE hull and sub-
+sets of its complement. The third section focuses on the random matrix techniques
+we use, as well as on the proof of Theorem 1.3. In subsection 3.4 we utilize a net
+argument that extends the previously obtained results for a fixed time t ∈ [0, T ], to
+all times simultaneously. In section 4, we prove Theorem 1.1 and in the Appendix
+we provide the stability part of the argument.
+2. Subset of the complement of the multiple SLE hull
+In this section, we provide probabilistic estimates for general β ≥ 1 that are
+useful in deducing the choice of the set G ⊂ H \ KT , where we establish the order
+of convergence of the family of maps. We present the estimates for general β ≥ 1,
+and specialize to the β = 1 and β = 2 cases in our application.
+Let ∂tgt(z) =
+1
+N
+�N
+i=1
+2
+gt(z)−λi
+t , where (λ(1)
+t , · · · , λ(N)
+t
+) is a Dyson Brownian
+motion (DBM) with parameter β ≥ 1. We first consider λi
+t ≡ 0, ∀t ∈ [0, T ],
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+5
+for all i = {1, 2, · · · , N}. Then, we have that ∂tgt(z) =
+2N
+Ngt(z) =
+2
+gt(z). Since
+gt(z) = Re(gt(z)) + iIm(gt(z)), we have that
+∂tIm(gt(z)) = −2Im(gt(z))
+|gt(z)|2
+≥
+2
+(Im(gt(z))2 .
+This allows us to conclude that
+Im (gt(z))2 ⩾ (Im(z))2 − 4t > 0,
+whenever Im(z) > 2
+√
+T.
+In order to control the real part, for a Dyson Brownian motion (λ(1)
+t , · · · , λ(N)
+t
+)
+with parameter β ≥ 1, we observe that
+∂tRe(gt(z)) = 1
+N
+N
+�
+i=1
+Re (gt(z)) − λi
+t
+��gt(z) − λi
+t
+��2
+> 0,
+whenever Re(gt(z)) > M = supt∈[0,T ] supi={1,2,...,N}
+��λi
+t
+�� . Then, combining the two
+estimates, we have that
+{z ∈ H| : |Re(z) > M or Im > 2
+√
+T} ⊂ H \ KT .
+We also note that for all t ∈ [0, T ], we have
+Kt ⊂ {z ∈ H : | Re z| ≤ M and Im z ≤ 2
+√
+T}.
+Next, we use the following probabilistic result on the behaviour of the extreme
+eigenvalues.
+Lemma 2.1 (Lemma 4.3.17 in [3]). Let λ∗
+N(t) := max1≤i≤N
+���λ(i)
+t
+��� = max
+�
+λ(N)
+t
+, −λ(1)
+t
+�
+.
+Let β ≥ 1. Then there exist finite constants α = α(β) > 0, C = C(β), and for all
+t ≥ 0 a random variable η∗
+N(t) with law independent of t, such that
+P (η∗
+N(t) ≥ x + C) ≤ e−αNx
+and, for all t ≥ 0,
+λ∗
+N(t) ≤ λ∗
+N(0) +
+√
+tη∗
+N(t).
+In the case of the DBM drivers, using Lemma 2.1, we have that for β ≥ 1 and
+for C = C(β) and α = α(β) some finite constants that
+P
+�
+sup
+t∈[0,T ]
+sup
+i={1,2,...,N}
+��λi
+t
+�� ≤ (C + x)
+√
+T
+�
+≥ 1 − e−αNx.
+For conformal maps, we have the following result.
+Lemma 2.2 (Lemma 4.5 in [36]). Let K be a hull and H = H\K. If K ⊂ B (x0, r),
+then gK maps H ∩ B (x0, 2r) into B (x0, 3r) and
+sup
+z∈H
+|gK(z) − z| ≤ 5r.
+For a box G ⊂ HT = H \ KT , we have that with overwhelming probability that
+(6)
+gN
+t (G) ⊂ {z :
+�
+Im(z0))2 − 4t ≤ Im(z) ≤ Im(z0); |Re(z)| ≤ f(N, T )},
+where f(N, T ) can be deduced from the following:
+
+6
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+(7)
+|RegK(z)| ≤ |gK(z)| ≤ |z| + 5r.
+In the case of the multiple SLE hull KT , we have r =
+�
+M 2 + (2
+√
+T)2.
+3. Random Matrix Techniques
+In this section we prove some random matrix results leading to the proof of
+Theorem 1.3. It is worth noting that for β = 1 and β = 2, DBM
+�
+λ(1)
+t , . . . , λ(N)
+t
+�
+defined as the solution to (1) starting from initial positions
+�
+λ(1)
+0 , . . . , λ(N)
+0
+�
+is equal
+in distribution to the eigenvalues of D−2
+√
+tA where D is an N ×N diagonal matrix
+of the initial positions and A is a matrix drawn from the Gaussian Orthogonal
+Ensemble for β = 1 or Gaussian Unitary Ensemble (GUE) for β = 2. We establish
+the results in this section for the case when A is drawn from the GOE, since the
+adjustments to the GUE model are straightforward.
+3.1. Tools. This section introduces the tools we will use throughout. We begin
+with a definition describing high probability events.
+Definition 3.1 (High probability events). Let E be an event that depends on n.
+• E holds asymptotically almost surely if P(E) = 1 − o(1).
+• E holds with high probability if P(E) = 1−O(n−c) for some constant c > 0.
+• E holds with overwhelming probability if, for every p > 0, P(E) ≥ 1 −
+Op(n−p).
+For z = E + iη ∈ C+, n × n Hermitian matrix H, and G(z) := (H − zI)−1 the
+Ward identity states that
+(8)
+n
+�
+j=1
+|Gij(z)|2 = 1
+η Im Gii(z).
+If A and B are invertible matrices, the resolvent identity states that
+(9)
+A−1 − B−1 = A−1(B − A)B−1 = B−1(B − A)A−1.
+If ξ is a Gaussian random variable with mean zero and variance σ2 and f : R → C
+is continuously differentiable, the Gaussian integration by parts formula states that
+(10)
+E[ξf(ξ)] = σ2E[f ′(ξ)],
+provided the expectations are finite.
+The next lemma is a convenient moment
+bound for a martingale difference sequence.
+Lemma 3.2 (Lemma 2.12 from [4]). Let {Xk} be a complex martingale difference
+sequence and Fk = σ(X1, . . . , Xk) be the σ-algebra generated by X1, . . . , Xk. Then,
+for any p ≥ 2,
+E
+�����
+n
+�
+k=1
+Xk
+�����
+p
+≤ Cp
+
+E
+� n
+�
+k=1
+Ek−1|Xk|2
+�p/2
++ E
+n
+�
+k=1
+|Xk|p
+
+ .
+where Cp is a constant that only depends on p and Ek−1[·] := E[·|Fk−1].
+The next concentration lemma is helpful in controlling the deviation of a qua-
+dratic form from its expectation.
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+7
+Lemma 3.3 (Equation (3) from [1]). Let X be an n-vector containing iid standard
+Gaussian random variables, A a deterministic n × n matrix and ℓ ≥ 1 an integer.
+Then
+E[X∗AX − tr A|2ℓ ≤ Cℓ(tr AA∗)ℓ
+where Cℓ is a constant that only depends on ℓ.
+Finally, we will require the following algebraic identity in Section 3.2.
+Lemma 3.4 (Theorem A.5 from [4]). Let A be an n × n symmetric matrix and
+Ak be the k-th major submatrix of size (n − 1) × (n − 1). If A and Ak are both
+invertible, then
+tr(A−1) − tr(A−1
+k ) =
+1 + α∗
+kA−2
+k αk
+Akk − α∗
+kA−1
+k αk
+where αk is obtained from the k-th column of A by deleting the k-th entry.
+3.2. Concentration of the Gaussian Orthogonal Ensemble. In this section
+we show that |M N
+t (z)−EM N
+t (z)| is small for a fixed z ∈ C+. To match the random
+matrix literature we will consider for fixed t > 0, mN(z) := − 1
+2M N
+t (z). We let At
+be
+√
+tA where A is drawn from the Gaussian Orthogonal Ensemble.
+We note that mN(z) − EmN(z) can be written as the following telescopic sum
+mN(z) − EmN(z) =
+n
+�
+k=1
+(EkmN(z) − Ek−1mN(z)) :=
+N
+�
+k=1
+γk
+Observe that
+mN(z) = 1
+N tr(At − z)−1 = 1
+N tr
+1
+√
+tA − z =
+1
+N
+√
+t tr
+1
+A − z/
+√
+t =
+1
+N
+√
+t tr
+1
+A − z′
+We define E′ = E/
+√
+t and η′ = η/
+√
+t. Let Ek denote the conditional expectation
+with respect to the σ-field generated by Aij with i, j ≤ k, so that ENmN(z) =
+mN(z) and E0mN(z) = EmN(z).
+
+8
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+γk =
+1
+N
+√
+t(Ek tr(A − z′)−1 − Ek−1 tr(A − z′)−1)
+=
+1
+N
+√
+t
+�
+Ek
+�
+tr(A − z′)−1 − (Ak − z′)−1�
+− Ek−1
+�
+tr(A − z′)−1 − tr(Ak − z′)−1��
+=
+1
+N
+√
+t(Ek − Ek−1)
+�
+a∗
+kG2
+kak − Eaka∗
+kG2
+kak
+Akk − z′ − a∗
+kGkak
++
+1 + Eaka∗
+kG2
+kak
+Akk − z′ − a∗
+kGkak
+−
+1 + Eaka∗
+kG2
+kak
+Akk − z′ − Eaka∗
+kGkak
+�
+=
+1
+N
+√
+t(Ek − Ek−1)
+�
+a∗
+kG2
+kak − Eaka∗
+kG2
+kak
+Akk − z′ − a∗
+kGkak
+−
+(1 + Eaka∗
+kG2
+kak)(a∗
+kGkak − Eaka∗
+kGkak)
+(Akk − z′ − a∗
+kGkak)(Akk − z′ − Eaka∗
+kGkak)
+�
+=
+1
+N
+√
+t(Ek − Ek−1)
+�
+a∗
+kG2
+kak − 1
+N tr G2
+k
+Akk − z′ − a∗
+kGkak
+−
+(1 + 1
+N tr G2
+k)(a∗
+kGkak − 1
+N tr Gk)
+(Akk − z′ − a∗
+kGkak)(Akk − z′ − 1
+N tr Gk)
+�
+where ak denotes the k-th row of A with the k-th entry removed. We define the
+following quantities,
+αk = a∗
+kG2
+kak − 1
+N tr G2
+k,
+βk =
+1
+Akk − z′ − a∗
+kGkak
+,
+¯βk =
+1
+Akk − z′ − 1
+N tr Gk
+,
+δk = a∗
+kGkak − 1
+N tr Gk,
+ǫk = 1 + 1
+N tr G2
+k,
+so that
+mN(z) − EmN(z) =
+1
+N
+√
+t
+N
+�
+k=1
+(Ek − Ek−1)αkβk −
+1
+N
+√
+t
+N
+�
+k=1
+(Ek − Ek−1)ǫkδkβk ¯βk
+:= 1
+√
+tS1 − 1
+√
+tS2.
+(11)
+For a fixed ε > 0, we will show that N 1−ε(η′)3|S1| = o(1) and N 1−ε(η′)3|S2| =
+o(1) with overwhelming probability. This will be done via the method of moments.
+We begin with S1. By Markov’s inequality, it suffices to bound E|N 1−ε(η′)3S1|2ℓ =
+E|N −ε(η′)3 �n
+k=1(Ek − Ek−1)αkβk|2ℓ for ℓ ∈ N.
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+9
+By Lemma 3.2, for any ℓ ≥ 1,
+E|N −ε(η′)3
+N
+�
+k=1
+(Ek − Ek−1)αkβk|2ℓ ≤ Cℓ
+�
+E
+� N
+�
+k=1
+Ek−1|N −ε(η′)3αkβk|2
+�ℓ
++
+N
+�
+k=1
+E|N −ε(η′)3αkβk|2ℓ
+�
+.
+We use Cℓ to indicate a constant that only depends on ℓ, but may change from line
+to line. Since Im a∗
+kGkak > 0,
+|βk| ≤ (η′)−1.
+Therefore,
+E
+�����N −ε(η′)3
+n
+�
+k=1
+(Ek − Ek−1)αkβk
+�����
+2ℓ
+≤ CℓN −2εℓ
+�
+E
+� N
+�
+k=1
+Ek−1|(η′)2αk|2
+�ℓ
++
+N
+�
+k=1
+E|(η′)2αk|2ℓ
+�
+.
+(12)
+By Lemma 3.3,
+E|(η′)2αk|2ℓ ≤ Cℓ(η′)4ℓN −2ℓE| tr G2
+kG∗2
+k |ℓ.
+We use the simple bound that
+tr G2
+kG∗2
+k =
+� N
+�
+i=1
+1
+((λi − E)2 + (η′)2)2
+�
+≤ N(η′)−4
+(13)
+We now have that
+E|(η′)2αk|2ℓ ≤ Cℓ(η′)4ℓN −2ℓE|N(η′)−4|ℓ
+≤ CℓN −ℓ
+Therefore, by equation (12),
+E
+�����N −ε(η′)3
+N
+�
+k=1
+(Ek − Ek−1)αkβk
+�����
+2ℓ
+≤ CℓN −2εℓ
+
+E
+� N
+�
+k=1
+Ek|(η′)2αk|2
+�ℓ
++ N −ℓ+1
+
+
+By the same reasoning as in (13), we also have that Ek|αk|2 ≤ N(η′)−4. Thus,
+Ek|(η′)2αk|2 ≤ KN −1
+so
+E
+� N
+�
+k=1
+Ek|(η′)2αk|2
+�ℓ
+≤ Cℓ.
+Finally, we can conclude that
+E
+�����N −ε(η′)3
+N
+�
+k=1
+(Ek−1 − Ek)αkβk
+�����
+2ℓ
+≤ CℓN −2εℓ.
+As ℓ is arbitrary, we have shown that |S1| = oη(t3/2/N 1−ε) with overwhelming
+probability.
+
+10
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Now we address S2. We first observe that
+����1 + 1
+N tr G2
+k
+���� ≤ 1 + 1
+N tr GkG∗
+k
+= (η′)−1 Im
+�
+−Akk + z′ + 1
+N tr Gk
+�
+Therefore,
+|ǫk ¯βk| =
+|1 + 1
+N tr G2
+k|
+|Akk − z′ − 1
+N tr Gk| ≤ (η′)−1
+Recalling that |βk| ≤ (η′)−1, we have that
+E|N 1−ε(η′)4S2|2ℓ = N −2εℓ(η′)2ℓ
+�����
+N
+�
+k=1
+(Ek − Ek−1)δk
+�����
+2ℓ
+Again, by Lemma 3.2
+E|N 1−ε(η′)4S2|2ℓ ≤ CℓN −2εℓ(η′)2ℓ
+
+E
+� N
+�
+k=1
+Ek−1|δk|2
+�ℓ
++
+N
+�
+K=1
+E|δk|2ℓ
+
+ .
+Note that by Lemma 3.3,
+E|δk|2ℓ ≤ CℓN −2ℓE| tr GkG∗
+k|ℓ.
+We have that
+tr GkG∗
+k ≤ N(η′)−2
+so
+E|δk|2ℓ ≤ CℓN −ℓ(η′)−2ℓ.
+Additionally,
+Ek−1|δk|2 ≤ N −1(η′)−2.
+Thus,
+E|N 1−ε(η′)4S2|2ℓ ≤ CℓN −2εℓ.
+We can then conclude that S2 is oη(t2/N 1−ε) with overwhelming probability. Re-
+turning to (11) we have shown that
+(14)
+|mN(z) − EmN(z)| = o
+�
+t
+N 1−ε
+�
+with overwhelming probability.
+3.3. Proof of Theorem 1.3. In this section we provide the proof of Theorem
+1.3. We will give begin the proof for generic initial starting positions of the Dyson
+Brownian motion, before specializing to the starting positions at the origin. Define
+the matrix
+(15)
+Lt = D − 2
+√
+tA
+where A is drawn from the Gaussian Orthogonal/Unitary Ensemble and D is an
+N × N deterministic diagonal matrix. Define the resolvent matrices
+Gt(z) := (Lt − zI)−1 ,
+and
+Q(z) := (D − zI)−1 .
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+11
+Next, we define the functions
+M N
+t (z) = − 2
+N tr Gt(z),
+and
+SN(z) = − 2
+N tr Q(z).
+Fix t, η > 0 and z such that Im(z) ≥ η. Additionally, define the matrices
+G := Gt(z),
+and
+Q := Q
+�
+z − 2tEM N
+t (z)
+�
+.
+In particular SN �
+z − 2tEM N
+t (z)
+�
+= − 2
+N tr Q. By the resolvent identity (9)
+EM N
+t (z) − SN �
+z − 2tEM N
+t (z)
+�
+= − 2
+N (tr Gt − tr Qt)
+(16)
+= −2E 1
+N tr
+�
+G ˜AQ
+�
++ 4tEM N
+t (z)E 1
+N tr (GQ)
+where ˜A = 2
+√
+tA. We now consider the term
+(17)
+− 2E 1
+N tr
+�
+G ˜AQ
+�
+= − 2
+N
+�
+i,j
+QiiE
+�
+Gij ˜Aji
+�
+.
+A computation involving the resolvent identity (9) shows that
+∂Gkl
+∂Aij
+=
+�
+GkiGji + GkjGil,
+if i ̸= j,
+GkiGjl,
+if i = j .
+Applying Gaussian integration by parts to (17) yields
+−2E 1
+N tr
+�
+G ˜AQ
+�
+= −8
+N 2 E
+�
+i,j
+QiiG2
+ij − 4t
+N EM N
+t (z) tr(QG),
+which when combined with (16) gives
+EM N
+t (z) − SN �
+z − 2tEM N
+t (z)
+�
+= −8
+N 2 E
+�
+i,j
+QiiG2
+ij − 4t
+N EM N
+t (z) tr(QG)
+(18)
++ 4tEM N
+t (z)E 1
+N tr (GQ) .
+We now fix z = E + iη ∈ C+. By the Ward identity (8)
+������
+8
+N 2 E
+�
+i,j
+QiiG2
+ij
+������
+≤ E 8
+N 2
+�
+j
+|Qii|
+�
+j
+|Gij|2
+≤ E
+8
+N 2η
+�
+i
+|Qii| Im Gii
+≤
+8
+Nη3 .
+
+12
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+For the difference 4tEM N
+t (z)E 1
+N tr (GQ) − 4t
+N EM N
+t (z) tr(QG), note that
+����
+4t
+N tr (GQ)
+���� =
+�����
+4t
+N
+�
+i
+QiiGii
+�����
+≤ 4t
+η2 .
+It then follows from (14) with D equal to the zero matrix that
+E
+�����4t
+�
+EM N
+t (z)
+� 1
+N tr (GQ) − 4t
+N M N
+t (z) tr(QG)
+����
+�
+≤ E
+���M N
+t (z)E − EM N
+t (z)
+��
+����
+4t
+N tr (GQ)
+����
+�
+= o
+�4 max(t, t2)
+N 1−εη2
+�
+.
+Thus, we conclude that
+(19)
+EM N
+t (z) − SN �
+z − 2tEM N
+t (z)
+�
+= O
+�4 max(t, t2)
+N 1−εη3
+�
+,
+where SN(z) = 2
+z for all N. Let M ∞
+t
+be defined as in Theorem 1.3, then
+M ∞
+t (z) − SN (z ��� 2tM ∞
+t (z)) = 0.
+Note for each z ∈ C+, st = − 1
+2M ∞
+t (z), ˜st = − 1
+2EM N
+t (z), and s0(z) = − 1
+2SN(z)
+satisfy the conditions of Proposition A.1, (see Appendix) and hence it follows from
+Proposition A.1 and (44) (see Appendix) that
+(20)
+EM N
+t (z) − M ∞
+t (z) = O
+�41/3 max(t, t2)1/3
+N 1/3−εη
+�
+.
+Applying (14) to (20) completes the proof of Theorem 1.3.
+3.4. Extension to uniform bound over [0, T ]. In this section we outline how to
+extend Theorem 1.3 uniformly in t ∈ [0, T ]. This relies on the continuity of DBM.
+Without loss of generality, we work with the interval [0, 1] instead of the interval
+[0, T ]. Let us consider a partition of the time interval [0, 1] into a uniform partition
+with tk = k
+n, k = 0, 1, . . . , n. The intervals of this partition are all equally-sized and
+their lengths are equal to 1
+n.
+Let us consider t ∈ (t1, t2) an intermediate time. We have that
+sup
+z∈G
+|M ∞
+t (z) − M N
+t (z)|
+≤ sup
+z∈G
+|M ∞
+t (z) − M ∞
+t1 (z)| + sup
+z∈G
+|M ∞
+t1 (z) − M N
+t1 (z)| + sup
+z∈G
+|M N
+t1 (z) − M N
+t (z)|,
+(21)
+with G being a particular subset of the complement of the hull as in the previous
+section. The first term can be controlled from the Burgers equation as the solution
+is locally Lipschitz in time.
+For the second term of the right hand side of (21), we have that from Theorem
+1.3, for any ǫ > 0
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+13
+(22)
+sup
+z∈G
+|M ∞
+t1 (z) − M N
+t1 (z)| = Oε
+�
+1
+N 1/3−ǫ
+�
+with overwhelming probability, that is with probability at least 1 − e−cN, for some
+constant c.
+By a union bound for any tj, j = 1, . . . , n in the net, we have that
+P
+��
+ti
+|M ∞
+ti (z) − M N
+ti (z)| = Ω
+�
+C
+N 1/3−ǫ
+��
+≤
+n
+�
+i=1
+P
+�
+|M ∞
+ti (z) − M N
+ti (z)| = Ω
+�
+C
+N 1/3−ǫ
+��
+≤ ne−CN,
+(23)
+where g = Ω(f) means g(x)
+f(x), as x → ∞.
+For the third term of the right hand side of (21), using the notation ˜ηi
+t = z − λi
+t,
+for i = 1, 2, . . ., N, we have that
+(24)
+|M N
+t1 (z) − M N
+t (z)| ≤ 2
+N
+N
+�
+i=1
+|λi
+t − λi
+t1|
+|˜ηi
+t1 ˜ηi
+t|
+≤
+˜C|t − t1|1/2−ǫ
+Im(z0)2
+,
+where we have used the regularity of the Dyson Brownian Motion driver ( [43])
+and the bound |˜ηi
+t| ≥ | Im(z)| ≥ | Im(z0)| where z0 ∈ H such that Im(z0) ≤
+minz∈G(Im(z)).
+Using the notation ˆC =
+˜
+C
+Im(z0)2 , if we want the error to not accumulate in our
+net we need
+ˆC
+1
+n1/2−ǫ ≤
+C
+N 1/3−ǫ .
+Thus, for our partition of the time interval we have
+n >
+ˆC2(N (1/3−ǫ))2
+C2
+,
+for ˆC and C some constants. It then follows from (21), that
+(25)
+sup
+t∈[0,1], z∈G
+��M N
+t (z) − M ∞
+t (z)
+�� = O
+�
+1
+N
+1
+3 −ε
+�
+.
+4. Proof of Theorem 1.1
+In this section we will complete the proof of Theorem 1.1. Fix ε > 0. Let G be
+a suitable compact subset of C+ and let ˜G be a compact subset of C+ such that
+gN
+t (G) ⊆ ˜G with overwhelming probability (see (6) for the existence of such a ˜G).
+
+14
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Begin by defining η := minz∈ ˜
+G(Im z) > 0. Note that
+|gN
+t (z) − g∞
+t (z)| =
+����
+� t
+0
+M N
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z))ds
+����
+(26)
+≤
+����
+� t
+0
+M N
+s (gN
+s (z)) − M ∞
+s (gN
+s (z))ds
+����
+(27)
++
+����
+� t
+0
+M ∞
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z))ds
+���� .
+For the term M N
+s (gN
+s (z)) − M ∞
+s (gN
+s (z)), observe that from Theorem 1.3
+sup
+z∈ ˜
+G
+��M N
+s (z) − M ∞
+s (z)
+�� = O
+� 4T 2
+N
+1
+3 −ε
+�
+,
+for fixed s ∈ [0, T ] with overwhelming probability. From the argument in Section
+3.4 this can be extended to
+(28)
+sup
+s∈[0,T ], z∈ ˜
+G
+��M N
+s (z) − M ∞
+s (z)
+�� = O
+� 4T 2
+N
+1
+3 −ε
+�
+.
+For the term M ∞
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z)), note that M ∞
+s
+is at most
+2
+η2 -Lipschitz on
+˜G, and hence
+(29)
+��M ∞
+s (gN
+s (z)) − M ∞
+s (g∞
+s (z))
+�� ≤ 2
+η2 |gN
+t (z) − g∞
+t (z)|.
+From (26), (28), and (29), we conclude that
+|gN
+t (z) − g∞
+t (z)| ≤ O
+� 4T 2
+N
+1
+3 −ε
+�
++
+� t
+0
+2
+η2 |gN
+s (z) − g∞
+s (z)|ds.
+Theorem 1.1 then follows from Gr¨onwall’s iequality.
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+15
+Appendix A. Stability
+The following is essentially a result of O’Rourke and Vu ( [44]). We provide the
+details for the time change for the convenience of the reader.
+Proposition A.1 (Stability for positive time). Let t > 0 and z, st, ˜st be elements
+of the upper half-plane such that
+(30)
+st = s0(z + 4tst),
+and
+(31)
+˜st = s0(z + 4t˜st) + O(ε),
+with s0(z) =
+�
+R
+dµ0
+x−z for some compactly supported probability measure µ0, some
+R ≥ z, and small ε > 0. Additionally assume there exists η > 0 such that Im(z) ≥ η.
+Then s, s′ = O(1) and
+(32)
+st = ˜st + O
+�
+ε1/3
+(4t)2/3η
+�
+.
+Proof. Showing st, ˜st = O(1) requires no change from O’Rourke and Vu.
+Let
+wt = z + 4tst and ˜wt = z + 4t˜st. We aim now to show |wt − ˜wt| is sufficiently
+small It follows from (30) and (31) that
+(33)
+s0(wt) − s0( ˜wt) = wt − ˜wt
+4t
++ O(ε).
+It additionally follows from the definition of s0 that
+s0(wt) − s0( ˜wt) = (wt − ˜wt)
+�
+R
+dµ0(z)
+(x − wt)(x − ˜wt),
+which when combined with (33) yields
+(34)
+�
+R
+dµ0(z)
+(x − wt)(x − ˜wt) = 1
+4t + O
+�
+ε
+|wt − ˜wt|
+�
+.
+On the other hand
+Im(st) = Im(s0(wt))
+= Im(wt)
+�
+R
+dµ0(x)
+|x − wt|2 ,
+(35)
+and rearranging yields
+(36)
+�
+R
+dµ0(x)
+|x − wt|2 =≤ 1
+4t.
+An identical argument yields
+(37)
+�
+R
+dµ0(x)
+|x − ˜wt|2 =≤ 1
+4t + O
+�ε
+η
+�
+.
+From the arithmetic mean-geometric mean inequality, we have
+����
+1
+(x − wt) (x − ˜wt)
+���� ≤ 1
+2
+1
+|x − wt|2 + 1
+2
+1
+|x − ˜wt|2 .
+
+16
+A. CAMPBELL, K. LUH, AND V. MARGARINT
+Since wt ̸= ˜wt, it follows that
+����Re
+�
+1
+(x − wt) (x − ˜wt)
+����� = (1 − δ)
+�
+1
+2
+1
+|x − wt|2 + 1
+2
+1
+|x − ˜wt|2
+�
+for some δ > 0. Then we have
+|x − wt| = (1 + O(δ)) |x − ˜wt| .
+and
+∠ (x − wt, x − ˜wt) = O
+�
+δ1/2�
+.
+Since x, wt, ˜wt = O(1), we obtain wt− ˜wt = O
+�
+δ1/2�
+. We obtain that Re
+�
+1
+(x−wt)(x− ˜
+wt)
+�
+≤
+�
+1 − C |wt − ˜wt|2� �
+1
+2
+1
+|x−wt|2 + 1
+2
+1
+|x− ˜wt|2
+�
+for some C > 0, and hence
+Re
+�
+R
+dµ(x)
+(x − wt) (x − ˜wt) ≤
+�
+1 − C |wt − ˜wt|2� � 1
+4t + O
+�ε
+η
+��
+.
+We have that
+(38)
+�
+1 − C|wt − ˜wt|2� � 1
+4t + O
+� ǫ
+η
+��
+= 1
+4t + O
+�
+ǫ
+|wt − ˜wt|
+�
+.
+Then,
+(39)
+1
+4t + O
+� ǫ
+η
+�
+− c
+4t|wt − ˜wt|2 − C|wt − ˜wt|2O
+� ǫ
+η
+�
+= 1
+4t + O
+�
+ǫ
+|wt − ˜wt|
+�
+.
+Furthermore, we obtain
+(40)
+O (ǫ) = |wt − ˜wt|O
+� ǫ
+η
+�
+− C
+4t|wt − ˜wt|3 − C|wt − ˜wt|3O
+� ǫ
+η
+�
+.
+Using that the first and the third term are bounded we obtain
+(41)
+O(ǫ) + O
+� ǫ
+η
+�
+= |wt − ˜wt|3 C
+4t.
+Thus, we have that
+(42)
+|wt − ˜wt| = O
+��4tǫ
+η
+�1/3�
+,
+and
+(43)
+|st − ˜st| = O
+�
+ǫ1/3
+(4t)2/3η1/3
+�
+.
+□
+For small t the following observation is useful. Fix η > 0, then s0 is Lipschitz
+with Lipschitz constant at most
+1
+η2 on {z : Im(z) ≥ η}. Thus for st and ˜st as in
+(30) and (31) one has
+(44)
+st − ˜st = 4t
+η2 (st − ˜st) + O (ε) .
+
+RATE OF CONVERGENCE IN MULTIPLE SLE USING RANDOM MATRIX THEORY
+17
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+Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO
+80309-0395, USA
+Email address: [email protected]
+Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO
+80309-0395, USA
+Email address: [email protected]
+Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO
+80309-0395, USA
+Email address: [email protected]
+

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+Graphene amplifier reaches the quantum-noise limit
+Kin Chung Fong1, ∗
+1Raytheon BBN Technologies, Quantum Engineering and Computing Group, Cambridge, Massachusetts 02138, USA
+(Dated: January 13, 2023)
+To make yourself heard in a noisy environment
+is no easy task.
+An amplifier, like a megaphone,
+can come to your rescue by increasing your voice’s
+volume over the background noise.
+Your speech
+can be heard clearly.
+This is analogous to mea-
+suring superconducting qubits. Since their energy
+quanta are a few orders of magnitude smaller than
+the thermal noise, amplifiers are necessary to boost
+the signal up before being registered by apparatus
+at room temperature. However, having a high gain
+from amplifiers is not nearly enough. The physical
+process of amplification is also subjected to fluctua-
+tions, resulting in added noise by the amplifier that
+can degrade the signal-to-noise ratio. For a phase-
+insensitive linear amplifier, the minimum amount of
+this added noise is half a quantum because of quan-
+tum fluctuations[1].
+Employing a parametric pro-
+cess to achieve this fundamental limit of amplifica-
+tion at radio and microwave frequencies has a long
+history: from using variable-capacitor diodes in the
+1960s to Josephson junctions in the 1980s[2]. With
+high gain and low noise, modern Josephson para-
+metric amplifiers (JPAs)[3] have quickly become a
+must-have in laboratories[4]: enabling high-fidelity
+qubit readouts, observing quantum jumps, tracking
+quantum trajectory, and even searching for the rare
+event when the axion dark matter converts into a
+microwave photon under a high magnetic field.
+Writing in Nature Nanotechnology, two indepen-
+dent reports by Guilliam Butseraen, et. al.[5] and
+Joydip Sarkar, et. al.[6], now further advance JPAs
+using a two-dimensional material—graphene.
+We can consult children on swings about the pro-
+cess of parametric amplification[7] (Fig. 1a). A child
+can amplify the pendulum oscillation by standing
+up and squatting down when the swing reaches its
+maximum and minimum height, respectively. This
+stand-and-squat action is pumping the pendulum
+motion at twice its resonant frequency, and doing
+work on the harmonic oscillator. The amplitude of
+the originally small oscillation (signal) will gradu-
+ally increase, i.e. amplify. Parametric amplification
+sets apart from other amplification mechanisms be-
+∗ [email protected]
+cause it modulates the reactance, rather than the
+resistance of the system. As such, it minimizes the
+noise that is inevitably brought into the amplifica-
+tion process according to the fluctuation-dissipation
+theorem.
+A JPA constructed by a superconducting LC res-
+onator, schematically shown in Fig.
+1b, operates
+under the similar principle. In addition to suppress-
+ing dissipation, superconductors can form Josephson
+junctions, which can provide an inductance—from
+the inertia of the Cooper pairs tunneling through the
+junction—to the circuit. When two Josephson junc-
+tions are made in the form of a loop, their supercur-
+rents will interfere and form a device known as super-
+conducting quantum interference device (SQUID)
+(Fig. 1c-d). Due to the Aharonov-Bohm phase, the
+total critical current of the SQUID depends on the
+magnetic flux through the loop. A magnetic field
+generated by an electrical current can control the
+Josephson inductance.
+Hence, we can pump the
+JPA by modulating its resonant frequency with a
+pump current at twice the frequency of the super-
+conducting resonators. Alternatively, JPAs can be
+powered up by feeding the pump tone directly into
+the input port for parametric amplification.
+The
+second method exploits the non-linear dependence
+of Josephson inductance on the magnitude of the
+current running through the junction, which allow
+for non-degenerate parametric amplifications. JPAs
+operate as a reflection amplifier: when the minute
+signal enters the JPA via the coupling capacitor, it
+is parametrically amplified before making its way
+back to the input port.
+The two research teams now replace the insula-
+tor traditionally used as the Josephson weak link
+with a layer of graphene encapsulated by hexagonal-
+boron nitride (Fig.
+1e-f).
+By doing so, the re-
+searchers can employ the gate voltage response of
+the graphene to tune the Josephson critical current
+and thus the Josephson inductance.
+Hence, they
+can control the resonant frequency of the JPA by
+voltage, rather than current. Butseraen, et. al. at-
+tain parametric amplification using a transmission-
+line resonator at ∼5 GHz and a pump tone through
+the gate; whereas Sarkar, et. al. exploit lump com-
+ponents and a pump tone applied at the input port,
+arXiv:2301.04730v1  [cond-mat.mes-hall]  11 Jan 2023
+
+FIG. 1. (a) Parametric amplification in a mechanical oscillator. The red dot marks the center of mass of the child
+in the swing. Its oscillation periodically modulates the effective length of the pendulum, and thus, the resonant
+frequency. (b) Schematic model of the parametric amplifier at radio or microwave frequencies. The signal will enter
+the LC resonator and be amplified by the modulation of the inductance before leaving the resonator. (c) A SQUID as
+an implementation of the variable inductor. The Josephson inductance depends inversely on the total critical current,
+which is controlled by the magnetic flux through the loop. (d) Josephson junctions based on superconductor-insulator-
+superconductor (SIS) heterostructure. (e) Graphene-based Josephson junction with a gate control can operate as
+a variable inductor. (f) Josephson junctions based on superconductor-normal metal-superconductor (SNS) lateral
+junctions. The latest reports demonstrate JPA by SNS junction with graphene, encapsulated in hexagonal-boron
+nitride (blue), as the weak link. This new JPA is voltage-tunable and can operate under a high-magnetic field.
+respectively.
+They both achieve similar figures of
+merit: ∼10 MHz bandwidth, ∼500 MHz tuning of
+the resonant frequency with a gain of >20 dB, and
+1-dB compression point at about -127 dBm of in-
+put power.
+Most importantly, both teams show
+that the added noise from the graphene JPA can
+reach the quantum limit. This is a pleasant surprise
+given that graphene Josephson junctions are more
+dissipative with a lower quality-factor at microwave
+frequency[8], than the conventional superconductor-
+insulator-superconductor (SIS) junctions.
+With
+their careful designs and implementations, the two
+teams demonstrate that a superconductor-normal
+metal-superconductor (SNS) junction can overcome
+its intrinsic dissipation to achieve quantum-limited
+amplification.
+Given the ubiquity of JPAs based on SQUIDs, it
+is natural to question the utility of graphene JPA
+beyond mere scientific curiosity. One of the poten-
+tial benefits may lie in the mitigation of cross-talks
+among different tunable components in the quantum
+circuitry due to the magnetic field generated by a
+current bias. Without the SQUID loop, graphene
+JPAs can also be more densely packed and oper-
+ate under a larger in-plane magnetic field.
+More-
+over, these latest results exemplify the renewed in-
+terest in applying SNS junctions to quantum sci-
+ence. Unlike their insulating counterpart, SNS junc-
+tions possess more variety in their physical proper-
+ties, arising from the interplay of the superconduc-
+tor and the materials in the weak link.
+Not only
+would it enable voltage control of quantum ampli-
+fiers as shown here, but, by developing this mo-
+tif, we can also create new hybrid superconducting
+electronics[9], qubits[10, 11], high-sensitivity pho-
+ton detectors[12], and topological superconductivity
+which may host Majorana zero modes[13]. With the
+rise of quantum technology, we shall expect more
+innovations from SNS junctions to come!
+The author declares no conflict of interest.
+[1] C. M. Caves, Physical Review D 26, 1817 (1982).
+[2] B. Yurke, P. G. Kaminsky, R. E. Miller, E. A. Whit-
+taker, A. D. Smith, A. H. Silver, and R. W. Simon,
+Physical Review Letters 60, 764 (1987).
+[3] M. A. Castellanos-Beltran and K. W. Lehnert, Ap-
+plied Physics Letters 91, 083509 (2007).
+[4] J. Aumentado, IEEE Microwave Magazine 21, 45
+(2020).
+[5] G. Butseraen, A. Ranadive, N. Aparicio, K. R.
+Amin,
+A.
+Juyal,
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+2
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+**
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+gateT. Taniguchi, N. Roch, F. Lefloch, et al., Nature
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+[6] J. Sarkar, K. V. Salunkhe, S. Mandal, S. Ghatak,
+A.
+H.
+Marchawala,
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+Watanabe,
+T. Taniguchi, R. Vijay, and M. M. Deshmukh, Na-
+ture Nanotechnology 17, 1147 (2022).
+[7] D. Rugar and P. Grtter, Physical Review Letters
+67, 699 (1991).
+[8] R. Haller,
+G. Flp,
+D. Indolese,
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+R. Kraft, L. Y. Cheung, J. H. Ungerer, K. Watan-
+abe, T. Taniguchi, D. Beckmann, et al., Physical
+Review Research 4, 013198 (2022).
+[9] J. J. A. Baselmans, A. F. Morpurgo, B. J. v. Wees,
+and T. M. Klapwijk, Nature 397, 43 (1999).
+[10] J. I.-J. Wang, D. Rodan-Legrain, L. Bretheau, D. L.
+Campbell, B. Kannan, D. Kim, M. Kjaergaard,
+P. Krantz, G. O. Samach, F. Yan, et al., Nature
+Nanotechnology 14, 120 (2019).
+[11] M. Hays, V. Fatemi, D. Bouman, J. Cerrillo, S. Dia-
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+grd, A. L. Yeyati, et al., Science 373, 430 (2021).
+[12] E. D. Walsh, W. Jung, G.-H. Lee, D. K. Efetov,
+B.-I. Wu, K. F. Huang, T. A. Ohki, T. Taniguchi,
+K. Watanabe, P. Kim, et al., Science 372, 409
+(2021).
+[13] A. Fornieri, A. M. Whiticar, F. Setiawan, E. Por-
+tols, A. C. C. Drachmann, A. Keselman, S. Gronin,
+C. Thomas, T. Wang, R. Kallaher, et al., Nature
+569, 89 (2019).
+3
+

TtE3T4oBgHgl3EQfzwsF/content/tmp_files/load_file.txt

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+page_content='Graphene amplifier reaches the quantum-noise limit  Kin Chung Fong1, ∗  1Raytheon BBN Technologies, Quantum Engineering and Computing Group, Cambridge, Massachusetts 02138, USA  (Dated: January 13, 2023)  To make yourself heard in a noisy environment  is no easy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  An amplifier, like a megaphone,  can come to your rescue by increasing your voice’s  volume over the background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Your speech  can be heard clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  This is analogous to mea-  suring superconducting qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Since their energy  quanta are a few orders of magnitude smaller than  the thermal noise, amplifiers are necessary to boost  the signal up before being registered by apparatus  at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' However, having a high gain  from amplifiers is not nearly enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The physical  process of amplification is also subjected to fluctua-  tions, resulting in added noise by the amplifier that  can degrade the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' For a phase-  insensitive linear amplifier, the minimum amount of  this added noise is half a quantum because of quan-  tum fluctuations[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Employing a parametric pro-  cess to achieve this fundamental limit of amplifica-  tion at radio and microwave frequencies has a long  history: from using variable-capacitor diodes in the  1960s to Josephson junctions in the 1980s[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' With  high gain and low noise, modern Josephson para-  metric amplifiers (JPAs)[3] have quickly become a  must-have in laboratories[4]: enabling high-fidelity  qubit readouts, observing quantum jumps, tracking  quantum trajectory, and even searching for the rare  event when the axion dark matter converts into a  microwave photon under a high magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Writing in Nature Nanotechnology, two indepen-  dent reports by Guilliam Butseraen, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' [5] and  Joydip Sarkar, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' [6], now further advance JPAs  using a two-dimensional material—graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  We can consult children on swings about the pro-  cess of parametric amplification[7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' A child  can amplify the pendulum oscillation by standing  up and squatting down when the swing reaches its  maximum and minimum height, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' This  stand-and-squat action is pumping the pendulum  motion at twice its resonant frequency, and doing  work on the harmonic oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The amplitude of  the originally small oscillation (signal) will gradu-  ally increase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' amplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Parametric amplification  sets apart from other amplification mechanisms be-  ∗ fongkc@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='com  cause it modulates the reactance, rather than the  resistance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' As such, it minimizes the  noise that is inevitably brought into the amplifica-  tion process according to the fluctuation-dissipation  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  A JPA constructed by a superconducting LC res-  onator, schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  1b, operates  under the similar principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' In addition to suppress-  ing dissipation, superconductors can form Josephson  junctions, which can provide an inductance—from  the inertia of the Cooper pairs tunneling through the  junction—to the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' When two Josephson junc-  tions are made in the form of a loop, their supercur-  rents will interfere and form a device known as super-  conducting quantum interference device (SQUID)  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' 1c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Due to the Aharonov-Bohm phase, the  total critical current of the SQUID depends on the  magnetic flux through the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' A magnetic field  generated by an electrical current can control the  Josephson inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Hence, we can pump the  JPA by modulating its resonant frequency with a  pump current at twice the frequency of the super-  conducting resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Alternatively, JPAs can be  powered up by feeding the pump tone directly into  the input port for parametric amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  The  second method exploits the non-linear dependence  of Josephson inductance on the magnitude of the  current running through the junction, which allow  for non-degenerate parametric amplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' JPAs  operate as a reflection amplifier: when the minute  signal enters the JPA via the coupling capacitor, it  is parametrically amplified before making its way  back to the input port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  The two research teams now replace the insula-  tor traditionally used as the Josephson weak link  with a layer of graphene encapsulated by hexagonal-  boron nitride (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  1e-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  By doing so, the re-  searchers can employ the gate voltage response of  the graphene to tune the Josephson critical current  and thus the Josephson inductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Hence, they  can control the resonant frequency of the JPA by  voltage, rather than current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Butseraen, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' at-  tain parametric amplification using a transmission-  line resonator at ∼5 GHz and a pump tone through  the gate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' whereas Sarkar, et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' exploit lump com-  ponents and a pump tone applied at the input port,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='04730v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='mes-hall]  11 Jan 2023  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (a) Parametric amplification in a mechanical oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The red dot marks the center of mass of the child  in the swing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Its oscillation periodically modulates the effective length of the pendulum, and thus, the resonant  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (b) Schematic model of the parametric amplifier at radio or microwave frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The signal will enter  the LC resonator and be amplified by the modulation of the inductance before leaving the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (c) A SQUID as  an implementation of the variable inductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The Josephson inductance depends inversely on the total critical current,  which is controlled by the magnetic flux through the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (d) Josephson junctions based on superconductor-insulator-  superconductor (SIS) heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (e) Graphene-based Josephson junction with a gate control can operate as  a variable inductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' (f) Josephson junctions based on superconductor-normal metal-superconductor (SNS) lateral  junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' The latest reports demonstrate JPA by SNS junction with graphene, encapsulated in hexagonal-boron  nitride (blue), as the weak link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' This new JPA is voltage-tunable and can operate under a high-magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  They both achieve similar figures of  merit: ∼10 MHz bandwidth, ∼500 MHz tuning of  the resonant frequency with a gain of >20 dB, and  1-dB compression point at about -127 dBm of in-  put power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Most importantly, both teams show  that the added noise from the graphene JPA can  reach the quantum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' This is a pleasant surprise  given that graphene Josephson junctions are more  dissipative with a lower quality-factor at microwave  frequency[8], than the conventional superconductor-  insulator-superconductor (SIS) junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  With  their careful designs and implementations, the two  teams demonstrate that a superconductor-normal  metal-superconductor (SNS) junction can overcome  its intrinsic dissipation to achieve quantum-limited  amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Given the ubiquity of JPAs based on SQUIDs, it  is natural to question the utility of graphene JPA  beyond mere scientific curiosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' One of the poten-  tial benefits may lie in the mitigation of cross-talks  among different tunable components in the quantum  circuitry due to the magnetic field generated by a  current bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Without the SQUID loop, graphene  JPAs can also be more densely packed and oper-  ate under a larger in-plane magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  More-  over, these latest results exemplify the renewed in-  terest in applying SNS junctions to quantum sci-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' Unlike their insulating counterpart, SNS junc-  tions possess more variety in their physical proper-  ties, arising from the interplay of the superconduc-  tor and the materials in the weak link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  Not only  would it enable voltage control of quantum ampli-  fiers as shown here, but, by developing this mo-  tif, we can also create new hybrid superconducting  electronics[9], qubits[10, 11], high-sensitivity pho-  ton detectors[12], and topological superconductivity  which may host Majorana zero modes[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content=' With the  rise of quantum technology, we shall expect more  innovations from SNS junctions to come!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
+page_content='  The author declares no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE3T4oBgHgl3EQfzwsF/content/2301.04730v1.pdf'}
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U9AyT4oBgHgl3EQf8vpk/content/tmp_files/2301.00861v1.pdf.txt

@@ -0,0 +1,916 @@
+Hardware Abstractions and Hardware Mechanisms to Support
+Multi-Task Execution on Coarse-Grained Reconfigurable Arrays
+Taeyoung Kong, Kalhan Koul, Priyanka Raina, Mark Horowitz, and Christopher Torng
+Stanford University
+{kongty,kkoul,praina,horowitz,ctorng}@stanford.edu
+Abstract
+Domain-specific accelerators are used in various com-
+puting systems ranging from edge devices to data centers.
+Coarse-grained reconfigurable arrays (CGRAs) represent
+an architectural midpoint between the flexibility of an
+FPGA and the efficiency of an ASIC and are a promising
+candidate for servicing multi-tasked workloads within an
+application domain. Unfortunately, scheduling multiple
+tasks onto a CGRA is challenging. CGRAs lack abstrac-
+tions that capture hardware resources, leaving workload
+schedulers unable to reason about performance, energy,
+and utilization for different schedules. This work first pro-
+poses a CGRA architecture that can flexibly partition key
+resources, including the global buffer memory capacity,
+the global buffer memory bandwidth, and the compute re-
+sources. Partitioned resources serve as hardware abstrac-
+tions that decouple compilation and resource allocation.
+The compiler uses these abstractions for coarse-grained
+resource mapping, and the scheduler uses them for flexi-
+ble resource allocation at run time. We then propose two
+hardware mechanisms to support multi-task execution.
+A flexible-shape execution region increases the overall
+resource utilization by mapping multiple tasks with dif-
+ferent resource requirements. Dynamic partial reconfig-
+uration (DPR) enables a CGRA to update the hardware
+configuration as the scheduler makes decisions rapidly.
+We show that our abstraction can help automatic and
+efficient scheduling of multi-tasked workloads onto our
+target CGRA with high utilization, resulting in 1.05x–
+1.24x higher throughput and a 23–28% lower latency in a
+multi-tasked cloud workload and 60.8% reduced latency
+in an autonomous system workload when compared to a
+baseline CGRA running single tasks at a time.
+1. Introduction
+Domain-specific accelerators have gained growing inter-
+est in recent years as they provide improved performance
+and energy efficiency over general-purpose processors.
+Application-specific integrated circuits (ASICs) [8, 18,
+21] show the highest performance and efficiency as they
+are specialized for target applications such as image pro-
+cessing or machine learning (ML). However, the ASIC
+design process can span multiple years, and fixed-function
+accelerators quickly become obsolete as applications con-
+tinue to evolve. Some works deploy applications on FP-
+GAs [12, 16, 17]. FPGAs enable reconfiguration of the
+underlying hardware and can accelerate diverse work-
+loads, but their bit-level flexibility incurs high area and
+energy overheads. Coarse-grained reconfigurable arrays
+(CGRAs) are promising architectures that lie between
+ASICs and FPGAs. A CGRA has arithmetic units and a
+routing system that are configurable in word-level gran-
+ularity, providing flexibility at a lower overhead than
+a FPGA. With its unique advantages, a CGRA can be
+widely adopted in domains with high performance, power,
+and flexibility requirements.
+As hardware accelerators are deployed in various sce-
+narios, the demand for multi-task execution support on
+hardware is growing. For example, many vendors [21, 13]
+offer INFerence-as-a-Service, where multiple tenants
+share the same hardware to run inference tasks. Also, an
+autonomous system handles concurrent tasks to process
+various types of data from numerous sensors. Some works
+have explored multi-task execution support in ASICs and
+FPGAs. PREMA [11] and Planaria [14] propose a sys-
+tolic array that supports multi-tenancy by temporal and
+spatial multiplexing, respectively. [35, 29, 34] propose an
+FPGA virtualization framework with multi-tenancy sup-
+port. However, multi-task execution support on CGRAs
+has not been explored much thus far. A noteworthy ex-
+ception is ChordMap [27] which schedules multiple tasks
+captured in synchronous data flow graphs onto a CGRA.
+However, it assumes that all tasks are known a priori,
+whereas in a multi-tenant cloud or multi-tasked edge work-
+load scenario, tasks may arrive dynamically and require
+schedulers to react to maximize utilization.
+Unfortunately, scheduling multiple tasks onto a CGRA
+is challenging as it lacks abstractions capturing hardware
+resources. In this paper, we propose hardware abstrac-
+tions of a CGRA by partitioning key hardware resources.
+Both compilers and schedulers can exploit the abstrac-
+arXiv:2301.00861v1  [cs.AR]  2 Jan 2023
+
+tions to reason about performance, energy, and utilization.
+We also develop hardware mechanisms that allow fast and
+flexible multi-task execution on a CGRA, which sched-
+ulers exploit to improve hardware utilization. We evaluate
+our CGRA with two different multi-tasked workload sce-
+narios to show the potential. Our key contributions are:
+• 1⃝ We propose a CGRA architecture that can flexibly
+re-partition key resources, including the Global Buffer
+(GLB) memory capacity, the GLB memory bandwidth,
+and the compute resources. Specifically, we partition
+the GLB into GLB-slices and the tile array into array-
+slices, which serve as hardware abstractions. The com-
+piler uses these abstractions for coarse-grain resource
+mapping, while the scheduler uses them for flexible
+resource allocation.
+• 2⃝ We propose two hardware mechanisms to support
+multi-task execution on the CGRA. First, the CGRA
+can form a flexible-shape execution region at run time.
+It improves resource utilization by enabling a scheduler
+to allocate GLB-slices and array-slices flexibly. Sec-
+ond, we propose a fast-DPR method to reconfigure the
+underlying hardware rapidly according to scheduler de-
+cisions. It also supports run time relocation of a task to
+any available array-slice without software intervention.
+• 3⃝ We quantify the benefits of our proposed mecha-
+nisms on two different examples. Our CGRA with
+flexible execution regions and fast-DPR shows 1.05x–
+1.24x higher throughput and 23–28% lower latency in a
+cloud system scenario and 60.8% reduced latency in an
+autonomous system scenario than the baseline CGRA.
+2. Architectural Support for Multi-Task Ex-
+ecution on a CGRA
+In this section, we explore the architectural support
+needed for multi-task execution on a CGRA. Section 2.1
+first introduces a baseline CGRA architecture with com-
+mon features present in many reconfigurable accelera-
+tors [7, 32, 15, 6, 1, 28]. Section 2.2 then introduces how
+we abstract the hardware resources in the CGRA for the
+scheduler by partitioning the global buffer (GLB) and
+the tile array into GLB-slices and array-slices, respec-
+tively. We further develop hardware mechanisms that
+enable multi-task execution on top of these abstractions
+(Section 2.3), including flexible-shape execution regions
+and dynamic partial reconfiguration (DPR).
+2.1. Baseline CGRA Architecture
+Our baseline CGRA consists of a tile array with pro-
+cessing element (PE) and memory (MEM) tiles and a
+global buffer (GLB) (Figure 1). We leverage almost the
+same hardware configuration used in the Amber SoC [7].
+The CGRA has 32x16 tiles with 384 PE tiles and 128
+MEM tiles, and tiles communicate through a statically
+Figure 1: Baseline CGRA block diagram corresponding to [23].
+App.
+Task
+Ver.
+Tpt.
+Array
+slices
+GLB
+slices
+ResNet-18
+conv2_x
+a
+64
+2
+7
+b
+256
+6
+7
+conv3_x
+a
+64
+2
+4
+b
+256
+6
+4
+conv4_x
+a
+64
+2
+6
+b
+256
+6
+6
+conv5_x
+a
+64
+2
+20
+b
+128
+6
+20
+MobileNet
+conv_dw
+_pw_2_x
+1
+a
+52
+2
+4
+b
+208
+5
+4
+conv_dw
+_pw_3_x
+a
+52
+2
+4
+b
+104
+3
+4
+conv_dw
+_pw_4_x
+a
+52
+2
+4
+b
+104
+3
+4
+Camera
+pipeline
+Camera
+pipeline
+a
+3
+4
+4
+b
+12
+6
+14
+Harris
+Harris
+a
+1
+2
+4
+b
+2
+4
+7
+c
+4
+7
+14
+Table 1: Variants of tasks with different resource usage and
+throughput. ResNet-18 and MobileNet consist of several lay-
+ers, and one or more layers form a single task. The unit of
+throughput (Tpt.) for ResNet-18 and MobileNet is MACs/cycle
+and for camera pipeline and harris it is pixels/cycle.
+configured mesh interconnect. A PE tile is extended from
+Amber version to support MAC operation. Each node
+in the interconnect has five incoming and five outgoing
+tracks in each direction, and switch boxes route data from
+incoming tracks to outgoing tracks. Connection boxes
+select data from incoming tracks and route it to the PE
+or MEM tile cores. The GLB consists of 32 banks, with
+each bank containing 128 KB of SRAM. Each GLB bank
+directly communicates with the tile array through IO tiles
+located at the top of the array.
+2.2. A Scheduler-Visible Abstraction of Hardware Re-
+sources
+We focus on three key hardware resources within the
+CGRA (Figure 1): the GLB memory capacity, the GLB
+1A conv_dw_pw refers to a merged task of a depth-wise convolu-
+tional layer and a point-wise convolutional layer.
+2
+
+CGRA
+GLB
+PE
+PE
+MEM
+Bank,
+Global Buffer (GLB)
+用
+PE
+PE
+用
+MEM
++
+CGRA Interconnect
+PE
+Tiles
+PE
+PE
+MEM
+MEM
+Tiles
+Routing
+Connection
+Switch
+Tracks
+Box
+Box(a) Baseline
+(b) Fixed-sized execution region
+(c) Variably sized execution region
+(d) Flexible-shape execution region
+Figure 2: Resource allocation in the baseline CGRA and a CGRA with three different execution regions. Resources colored grey
+represent the blocks occupied by a current-running task, and those colored red represent blocks occupied by a next-running task.
+memory bandwidth, and the compute resources within
+the tile array. When a task is compiled in the Amber
+toolchain [23], a compiler converts it into a dataflow
+graph where each node and edge represents a hardware
+resource and communication, respectively. Specifically,
+GLB banks are used for medium-sized storage and com-
+munication to the host and tile array, and PE and MEM
+tiles are used for computation and as small scratchpads.
+The dataflow graph can derive the usage of memory capac-
+ity, memory bandwidth, compute units, and throughput.
+We abstract the hardware resources by partitioning the
+GLB and tile array into homogeneous GLB-slices and
+array-slices, respectively. For example, we can abstract
+each GLB bank within our CGRA as a GLB-slice and
+every set of four columns in the tile array (48 PE tiles and
+16 MEM tiles) as an array-slice. This abstraction serves
+as a middle layer that decouples offline bitstream genera-
+tion by a compiler and run time resource allocation by a
+scheduler. During compilation, we represent the resource
+usage of each task using these abstracted GLB-slices and
+array-slices. For instance, a conv2_x layer in [19] utilizes
+750KB of GLB memory capacity, 17.3MB/s of memory
+bandwidth, 80 PE tiles, and 17 MEM tiles and achieves
+64 OPs/cycle throughput at a 500MHz clock frequency.
+The task is abstracted as seven GLB-slices and two array-
+slices in coarse-grain resource slice usage. It is possible to
+produce variants of the same task with different resource
+usage and throughput by tweaking the compiler. For ex-
+ample, increasing the unroll factor of the same task by
+four would achieve 4x throughput (256 OPs/cycle) with
+288 PE tiles, 33 MEM tiles, and the same GLB mem-
+ory capacity and bandwidth, which is abstracted as seven
+GLB-slices and six array-slices. Our approach allows
+for pre-computation of bitstreams that support different
+resource usage and throughput to be cached in on-chip
+storage to support fast dynamic partial reconfiguration, as
+discussed later. Table 1 summarizes the resource usage
+and throughput for several different variants of tasks. At
+run time, a scheduler leverages the hardware slice abstrac-
+tion to decide which variant of tasks to choose, which
+resources to allocate, and when to execute.
+2.3. Hardware Mechanisms
+Flexible-Shape Execution Regions. To manage multi-
+ple tasks that are concurrently running, we need a way
+to monitor hardware resources and the status of tasks,
+that are build upon the abstractions described above. We
+introduce an execution region, a sub-region of the CGRA
+on which a single task is mapped and executed. An ex-
+ecution region consists of one or more GLB-slices and
+array-slices. The flexibility to form different sizes and
+shapes of execution regions gives the scheduler a sim-
+plified and quantized view of hardware resources while
+providing enough information to allocate resources to
+each task to maximize resource utilization in multi-tasked
+workloads.
+Figure 2 compares different mechanisms to form an
+execution region and how they affect resource allocation.
+3
+
+CGRA
+GlobalBuffer（GLB)
+GLB-slice
+Array-slice
+Tile-ArrayCGRA
+GlobalBuffer (GLB)
+1111111111
+Tile-ArrayCGRA
+GlobalBuffer(GLB)
+Tile-ArrayCGRA
+Global Buffer (GLB)
+GLB-slice
+[available
+Multi-stageNetwork
+Array-slice
+Tile-Array
+[available]The blocks colored in gray represent resources occupied
+by the currently running task, and those colored in red rep-
+resent resources allocated to the next-running task. The
+baseline CGRA (Figure 2a) is unaware of our hardware
+slice abstraction, and the entire CGRA serves as a single
+large execution region. Since an existing task is already
+mapped onto the CGRA, subsequent tasks are always
+forced to wait until the previous tasks finish and release
+the single execution region.
+The simplest mechanism to form an execution region
+is only to support fixed-sized regions. For example, all
+execution regions in Figure 2b consist of two GLB-slices
+and one array-slice. Fixed-sized regions are not optimal.
+Since each task must fit within the fixed-sized execution
+region, the largest task with the highest resource usage
+determines the size. On the other hand, when there are
+several available execution regions, a task can be unrolled
+and mapped in parallel to achieve higher throughput (e.g.,
+the next-running task is unrolled by three in Figure 2b).
+This method does not require much architectural change,
+and the implementation of a scheduling algorithm can
+be straightforward given the assumption that all target
+tasks fit within an execution region. However, although
+unrolling increases throughput, optimization across the
+unrolled dimension can be challenging to support.
+Another method is to support variably sized execution
+regions by merging multiple fixed-sized regions. We de-
+fine the unit size of a region as in the fixed-sized region
+case, but we can merge multiple unit regions to form a
+larger execution region. For example, in Figure 2c, three
+unit-sized regions are merged to execute the next-running
+task (colored in red). The benefit of variably sized execu-
+tion regions is to allow compilation optimization across
+the unrolled dimension. For example, a camera pipeline
+task with 3 pixels/cycle throughput uses four array-slices
+(Table 1). Naively unrolling it by four achieves 12 pix-
+els/cycle throughput using 16 array-slices. However, the
+compiler can optimize to time-multiplex PE tiles and
+achieve 12 pixels/cycle throughput with only six array-
+slices. Support for a variably sized region still allows for
+the pre-computation of bitstreams for multiple variants of
+tasks with different resource usage and throughput. How-
+ever, this approach may still suffer from low resource
+utilization since the ratio of GLB-slices and array-slices
+within an execution region always remains the same.
+Therefore, we propose flexible-shape execution regions
+in which GLB-slices and array-slices are no longer cou-
+pled. Decoupling of GLB-slices and array-slices enables
+finer-grained resource allocation. For example, Figure 2d
+shows how an execution region can be allocated any
+number of GLB-slices and array-slices, forming a non-
+rectangular shape, with remaining array-slices and GLB-
+slices available to be used by other tasks. The support
+for flexible-shape execution regions improves resource
+utilization, especially for multi-tasked workloads where
+memory-intensive and compute-intensive tasks are mixed.
+However, it may require additional communication be-
+tween the GLB-slices and the array-slices. In this work,
+we limit the placement of GLB-slices and array-slices
+within an execution region to be contiguous to simplify
+our study. Design space exploration on flexible placement
+support and the required network remains as future work.
+Section 3.1 describes the benefits of these mechanisms in
+more detail with a cloud system example.
+Dynamic Partial Reconfiguration. Dynamic partial re-
+configuration (DPR) is a mechanism to update the hard-
+ware configuration in reconfigurable architectures. We
+propose fast-DPR following the DPR mechanism pro-
+posed in Amber SoC [7], but with added features to
+exploit hardware abstractions. In Amber, every other
+GLB bank stores the configuration bitstreams and inde-
+pendently streams configuration into two columns of the
+tile array. Also, clocks and configuration signals are dis-
+tributed down each column together, enabling reconfigur-
+ing the tile array at high clock frequency without pipeline
+stages. In our CGRA, we also reuse GLB blocks to store
+and stream bitstreams to the tile array and follow the same
+clock distribution network. Unlike Amber, however, one
+GLB bank streams configuration into one array-slice (in
+turn, four columns of the tile array) as an array-slice is
+the minimum unit of execution regions.
+We added a feature to relocate bitstreams at run time to
+exploit hardware abstractions further. In Amber, the com-
+piler generates region-aware bitstreams; the bitstreams
+for one region cannot be reused in different regions even
+though the two regions are homogeneous. This limitation
+comes from the fact that the address of each configuration
+register in different columns has a distinct column #id. On
+the other hand, our compiler generates region-agnostic
+bitstreams by assuming that the task is always mapped to
+the leftmost region. We also added a register indicating
+the destination region of DPR to GLB banks. When the
+host processor triggers DPR, GLB banks read the register
+and stream bitstreams to the target region via the network
+between the GLB and the tile array. With this bitstream
+relocation feature, a user can pre-load bitstreams of the
+next task to the GLB in advance and rapidly map it to any
+next available region just by writing to a single register.
+3. Evaluation
+We evaluate the benefits of multi-task execution support
+under two different workload scenarios. In a cloud sys-
+tem example scenario (Section 3.1), our CGRA with
+flexible-shape execution regions enables 1.05x-1.24x
+higher throughput and 23-28% lower normalized turn-
+around time (NTAT) over the baseline CGRA. In an au-
+tonomous system example scenario (Section 3.2), our
+CGRA enables 60.8% reduced total latency.
+4
+
+(a) Cloud system example
+(b) Autonomous system example
+Figure 3: (a) Cloud system example scenario with four tenants
+submitting requests to the CGRA. Each tenant is assigned with
+a task from MobileNet, ResNet-18, camera pipeline, and Harris,
+respectively. (b) Autonomous system example with tasks that
+may be triggered under conditions.
+3.1. Example 1: Cloud System
+Overview. In this example, we construct a synthetic
+cloud computing scenario that models real-world exam-
+ples in which the CGRA serves application requests from
+multiple users (Figure 3a). We construct the multi-tasked
+workload using kernels from machine learning (ML) and
+image processing domains, including ResNet-18 [19]
+and MobileNet [20] from the ML domain, and camera
+pipeline and Harris corner detector from the image pro-
+cessing domain. Table 1 summarizes the benchmark tasks
+and their resource requirements.
+To generate the multi-tasked workload, we assume four
+tenants share the CGRA and are assigned one of the four
+target applications. Each tenant sends a request to the
+CGRA following a Poisson distribution. Whenever a new
+task arrives, or an existing task finishes, the scheduler
+is triggered and runs a greedy algorithm to schedule the
+next available task. The scheduler checks if dependencies
+are met before scheduling the task (e.g., in ResNet-18,
+conv2_x depends on conv1_x). If there is more than one
+version of a task that can be mapped onto the available
+resources, the greedy scheduler always chooses the one
+with the highest throughput.
+Metrics. We measure Normalized Turn-Around Time and
+throughput to compare the baseline CGRA and the three
+partitioning mechanisms described in Section 2.3. Turn-
+(a) NTAT
+(b) Throughput
+Figure 4: Evaluation in a cloud system example. (a) NTAT and
+(b) throughput for each task with fixed-sized, variably sized, and
+flexible-shape resource partitioning, normalized to the baseline
+CGRA. Flexible-shape partitioning decreases NTAT by 23-28%
+and increases throughput by 1.05x-1.24x.
+Around Time (TAT) is the interval from the time of request
+to submit a task to the time of task completion. Normal-
+ized Turn-Around Time (NTAT) is the ratio of the TAT to
+the execution time, which represents the relative delay of
+a task (Equation (1) - (2)). We calculate NTAT for each re-
+quest and the arithmetic average for each application. We
+also measure the average throughput for each application
+to demonstrate the performance benefit.
+TAT = wait_time + execution_time
+(1)
+NTAT = TAT / execution_time
+(2)
+Results. Figure 4 illustrates the relative improvements
+in NTAT and throughput for flexible-shape execution re-
+gions compared to fixed- and variably-sized execution
+regions. Even with a simple greedy scheduling algo-
+rithm, we achieve 23–28% decreased NTAT and 1.05x–
+1.24x higher throughput. Note that we only pre-compile
+each task to two different variants in this case study (Ta-
+ble 1), and a scheduler greedily selects the one with higher
+throughput if resources are available. Co-optimizing com-
+pilation and scheduling policy may improve NTAT and
+throughput further, which remains future work.
+3.2. Example 2: Autonomous System
+Overview. In this case study, we construct a synthetic
+edge system scenario modeling the real world in which
+multiple tasks from image processing and ML domains ex-
+ecute in parallel and can dynamically trigger. Specifically,
+we develop an autonomous system scenario as described
+in Figure 3b following a methodology used in [30]. 2 The
+system takes a RAW image in Bayer encoding format
+(RGGB) from sensors at 30 fps and first runs a camera
+2We also changed the tasks to simplify the example.
+5
+
+User 1
+User 2
+MobileNet
+ResNet-18
+CGRA
+User 3
+User 4
+Camera
+Harris Corner
+Pipeline
+DetectorBackground detected
+Senddatatothecloud
+Depth estimation
+Image compression
+(stereo)
+(gaussian)
+CGRA
+Foreveryframe
+Sign detected
+Camera pipeline
+Sign classifier
+(camerapipeline)
+(ResNet)baseline
+Fixed
+Variable
+Flexible
+1.250
+1.000
+NTAT
+hh
+0.750
+0.500
+ResNet-18
+MobileNet
+Camera
+Harrisbaseline
+Fixed
+Variable
+Flexible
+1.25
+Throughput
+1.00
+川
+0.75
+0.50
+ResNet-18
+MobileNet
+Camera
+HarrisFigure 5: The average latency of an autonomous system
+example with different execution regions. The values are nor-
+malized to the result of the baseline. A red bar indicates the
+time spent for reconfiguration, and a blue bar indicates the
+sum of wait time and execution time. To show the benefit of
+fast-DPR (Section 2.3), we assume the baseline CGRA uses
+AXI4-Lite interface for DPR, while others use fast-DPR.
+pipeline task on the CGRA to convert to an RGB image.
+Once the CGRA generates an RGB image, the system
+runs object detection and dynamically decides on the next
+tasks. 3 When an event happens (e.g., detection of a spe-
+cific background), it processes the event and executes the
+corresponding tasks (e.g., depth estimation). Except for a
+camera pipeline that runs every frame, we set the period
+from one event to the next same event to follow a uniform
+random distribution between 3–7 frames.
+Results. We evaluate the benefit of hardware resource
+partitioning and fast DPR by comparing our proposed
+CGRA to the baseline CGRA with AXI4-Lite-based DPR.
+Specifically, the baseline CGRA maps only one task at
+a time. When more than one event occurs, the base-
+line handles each task one by one and reconfigures using
+sequential AXI4-Lite configuration transactions. In the
+proposed CGRA with multi-task execution support, we ex-
+ploit flexible-shape resource partitioning to concurrently
+run more than one task on the CGRA when possible. Also,
+we use the parallel and high-frequency DPR mechanisms
+in Section 2.3 to configure bitstreams. We compute the
+arithmetic average of the latency over all frames. As de-
+scribed in Figure 5, our techniques enable a 60.8% latency
+reduction compared to the baseline. With fast DPR, re-
+configuration takes less than 5% of the total latency, an
+appreciable reduction from 14.4% in the baseline.
+4. Related Work
+As Deep Neural Networks (DNNs) are widely used in vari-
+ous domains, DNN accelerators [18, 17, 8, 9, 10, 25] have
+emerged and been deployed in the cloud system [21, 13].
+To that end, many prior works have explored multi-
+tenancy support on DNN accelerators in cloud systems.
+3This work assumes that object detection is executed in another
+hardware in the system (e.g. GPU or ASIC).
+Multi-task execution support is also studied in FPGAs
+targeting both cloud and edge computing. However, a non-
+negligible portion of FPGA resources is typically reserved
+for controlling multi-task execution, ultimately decreas-
+ing the available computing resources. ChordMap [27]
+explores the automated mapping of multi-tasked applica-
+tions onto a CGRA, but it is limited to mapping multiple
+tasks within streaming applications with all tasks known
+a priori. Our work proposes hardware abstractions and
+mechanisms, which both compilers and schedulers can
+exploit and co-optimize to improve resource utilization in
+both cloud and edge systems.
+Multi-Task Execution on DNN Accelerators. Some
+DNN accelerators service multi-DNN tasks at the soft-
+ware level. AI-MT [2] and Layerweaver [31] propose a
+scheduling policy to mix compute- and memory-intensive
+tasks to increase hardware utilization. PREMA [11] im-
+plements preemptible NPUs to support multi-tenancy
+via temporal multiplexing.
+Many works add flexibil-
+ity to an accelerator to accommodate multiple DNN
+tasks. Planaria [14] introduces a flexible systolic array
+with dynamic architecture fission to map multiple DNN
+tasks. [26] suggests a multi-directional network to sup-
+port up to four DNN tasks with different dataflow. Other
+works [24, 3] explore a computing system with multiple
+DNN accelerators with different hardware characteristics.
+While these works only support DNN workloads, our
+work can support any applications that can be mapped
+onto a CGRA.
+Multi-Task Execution on FPGAs. In FPGAs, multi-
+task execution support has been explored in the context of
+virtualization. Some works divide an FPGA into a static
+region, a shell, which serves as glue logic between the
+host and the FPGA, and a dynamic region, a role, which
+handles the computation of tasks. [4, 5, 33] partition
+a physical FPGA into several fixed-size virtual blocks
+and share them across multiple tasks. AmorphOS [22]
+presents a hardware abstraction of an FPGA, Morphlet,
+which dynamically alters its size based on resource re-
+quirements. ViTAL [35] provides a full-stack framework
+to run multiple tasks with different sizes on homogeneous
+regions. [34] supports running multi-DNN tasks on an
+FPGA by dividing hardware resources into multiple PE
+cores and spatially multiplexing them, while [30] eval-
+uates the benefits of temporal multiplexing of FPGAs
+using DPR for vision applications on embedded devices.
+While these works only target scenarios where underlying
+applications change infrequently because of long reconfig-
+uration time of FPGAs, our work can support both cloud
+systems and real-time edge systems due to rapid DPR.
+5. Conclusion
+Multi-task execution support on accelerators is becoming
+increasingly relevant in both cloud and edge systems and
+6
+
+DPR
+Waiting + Execution
+1.000
+1.0.0.0
+0.750
+latency
+0.500
+0.447
+0.445
+0.392
+0.250
+0.000
+baseline
+fixed
+variable
+flexiblehas the potential to improve performance through bet-
+ter hardware utilization. This work proposes abstracting
+hardware resources within a CGRA into coarser-grained
+units with which a workload scheduler can quickly make
+decisions. Based on the proposed abstraction, we develop
+hardware mechanisms to support multi-task execution
+through flexible-shape hardware partitioning and high-
+throughput dynamic partial reconfiguration. Our evalua-
+tions modeling both a cloud and an edge system scenario
+suggest that the abstraction and hardware mechanisms can
+enable automatic schedulers to achieve high performance
+in multi-tasked workloads on future CGRAs.
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+Topological order in interacting semimetals
+Jinmin Yi,∗ Xuzhe Ying,∗ Lei Gioia, and A.A. Burkov
+Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada and
+Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
+(Dated: January 11, 2023)
+It has recently been demonstrated that it is possible to open a gap in a magnetic Weyl semimetal,
+while preserving the chiral anomaly along with the charge conservation and translational symmetries,
+which all protect the gapless nodes in a weakly interacting semimetal. The resulting state was shown
+to be a nontrivial generalization of a nonabelian fractional quantum Hall liquid to three dimensions.
+Here we point out that a second fractional quantum Hall state exists in this case.
+This state
+has exactly the same electrical and thermal Hall responses as the first, but a distinct (fracton)
+topological order. Moreover, the existence of this second fractional quantum Hall state necessarily
+implies a gapless phase, which has identical topological response to a noninteracting Weyl semimetal,
+but is distinct from it. This may be viewed as a generalization (in a weaker form) of the known
+duality between a noninteracting two-dimensional Dirac fermion and QED3 to 3 + 1 dimensions. In
+addition we discuss a (3 + 1)-dimensional topologically ordered state, obtained by gapping a nodal
+line semimetal without breaking symmetries.
+I.
+INTRODUCTION
+Topological order, a concept that originated in the
+study of the fractional quantum Hall effect (FQHE) in
+two dimensional (2D) electron gas systems,1 continues
+to be a subject of intense interest.
+From the funda-
+mental physics prospective, topologically ordered states
+provide perfect examples of emergent macroscopic quan-
+tum phenomena, with fractionally-quantized electromag-
+netic and thermal responses, that are impossible to ex-
+plain based on textbook models of weakly-interacting
+electrons.
+Instead, such fractionally-quantized observ-
+able responses necessarily imply excitations with frac-
+tional charges, fractional and nonabelian statistics, which
+can not be constructed out of any finite number of ele-
+mentary constituents.2 In addition, such exotic excita-
+tions may have future potential practical uses in quan-
+tum computing and quantum simulation, as their non-
+local topological nature makes them highly resistant to
+decoherence and noise.3
+Topologically-ordered states in 2D are by now well-
+understood.
+Various theoretical models,4–7 as well as
+complete formal classifications of 2D topological orders
+exist.8 Although significant progress has been made in re-
+cent years,9–19 less is known about topologically-ordered
+states in three dimensions (3D). 3D topologically ordered
+states are significantly different from the 2D ones. On
+the one hand, fractional statistics is impossible in 3D and
+quasiparticle excitations may only be bosons or fermions.
+This could make one doubt that, for example, fractional
+quantum Hall (FQH) states may even in principle be gen-
+eralized to 3D, as the existence of anyons, i.e. quasipar-
+ticles with fractional statistics, is an essential feature of
+the 2D FQHE. On the other hand, in addition to point
+quasiparticles, one-dimensional loop excitations exist in
+3D, which both adds complexity and opens up new in-
+teresting possibilities.
+We recently demonstrated that a promising way to
+achieve 3D topologically ordered states is through gap-
+ping topological semimetals without breaking the pro-
+tecting symmetries20–22 (see Refs. 23–29 for related
+work).
+Topological semimetals30–35 are intermediate
+phases between insulators of different electronic structure
+topology.
+They may be characterized by unquantized
+anomalies,36,37 i.e. topological terms with noninteger and
+continuously-tunable coefficients, similar to the electron
+filling parameter, characterizing ordinary Fermi liquids.
+Much like fractional filling in a Fermi liquid mandates the
+existence of a Fermi surface of gapless particle-hole ex-
+citations,38 these unquantized anomalies necessarily im-
+ply gapless modes and corresponding long-range entan-
+glement. The only way gaplessness may be circumvented
+in the absence of broken symmetries is through the for-
+mation of a topologically-ordered state, which preserves
+the anomaly and the long-range entanglement of the gap-
+less semimetal.
+Specifically, in Ref. 20 we presented an explicit con-
+struction of a 3D topologically-ordered state in a gapped
+magnetic Weyl semimetal, which exhibits a nontrivial
+generalization of the FQHE to 3D. This state is ob-
+tained starting from a magnetic Weyl semimetal with
+a single pair of nodes, separated by half a reciprocal lat-
+tice vector.
+These nodes may be gapped by breaking
+the U(1) charge conservation symmetry while forming a
+superconducting state with intra-nodal pairing. In gen-
+eral, such states break translational symmetry since the
+Weyl nodes exist at nontrivial momenta in the first Bril-
+louin zone (BZ). However, when the nodes are separated
+by exactly half a reciprocal lattice vector, such a pair-
+ing leads to density modulation at the reciprocal lattice
+vector, which does not break the crystal translational
+symmetry.
+Restoring the charge conservation symme-
+try by proliferating flux 2hc/e = 4π (we will be using
+ℏ = c = e = 1 units throughout this paper) vortices in
+the superconducting order parameter leads to a feature-
+less fractionalized insulator with Z4 topological order,
+that has the same electrical and thermal Hall conductiv-
+ities as the original noninteracting Weyl semimetal, i.e.
+arXiv:2301.03628v1  [cond-mat.str-el]  9 Jan 2023
+
+2
+exhibits FQHE in 3D. Unlike in 2D FQH liquids, quasi-
+particle excitations in this state are bosons and fermions.
+What plays the role of the anyons in the 2D FQHE are
+intersections of the vortex-loop excitations with atomic
+planes.
+These behave as fractionally-charged particles
+with semionic statistics, which may be sharply defined
+by considering three-loop braiding processes,15 involving
+a line defect of translational symmetry, i.e. an edge dis-
+location.
+In this paper we show that, in addition to the 3D FQH
+state of Ref. 20, another state exists, which has identi-
+cal topological response, but distinct topological order,
+which turns out to be of a fracton type. The existence
+of these two distinct states turns out to be closely re-
+lated to a very similar property of gapped symmetric
+2D Dirac surface states of 3D time-reversal (TR) in-
+variant topological insulators (TI).39–43 In this case, two
+distinct topologically-ordered states exist.
+One, called
+Pfaffian-antisemion,40,42 is closely related to the 3D FQH
+states of Ref. 20 (more precisely, the relation is with the
+TR-broken version of this state).
+The second one, T-
+Pfaffian,39,41 is related to the other 3D state we will con-
+struct in the present paper (again, more precisely, the
+relation is with the TR-breaking version of this state,
+which is usually called PH, which stands for particle-
+hole-symmetric, Pfaffian44,45).
+Another interesting consequence emerges from these
+analogies to the 2D TR-invariant TI surface state topo-
+logical orders. It is well-known that the PH-Pfaffian is
+closely related to the recently discovered duality relation
+between a massless noninteracting 2D Dirac fermion and
+QED3.44,46–51 Namely, the PH-Pfaffian state is obtained
+when the dual Dirac fermion of QED3 is gapped by pair-
+ing, which does not break the charge conservation sym-
+metry since the dual fermion is neutral. The existence of
+the analog of the PH-Pfaffian state in our 3D system then
+also implies the existence of a gapless state, which is re-
+lated to the noninteracting Weyl semimetal via a duality
+relation, somewhat similar to the 2D Dirac duality. We
+demonstrate that this is indeed the case. However, we
+find that the duality only applies to topological response
+in this case and not to the dynamics and is weaker than
+the 2D duality in this sense.
+The path to topologically ordered insulators through
+gapping topological semimetals is quite general and is
+not limited to the magnetic Weyl semimetal case.
+To
+emphasize this point, here we also discuss a topologically-
+ordered state, which is obtained by gapping a nodal
+line semimetal without breaking symmetries. This state
+has a topological order, distinct from a gapped Weyl
+semimetal, and is characterized by a fractional elec-
+tric polarization, impossible in an ordinary weakly-
+interacting insulator.
+The rest of the paper is organized as follows.
+In
+section II, after a preliminary discussion of topological
+field theory description of the electromagnetic response
+of Weyl semimetals, we recap the construction of the 3D
+analog of the Pfaffian-antisemion state of Refs. 20 and
+21. In section III, we demonstrate the existence of a du-
+ality relation (which applies to topological response only)
+between a noninteracting Weyl semimetal and a QED4,
+which describes a time-reversed Weyl semimetal, coupled
+to a dynamical gauge field. In section IV we discuss a
+topologically-ordered state, obtained by gapping a nodal
+line semimetal without breaking symmetries. This state
+is characterized by a fractional electric polarization, im-
+possible in an ordinary insulator. We conclude in sec-
+tion V with a brief discussion of our results.
+II.
+GAPPED SYMMETRY-PRESERVING
+STATES IN WEYL SEMIMETALS
+A.
+Preliminaries
+To keep the paper self-contained, we will start by re-
+capping the construction of the 3D FQH state of Ref. 20
+and 21, which, as will be explained below, may be viewed
+as a TR-breaking 3D analog of the Pfaffian-antisemion
+state on a strongly-interacting 3D TI surface. We will
+also put the theory of Ref. 21 on a more rigorous foot-
+ing by introducing the language of translation gauge
+fields,36,52–55 which allows one to use proper coordinate-
+free notation for the corresponding topological field the-
+ories.
+We start from the simplest cubic lattice model of a
+magnetic Weyl semimetal with a pair of nodes34
+H =
+�
+k
+ψ†
+k [σx sin(kxd) + σy sin(kyd) + σzm(k)] ψk.
+(1)
+Here σi are Pauli matrices, describing the pair of touching
+bands and
+m(k) = cos(kzd) − cos(Qd) − ˜m[2 − cos(kxd) − cos(kyd)],
+(2)
+where d is the lattice constant, ˜m > 1 and m(k) van-
+ishes at two points on the z-axis with kz = ±Q, which
+correspond to the locations of the Weyl nodes.
+Such a Weyl semimetal is characterized by the anoma-
+lous Hall conductivity
+σxy = e2
+h
+2Q
+2π = 1
+2π
+2Q
+2π .
+(3)
+This may be expressed as a topological term in the effec-
+tive action for probe electromagnetic gauge fields when
+the fermions are integrated out56
+L = i2Q
+8π ϵzναβAν∂αAβ.
+(4)
+In its primitive form above, Eq. (4) does not actually
+look like a topological term, since it explicitly contains a
+preferred direction in space (z) and depends on a nonuni-
+versal microscopic lattice constant d through the Weyl
+node separation 2Q.
+
+3
+To fix these issues, it proves useful to introduce the
+concept of a translation gauge field.36,52–55 Recall that
+Bravais lattice points R of a perfect crystal may be
+described as intersections of families of crystal planes,
+perpendicular to primitive reciprocal lattice vectors bi,
+where i = 1, 2, 3 (or x, y, z for a cubic crystal). Mathe-
+matically, this is expressed by the equation
+θi(r, t) = bi · r = 2πni,
+(5)
+where ni are sets of integers, labeling the crystal planes
+in a family i and the Bravais lattice vectors r = R are the
+solutions of this equations. Eq. (5) implies that the recip-
+rocal lattice vectors in a perfect crystal may be expressed
+as gradients of the phases bi
+j = ∂jθi. This may be gen-
+eralized to a distorted crystal, including time-dependent
+distortions, by introducing translation “gauge fields”
+ei
+µ = 1
+2π ∂µθi.
+(6)
+The fields ei
+µ may in fact be viewed as true (strictly
+speaking, integer valued) gauge fields, if one explicitly
+takes account of the fact that the phases θi on crys-
+tal planes may be relabelled in arbitrary 2π × Z incre-
+ments.54,55 This will not make a significant difference in
+our case and either viewpoint is acceptable.
+In a convenient differential form language, we may view
+ei as a one-form
+ei = ei
+µdxµ.
+(7)
+In a crystal without dislocations,
+dei = 1
+2(∂µei
+ν − ∂νei
+µ)dxµ ∧ dxν = 0,
+(8)
+as clearly follows from the definition Eq. (6).
+On the
+other hand, if a dislocation with a Burgers vector along
+bi is present, the integral of ei around a loop, enclosing
+the dislocation line is
+�
+ei = 1.
+The benefit of introducing translation gauge fields be-
+comes apparent if we now replace a reciprocal lattice vec-
+tor along the z-direction in Eq. (4) with the correspond-
+ing translation gauge field
+2π
+d δz
+µ → 2πez
+µ.
+(9)
+Then Eq. (4) becomes
+L = i λ
+4π ϵµναβez
+µAν∂αAβ = i λ
+4π ez ∧ A ∧ dA,
+(10)
+where λ = 2Q/(2π/d) is a dimensionless separation be-
+tween the Weyl nodes in units of the reciprocal lattice
+vector. Now Eq. (10) looks like a proper topological term,
+which only contains gauge fields and a universal coeffi-
+cient. The nonuniversal and variable lattice constant d
+has been absorbed into the definition of the translation
+gauge field and we will henceforth set d = 1 for simplicity.
+Varying Eq. (10) with respect to ez
+z produces response
+per atomic xy-plane, which is determined by a universal
+numerical coefficient λ. A noninteger value of the coef-
+ficient λ requires gapless modes in the form of a pair of
+Weyl nodes to be present,36,37 since a fractional value (in
+units of e2/h) of the Hall conductance per atomic plane
+is impossible in a noninteracting gapped insulator.
+B.
+3D analog of the Pfaffian-antisemion state
+To derive the field theory of the gapped 3D FQH state
+of Ref. 20 we first move to a dual description of the nonin-
+teracting Weyl semimetal of Eq. (1), in which the electric
+charge is separated from the fermions and is represented
+in terms of a two-form gauge potential, which couples
+to the vortex loop excitations.21,57,58 This approach is
+essentially equivalent to what is known as “functional
+bosonization”,59–62 apart from unimportant technical de-
+tails. We start by representing the fermion operators in
+Eq. (1) (after Fourier transforming them to real space)
+as
+ψr = eiθrfr,
+(11)
+where r label the sites of a cubic lattice, eiθr represents
+a spinless charged boson (chargon) and fr is a neutral
+fermion (spinon).
+After straightforward and standard
+manipulations,21,63 one obtains the following exact rep-
+resentation of the Weyl semimetal Lagrangian L
+L = Lf + Lb
+(12)
+where Lf is the Lagrangian of the spinons fr, which has
+a form, identical to the lattice Lagrangian of the original
+electrons ψr, except that fr are coupled to a compact
+statistical gauge field aµ rather than the probe electro-
+magnetic field Aµ. The statistical field expresses U(1)
+gauge invariance of the parton decomposition Eq. (11)
+and serves the purpose of gluing together the spinons
+and the chargons. The chargon Lagrangian has the form
+Lb = i
+4π (Aµ − aµ)ϵµναβ∆νbαβ +
+1
+8π2χ(ϵµναβ∆νbαβ)2.
+(13)
+Here bµν = −bνµ is a two-form 2π×Z valued lattice gauge
+field, which represents integer chargon currents Jµ as
+Jµ = 1
+4π ϵµναβ∆νbαβ,
+(14)
+∆µ is a lattice derivative and χ is a positive constant.
+Lattice site indices r have been suppressed everywhere
+for brevity.
+To avoid dealing with discrete variables, we may imple-
+ment the 2πZ constraint on bµν by adding a term i
+2 ˜Jµνbµν
+to Lb and summing over integer-valued variables ˜Jµν,
+which have the meaning of vortex loop currents. Gauge
+invariance of Eq. (14) with respect to a transformation
+bµν → bµν + ∆µgν − ∆νgµ implies a conservation law
+∆µ ˜Jµν = 0,
+(15)
+
+4
+which may be solved as
+˜Jµν = 1
+2π ϵµναβ∆αcβ,
+(16)
+where cµ are 2πZ valued one-form gauge fields. The con-
+straint on cµ may, in turn, be softened by adding a term
+−t cos(∆µφ + cµ), where the presence of a new compact
+angular variable φ takes account of the gauge invariance
+of Eq. (16) with respect to cµ → cµ + ∆µφ. In essence,
+the particle created by eiφ, is the original chargon.
+Then, after taking the continuum limit, the chargon
+Lagrangian takes the dual form
+Lb = i
+4π (Aµ − aµ + cµ)ϵµναβ∂νbαβ + . . . ,
+(17)
+where . . . contain both the higher-derivative terms for
+bµν and the additional terms for cµ whose form depends
+on the value of the parameter t. In particular, when t
+is large, eiφ boson is condensed, leading to a mass term
+for cµ (i.e. gap for vortices), which may then be ignored.
+Integration over bµν the simply sets Aµ = aµ, i.e. the
+electric charge is re-attached to the spinons and we re-
+cover the original Weyl semimetal. In contrast, when t
+is small, eiφ particle is gapped, which leads to a Maxwell
+term, (ϵ∂c)2, for the gauge field cµ. In this case, integra-
+tion over cµ produces a mass term b2 for the two-form
+gauge field, which corresponds to a charge gap.
+This
+state is a Mott insulator, which has gapless spinons that
+retain the Weyl semimetal band structure.
+To obtain a fully gapped state, which preserves topo-
+logical response of the Weyl semimetal Eq. (10) and does
+not break any symmetries, we place the spinons into
+a paired state. For weak pairing, only the intra-nodal
+pairing state opens a gap.64–67 Such a pairing generally
+breaks translational symmetry, except when 2Q = π or
+λ = 1/2,20 to which we now specialize. With such an
+intra-nodal pairing term, the spinon Hamiltonian may
+be brought to the form
+H = 1
+2
+�
+k
+f †
+k {σx sin(kx) + σy sin(ky)
++
+��
+∆2 + cos2(kz) − ˜m(2 − cos(kx) − cos(ky))
+�
+σz
+�
+fk,
+(18)
+where ∆ is the pairing amplitude. This Hamiltonian de-
+scribes a 3D topological p-wave superconductor, which
+has a chiral Majorana mode, spanning the full extent of
+the BZ. This may also be viewed as a stack of 2D topo-
+logical superconductors, since the pairing gap does not
+close at any value of kz.
+The spinon pairing produces a term ∝ − cos(2aµ)
+for the statistical gauge field, which leaves only aµ =
+0, π mod 2π possible values at low energies and makes it
+a Z2 gauge field. While nontrivial π-flux configurations of
+aµ (visons68) are still possible, these may be easily shown
+to bind gapless 1D Majorana mode in their cores, which
+is a direct consequence of the fact that the spinon super-
+conductor is topologically nontrivial. This means that in
+any fully gapped symmetry-preserving state such vison
+loop excitations must be gapped, which means that we
+may set aµ = 0 mod 2π at low energies. This detaches
+the boson and fermion sectors of the theory. The fermion
+sector thus contributes the same thermal Hall response
+as the noninteracting Weyl semimetal at λ = 1/2, which
+arises from the chiral Majorana mode, spanning the full
+BZ. The electrical response must entirely come from the
+boson sector.
+In order to reproduce the electrical response of the non-
+interacting Weyl semimetal, it is necessary to condense
+double (i.e. flux 4π) vortices of the boson field eiθ. This
+is accomplished by the following modification of the field
+theory Eq. (17)
+Lb = i
+4π (Aµ + 2cµ)ϵµναβ∂νbαβ + 2i
+4π ϵµναβez
+µcν∂αcβ
++ 1
+2g (ϵµναβ∂αcβ)2 + i
+2bµν˜jµν + icµjµ.
+(19)
+The extra factor of 2 in front of cµ, compared to Eq. (17),
+expresses the fact that double (flux 4π) vortices are be-
+ing condensed. This also means that the quasiparticle,
+which is minimally coupled to the gauge field cµ, carries
+a charge 1/2.
+The second term is a topological term,
+which will give rise to the correct electrical Hall con-
+ductivity, as will be shown below.
+This term may be
+viewed as describing a layered integer quantum Hall state
+of the charge-1/2 bosonic quasiparticles. The third term
+is the Maxwell term. It is subdominant to the topological
+term at long wavelengths, but has been included explic-
+itly since the topological term only contains components
+of cµ, transverse to the translation gauge field ez. In par-
+ticular, if ez
+µ = δz
+µ, cz does not enter into the topological
+term and its dynamics is governed by the Maxwell term.
+Let us now demonstrate that Eq. (19) indeed describes
+the correct physics. Let us set ˜jµν = 0 and integrate out
+bµν. This gives cµ = −Aµ/2. Substituting this back into
+Eq. (19), we obtain
+Lb = i
+8π ϵµναβez
+µAν∂αAβ − i
+2Aµjµ.
+(20)
+The first term in Eq. (20) correctly reproduces the electri-
+cal Hall conductivity of a noninteracting Weyl semimetal
+with λ = 1/2, which is half conductance quantum σxy =
+1/4π per atomic plane. The second term tells us that
+quasiparticle excitations in the gapped state, described
+by Eq. (19), are bosons with electric charge 1/2. To es-
+tablish gapped nature of this state it is important to note
+the following. If we reinsert the statistical gauge field aµ
+into Eq. (19), it is clear that fluctuations of bµν effec-
+tively constrain cµ = (aµ − Aµ)/2. This implies that,
+since aµ is made a Z2 gauge field by spinon pairing, cµ
+becomes a discrete Z4 gauge field.
+This is important,
+since, unlike in 2 + 1 dimensions, a (3 + 1)-dimensional
+Maxwell-Chern-Simons theory with U(1) gauge fields is
+gapless.69,70
+The most straightforward way to see that this the-
+ory also correctly captures the thermal Hall conductivity
+
+5
+κxy = 0 is to consider the boundary theory, that cor-
+responds to Eq. (19).
+To derive the boundary theory
+we follow the standard method.2 We choose a gauge, in
+which on the boundary, taken to be in the xz-plane, we
+set the temporal components of all the gauge fields to
+zero, i.e. c0 = 0, b0µ = 0. Then, integrating out c0, we
+obtain
+ϵ0νλρ∂νbλρ = ϵ0νλρ∂ν(ez
+λcρ − ez
+ρcλ),
+(21)
+while integrating b0ν gives
+ϵ0νλρ∂λcρ = 0.
+(22)
+Eqs. (21) and (22) along with dez = 0 imply that
+ϵ0νλρ∂νbλρ = 0.
+(23)
+Eq. (23) may then be solved as
+bij = ∂igj − ∂jgi,
+(24)
+where i, j = x, z refer to spatial coordinates on the
+boundary, while Eq. (22) is solved as
+ci = ∂iϕ.
+(25)
+Plugging this back into what remains of Eq. (19) after
+integrating out c0 and b0µ, we obtain
+Lb = i
+2π ϵ0νλρez
+ν∂λϕ∂τ∂ρϕ − i
+π ϵ0νλρ∂νϕ∂τ∂λgρ,
+(26)
+where ∂τ ≡ ∂0.
+Integrating this in the presence of a
+boundary, perpendicular to the y-direction, gives
+Lsurf = i
+2π ϵijez
+i ∂τϕ∂jϕ − i
+π ϵij∂τϕ∂igj,
+(27)
+where i, j
+=
+x, z.
+Adding symmetry-allowed non-
+topological terms and the electromagnetic field, we finally
+obtain the following surface state Lagrangian
+Lsurf = i
+2π ϵijez
+i ∂τϕ∂jϕ − i
+π ϵij∂τϕ∂igj + vϕ
+2π (∂iϕ)2
++ vg
+2π (∂igj − ∂jgi)2 + i
+2π ϵµνλAµ∂νgλ.
+(28)
+Setting ez
+µ = δz
+µ and Fourier transforming, we obtain the
+following expression for the excitation spectrum of the
+surface modes
+ϵ(k) = −vgkx
+2
++
+��v2g
+4 + vgvϕ
+�
+k2x + vgvϕk2z.
+(29)
+This looks like an ordinary anisotropic 2D superfluid dis-
+persion, except for a “tilt” in the x-direction due to the
+first term. However, the dispersion is still nonchiral, since
+there is always a pair of left- and right-handed modes for
+every value of kz. Consequently, a straightforward calcu-
+lation gives a vanishing thermal Hall conductivity in this
+state
+κxy ∼
+�
+dkxdkzvx(k)ϵ(k)∂nB[ϵ(k)]
+∂T
+= 0,
+(30)
+where vx(k) = ∂ϵ(k)
+∂kx
+and nB(ϵ) is the Bose-Einstein dis-
+tribution. The integral over kx in Eq. (30) vanishes since
+the left-handed (kx < 0) and right-handed (kx > 0)
+modes give a contribution that is equal in magnitude but
+opposite in sign.
+By construction, this state is a fully gapped symmet-
+ric state, which has an identical topological response to
+a noninteracting Weyl semimetal at λ = 1/2. Note again
+that, while there does exist a close connection between
+this state and the 2D Pfaffian-antisemion state, it may
+not be viewed as a simple stack of such 2D states. In par-
+ticular, there are no semion quasiparticles, but isolated
+intersections of 2π vortex loop excitations with atomic
+xy-planes do behave as semions.
+C.
+3D analog of the PH-Pfaffian state
+Now we note that a second distinct gapped symmet-
+ric state, reproducing topological response of a nonin-
+teracting Weyl semimetal, actually exists. This state is,
+in a way, simpler than the 3D analog of the Pfaffian-
+antisemion above and, as we will demonstrate, may be
+viewed as a 3D analog of the PH-Pfaffian.39,41,44,45
+To construct this state, we take a time-reversed copy
+of our Weyl semimetal with λ = 1/2. Writing its La-
+grangian in terms of spinon and chargon variables, we
+have
+L = ¯fγµ(∂µ+iaµ)f − i
+8π ez∧a∧da+ i
+4π (A−a)∧db, (31)
+where the first term is the contribution of the gapless
+Weyl fermions while the second term is the topological
+contribution from all the filled negative-energy states.
+We will switch to the index-free notation henceforth.
+We now place the chargons into a stack of independent
+ν = 1/2 quantum Hall states in each xy-atomic plane.
+Technically, this means that we take the two-form gauge
+field b to be “foliated”71–74
+b = ez ∧ ˜b,
+(32)
+where ˜b = ˜b0dτ +˜bxdx+˜bydy is a one-form field that lacks
+the z-component, and add a term − 2i
+4πez ∧ ˜b ∧ d˜b to the
+Lagrangian Eq. (31). Furthermore, we place the spinons
+into the intra-nodal pairing state of Eq. (18), which leads
+to a 3D p + ip topological superconductor with a chiral
+Majorana mode, spanning the surface BZ, whose chirality
+is, however, opposite to the chirality of the Fermi-arc
+state of the original noninteracting Weyl semimetal. This
+gaps out the gauge field aµ and decouples the boson and
+fermion sectors.
+The boson sector Lagrangian now reads
+Lb = − 2i
+4π ez ∧ ˜b ∧ d˜b + i
+2π ez ∧ A ∧ d˜b.
+(33)
+Integrating over ˜b leaves the effective action
+Lb = i
+8π ez ∧ A ∧ dA,
+(34)
+
+6
+which describes topological electrical response, which is
+identical to the original noninteracting Weyl semimetal.
+The thermal Hall effect, coming from Lb, is twice that
+of the noninteracting Weyl semimetal, however a mi-
+nus a half is contributed by the opposite-chirality Ma-
+jorana surface state of the paired time-reversed spinons.
+Thus we fully reproduce both electrical and thermal
+topological responses of the noninteracting gapless Weyl
+semimetal.
+This state may be viewed as a 3D generalization of
+the 2D PH-Pfaffian state. Note that, unlike the 3D ana-
+log of the Pfaffian-antisemion state, described above, this
+state is not a 3D incompressible liquid, but exhibits a
+fracton-type order.71–74 If we ignore fermions, Eq. (33)
+describes a stack of independent 2D PH-Pfaffian states.
+The charge-1/2 anyon excitations in these 2D states are
+only able to move within a given plane and can not tun-
+nel between the planes. Neutral fermions propagate in
+3D and connect the individual layers together, but the
+anyons remain confined within 2D layers.
+III.
+“DUAL” WEYL SEMIMETAL
+The existence of a 3D analog of the PH-Pfaffian has an
+important implication, which we will now discuss. Let us
+first return back to the 3D Pfaffian-antisemion state. Let
+us note that, in this case, the topological response of a
+noninteracting Weyl semimetal is only reproduced when
+the fermionic spinons are gapped by pairing and vison
+vortex loops excitations are gapped. If the pairing gap is
+taken to zero, the statistical field a is no longer massive
+and its coupling to the gapless Weyl spinons produces a
+topological term
+i
+8πez ∧ a ∧ da, so that the Lagrangian
+may be written as
+L = ¯fγµ(∂µ + iaµ)f + i
+8π ez ∧ a ∧ da
++ i
+4π (A − a + 2c) ∧ db + i
+2π ez ∧ c ∧ dc.
+(35)
+Integrating out b and c gives
+L = ¯fγµ(∂µ + iaµ)f + i
+4π ez ∧ a ∧ da
+− i
+4π ez ∧ A ∧ da + i
+8π ez ∧ A ∧ dA.
+(36)
+To obtain the electromagnetic response, we now integrate
+out a. This may be done perturbatively, treating the re-
+sponse of the gapless low-energy modes, i.e.
+the first
+term in Eq. (36) as a perturbation, compared to the sec-
+ond term. This is possible, because the response of the
+gapless modes, treated in the random phase approxima-
+tion (RPA), is given by
+Sf = 1
+2
+�
+q
+aµ(q)Πµν(q)aν(−q),
+(37)
+where
+Πµν(q) = (q2δµν − qµqν)f(q2),
+(38)
+is the polarization operator of the massless 3D Dirac
+fermion and
+f(q2) =
+1
+12π2 ln
+�4Λ2
+q2
+�
++ O(1).
+(39)
+Here Λ ≫ q is the cutoff momentum, and a convention
+q0 = −Ω is used (Ω is the Matsubara frequency). Note
+that Πµν(q) is almost the same as the polarization op-
+erator of the massive 3D Dirac fermion, in which case
+f(q2) would have been a constant at small q. Even with
+the log nonanalyticity, Πµν(q) is still much smaller, in
+the long wavelength limit, than the topological contribu-
+tions, which are of first order in q.
+At leading order we may then ignore the gapless
+fermions and vary the Lagrangian with respect to a,
+which gives at the saddle point a = A/2 and leaves the
+Lagrangian
+L = ¯fγµ(∂µ + iAµ/2)f +
+i
+16π ez ∧ A ∧ dA,
+(40)
+which clearly corresponds to half of the Hall conductivity
+of a noninteracting Weyl semimetal, i.e. the theory with
+gapless spinons does not reproduce topological response
+of the noninteracting Weyl semimetal.
+In contrast, let us return to Eq. (31), which describes a
+time-reversed Weyl semimetal and add to it the foliated
+topological term of Eq. (33), without opening the spinon
+pairing gap
+L = ¯fγµ(∂µ + iaµ)f − i
+8π ez ∧ a ∧ da
++ i
+2π ez ∧ (A − a) ∧ d˜b − 2i
+4π ez ∧ ˜b ∧ d˜b.
+(41)
+Integrating out ˜b now, we obtain
+L = ¯fγµ(∂µ+iaµ)f − i
+4π ez∧A∧da+ i
+8π ez∧A∧dA. (42)
+This has identical electrical and thermal Hall responses to
+the original noninteracting Weyl semimetal. This means
+that Eq. (41) describes a distinct gapless state, which
+reproduces the topological response of a noninteracting
+Weyl semimetal. This statement is very closely analo-
+gous to the statement of duality between noninteract-
+ing 2D Dirac fermion and QED3.44,46–51 However, note
+that, in contrast to the 2D Dirac duality case, dynami-
+cally this system is quite different from a noninteracting
+Weyl semimetal. Indeed, integrating out f and then a
+in Eq. (42) using RPA produces a Meissner term for the
+components of A, transverse to z. The coefficient of the
+Meissner term, however, vanishes in the long-wavelength
+limit (it is equal to the inverse of the function f(q2), in-
+troduced in Eq. (39)).
+The system thus behaves as a
+superconductor at finite length scales and in directions,
+transverse to z, but with a phase stiffness that vanishes
+in the thermodynamic limit. In contrast, it behaves as
+an insulator along z.
+
+7
+IV.
+TOPOLOGICAL ORDER IN A GAPPED
+NODAL LINE SEMIMETAL
+We will now extend the ideas, developed above, to
+the case of nodal line semimetals, which realize a dis-
+tinct kind of (3 + 1)-dimensional topological order, when
+gapped without breaking the protecting symmetries. In
+the nodal line semimetals, only nodal lines which arise
+from touchings of pairs of nondegenerate bands, are topo-
+logically nontrivial. In this case, TR symmetry may be
+taken to be broken, while the nodal line is then protected
+by the mirror reflection symmetry in the plane, contain-
+ing the line.75 This may be described by the following
+two-band cubic-lattice Hamiltonian76,77
+H(k) = [6 − t1 − 2(cos kx + cos ky + cos kz)] σx
++ 2t2 sin kzσy.
+(43)
+The nodal line in this model appears in the xy-plane
+of the momentum space and is protected by the mirror
+reflection symmetry within this plane, where the mirror
+reflection operator is σx. The band-touching line in the
+xy-plane is given by the solution of the equation
+4 − t1 − 2(cos kx + cos ky) = 0.
+(44)
+In order construct a gapped symmetric state, it is use-
+ful to reinterpret Eq. (43) as a stacking of alternating
+electron and hole-like Fermi liquids with the band dis-
+persions (see Fig. 1)
+ϵ±(k) = ±[4 − t1 − 2(cos kx + cos ky)],
+(45)
+where ± are the two eigenvalues of the mirror reflection
+operator σx.37 The Luttinger volumes of the two Fermi
+liquids ±VF are equal in magnitude to the area in mo-
+mentum space, enclosed by the nodal line. For the two
+Fermi liquids, the topological response describes the fill-
+ing of the charged particles
+L = ±iVF
+4π2 ex ∧ ey ∧ A,
+(46)
+Consequently, the topological response of the nodal line,
+takes the form of fractional electric polarization37,78,79
+L = ±iVF
+8π2 ex ∧ ey ∧ dA,
+(47)
+impossible in an ordinary insulator without topological
+order.
+The simplest way to obtain a gapped mirror-symmetric
+insulator with the same topological response Eq. (47)
+is to stack gapped 2D Fermi liquid states in a mirror-
+symmetric fashion.
+To gap the 2D Fermi liquids, we
+follow the same procedure as above.
+We represent an
+electron as a product of a neutral spinon f and a bosonic
+chargon eiθ and place the spinons into a fully-gapped
+paired state.
+The simplest fully gapped paired spinon
++
+-
++
++
++
++
+-
+-
+-
+-
+d
+d
+FIG. 1.
+(Color online) Construction of the nodal line
+semimetal as a stack of alternating coupled electron and hole-
+like Fermi liquids. The Luttinger volume of each 2D Fermi
+liquid is equal in magnitude to the area in momentum space,
+enclosed by the nodal line. The lattice constant d is set equal
+to unity in all the equations.
+state is p-wave (since the Fermi liquids are spinless), de-
+scribed by the following Hamiltonian
+Hf =
+�
+k
+�
+ϵ±(k)f †
+kfk + ∆
+2 (sin kx + i sin ky)f †
+kf †
+−k + h.c.
+�
+.
+(48)
+Introducing Nambu spinor notation ψk = (fk, f †
+−k), this
+may be represented as a massive 2D Dirac Hamiltonian
+H = 1
+2
+�
+k
+ψ†
+k [ϵ±(k)τz + ∆(τx sin kx − τy sin ky)] ψk,
+(49)
+where τa are Pauli matrices in the particle-hole space.
+This is the Hamiltonian of a Read-Green topological su-
+perconductor,80 which hosts chiral Majorana modes at
+the edges, with opposite chirality for electron and hole-
+like Fermi liquid states. Consequently, an elementary flux
+hc/2e = π vortex hosts a zero-energy localized Majorana
+bound state and can not be condensed.
+To condense higher-flux vortices, we need to consider
+the chargon sector of the theory. Suppose we attempt
+to condense flux-2π vortices. The chargon sector will be
+described by the following theory81,82
+Lb = i
+2π (A + c) ∧ db ± iVF
+(2π)2 ex ∧ ey ∧ c.
+(50)
+Here b is a one-form gauge field, which determines the
+charge current
+Jµ = 1
+2π ϵµνλ∂νbλ,
+(51)
+while c is a one-form gauge field, which determines the
+vortex current
+˜Jµ = 1
+2π ϵµνλ∂νcλ.
+(52)
+The last term of the Lagrangian produces the correct
+electromagnetic response of a system with charge ν =
+
+8
+±VF /(2π)2 per unit cell when b is integrated out, setting
+c = −A. However, when the filling ν is not an integer,
+Eq. (50) can not be the correct theory of a featureless
+insulator since the last term is not gauge invariant. With
+ν = ±p/q, a featureless insulator may be obtained only
+by condensing flux 2πq vortices, which is described by
+the theory
+Lb = i
+2π (A + qc) ∧ db ± ipex ∧ ey ∧ c,
+(53)
+where all terms now have properly quantized integer co-
+efficients and are gauge invariant. This is because the
+quasiparticle, minimally coupled to cµ, carries charge
+1/q, as seen from the first term. Therefore, the filling
+of such quasiparticles is qν = p (i.e. an integer), which is
+what the second term expresses.
+Stacking such insulators with alternating sign of ν in
+the z-direction in a mirror-symmetric fashion, we obtain
+Lb = i
+4π (A + qc) ∧ db ± ip
+2 ex ∧ ey ∧ dc,
+(54)
+where the factor of 1/2 in front of the last term arises
+due to the fact that the unit cell of the stack contains a
+pair of electron and hole-like 2D Fermi liquids and the
+mirror symmetry requires that all neighboring 2D Fermi
+liquids in the stack are separated by an equal distance.
+The gauge field b in Eq. (54) has now been promoted to a
+two-form field, such that the (3 + 1)-dimensional charge
+current is given by
+Jµ = 1
+4π ϵµναβ∂νbαβ,
+(55)
+while the two-form vortex current is
+˜Jµν = 1
+2π ϵµναβ∂αcβ.
+(56)
+Integrating out b in Eq. (54) gives c = −A/q and the
+electromagnetic response described by
+L = ± ip
+2q ex ∧ ey ∧ dA = ±iVF
+8π2 ex ∧ ey ∧ dA,
+(57)
+which coincides with Eq. (47). Thus we obtain a feature-
+less insulator with topological order, which has an iden-
+tical topological response to a weakly-interacting nodal
+line semimetal. Note that the nodal line semimetal has
+no topological thermal response, which is also the case in
+the fractionalized insulator that we have constructed.
+V.
+DISCUSSION AND CONCLUSIONS
+In this paper we have presented a theory of (3 + 1)-
+dimensional topologically ordered states, obtained by
+gapping 3D topological semimetals without breaking pro-
+tecting symmetries.
+We started by pointing out that
+a second gapped symmetric topologically-ordered state,
+preserving the chiral anomaly of magnetic Weyl semimet-
+als, exists, in addition to the state, originally proposed
+in Ref. 20.
+We have shown that, while the state of
+Ref. 20 may be viewed as a 3D TR-breaking analog of
+the Pfaffian-antisemion state in gapped 3D TI surface
+states, the new state is the 3D analog of the PH-Pfaffian.
+In contrast to the 3D Pfaffian-antisemion state, the 3D
+PH-Pfaffian does not exhibit a true 3D topological order,
+but a fracton-like order instead, with independent layers
+of 2D PH-Pfaffian liquid immersed in a 3D p + ip topo-
+logical superconductor of neutral composite fermions.
+We then demonstrated that an interesting consequence
+of the existence of the 3D PH-Pfaffian, is a duality
+relation between a noninteracting Weyl semimetal and
+QED4, in which a time-reversed electrically-neutral Weyl
+semimetal is coupled to a dynamical gauge field, whose
+topological defects (intersections of flux lines with atomic
+planes) carry the electric charge. This duality relation
+may be viewed as a 3D generalization of the known Dirac
+fermion to QED3 duality relation, but is weaker than in
+the 2D case, since the duality only applies to the topo-
+logical response and not to the dynamics.
+Finally, we have extended the theory to include topo-
+logical orders in a gapped nodal line semimetal. Other
+extensions, in particular to TR-invariant Weyl and Dirac
+semimetals are also possible, but do not lead to any fun-
+damentally new structure. One lesson we may highlight
+is that gapped symmetric topological semimetals provide
+a very simple and natural setting for (3 + 1)-dimensional
+topologically-ordered states to appear.
+The simplicity
+stems, in part, from the fact that, due to the existence
+of a preferred direction, selected by either the separa-
+tion between the Weyl points in momentum space, or
+the plane of the nodal line, there exists a natural con-
+nection to well-studied (2 + 1)-dimensional topological
+orders. The connection manifests either directly, in the
+form of layered fracton-like order, or less directly, when
+intersections of (3 + 1)-dimensional vortex-loop excita-
+tions with atomic planes behave as fractionally-charged
+and sometimes anyonic (2 + 1)-dimensional quasiparticle
+excitations.
+ACKNOWLEDGMENTS
+We acknowledge useful discussions with Chong Wang.
+Financial support was provided by the Natural Sciences
+and Engineering Research Council (NSERC) of Canada.
+AAB was also supported by Center for Advancement
+of Topological Semimetals, an Energy Frontier Research
+Center funded by the U.S. Department of Energy Office
+of Science, Office of Basic Energy Sciences, through the
+Ames Laboratory under contract DE-AC02-07CH11358.
+Research at Perimeter Institute is supported in part by
+the Government of Canada through the Department of
+Innovation, Science and Economic Development and by
+the Province of Ontario through the Ministry of Eco-
+nomic Development, Job Creation and Trade.
+
+9
+∗ These authors contributed equally to this work.
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+Unsupervised Manifold Linearizing and Clustering
+Tianjiao Ding1 Shengbang Tong2 Kwan Ho Ryan Chan1 Xili Dai3 Yi Ma2 Benjamin D. Haeffele1
+Abstract
+Clustering data lying close to a union of low-
+dimensional manifolds, with each manifold as a cluster,
+is a fundamental problem in machine learning. When the
+manifolds are assumed to be linear subspaces, many meth-
+ods succeed using low-rank and sparse priors, which have
+been studied extensively over the past two decades. Un-
+fortunately, most real-world datasets can not be well ap-
+proximated by linear subspaces. On the other hand, several
+works have proposed to identify the manifolds by learning
+a feature map such that the data transformed by the map
+lie in a union of linear subspaces, even though the original
+data are from non-linear manifolds. However, most works
+either assume knowledge of the membership of samples to
+clusters, or are shown to learn trivial representations. In
+this paper, we propose to simultaneously perform cluster-
+ing and learn a union-of-subspace representation via Max-
+imal Coding Rate Reduction. Experiments on synthetic and
+realistic datasets show that the proposed method achieves
+clustering accuracy comparable with state-of-the-art alter-
+natives, while being more scalable and learning geometri-
+cally meaningful representations.
+1. Introduction
+1.1. Motivation and Contributions
+Clustering is a fundamental problem in machine learn-
+ing, allowing one to group data into clusters based on as-
+sumptions about the geometry of clusters. For example,
+when data are concentrated around distinct centroids, classi-
+cal k-means clustering [14,17,26,29] and its variants [2,3,5]
+are able to find the cluster centroids and assign membership
+to each data point. More generally4, subspace clustering
+methods [11, 13, 16, 25, 27, 50] are designed to cluster data
+that lie close to a union of different low-dimensional linear
+(or affine) subspaces, where each subspace defines a cluster.
+1Mathematical Institute for Data Science, Johns Hopkins University,
+USA 2Department of Electrical Engineering and Computer Sciences, Uni-
+versity of California, Berkeley, USA 3The Hong Kong University of Sci-
+ence and Technology (Guangzhou), PRC
+4This includes k-means-based methods, since a centroid is a 0-
+dimensional affine subspace.
+Overall, those methods often enjoy theoretical guarantees of
+correct clustering [13,22,27,37,40,41,44,47–51] and find
+applications in various problems such as image clustering,
+face recognition, motion segmentation, and recently in pop-
+ular Transformer architectures in deep learning [38].
+Despite the wide range of applications and theoretical
+guarantees, subspace clustering methods rely on a crucial
+assumption that each cluster can be well approximated by
+a linear/affine subspace, which is often not valid for many
+real-world datasets. For instance, even in a dataset as sim-
+ple as MNIST hand-written digits, images of a single digit
+do not lie close to a low-dimensional linear subspace, thus
+directly applying subspace clustering will fail. Instead, it
+is more natural to assume the clusters are from non-linear
+low-dimensional manifolds (one manifold per cluster), and
+attempt to learn or design a non-linear embedding of the
+data so that the transformed data lies close to distinct linear
+subspaces, with points from one manifold mapped to the
+same subspace. For example, [24] shows that a subspace
+clustering method can achieve 99% clustering accuracy on
+MNIST images after embedding the data with the scattering
+transform [6].
+Beyond the above example, numerous other subspace
+clustering methods have explored hand-designing an appro-
+priate feature embedding (or kernel) such as polynomial or
+exponential mappings [34]. However, these embeddings as-
+sume specific families of manifolds, thus they need to be
+hand-crafted for various tasks and datasets using domain
+knowledge, which makes their application challenging for
+complicated data such as natural images.
+On the other
+hand, [12] proposes to cluster data based on treating a lo-
+cal neighborhood of the manifold approximately as a linear
+subspace. However, for this to succeed sufficient sampling
+density is required, which implies a prohibitive number of
+samples when the manifolds are of moderate dimension or
+are highly curved. Further, for a new sample unseen at train-
+ing time one needs to run the algorithm with all samples
+to embed it or assign a membership to it, which is expen-
+sive computationally. More recently, numerous works pro-
+pose to learn an appropriate linear embedding of the data
+via deep networks and then perform subspace clustering in
+the feature space [1, 18, 20, 35, 54]. Unfortunately, it has
+been shown that many of these formulations are provably
+arXiv:2301.01805v1  [cs.LG]  4 Jan 2023
+
+(a)
+(b)
+(c)
+(d)
+Figure 1. (a) Input data X of two manifolds each containing 100 points. (b) Features Zθ at random initialization. (c) Zθ after self-
+supervised initialization. (d) Zθ after MLC (4) training. Details are in the Appendix.
+ill-posed and learn trivial representations5, with much of the
+claimed benefit coming from ad-hoc post-processing rather
+than the method itself [15].
+This leads to the following
+question:
+Question 1. For data approximately supported on an un-
+derlying union of manifolds, can we learn a transformation
+of the data, so that the transformed data lie in distinct linear
+subspaces to be easily clustered?
+Meanwhile, learning a representation from multi-modal
+data has been a topic of its own interest in machine learning.
+An ideal property of the learned representation often pur-
+sued is between-cluster discrimination, i.e., features from
+different clusters should be well separated. Further, an im-
+portant yet often ignored property of the learned representa-
+tion is that it maintains within-cluster diversity. This is de-
+sirable as it allows distances of samples within a cluster to
+be preserved under the learned transformation, which could
+facilitate downstream tasks such as denoising, hierarchical
+clustering and semantic interpretation. In the supervised
+setting, training with the cross-entropy (CE) classification
+objective fails to achieve the second property, as it has been
+shown empirically [32] and theoretically [43, 56] that the
+representation learned by CE has the property that features
+from one cluster tend to collapse to a single point. On the
+other hand, recent work has proposed the principle of Max-
+imal Coding Rate Reduction (MCR2) [53] as one of the few
+methods that are able to achieve the two ideal properties
+by learning a representation where features from each clus-
+ter are expected to lie close to a low-dimensional subspace
+(within-cluster diverse), and the subspaces from different
+clusters are orthogonal to each other (between-cluster dis-
+criminative). However, for MCR2 to learn such orthogonal
+subspaces each corresponding to one cluster, one needs the
+annotation of which sample belong to which cluster. Such
+annotation might be expensive or impossible to acquire for
+5In this paper, we use ‘representation’ and ‘feature’ interchangeably to
+mean the image of data under a (learned) transformation.
+large-scale datasets. This motivates another question of in-
+terest:
+Question 2. Can we learn a union-of-orthogonal-subspace
+representation of data coming from an underlying union of
+manifolds without access to the labels?
+This paper gives positive answers to the two interrelated
+questions by making the following contributions.
+1. We propose to simultaneously cluster the data and
+learn a union-of-orthogonal-subspace representation
+via MCR2, when data is assumed to lie close to a
+union of manifolds. This is achieved by formulation
+(4), which optimizes over both the representation and
+a doubly stochastic membership formulation inspired
+by the state-of-the-art subspace clustering result [24].
+2. Since the membership has as many entries as the
+square of the batch size of the input data, we give a
+parameterization of the membership (Figure 2). Fur-
+ther, as problem (4) is highly non-convex, we give a
+meta-algorithm (Algorithm 1) on how to initialize the
+variables and to optimize it.
+3. We conduct experiments on simulation and CIFAR10
+to demonstrate some desirable properties of the pro-
+posed method.
+We further experiment on datasets
+with larger number of clusters and imbalanced clus-
+ters such as CIFAR100-20, CIFAR100-100, and Tiny-
+ImageNet200, and show that the proposed method
+achieves state-of-the-art performance.
+1.2. Additional Related Work
+Beyond the above, we make connections to a few impor-
+tant works that are related to this paper.
+Deep Clustering and Representation Learning.
+Re-
+cently, there is an interesting line of research in representa-
+tion learning and clustering that takes advantage of pseudo-
+labelling and semi/self-supervised learning [7, 30, 33, 45].
+
+1.0
+0.5
+0.0
+-1.0
+-0.5
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+1.0Specifically, one first identifies a subset of samples (often
+termed reliable samples) based on geometric or statistical
+criteria in the learned representation and cluster prediction,
+and then uses the predicted labels for those reliable sam-
+ples as if they are ground-truth labels to refine the rep-
+resentation and cluster prediction of other samples.
+De-
+spite the promising clustering performance, the represen-
+tation learned by these methods are not constrained to be
+both between-cluster discriminative and within-cluster di-
+verse. In contrast, the proposed method learns a represen-
+tation with these two ideal properties (see Figure 4) and
+also achieves state-of-the-art clustering performance (see
+Tables 2 and 4).
+Neural Manifold Clustering and Embedding (NMCE).
+A recent preprint [23] also proposes a solution to the same
+problem we study, i.e., clustering the data and learning an
+union-of-orthogonal-subspace representation.
+In particu-
+lar, [23] proposes to model the point-to-cluster membership
+and optimize MCR2 [53] over both the representation and
+the membership. In this paper, we adopt a similar formula-
+tion, but we propose to model the point-to-point affinity us-
+ing a doubly stochastic matrix, inspired by the state-of-the-
+art subspace clustering methods (§2.2). Aside from having
+different conceptual formulations and algorithms, our for-
+mulation is much more stable with respect to initialization
+and is naturally suitable for hierarchical clustering. We de-
+tail these distinctions in §2.2. Experiments (Table 2) further
+show that the proposed method (MLC) achieves higher ac-
+curacy than [23] (NMCE) on large scale realistic datasets.
+2. Problem Formulation
+We start by defining the problem that we study. Suppose
+X = [x1, . . . , xn] ∈ RD×n is a dataset of n samples drawn
+from a union of k underlying manifolds �k
+j=1 Mj and y ∈
+Rn their memberships to the manifolds, i.e., xi ∈ My(i).
+Problem 1 (Unsupervised Manifold Linearizing and Clus-
+tering). Given the dataset X, can we simultaneously 1)
+cluster the samples, i.e., estimate y, and 2) learn a lin-
+ear representation for manifolds, i.e., find a transformation
+f : RD → Rd, such that the image of each manifold f(Mi)
+is a low-dimensional linear subspace of Rd, and the sub-
+spaces satisfy desired properties (§1), i.e., they are between-
+cluster discriminative and within-cluster diverse?
+Here we base our approach on the principle of Maximal
+Coding Rate Reduction (MCR2) which is designed to learn
+ideal representations in the supervised case, i.e., when the
+membership y is given (§2.1). Then we discuss the chal-
+lenges of simultaneously clustering and learning represen-
+tation (Problem 1), and propose our MCR2 clustering ob-
+jective to solve Problem 1 with those challenges in mind
+(§2.2). We further give an algorithm to optimize the pro-
+posed objective (§2.3).
+2.1. Supervised Manifold Linearizing via MCR2
+In the case when the labels y are given as supervision,
+MCR2 [53] aims to address part 2) of Problem 1.
+Let
+fθ : RD → Sd−1 be a featurizer parameterized by a
+neural network, which in turn gives an embedding Zθ :=
+[z1, . . . , zn] ∈ Rd×n of data with zi := fθ(xi) ∈ Sd−1.
+MCR2 aims to learn an ideal representation by optimizing
+max
+θ
+R(Zθ; ϵ) − Rc(Zθ, Π; ϵ)
+s.t.
+Zθ ∈ S
+(1)
+where R(Zθ; ϵ) := log det
+�
+I +
+d
+nϵ2 ZθZ⊤
+θ
+�
+,
+and Rc(Zθ, Π; ϵ) :=
+k
+�
+j=1
+⟨Πj, 1⟩
+n
+log det
+�
+I +
+d
+⟨Πj, 1⟩ϵ2 Zθ Diag(Πj)Z⊤
+θ
+�
+.
+Here S is the set of matrices whose columns all have unit
+ℓ2 norm6, Π ∈ Rn×k is a given membership matrix such
+that Πij = 1 if j = y(i) and Πij = 0 otherwise, ϵ > 0
+is a prescribed precision parameter, Πj ∈ Rn denotes the
+jth column of Π, 1 is a vector of all ones so that ⟨Πj, 1⟩
+is the number of points in cluster j, and finally for v ∈
+Rn, Diag(v) denotes a diagonal matrix with the entries of
+v along the diagonal.
+Intuitively7, the R(Zθ; ϵ) term of (1) measures the vol-
+ume of Zθ, and maximizing it would diversify features from
+all samples, which we will refer to as the expansion term
+Likewise, the Rc(Zθ, Π; ϵ) term measures the sum of vol-
+umes of each cluster of Zθ and is referred to as the compres-
+sion term, since minimizing it would push features within
+each cluster to stay close. It has been shown that given
+Π, the features obtained by maximizing the rate reduction
+R(Zθ; ϵ)−Rc(Zθ, Π; ϵ) has the property that the features
+of each cluster spread uniformly within a subspace (within-
+cluster diverse), and the subspaces from different clusters
+are orthogonal (between-cluster discriminative), under rel-
+atively mild assumptions [53].
+2.2. Unsupervised Manifold Linearizing and Clus-
+tering via MCR2
+While the MCR2 formulation is designed to learn ideal
+representations (§1) when the membership y (or equiva-
+lently Π) is given, here we are interested in the unsuper-
+vised setting where one does not have access to membership
+annotations. Thus, we propose to simultaneously perform
+both parts 1) and 2) of Problem 1 by also optimizing over
+6This can be easily achieved by having the last layer of the neural net-
+work fθ be a normalization layer.
+7More formally, terms of the form log det
+�
+I +
+d
+nϵ2 W W ⊤�
+esti-
+mate the average number of bits needed to code n i.i.d. samples W ∈
+Rd×n from a zero-mean d-dimensional Gaussian up to a distortion ϵ [28],
+hence the name coding rate.
+
+the membership Π of data. This naturally leads to
+max
+θ,Π∈Ω◦ R(Zθ; ϵ) − Rc(Zθ, Π; ϵ)
+s.t.
+Zθ ∈ S,
+(2)
+where Ω◦ := {Π ∈ Rn×k : ∀i ∈ [n], ∃ˆy(i)
+s.t. Πiˆy(i) =
+1 and Πij = 0 for j ̸= ˆy(i)} is the set of all ‘hard’ assign-
+ments, i.e., each row of Π is a one-hot vector. However,
+this optimization is in general combinatorial: its complex-
+ity grows exponentially in n and k, and it does not allow
+smooth and gradual changes of Π. Further, a second chal-
+lenge is the chicken-and-egg nature of this problem: If one
+already has an ideal representation Z, then existing sub-
+space clustering methods can be applied on Z to estimate
+the membership. Likewise, if one is given the membership
+Π of clusters, then solving (1) would lead to an ideal rep-
+resentation. However, the Zθ and Π at the beginning of
+optimization is typically far from ideal.
+Doubly Stochastic Subspace Clustering. To address the
+combinatorial of estimating the cluster memberships, we
+draw inspiration from the closely related problem of sub-
+space clustering, where the goal is to cluster n samples as-
+sumed to lie close to a union of k low-dimensional sub-
+spaces (§1). In this case, one typically does not directly
+learn an n × k matrix denoting memberships of n points
+into k subspaces. Instead, one first learns an affinity ma-
+trix Π ∈ Rn×n signaling the similarity between pairs of
+points, and then applies spectral clustering on the learned
+Π to obtain a final clustering [11,13,16,25,27,50]. In par-
+ticular, requiring doubly-stochastic constraints on the affin-
+ity Π is shown theoretically to suppress false inter-cluster
+connections for clustering problems [9] along with state-of-
+the-art empirical performance for subspace clustering prob-
+lems [24].
+Inspired by the above, we propose a constraint set Ω for
+matrix Π to be the set of n × n doubly stochastic matrices,
+Ω = {Π ∈ Rn×n : Π ≥ 0,
+Π1 = Π⊤1 = 1}.
+(3)
+However, this constraint alone is insufficient for strong clus-
+tering performance: Consider the optimization of (2) with
+respect to Π ∈ Ω only, and note that the objective is
+strongly convex with respect to Π. Since we maximize a
+convex function with respect to convex constraints Ω, an
+optimal Π would lie at an extreme point of Ω, which for
+doubly stochastic matrices is a permutation matrix. This is
+not ideal for clustering, as it implies that every point is as-
+signed to its own distinct cluster, and there is no incentive
+to merge points into larger clusters. To resolve this issue,
+we follow the approach in [24] and add ℓ2 regularization8
+γ
+2 ∥Π∥2
+F to Π which biases Π toward the uniform matrix
+1
+n11⊤, so by tuning γ we can also tune the sparsity level of
+8Other choices of regularization are also possible: Essentially any func-
+tion which achieves its minimum over Ω at the uniform matrix could po-
+tentially be used, e.g., the negative entropy function �
+ij Πij log(Πij).
+Π. This results in our final proposed formulation, dubbed
+Manifold Linearizing and Clustering (MLC):
+max
+θ
+R(Zθ; ϵ) − Rc(Zθ, Πθ; ϵ) − γ
+2 ∥Πθ∥2
+F
+(4)
+s.t.
+Zθ ∈ S, Πθ ∈ Ω,
+where R(Zθ; ϵ) = log det
+�
+I +
+d
+nϵ2 ZθZ⊤
+θ
+�
+, and
+Rc(Zθ, Πθ; ϵ) = 1
+n
+n
+�
+j=1
+log det
+�
+I + d
+ϵ2 Zθ Diag((Πθ)j)Z⊤
+θ
+�
+.
+Note that here Πθ = Πθ(X) is now also parameterized
+by a neural network. While this is constrained optimization
+which may appear difficult to handle, we explain in §2.3
+how we parameterize Zθ and Πθ via neural networks so
+that the constraints are satisfied by construction. Below, we
+note a few advantages of the proposed formulation.
+Parameterizing Π via a Neural Network versus Free
+Variables. An alternative way to parameterize the mem-
+bership would be to directly take Π as decision variables in
+Ω, as opposed to outputs of a neural network. However, this
+leads to maintaining O(n2) variables which is prohibitive
+for large datasets (e.g., n = 106 for ImageNet). In contrast,
+this is not the case if one parameterizes Π as a neural net-
+work, since one can do stochastic gradient descent such that
+for each batch both the memory and computational com-
+plexity is at most square of the batch size (Figure 2).
+Comparison with NMCE. As mentioned in §1.2, NMCE
+[23] approaches Problem 1 also by optimizing MCR2 over
+both the representation and membership.
+However, in
+NMCE the membership is parameterized by an n × k ma-
+trix Πn×k that models the point-cluster membership, which
+is different from our doubly stochastic point-point member-
+ship matrix Πθ inspired from the state-of-the-art subspace
+clustering. Note further that for NMCE the initialization
+of Πn×k is arbitrary and has nothing to do with the struc-
+tures in the initialized representation Πθ, and a bad initial-
+ization of Πn×k could lead to the features from different
+true clusters being compressed. On the other hand, the pro-
+posed doubly stochastic membership Πθ can be initialized
+deterministically using structures from self-supervised ini-
+tialized features Zθ (§2.3).
+Interestingly, optimizing (4)
+allows an interpretation of linearizing each point with its
+neighbors. Empirically as seen in (Table 2), the proposed
+MLC yields a higher clustering accuracy than NMCE [23].
+2.3. Algorithms
+As mentioned, in the MLC objective (4), we parameter-
+ize both the representation Zθ and doubly stochastic mem-
+bership Πθ via a neural network. Below we elaborate on
+how this is done. We summarize the network architecture
+in Figure 2, and the meta algorithm in Algorithm 1.
+Parameterizing Zθ. We follow [53] and take some existing
+network architecture as the backbone. We append a few
+
+Figure 2. Overall architecture for optimizing the proposed manifold linearizing and clustering (MLC) objective (4). Given n input samples
+X each lying in RD, their d-dimensional representation is given by Zθ(X), where θ denotes network parameters. Further, their doubly
+stochastic membership matrix Πθ(X) is given by taking an inner product kernel of the output of the cluster head Cθ(X) followed by a
+doubly stochastic projection.
+affine layers with non-linearity as the representation head to
+further transform the output in Rd, followed by a projection
+layer to respect the unit sphere Sd−1 constraint.
+Parameterizing Πθ. In subspace clustering, the member-
+ship Π given data X often takes the form of g(X)⊤g(X)
+for some (linear) transformation g, such as in the inner
+product kernel [9, 16] where g = I or the least square re-
+gression [27] where g = (I + λX⊤X)−1/2. This moti-
+vates us to parameterize gθ by a neural network, and take
+C⊤
+θ Cθ ∈ Rn×n as the membership where Cθ is shorthand
+for gθ(X). Nevertheless, such an n×n matrix is in general
+not doubly stochastic, i.e., C⊤
+θ Cθ /∈ Ω. To obtain a doubly
+stochastic membership, we further apply a Sinkhorn projec-
+tion layer PΩ,η(·) [10,39], which gives our final parameter-
+ization of the membership as Πθ = PΩ,η(C⊤
+θ Cθ) ∈ Ω.
+Initializing Zθ: Self-supervised Representation Learn-
+ing via MCR2. Since the proposed MCR2 clustering objec-
+tive (4) is non-convex, it is important to properly initialize
+both Z and Π to converge to good (local) minimum. On
+the other hand, randomly initialized features are typically
+far from being ideal, since they may not satisfy the idealized
+properties (§1), and further may not respect the invariance
+to augmentation, i.e., the augmented samples should have
+their representation close to each other. Thus, we adopt the
+self-supervised training strategy [23]
+max
+θ
+R
+�Zθ + Z′
+θ
+2
+; ϵ
+�
++ λ
+n
+�
+i=1
+|z⊤
+i z′
+i|,
+s.t.
+z′
+i, zi ∈ Sd−1,
+∀i ∈ [n],
+(5)
+where for every i, zi and z′
+i are features of different aug-
+mentations of the i-th sample. This essentially requires that
+features from different augmentations of the same sample
+should be as close as possible, whereas features from dif-
+ferent samples should be as uncorrelated as possible.
+Initializing Πθ. An ideal initialization of Πθ would be
+such that if (Πθ)ij has a high value then points i, j are
+likely to be from the same true cluster and vice versa. On the
+other hand, after the self-supervised feature initialization
+mentioned above, Zθ already have some structures which
+we can utilize.
+Thus, we propose to initialize Πθ with
+PΩ,η(Z⊤
+θ Zθ), which is easily implemented by copying the
+parameters from Zθ to Cθ once after the self-supervised
+initialization of the former, i.e., from the feature head to the
+cluster head in Figure 2.
+Data Augmentation. Beyond initializing Zθ, it is often
+desirable to incorporate augmentation in optimizing (4).
+Specifically, from {X(a) ∈ RD×n}A
+a=1 the dataset X
+under A different augmentations, one computes (Z(a)
+θ
+∈
+Rd×n, Π(a)
+θ
+∈ Rd×n) for each augmentation a, and use in
+(4)
+Zθ = PSd−1
+�
+1
+A
+A
+�
+a=1
+Z(a)
+θ
+�
+,
+Πθ = 1
+A
+A
+�
+a=1
+Π(a)
+θ
+∈ Ω.
+(6)
+Note that one can benefit from parallelization by putting
+X(a), Z(a)
+θ , Π(a)
+θ
+for each augmentation a on one comput-
+ing device, since Π(a)
+θ
+only depends on X(a) but not from
+other augmentations.
+3. Experiments on Real Datasets
+Metrics. To evaluate the clustering quality, we run spec-
+tral clustering on learned membership matrix Π, and re-
+port the normalized mutual information (NMI, [42]) and
+clustering accuracy (ACC, [21]), as are commonly used in
+clustering tasks.
+To evaluate the learned representation,
+we define the following metric: for a collection of points
+W = [w1, . . . , wl] ∈ Rd×l (l > d) with associated sin-
+gular values {σi}d
+i=1, define the numerical rank of W as
+arg minr
+�
+r : �r
+i=1 σ2
+i / �d
+i=1 σ2
+i > 0.95
+�
+. Now, one can
+measure the numerical rank of the learned representaion Z,
+
+BackboneAlgorithm 1 MLC: Unsupervised Manifold Linearizing
+and Clustering
+Input: X ∈ RD×n,
+ϵ, γ, η, λ > 0,
+d, k, nb, T, A ∈ Z≥0
+1: initialize Zθ by self-supervised representation learning
+via MCR2
+▷ (5)
+2: initialize Πθ
+3: for t = 1, . . . , T do
+4:
+¯
+X ∈ RD×nb ← sample a batch from X
+5:
+¯
+X(1), . . . , ¯
+X(A) ← apply A augmentations to ¯
+X
+6:
+¯Zθ, ¯Πθ ← forward pass with { ¯
+X(a)}A
+a=1 and net-
+work parameters θ
+▷
+(6)
+7:
+∇θ(4) ← backward pass with respect objective (4)
+8:
+θ ←update θ using some optimizer on ∇θ(4)
+9: end for
+10: run spectral clustering on Πθ to estimate labels ˆy of
+samples
+Output: Zθ, ˆy
+as well as that of each ground-truth cluster9 of Z. A low
+numerical rank of W implies that points in W lie close to
+a low-dimensional subspace. We further report the cosine
+similarity of learned representation, which is simply |z⊤
+i zj|
+for points i and j, since ∥zi∥ = 1 by construction in (4).
+Finally, to compare the efficiency of methods we report the
+training time in §3.2, where the experiments are run on 2
+Nvidia RTX3090 GPUs.
+3.1. Comparison with Subspace Clustering
+To demonstrate the ability of MLC to cluster the sam-
+ples and linearize the manifolds, we conduct experiments on
+CIFAR10, which consists of RGB images from 10 classes
+such as planes, birds, and deers. As mentioned in §1 sub-
+space clustering methods rely crucially on the assumption
+that data lie close to a union of linear subspaces, which
+many real-world dataset may not satisfy. To show that this
+is the case, we additionally compare the proposed method
+with subspace clustering methods. As we shall see, apply-
+ing subspace clustering directly on self-supervised features
+of CIFAR10 will yield low clustering accuracy. In contrast,
+MLC is able to achieve high clustering accuracy, and more-
+over, produce a union-of-orthogonal-subspace representa-
+tion on which subspace clustering methods can also achieve
+high accuracy.
+Data.
+We use the training split of CIFAR10 containing
+50000 RGB images, each of size 3 × 32 × 32. We use
+the augmentation specified in the Appendix to perform self-
+supervised representation learning (5) and get Zself. For
+a fair comparison, the so-learned Zself are used both as ini-
+tialization for MLC (line 1 of Algorithm 1), and as the input
+9They are defined by the true labels y (§2), so that the numerical rank
+metric is decoupled from the quality of learned membership Π.
+Table 1. Clustering accuracy and normalized mutual information
+for subspace clustering (EnSC, SSC-OMP) on self-supervised fea-
+tures Zself, features ZMLC learned by MLC, and manifold clus-
+tering (MLC) on X, where X is 6 · 104 images from 10 classes
+of CIFAR10.
+Method
+Input Data
+ACC
+NMI
+EnSC
+Zself
+72.2
+67.9
+ZMLC
+81.5
+79.2
+SSC-OMP
+Zself
+67.8
+64.5
+ZMLC
+78.4
+76.3
+MLC
+X
+86.3
+78.3
+for subspace clustering methods10. In MLC, for each image
+in each batch we randomly sample A = 2 augmentations
+to apply on the image. As an additional comparison, we
+also run subspace clustering methods on the features ZMLC
+learned by MLC.
+Methods. We compare with the elastic-net subspace clus-
+tering with active-set solver (EnSC, [50]) and sparse sub-
+space clustering with orthogonal matching pursuit solver
+(SSC-OMP, [51]), using off-the-shelf implementation pro-
+vided by the authors11.
+We search the parameters of
+EnSC over (γ, τ) ∈ {1, 5, 10, 50, 100} × {0.9, 0.95, 1}
+and those of SSC over (kmax, ϵ)
+∈
+{3, 5, 10, 20} ×
+{10−4, 10−5, 10−6, 10−7}, and report the run with the
+highest clustering accuracy for each method. We summa-
+rize detailed parameters for MLC in the Appendix.
+Results. Figure 3 reports the coding rates (as loss terms
+in (4) and numerical ranks of features learned by MLC as
+epoch varies. As a first note, the coding rate R of all fea-
+tures (the blue curve in 3a) decreases only slightly as epoch
+goes, indicating that the overall representation is diverse in
+the feature space. Indeed, the numerical rank of all features
+(the dark curve in Figure 3b) stays 118 which is close to the
+dimension 128 of the feature space. This is in sharp contrast
+to the deep subspace clustering methods where all the fea-
+tures collapse to a one-dimensional subspace [15]. More-
+over, as the coding rate Rc of clustered features (the orange
+curve in Figure 3a) goes down, the numerical ranks of fea-
+tures from each ground-truth cluster decrease. For instance,
+the representation from true cluster 3 has a numerical rank
+of 37 in the first step and 24 in the last step. This implies
+that most representation gets linearized better and clustered
+more accurately, even though the MLC objective (4) is un-
+supervised, i.e., it does not use ground-truth labels y. Last
+but not the least, note that the features within each ground-
+10The self-supervised features Zself empirically exhibit some union-of-
+subspace structure, and are typically used for subspace clustering, as also
+seen in [53, §3.2] and [55, §4.2].
+11https : / / github . com / ChongYou / subspace -
+clustering
+
+(a) Coding rate of all features R, that of clustered
+features Rc, and the rate reduction ∆R = R−Rc.
+(b) Numerical ranks of all features Zθ and features
+from each ground-truth cluster i, {zj : y(j) = i}.
+Figure 3. Coding rates (as loss terms in (4)) and numerical ranks (§3.1) of the features
+learned by MLC on CIFAR10 as epoch varies.
+Figure 4. Cosine similarity |Z⊤
+MLCZMLC|
+of the features ZMLC learned by MLC.
+truth cluster spread well in a low-dimensional subspace,
+e.g., the numerical ranks for the true clusters at the last step
+are within [13, 23]. This achieves the desired within-cluster
+diverse property (§1), as opposed to the neural collapse phe-
+nomenon that appears with the cross-entropy loss.
+To compare MLC with subspace clustering methods, we
+report clustering accuracy and normalized mutual informa-
+tion for EnSC, SSC-OMP on self-supervised features Zself,
+features ZMLC learned by MLC, and MLC on X, where X
+is 6 · 104 images from 10 classes of CIFAR10. In addition
+we plot the cosine similarity of the features learned by MLC
+in Figure 4. Remarkably, the highest clustering accuracy
+is 86.3% achieved by MLC on X, which surpasses EnSC
+(72.2%) and SSC-OMP (67.8%) on Zself by a large margin,
+even though Zself is used both as initialization for MLC and
+input for EnSC and SSC-OMP. Interestingly, using instead
+the features ZMLC learned by MLC, the clustering perfor-
+mance of EnSC and SSC-OMP increases and even becomes
+comparable to MLC, e.g., EnSC achieves 79.2% normal-
+ized mutual information compared to 78.3% of MLC. This
+suggests that ZMLC has a union-of-subspace structure that
+can be utilized by subspace clustering. Indeed, as seen in
+Figure 4, features from different clusters tend to have a
+small similarity, i.e., being orthogonal to each other. This
+demonstrates the between-cluster discrimination (§1) as de-
+sired.
+3.2. Comparison with Deep Clustering Methods
+We further compare the proposed MLC with state-of-
+the-art deep clustering methods. Note that most methods
+reported (all except NMCE which is discussed in §2.2) do
+not aim to learn a union-of-orthogonal-subspace represen-
+tation, in contrast to MLC. As we will see, MLC achieves
+clustering accuracy comparable to state-of-the-art methods
+on large scale datasets with faster computational time, and
+further surpasses them on extreme yet realistic cases like
+datasets of imbalanced clusters.
+Compared Methods. We conduct experiments with MLC,
+SCAN [45], and IMC-SWAV [31].12 Training details can
+be found in the Appendix. In addition we include the num-
+bers reported from DeepCluster [7], IIC [19], RUC [33] and
+NMCE [23]. For a fair comparison, all methods reported
+use ResNet-18 as the backbone, which is also commonly
+adopted by other methods.
+Datasets.
+Beyond CIFAR10 (§3.1), we further use
+CIFAR100-20, CIFAR100-100 and Tiny Imagenet-200 to
+evaluate the performance of our method. Both CIFAR100-
+100 and CIFAR100-20 contain the same 50000 train images
+and 10000 test images with size 32 × 32 × 3, while the for-
+mer are split into 100 clusters and the latter 20 super clus-
+ters. Finally, Tiny ImageNet contains 100000 train images
+and 10000 test images with size 64 × 64 × 3 split into 200
+clusters.
+Results on Large-scale Datasets. We report clustering ac-
+curacy and normalized mutual information on CIFAR10,
+CIFAR100-20, CIFAR100-100, and TinyImageNet in Ta-
+ble 2, and we further report running time in minutes for
+CIFAR100-100 in Table 3. As seen, the highest cluster-
+ing performance on CIFAR10 is achieved by RUC+SCAN
+(90.3% ACC) and IMC-SWAV (81.1% NMI), where MLC
+yields a slightly lower ACC of 86.3% and NMI of 78.3%.
+We note some interesting semantic interpretation for the
+clustering obtained by MLC in the Appendix. On the other
+hand, MLC performs comparably with other methods on
+CIFAR100-20 by achieving an ACC of 52.2% and NMI of
+54.6%. Notably, MLC outperforms SCAN and IMC-SWAV
+on CIFAR100-100 and TinyImageNet-200 by a large mar-
+gin, while using lower running time: E.g., on CIFAR100-
+12The authors are aware of a preprint [30] which approaches image
+clustering via a combination of self/semi-supervised learning and pseudo-
+labeling. However, to the best of our effort we are unable to reproduce the
+numbers reported in this paper using the implementation provided by the
+authors. We discuss the details in the Appendix and thus do not report their
+numbers here.
+
+Loss terms
+140
+120
+100
+Loss terms
+80
+60
+40 
+△R
+R
+Rc
+20
+0
+500
+1000
+1500
+2000
+2500
+StepNumerical rank of Z
+120
+100
+All
+Class 0
+Class 1
+80
+Class 2
+Class 3
+Class 4
+60
+Class 5
+Class 6
+Class 7
+Class 8
+40
+Class 9
+20
+0
+500
+1000
+1500
+2000
+2500
+Step1.0
+0
+0.8
+2000
+0.6
+4000
+6000
+0.4
+8000
+0.2
+0
+2000
+4000
+6000
+8000
+0.0Table 2. Clustering accuracy and normalized mutual information on large scale datasets. For a fair comparison, all methods use ResNet-18
+as backbone.
+Method / Dataset
+CIFAR10-10
+CIFAR100-20
+CIFAR100-100
+Tiny ImageNet-200
+Metrics
+ACC
+NMI
+ACC
+NMI
+ACC
+NMI
+ACC
+NMI
+DeepCluster (ECCV′18)
+37.4
+-
+18.9
+-
+-
+-
+-
+-
+IIC (ICCV′19)
+61.7
+51.1
+25.7
+22.5
+-
+-
+-
+-
+SCAN (ECCV′20)
+87.6
+78.7
+46.8
+45.9
+34.3
+55.7
+-
+-
+RUC+SCAN (CVPR′21)
+90.3
+-
+53.3
+-
+-
+-
+-
+-
+IMC-SWAV (Arxiv′21)
+89.1
+81.1
+49.0
+50.3
+43.9
+58.3
+28.2
+52.6
+NMCE (Arxiv′22)
+83.0
+76.1
+43.7
+48.8
+-
+-
+-
+-
+MLC
+86.3
+78.3
+52.2
+54.6
+49.4
+68.3
+33.5
+67.5
+Table 3.
+Running time in minutes and clustering accuracy on
+CIFAR100-100. For a fair comparison, all methods use ResNet-18
+as backbone.
+Method / Metric
+Running Time
+ACC
+Stage
+I
+II
+III
+Total
+SCAN (ECCV′20)
+308.3
+33.3
+54.7
+396.3
+34.3
+IMC-SWAV (Arxiv′21)
+529.4
+-
+-
+529.4
+43.9
+MLC
+266.7
+17.7
+-
+284.4
+48.3
+Table 4. Clustering accuracy on imbalanced datasets: (a) Imb-
+CIFAR10, (b) Imb-CIFAR100-100.
+For a fair comparison, all
+methods use ResNet-18 as backbone.
+Method / Dataset
+(a)
+(b)
+IMC-SWAV (Arxiv′21)
+65.7
+38.2
+SCAN (ECCV′20)
+62.9
+31.1
+MLC
+80.0
+46.1
+100, MLC yields an accuracy of 49.4% in 291 minutes,
+whereas IMC-SWAV has 43.9% using 529 minutes, and
+SCAN has 34.3% in 396 minutes.
+Imbalanced Clusters.
+Note that for CIFAR10 or CI-
+FAR100 each cluster contains approximately the same num-
+ber of samples. On the other hand, natural images are typ-
+ically imbalanced, i.e., the clusters have unequal number
+of samples.
+To mimic this setting, we take a naive ap-
+proach to construct the following imbalanced datasets. For
+the 10 clusters of CIFAR10, we remove half of the sam-
+ples from odd-numbered clusters (i.e., clusters 1, 3, . . . , 9)
+from both the training and test split. We refer to the re-
+duced dataset Imb-CIFAR10. Likewise we construct Imb-
+CIFAR100-100. We run two state-of-the-art methods IMC-
+SWAV and SCAN as well as the proposed MLC on Imb-
+CIFAR10 and Imb-CIFAR100-100.
+Table 4 shows clustering accuracy on the imbalanced
+datasets Imb-CIFAR10 and Imb-CIFAR100-100. As a first
+observation, the clustering accuracy of all methods is lower
+on the imbalanced datasets than on the balanced counter-
+parts, as expected. Notably, MLC suffers from the least
+performance drop, e.g., when moving from CIFAR10 to
+Imb-CIFAR10 the accuracy of MLC drops from 86% to
+80%, whereas that of SCAN and IMC-SWAV decreases
+from above 87% to below 66%.
+4. Conclusion
+This paper studies the problem of simultaneously clus-
+tering and learning an union-of-orthogonal-subspace repre-
+sentation for data, when data lies close to a union of low-
+dimensional manifolds. To address the problem we pro-
+pose an objective based on maximal coding rate reduction
+and doubly stochastic membership inspired by the state-of-
+the-art subspace clustering results. We provide an efficient
+and effective parameterization of the membership variables
+as well as a meta-algorithm to optimize the representation
+and membership jointly. We further conduct experiments
+on datasets with larger number of clusters and imbalanced
+clusters and show that the proposed method achieves state-
+of-the-art performance. We believe that our work provides
+a general and unified framework for unsupervised learning
+of structured representations for multi-modal data.
+
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+
+(a) Learned cluster 1
+(b) Learned cluster 2
+(c) Learned cluster 3
+(d) Learned cluster 4
+(e) Learned cluster 5
+(f) Learned cluster 6
+(g) Learned cluster 7
+(h) Learned cluster 8
+(i) Learned cluster 9
+Figure 5. Principal images (defined in §A) of clusters learned by MLC on CIFAR10.
+
+A. Semantic Interpretability of the Learned
+Representation and Clusters on CIFAR10
+Recall that MLC is designed to perform clustering
+while learning a union-of-orthogonal-subspace representa-
+tion (§1), where each cluster defines a low-dimensional sub-
+space. Therefore, we further visualize the different direc-
+tions within each learned cluster or subspace. Specifically,
+after a final clustering is obtained (line 10 of Algorithm 1),
+we take the features from each learned cluster and apply
+Principal Component Analysis (PCA) to them to obtain the
+first 8 principal components. These correspond to the 8
+rows for each cluster in Figure 5. Recall that the princi-
+pal components are mutually orthogonal, indicating uncor-
+related directions within one cluster. To visualize those di-
+rections or principal components in images, we take the fea-
+tures that are closest to the principal components and visu-
+alize the corresponding original images.
+Interestingly, the rows of images corresponding to prin-
+cipal components appear to exhibit some semantic ‘con-
+cepts’. For example in Figure 5b, row 1 and 8 are respec-
+tively white and red trucks, while row 4 are the trucks that
+ship sand or mud; row 1 of Figure 5d are deers with trees
+as background. This further suggests that the learned em-
+bedding seems to preserve distance within each cluster (as
+desired in §1), i.e., images that are close/far in semantic
+meaning will be close/far in the feature space. Note how-
+ever, that some learned clusters do not align fully with the
+ground-truth labels. For instance, rows 1 and 3 of Figure 5h
+are cats while all other rows in this cluster are dogs. On the
+other hand, one may argue that Figure 5h are a cluster of
+cats and dogs of light colors, whereas Figure 5c is a clus-
+ter of those of brown colors, which could be a semantically
+meaningful clustering even though it does not align with the
+ground-truth labels. We believe it would be an interesting
+future work to use MLC to discover new semantics that are
+not present in the given labels.
+Table 5. Ablation study on the roles of different parts of Algo-
+rithm 1 and on using augmentation.
+Ablation Study on CIFAR-10
+Clustering
+Accuracy
+Full Algorithm 1
+86.3%
+Replacing self-supervised initialization (line 1)
+with random initialization
+20.0%
+Replacing updating MLC objective (4) (lines 3-9)
+with subspace clustering (EnSC)
+73.4%
+Not using augmentation in lines 3-9
+80.0%
+B. Role of Augmentation
+Recall that data augmentation was used both in the self-
+supervised initialization (line 1 of Algorithm 1, see ‘Initial-
+izing Zθ’ in §2.3) and in updating the MLC objective (lines
+3-9, ‘Data Augmentation’ in §2.3). Below we give addi-
+tional clarification on the role of augmentation therein.
+B.1. Augmentation for Initializing the Features
+Since the proposed MLC objective (4) is highly non-
+convex, the (local) solution that a first-order optimizer con-
+verges to in general depends on the initialization. However,
+before line 1 of Algorithm 1 is executed, the features Z at
+initialization could be very far from union-of-orthogonal-
+subspace (as desired by Problem 1), since the neural net-
+work has an arbitrary architecture and initialization. To at
+least promote some ideal structures in the features, we con-
+duct line 1 of Algorithm 1 so that the features from an origi-
+nal sample and its augmented copy are close, while features
+from different samples spread out in the feature space. This
+is a common idea used in contrastive learning, and more re-
+lated in [53, §3.2] [23, §3.6] that are both based on MCR2
+as in this paper (even though the formulations are different,
+as argued in §1.2). Empirically, initializing the features us-
+ing augmentation (line 1 of Algorithm 1) is important for
+the following-up steps: as seen in Table 5, on CIFAR-10, if
+one uses random initialization to replace this step, then the
+final clustering accuracy is 20%, in sharp contrast to 86.3%.
+B.2. Augmentation for Updating MLC Objective (4)
+In optimizing MLC (lines 3-9 of Algorithm 1), augmen-
+tation empirically improves clustering performance. As one
+can see in Table 5, on CIFAR-10 using the sample self-
+supervised initialization of the features, MLC achieves only
+80% clustering accuracy without augmentation, in contrast
+to 86.3% with augmentation. We attribute this difference to
+the fact that augmentation enriches the diversity of samples
+the algorithm sees.
+C. Role of Different Parts of Algorithm 1
+C.1. Initialization of the Features (Line 1)
+Please kindly refer to §B.1.
+C.2. Updating the MLC Objective (4) (Lines 3-9)
+The main novelty of this paper lies in updating the MLC
+objective that learns both the representation Zθ and a dou-
+bly stochastic membership Πθ. Note that in this step, clus-
+tering is pursued by modeling the membership Πθ, as op-
+posed to the self-supervised feature initialization step where
+no membership is explicitly pursued. This step is indeed im-
+portant for clustering: as seen in Table 5, on CIFAR-10, the
+
+clustering accuracy on the self-supervised initialized fea-
+tures Zθ is only 73.4%, in contrast to 86.3% obtained after
+updating the MLC objective (4).
+C.3. Spectral Clustering (Line 10)
+Since the proposed MLC learns a doubly stochas-
+tic membership that signals pair-wise similarity between
+points, it is standard to run spectral clustering [46] to com-
+pute a final set of clusters from the learned membership.
+This is done once at the very end of Algorithm 1, and is
+rather efficient compared to the other parts of Algorithm 1:
+for instance, using an unaccelerated implementation from
+SciPy, it takes less than 30 seconds to perform spectral clus-
+tering on a 10000 × 10000 matrix.
+D. Details on Experiment Settings
+D.1. Synthetic Union-of-Manifold Data
+We perform simulations to visualize the properties of the
+proposed manifold learning and clustering method. As seen
+in Figure 1a, we generate data X from two manifolds on
+the sphere S2, each consisting of 200 samples. The points
+from the first manifold (green) take the form
+xi =
+�
+�
+cos
+�
+A sin(ωφi)
+�
+cos φi
+cos
+�
+A sin(ωφi)
+�
+sin φi
+sin
+�
+A sin(ωφi)
+�
+�
+� + ϵi,
+(7)
+where A = 0.2 and ω = 5 sets the curvature of the mani-
+fold, ϵi ∼ N(0, 0.05I3) is the additive noise, and we take
+φi = 2πi
+100 for i = 1, . . . , 100 to generate 100 points. On the
+other hand, the points from the second manifold (blue) are
+simply 100 samples from N([0, 0, 1]⊤, 0.05I3). We take
+the feature dimension d = 3 to be equal to he input di-
+mension D = 3. We paramterize both the feature head
+fθ and the cluster head gθ to be a simple fully-connected
+network with 100 hidden neurons, followed by a Rectified
+Linear Unit as non-linearity and a projection operator onto
+the sphere S2. Figures 1b to 1d report the features Zθ with
+random initialization (i.e., before line 1 of Algorithm 1),
+with self-supervised initialization, and at convergence of
+MLC. Notably, despite Zθ being noisy and only approxi-
+mately piece-wise linear, as epoch goes Zθ gradually trans-
+form to two linear subspaces: the green points converge to
+a 2-dimensional subspace (intersected with S2) and the blue
+points converge to a 1-dimension subspace.
+D.2. Training Details on Real Datasets
+MLC. As said, we use ResNet-18 as the backbone for ex-
+periments on CIFAR10, CIFAR100-20, CIFAR100-100 and
+Tiny-ImageNet-200, and the imbalanced counterparts Imb-
+CIFAR10, Imb-CIFAR100-100. We also fix the batch size
+to be 1024 in all experiments. In self-supervised initializa-
+tion of Zθ (line 1 of Algorithm 1), we use the precision
+(§2.1) parameter ϵ2 = 0.2, a LARS optimizer [52] (as is
+also done in [8, 23]) with a learning rate of 0.3 and trained
+MLC for 1000 epochs. On the other hand, in the train-
+ing of MLC objective, we use ϵ2 = 0.1, γ = 0.05, and
+η = 0.175 for the entropy regularization in the Sinkhorn
+projection [10] layer PΩ,η(·). We fix the backbone and for
+each batch, we perform one update for parameters in the
+feature head Zθ and one update for parameters in the clus-
+ter head Cθ. For each head we use one SGD optimizer [36]
+with a learning rate of 10−2, momentum of 0.9, and weight
+decay of 5 · 10−4. Finally, for all experiments, we use the
+augmentation from [4] detailed below in PyTorch code.
+Augmentation 1 Augmentations for real datasets
+import torchvision.transforms as t
+t.Compose([
+t.RandomResizedCrop(32,scale=(0.04,
+1.0)),
+t.RandomHorizontalFlip(p=0.5),
+t.RandomGrayscale(p=0.2),
+t.RandomApply([t.ColorJitter(0.4,
+0.4, 0.4, 0.1)], p=0.8),
+GaussianBlur(p=0.1)
+])
+SCAN and IMC-SWAV. Recall that we conduct ex-
+periments on CIFAR100-100, Imb-CIFAR10, and Imb-
+CIFAR100-100 with SCAN [45], IMC-SWAV [31] and
+MLC, and report clustering and running time in Tables 3
+and 4. We use off-the-shelf implementation13 provided by
+the authors. For a fair comparison, SCAN, IMC-SWAV and
+MLC all use ResNet-18 as the backbone. Finally, the hyper-
+parameters of SCAN and IMC-SWAV are set to be the ones
+optimally chosen for CIFAR10 and CIFAR100 respectively
+provided by the authors.
+SPICE. As mentioned, the preprint [30] proposed a method
+SPICE that appears to achieve state-of-the-art performance
+in image clustering. We tried to reproduce their results on
+CIFAR-100-20 using the official implementation14. How-
+ever, the provided implementation ran into a few errors,
+which are also observed15. Despite our best effort to fix
+those issues, the experiments yield only 14% clustering ac-
+curacy on CIFAR100-20 as opposed to the 53% reported in
+the paper [30]. Therefore, we note this observation and do
+not include SPICE in the main text.
+13https://github.com/wvangansbeke/Unsupervised-
+Classification,
+https : / / github . com / foiv0s / imc -
+swav-pub
+14https : / / github . com / niuchuangnn / SPICE,
+commit
+5eba538.
+15https://github.com/niuchuangnn/SPICE/issues/
+27,https://github.com/niuchuangnn/SPICE/issues/31
+
+Table 6. Clustering accuracy and normalized mutual information of MLC and NMCE on CIFAR10 over 10 random seeds, using the same
+self-initialized features. For the purpose of comparison, both methods use the same optimization strategy and hyper-parameters optimally
+tuned for NMCE. Consequently, the clustering performance of MLC reported here is lower than that in Table 2.
+Method
+Metric
+Seed
+Mean
+Std.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+MLC
+ACC
+84.5
+84.8
+84.8
+84.6
+84.4
+84.4
+84.0
+84.3
+84.4
+84.6
+84.5
+0.24
+NMI
+76.6
+77.1
+76.8
+76.8
+76.5
+76.4
+76.1
+76.4
+76.4
+76.5
+76.6
+0.28
+NMCE
+ACC
+83.7
+82.1
+81.6
+73.7
+80.4
+77.9
+81.7
+81.4
+72.7
+80.9
+79.6
+3.69
+NMI
+74.4
+71.2
+70.4
+65.2
+70.0
+68.1
+72.7
+70.8
+69.2
+69.8
+70.2
+2.49
+(a) CIFAR100-20
+(b) CIFAR100-100
+(c) Tiny ImageNet-200
+Figure 6.
+Cosine similarity |Z⊤
+MLCZMLC| of the features ZMLC learned by MLC on more complicated datasets: CIFAR100-20,
+CIFAR100-100, Tiny ImageNet-200.
+E. Additional Comparison of MLC and NMCE
+on Stability with Respect to Random Seeds
+As detailed in §2.2, one of the advantages of the pro-
+posed MLC over NMCE [23] is that MLC has a more sta-
+ble performance with respect to random seeds, since MLC
+is able to initialize the membership deterministically using
+structures from the self-supervised initialized features. Be-
+low we conduct extra experiments to provide empirical ev-
+idence. We first fix a self-supervised initialization of fea-
+tures that is in turn used for both NMCE and MLC. Then,
+using this very same initialization of features, we update
+NMCE and MLC objective respectively with 5 different
+seeds: recall that NMCE initializes the membership ran-
+domly whereas MLC initializes the membership determin-
+istically using the initialized features. To make a valid com-
+parison, for both methods we further use the same opti-
+mization strategy and hyper-parameters that are optimally16
+tuned for NMCE (which are not optimal for MLC): pre-
+cision ϵ2 = 0.2, # epochs 100, LARS optimizer for Zθ
+with an initial learning rate 0.3 decayed to 0 in a cosine
+annealing manner. Table 6 reports clustering accuracy and
+normalized mutual information of MLC and NMCE over
+16For NMCE, we use the implementation as well as the parameters pro-
+vided in https://github.com/zengyi-li/NMCE-release.
+10 random seeds. As expected, MLC has a more stable
+clustering performance by having a standard deviation of
+clustering accuracy and normalized mutual information less
+than 0.28, in contrast to more than 2.49 achieved by NMCE.
+Further, MLC achieves higher mean clustering performance
+than NMCE, as also observed in Table 2. Last but not least,
+we note that the numbers in Table 6 are not comparable to
+those in Table 2, since for MLC the hyper-parameters and
+optimizers are different, and for NMCE an additional step
+that fine tunes the backbone is used in Table 2.
+F. Addtional Visualization on Learned Repre-
+sentation and Clusters
+Figure 6 presents the cosine similarity (as defined in the
+preamble of §3) of the representation learned by MLC on
+CIFAR100-20, CIFAR100-100 and TinyImageNet-200 (for
+the counterpart on CIFAR10 see §3.1). As seen, the cosine
+similarity maps form approximately block diagonal struc-
+tures, showing that the features from different clusters are
+roughly orthogonal to each other. This is desired by the
+between-cluster discrimination (§1).
+Finally, we provide additional visualization of principal
+images on CIFAR100-20 (see §A for definition) in Figure 8.
+
+ZtZepoch:{epoch}batch:{step}
+0
+1.0
+200
+0.8
+400
+0.6
+600
+0.4
+800
+0.2
+1000
+0
+200
+400
+600
+800
+1000ZtZepoch:{epoch}batch:{step)
+0
+1.0
+200
+0.8
+400
+600
+0.6
+800
+0.4
+1000
+1200
+0.2
+1400
+0
+200
+400
+600
+800
+1000
+1200
+1400ZtZepoch:fepoch}batch:(step}
+0
+1.0
+200
+0.8
+400
+600
+0.6
+800
+0.4
+1000
+1200
+0.2
+1400
+0
+200
+400
+600
+800
+1000
+1200
+1400(a) Learned cluster 1
+(b) Learned cluster 2
+(c) Learned cluster 3
+(d) Learned cluster 4
+(e) Learned cluster 5
+(f) Learned cluster 6
+(g) Learned cluster 7
+(h) Learned cluster 8
+(i) Learned cluster 9
+(j) Learned cluster 10
+(k) Learned cluster 11
+(l) Learned cluster 12
+
+9d(m) Learned cluster 13
+(n) Learned cluster 14
+(o) Learned cluster 15
+(p) Learned cluster 16
+(q) Learned cluster 17
+(r) Learned cluster 18
+(s) Learned cluster 19
+(t) Learned cluster 20
+Figure 8. Principal images (defined in §A) of clusters learned by MLC on CIFAR100-20.
+

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+Continuous optical-to-mechanical quantum state transfer in the unresolved sideband
+regime
+Amy Navarathna1,2, James S. Bennett1,2,3, and Warwick P. Bowen1,2∗
+1 ARC Centre of Excellence for Engineered Quantum Systems, St Lucia, Queensland 4072, Australia
+2 School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia
+3 Centre for Quantum Dynamics, Griffith University, Nathan, QLD 4222, Australia
+(Dated: January 11, 2023)
+Optical-to-mechanical quantum state transfer is an important capability for future quantum net-
+works, quantum communication, and distributed quantum sensing. However, existing continuous
+state transfer protocols operate in the resolved sideband regime, necessitating a high-quality optical
+cavity and a high mechanical resonance frequency. Here, we propose a continuous protocol that
+operates in the unresolved sideband regime. The protocol is based on feedback cooling, can be im-
+plemented with current technology, and is able to transfer non-Gaussian quantum states with high
+fidelity. Our protocol significantly expands the kinds of optomechanical devices for which continu-
+ous optical-to-mechanical state transfer is possible, paving the way towards quantum technological
+applications and the preparation of macroscopic superpositions to test the fundamentals of quantum
+science.
+The ability to transfer quantum states between op-
+tical communication channels and quantum computing
+nodes is a necessary ingredient of the emerging quan-
+tum internet [1]. Quantum state transfer also has im-
+portant applications in quantum-enhanced sensing [2, 3],
+quantum-secure communications [4], and fundamental
+tests of macroscopic quantum mechanics [5–10]. A lead-
+ing approach is to mediate the transfer using an optome-
+chanical resonator [11–16]. This is attractive because me-
+chanical resonators interact via radiation pressure with
+electromagnetic fields of all frequencies [1] and can also
+be functionalized to interact with most quantum com-
+puting nodes, such as spins [18–20], superconducting de-
+vices [21–23] and atomic ensembles [24].
+The first step in the transfer process is an optical-
+to-mechanical state transfer, with a subsequent transfer
+to the final computing node [25–27].
+An optical cav-
+ity is employed to enhance the radiation pressure during
+the optical-to-mechanical state transfer.
+Leading pro-
+posals work only in the resolved sideband regime, where
+the decay rate of this cavity is lower than the mechani-
+cal resonance frequency [12, 28]. By contrast, most op-
+tomechanical systems operate in the unresolved sideband
+regime [29].
+In many cases this is due to the benefits
+that low mechanical frequencies convey for applications,
+for instance in precision sensing [30–32].
+In others, it
+is because of the difficulty of simultaneously achieving a
+low decay rate, a high resonance frequency, and sufficient
+radiation pressure coupling [33].
+To date, the only proposals for optical-to-mechanical
+state transfer in the unresolved sideband regime have
+used pulsed, rather than continuous, optomechanical in-
+teractions [34–36].
+This narrows the range of applica-
+tions, introduces significant technical challenges due to
+the additional timing and phase accuracy required [36–
+∗ [email protected]
+38], and involves large radiation pressure impulse forces
+that can be problematic [35, 39, 40].
+It is well known that a mechanical resonator can be
+feedback cooled close to its motional ground state in
+the unresolved sideband regime [41]. Here we propose
+a continuous optical-to-mechanical state transfer proto-
+col based on the same concept. By modelling the open
+quantum system dynamics, we show that feedback cool-
+ing can be understood as the transfer of a vacuum state
+of light onto the mechanical resonator. We find that ap-
+propriate choice of the feedback parameters allows the
+transfer of arbitrary quantum states. The requirements
+for successful transfer closely match those for ground-
+state cooling – once the optomechanical cooperativity ex-
+ceeds the thermal occupancy of the mechanical resonator,
+a coherent state can be transferred with near unity fi-
+delity and the Wigner-negativity of non-Gaussian states
+can be preserved.
+Moreover, the feedback parameters
+can be used to phase-sensitively amplify (or squeeze) the
+transferred state, to engineer its temporal profile, and –
+in direct analogy to state-transfer via resolved sideband
+cooling [42] – to achieve the transfer of a single optical
+sideband.
+Our work extends continuous optomechanical state
+transfer beyond the resolved sideband limit, to low-
+quality optical cavities and low frequency mechanical res-
+onators. Feedback cooling of a mechanical resonator to
+near its motional ground state has recently been demon-
+strated, both in cryogenic [43] and room temperature en-
+vironments [44]. As such, our proposal can be directly
+implemented with existing technology, providing a new
+tool for quantum networks and opening a new pathway
+to create and study macroscopic quantum systems. Our
+work also provides new insights into feedback cooling,
+showing that the process is in fact a quantum state trans-
+fer from light to mechanical motion.
+We consider an optomechanical system in the unre-
+solved sideband, high mechanical quality regime (κ ≫
+Ω ≫ Γ) with resonant optical driving, where κ (Γ) is
+arXiv:2301.03855v1  [quant-ph]  10 Jan 2023
+
+2
+FIG. 1.
+Schematic optomechanical system with feedback.
+Light is coupled into an optomechanical cavity. The reflected
+light is measured through homodyne detection. The detected
+photocurrent (Yout(t)) is convolved with a filter f(t) and di-
+rectly fed back to the momentum of the mechanical resonator.
+the optical (mechanical) energy decay rate, and Ω the
+mechanical resonance frequency.
+In this scenario, the
+amplitude quadrature of the input optical field Xin is di-
+rectly imprinted on the mechanical motion via radiation
+pressure. The phase quadrature Yin is not, but is encoded
+on the phase quadrature of the output optical field as [1]:
+Yout = −√ηYin + 2
+�
+ηΓCQ +
+�
+1 − ηYv,
+(1)
+where η is the detection efficiency, C
+=
+4g2
+om/Γκ
+is the optomechanical cooperativity with gom being
+the
+coherent-amplitude-boosted
+optomechanical
+cou-
+pling rate, Yv is the vacuum noise introduced due to
+detection loss, Q (P) is the dimensionless mechanical
+position (momentum) operator with [Q, P] = i, and all
+optical quadrature operators are normalised such that
+[X(t), Y (t′)] = iδ(t−t′). We propose to detect the output
+phase quadrature and use continuous feedback to trans-
+fer it to the mechanical resonator, as shown in Fig. 1. We
+note that feed-forward, similar to our feedback, has been
+applied to improve microwave-to-optical state transfer in
+the resolved sideband regime [45]. In contrast, the feed-
+forward functioned in that experiment to suppress cor-
+related noise terms, while both optical quadratures were
+transferred by radiation pressure.
+Our scheme is analogous to feedback cooling [41, 43,
+44, 46–50], with the detected signal applied as a force
+onto the mechanical resonator. Using quantum Langevin
+equations, we find that it is described by the following
+equations of motion:
+˙Q = ΩP − Γ
+2 Q +
+√
+ΓQin,
+(2)
+and
+˙P = − ΩQ − Γ
+2 P +
+√
+ΓPin − 2
+√
+ΓCXin
+(3)
+− ΓG
+2 f(t) ⊛
+�
+−
+�
+Yin −
+�1 − η
+η
+Yv
+�
+1
+2
+√
+ΓC
++ Q
+�
+,
+where Pin and Qin are white thermal noise operators that
+satisfy [Qin(t), Pin(t′)] = iδ(t−t′), and we have made the
+rotating wave approximation (RWA) with respect to the
+mechanical bath [1, 51]. The last term of Eq. (3) repre-
+sents the feedback force, where the measured photocur-
+rent is convolved with an arbitrary causal filter function
+f(t) ∈ R and amplified by the gain factor G. The fil-
+ter function is normalised so that |f(Ω)| = 1, where
+f(ω) =
+� ∞
+−∞ f (t) eiωtdt is the Fourier transform of f(t).
+The steady-state solutions to Eqs (2) and (3) are found
+by moving into frequency space and adiabatically elimi-
+nating the dynamics of the optical cavity field (Supple-
+mentary Material, Section I [56]).
+This results in the
+quadratures
+Q (ω) =
+√
+Γχ(ω)
+�
+Qin + φ(ω)Pin − 2
+√
+Cφ(ω)Xin + Gf(ω)φ(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(4)
+P (ω) =
+√
+Γχ(ω)
+�
+Pin −
+�Gf(ω)Γ
+2Ω
++ 1
+�
+φ(ω)Qin − 2
+√
+CXin + Gf(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(5)
+where
+φ(ω) =
+Ω
+Γ/2 − iω ,
+(6)
+the feedback-broadened mechanical susceptibility is
+χ(ω) =
+1
+Ωφ(ω)−1 + (Ω + GΓ f(ω)
+2 )φ(ω)
+,
+(7)
+and the adiabatic elimination is valid in the unresolved
+sideband regime ({Ω, CΓ} ≪ κ) taken throughout this
+paper. From Eq. (7), we see that the mechanical suscep-
+tibility decreases as G increases. This suppresses most
+of the mechanical terms in Eqs (4) and (5). The only
+term that remains is Qin in P(ω), but this is suppressed
+by the large mechanical quality factor (Ω/Γ ≫ 1). It is
+this combined suppression of all mechanical terms that
+enables optical state transfer with high fidelity.
+The optical input field consists of a continuum of op-
+tical modes. To build insight into which of these modes
+is best transferred to the single mechanical mode, as well
+as the gain and noise of the transfer process, we re-write
+
+3
+Eqs (4) and (5) as:
+Q = gXXtrans + Qnoise,optical + Qnoise,mechanical
+(8)
+P = gY Ytrans + Pnoise,optical + Pnoise,mechanical.
+(9)
+Here, Xtrans and Ytrans are the optical quadratures trans-
+ferred to position and momentum, respectively, and gX
+and gY are the transfer gains.
+Terms labelled with a
+subscript ‘noise’ encompass the residual thermal variance
+remaining after feedback, and any optical terms not aris-
+ing from the temporal mode of interest (i.e., inefficient
+detection, mode mismatch).
+The input optical quadratures transferred to Q and P
+in Eqs. (4) and (5) are not perfectly conjugate observ-
+ables. The difference is embodied in φ, and is a result
+of the retarded response of the mechanical position to an
+applied force. The imperfection introduces an ambigu-
+ity in the optical mode that is optimally transferred –
+a different mode is best transferred to P and Q. Here,
+we choose to assess the transfer of the mode that is op-
+timally transferred to P. This mode is described by the
+annihilation operator
+atrans(ω) = u(ω)ain(ω)
+(10)
+and spectral modeshape
+u(ω) = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+− i
+�
+,
+(11)
+where ain(ω) = (Xin(ω) + iYin(ω))/
+√
+2.
+Using the
+relations Xtrans = (a†
+trans + atrans)/
+√
+2 and Ytrans =
+i(a†
+trans − atrans)/
+√
+2, its amplitude and phase quadra-
+tures are found to be
+Xtrans = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+Xin + Yin
+�
+(12)
+Ytrans = 2
+√
+ΓC
+gY
+χ(ω)
+�
+−Xin + Gf(ω)
+8C
+Yin
+�
+.
+(13)
+Comparison of Eq. (13) with Eq. (5) confirms that Ytrans
+is reproduced exactly in P(ω), scaled by the momentum
+gain gY .
+The phase quadrature transfer gain, gY , can be de-
+termined by enforcing the boson commutation relation
+[atrans(t), a†
+trans(t)] = 1 on Eq. (10); while that for
+the amplitude quadrature, gX, can be found by requir-
+ing that the optical noise on position commutes with
+both Xtrans and Ytrans, i.e., [Qnoise,optical(t), Xtrans(t)] =
+[Qnoise,optical(t), Ytrans(t)] = 0, where Qnoise,optical is ob-
+tained by rearranging Eq. (8). Together, these give
+gY =
+�4ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2 �
+|f(ω)|2 + 1
+�
+dω
+�1/2
+(14)
+gX = − 1
+gY
+8ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2ℑ(φ(ω))ℑ(f(ω))dω.
+(15)
+The spectral modeshape and quadratures of the trans-
+ferred mode depend on both the feedback-broadened me-
+chanical susceptibility χ(ω) and the feedback filter func-
+tion f(ω), so that the transferred state can be controlled
+through appropriate choice of the filter properties. Thus
+far our results are valid for an arbitrary real-valued causal
+filter function. In the remainder of the paper we choose
+the generalized-Lorentzian filter
+f(ω) =
+Γ′Ω
+ω2 − Ω2 + iΓ′ω ,
+(16)
+where Γ′ is the filter bandwidth. This filter is commonly
+used for feedback cooling [41, 48, 50, 52] and is close
+to the known optimal filter for both momentum estima-
+tion [53] and feedback cooling [54]. Γ′ is chosen to be
+much larger than Ω, so that the filter acts as an integra-
+tor near the mechanical resonance frequency. The gain
+factor G can then be understood as the fractional increase
+in the mechanical decay rate due to the feedback.
+With the filter in Eq. (16) and in the limit of large filter
+bandwidth and mechanical quality factor (Ω/Γ ≫ 1), the
+amplitude and phase quadrature transfer gains can be
+approximated as
+gY = 2
+�
+�
+�
+�C
+�
+1 +
+G2
+64C2
+2 + G
+�
+and gX =
+1
+gY (1 + 2/G).
+(17)
+We define the overall gain of the transfer process as
+√gXgY , so that it is independent of unitary squeezing
+operations on the transferred state [55], and define the
+level of squeezing applied during the transfer as gX/gY .
+The overall gain and squeezing level are plotted as a
+function of the feedback gain factor G in Fig. 2 using
+both numerical calculations and the analytic approxima-
+tions of Eqs (17). For these plots and throughout the
+paper we use the system parameters Ω/2π = 1 MHz,
+Γ/2π = 1 Hz, Γ′/2π = 1.59 MHz, κ/2π = 100 MHz, and
+gom/2π = 395 kHz, T = 30 mK which have been achieved
+in a range of optomechanical experiments [33, 43, 44].
+The overall transfer gain approaches unity for G ≫ 1,
+and the transfer generally involves amplitude quadrature
+squeezing (gX/gY < 1).
+Only at G = 8C do we find
+that the input state is transferred without any squeezing
+(gX/gY = 1). Comparison of Eq. (12) with Eq. (4) shows
+that, in the high quality limit for which f(ω) can be sub-
+stituted with f(±Ω) = ∓i, this choice of gain also results
+in near-agreement between Xtrans and the optical input
+terms in Q. The remaining discrepancy arises from the
+retardation factor φ(ω), and this discrepancy approaches
+zero in the high-quality-factor limit. We therefore select
+G = 8C for the remainder of the paper.
+It is illustrative to consider how our choice of filter
+function and gain factor influences the spectral mode-
+shape u(ω). The frequency dependence of the prefactor
+in Eq. (11) depends only on χ(ω), and is sharply peaked
+at both ±Ω. However, since f(±Ω) = ∓i, for G = 8C the
+term in parentheses is precisely zero at −Ω and equals
+
+4
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+G/C
+0
+0.50
+1.0
+√gXgY
+1
+0.5
+0
+gX/gY
+FIG.
+2.
+Transfer
+gain
+(√gXgY , red)
+and
+squeezing
+(gX/gY , blue) as a function of the feedback strength by co-
+operativity (G/C). The dashed line indicates G = 1 and the
+full grey line indicates the optimal gain (G = 8C), where
+gX/gY = 1. The dots are numerically obtained, and the lines
+are using the analytic expressions derived in the high quality
+factor limit.
+−2i at Ω.
+Our particular choice, therefore, enables a
+single-sideband state transfer, transferring only the lower
+optical sideband and doing this with a modeshape given
+approximately by χ(ω) (see also Supplementary Mate-
+rial, Section II [56]).
+To quantitatively assess the quality of transfer we
+first consider an input vacuum state. We calculate the
+contributions to the position and momentum variances
+from this input and from the noise sources specified in
+Eqs (8) and (9) (see Supplementary Material, Sections
+II & III [56]). We separate the optical noise into con-
+tributions arising from inefficiences and mode mismatch,
+so that the non-ideality of the transfer that arises due
+to φ(ω) can be assessed.
+The results are plotted in
+Fig. 3 (a) as a function of C/nth (with G = 8C). The
+variance of the transferred optical mode increases with C,
+asymptoting to the vacuum variance of 1/2 once C ≫ 1.
+Conversely, the mechanical noise contribution decreases,
+dropping below the vacuum level for C ≫ nth. The vari-
+ance of the optical inefficiency noise has a cooperativity
+dependence that is similar to the optical signal, increas-
+ing with C and asymptoting to a constant value once
+C ≫ 1. As expected, this noise increases as the detec-
+tion efficiency degrades. However, even for η as low as
+0.5 the transferred signal variance still dominates ineffi-
+ciency noise for the whole range of C/nth. The mode-
+mismatch noise on Q is very low for small C, increases
+approximately linearly with C, and eventually exceeds
+the signal variance. Thus, the mode-mismatch ultimately
+constrains the performance of the state transfer.
+Using the analytic expressions for the gains in Eqs (17),
+we derive analytic expressions for the different variance
+contributions that are valid in the same high-quality,
+high-bandwidth limit (see Supplementary Material [56],
+Section III). With the exception of the mismatch noise,
+which is zero in the limit of high quality, these expres-
+sions agree well with the numerical results in Fig. 3 (a).
+From them, we find that when C ≫ 1 the noise variance
+introduced by optical inefficiency is Vη = (1−η)/4η, and
+that the mechanical noise variance is suppressed below
+the vacuum noise level once C > ¯nth/2.
+10−3
+10−2
+10−1
+100
+101
+C/nth
+10−2
+100
+102
+Variance
+(a)
+0
+0.5
+1
+F
+(b)
+Coherent→
+(c)
+↑→
+P
+Q
+Cat→
+Fock→
+0.0 0.2 0.4 0.6 0.8 1.0
+η
+0.0
+0.5
+1.0
+F
+−0.30
+−0.15
+0.00
+0.15
+0.30
+Wtransferred
+FIG. 3.
+(a) Contributions to the variance as a function of
+interaction strength of mechanical noise (blue), optical signal
+(yellow), and two contributions of optical noise: mode mis-
+match on Q (black), and inefficiency (red). The size of the
+markers correspond to the inefficiency (η = 0.9, 0.75, 0.5) for
+decreasing size respectively. (b) The transfer fidelity (F) as a
+function of interaction strength for a coherent state (black),
+cat state (green) (α = 2) and single photon Fock state (dark
+blue). Inset shows F as a function of η for the coherent state,
+at a fixed value of C/nth = 10. (c) Corresponding plots of the
+Wigner distributions for a coherent state (top row), cat state
+(middle row) and Fock state (bottom row) at the interaction
+strengths indicated by the grey lines connected to subplot (b).
+The black dotted circle in the top right indicates the length
+scale of the contour of the ground state. The orientation of
+the plots is indicated by the black arrows in the top right plot.
+Since the feedback process is linear and all noise
+sources are Gaussian, it is straightforward to extend our
+analysis beyond the transfer of vacuum states, to more
+elaborate states such as Schr¨odinger cat states. This can
+be achieved using Wigner functions (Supplementary Ma-
+terial, Section IV [56]). Imperfections introduced by the
+thermal noise, mode mismatch, and inefficiency tend to
+‘smear out’ quantum features of the transferred optical
+mode’s Wigner function. Mathematically, this is repre-
+sented by convolving the signal’s Wigner function with a
+
+5
+Gaussian noise kernel G(r) (with r = (Q P)T ) [57]:
+Wtransferred(r) = (W ⊛ G) (r).
+(18)
+In the regime relevant to this paper, G is typically close to
+symmetric, with a slight wider spread in the Q direction
+due to mode mismatch. The transfer fidelity can then be
+determined for any pure input state as
+F = 2π
+� ∞
+−∞
+� ∞
+−∞
+W(r)Wtransferred(r)d2r.
+(19)
+We plot the fidelity for input coherent, Fock, and cat
+states in Fig. 3 (b) as a function of C/nth and assum-
+ing that η = 1. The coherent state fidelity exceeds the
+classical limit of 1/2 at C/nth = 0.25 and the no-cloning
+bound of 2/3 at C/nth = 0.50. The fidelity for the non-
+Gaussian states also reach fidelities greater than 0.5 at
+similar, experimentally accessible [33, 43, 58] coopera-
+tivities.
+For the chosen experimental parameters, the
+maximum achievable fidelities are 0.98, 0.93, and 0.82
+for coherent, Fock, and cat states, respectively, and are
+limited by the mode-mismatch noise. The fidelity is ro-
+bust against measurement inefficiencies as visible in the
+inset of Fig. 3 (b), which shows that the coherent state
+fidelity can exceed 1/2 even with a detection efficiency as
+low as η = 0.2. Fig. 3 (c) plots the Wigner distributions
+of transferred coherent, Fock, and cat states at three dif-
+ferent values of C/nth, showing that the negativity of
+the Fock and cat states can be transferred, and therefore
+non-classical properties of the input state preserved.
+In conclusion, we have identified that feedback can be
+used to achieve continuous optical-to-mechanical state
+transfer in the unresolved sideband regime. We predict
+that state transfer can be achieved with high fidelity and
+whilst preserving non-classical features such as Wigner
+negativity.
+The ability to implement continuous state
+transfer in the unresolved sideband regime significantly
+widens the class of optomechanical systems that can be
+used as interfaces in quantum networks.
+ACKNOWLEDGEMENTS
+The authors thank Mr S. Khademi and Dr C. Meng
+for useful discussions. This research was primarily sup-
+ported by the Australian Research Council Centre of
+Excellence for Engineered Quantum Systems (EQUS,
+CE170100009). Support was also provided by the by the
+Air Force Office of Scientific Research under award num-
+ber FA9550-20-1-0391.
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+
+1
+Supplemental Material: Continuous optical-to-mechanical quantum state transfer in
+the unresolved sideband regime
+I.
+THEORETICAL MODEL
+The Quantum Langevin equation used to derive the equation of motion for an arbitrary quadrature (O) is [S1]:
+dO
+dt = 1
+iℏ [O, H] −
+�
+O, a†� �γ
+2 a − √γain(t)
+�
++
+�γ
+2 a† − √γa†
+in(t)
+�
+[O, a] ,
+(S1)
+with γ the linewidth and a† (a) the creation (annihilation) operator associated with the operator O.
+The full set of equations of motion (EOM) in frequency space, after including adiabatic elimination on the optical
+quadratures, is:
+−iωP = −ΩQ − Γ
+2 P +
+√
+ΓPin − 2αgom,0X
+−iωQ = ΩP − Γ
+2 Q +
+√
+ΓQin
+0 = −κ
+2 X + √κXin
+0 = −κ
+2 Y − √κYin − 2αgom,0Q,
+(S2)
+where Γ (κ) is the mechanical (optical) linewidth, and gom,0 the single-photon optomechanical interaction strength,
+which is boosted by α from the intracavity photon amplitude, P(in) and Q(in) are the (intracavity) momentum and
+position quadratures of the mechanical oscillator, respectively, Ω is the mechanical resonance frequency and X(in) and
+Y(in) are the (intracavity) amplitude and phase quadratures of the optical cavity respectively.
+Using input-output relations (Oout = Oin − √γO), we can determine an estimate for the mechanical position
+quadrature, by rescaling the detected photocurrent:
+Yout = −√ηYin + 2
+�
+ηΓCQ +
+�
+1 − ηYv,
+(S3)
+such that
+Qest = Q −
+√κ
+4αgom,0
+�
+Yin −
+�1 − η
+η
+Yv
+�
+,
+(S4)
+which includes the added noise (Yv) from detection inefficiencies (η). After convolution with an arbitrary causal
+real-valued filter function (f(t) ⊛ Qest) this is equivalent to an estimate of the momentum quadrature (Pest). To
+model applying a feedback force we subtract G Γ
+2 Pest from the equation of motion for P, where G is the strength of
+the feedback force.
+The steady state solutions for the mechanical quadratures are then as included in the main text (reproduced here
+for ease of reading):
+Q (ω) =
+√
+Γχ(ω)
+�
+Qin + φ(ω)Pin − 2
+√
+Cφ(ω)Xin + Gf(ω)φ(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(S5)
+and
+P (ω) =
+√
+Γχ(ω)
+�
+Pin −
+�Gf(ω)Γ
+2Ω
++ 1
+�
+φ(ω)Qin − 2
+√
+CXin + Gf(ω)
+4
+√
+C
+�
+Yin −
+�1 − η
+η
+Yv
+� �
+,
+(S6)
+with
+φ(ω) =
+Ω
+Γ/2 − iω ,
+(S7)
+
+2
+and
+χ(ω) =
+1
+Ωφ(ω)−1 + (Ω + GΓ f(ω)
+2 )φ(ω)
+.
+(S8)
+Following the main text, we re-write Eqs. (S5) and (S6) in the form:
+Q = gXXtrans + Qnoise, optical + Qnoise, mechanical
+(S9)
+P = gY Ytrans + Pnoise, optical + Pnoise, mechanical.
+(S10)
+Here, Xtrans and Ytrans are the quadratures of the transferred optical mode that are imprinted on the position and
+momentum, respectively, and gX and gY are the transfer gains allowing for possible differences in gain between
+position and momentum. Terms labelled with a subscript ‘noise’ encompass the residual mechanical noise remaining
+after feedback, and the optical noise imprinted on the mechanical oscillator by both feedback and radiation pressure.
+Following the main text, we can derive the susceptibility of the transferred mode, by using the relations Xtrans =
+(a†
+trans + atrans)/
+√
+2 and Ytrans = i(a†
+trans − atrans)/
+√
+2 to find:
+u(ω) = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+− i
+�
+(S11)
+or in terms of Xtrans and Ytrans:
+Xtrans = 2
+√
+ΓC
+gY
+χ(ω)
+�Gf(ω)
+8C
+Xin + Yin
+�
+(S12)
+Ytrans = −2
+√
+ΓC
+gY
+χ(ω)
+�
+Xin − Gf(ω)
+8C
+Yin
+�
+.
+(S13)
+The gain parameters are found through the method described in the main text:
+gY =
+�4ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2 �
+|f(ω)|2 + 1
+�
+dω
+�1/2
+(S14)
+gX = − 1
+gY
+8ΓC
+2π
+� ∞
+−∞
+|χ(ω)|2ℑ(φ(ω))ℑ(f(ω))dω.
+(S15)
+II.
+MECHANICAL POWER SPECTRAL DENSITY
+We construct the symmetrised power spectral densities for all separate contributions listed in Eqs. (S9) and (S10)
+using the following relations:
+¯SOO = SOO(ω) + SOO(−ω)
+2
+,
+(S16)
+and
+⟨Qin (t)Qin (t′)⟩ = ⟨Pin (t) Pin (t′)⟩ = (¯nth + 1/2) δ (t − t′) ,
+(S17)
+where ¯nth ≈ kBT/ℏΩ is the mean thermal occupancy of the mechanical resonator and T is its temperature. Due to
+the typically high frequency of the optical field we approximate the optical field to have no thermal occupancy, with
+⟨Xin (t)Xin (t′)⟩ = ⟨Yin (t) Yin (t′)⟩ = δ (t − t′) /2.
+(S18)
+
+3
+10−12
+10−10
+10−8
+10−6
+10−4
+PSD (Hz−1)
+(a)
+C/nth = 0.01
+10−1
+100
+101
+ω/Ω
+10−6
+10−10
+10−8
+PSD (Hz−1)
+C/nth = 10.0
+(b)
+10−1
+100
+101
+|ω/Ω|
+10−14
+10−9
+10−4
+|u(ω)|2 (Hz−1)
+10−1
+100
+101
+|ω/Ω|
+10−10
+10−6
+10−8
+|u(ω)|2 (Hz−1)
+FIG. S1. (a) Four separate contributions to the power spectral density (PSD) with C/nth = 0.01 of the mechanical quadratures:
+transferred optical mode (yellow), mechanical noise (blue), optical noise from mode mismatch on Q (black) and optical noise
+from inefficiencies (η) (red). The width of the red lines correspond to η = 0.9, 0.75, and 0.5 (decreasing width respectively).
+The inset shows the spectral mode of the signal for ω < 0 (green) and ω > 0 (purple). (b) Same as in (a) but with a larger
+interaction strength, C/nth = 10.
+Using these, we find:
+SQQ, optical signal =g2
+Y
+1
+2π
+1
+24Γ4g2
+om
+Γκ |χ(ω)|2 �
+|f(ω)|2 + 1
+�
+SQQ, optical noise =1
+24Γ4g2
+om
+Γκ |χ(ω)|2
+� �1 − η
+η
+�
+|f(ω)φ(ω)|2 + |φ(ω)f(ω) − gXgY |2 + |−φ(ω) − gY gXf(ω)|2
+�
+SQQ, mechanical noise =Γ|χ(ω)|2 �
+1 + |φ(ω)|2�
+(¯nth + 1
+2)
+SPP, optical signal =g2
+Y
+1
+24Γ4g2
+om
+Γκ |χ(ω)|2 �
+|f(ω)|2 + 1
+�
+SPP, optical noise =1
+24Γ4g2
+om
+Γκ |χ(ω)|2
+�1 − η
+η
+� �
+|f(ω)|2 + 1
+�
+SPP, mechanical noise =Γ|χ(ω)|2
+�
+1 +
+����4φ(ω)f(ω) 1
+Ω
+4g2
+om
+κΓ + 1
+����
+2� �
+¯nth + 1
+2
+�
+SPQ =4g2
+om
+πκ |χ(ω)|2
+�
+1 + |f(ω)|2
+4
+�
+1 −
+�1 − η
+η
+�2�
+ℜ(φ(ω))+
+Γ
+2π |χ(ω)|2 8g2
+om
+κΩ2 (ℑ (f(ω))ℑ (φ(ω)) − ℜ(f(ω))ℜ(φ(ω)))
+�
+¯nth + 1
+2
+�
+,
+(S19)
+Each contribution is plotted for two different optomechanical cooperativities in Fig. S1 (a) and (b), including
+inefficiency noise for three different detection efficiencies. Apart from the additional mode mismatch noise on Q, the
+contributions are near identical for Q and P in the limit that C ≪ Ω/Γ. We therefore only plot them for Q.
+Apart from the mode-mismatch noise, all contributions to the power spectral density are peaked at the mechanical
+
+4
+resonance frequency and have a roughly Lorenzian shape. The mismatch-noise, by contrast, is peaked at ω = 0 and
+approximates the shape of a low pass filter. At low cooperativity (Fig. S1 (a)), the noise dominates the optical signal,
+preventing an effective quantum state transfer. At higher cooperativity (Fig. S1 (b)), the optical signal rises above
+the noise contributions, suggesting that quantum state transfer is possible. The inset in both figures plots the spectral
+modeshape of the signal mode (u(ω)). The optical signal peaks at −Ω (blue) and is suppressed relative to this peak
+by several orders of magnitude at Ω (green), evidencing that the chosen filter enables a single-sideband transfer.
+III.
+VARIANCES
+We obtain numerical values for the variances of each component of the power spectral density by integrating over
+frequency. To obtain the variance plot of the main text we sweep the interaction strength gom, while keeping the
+other system parameters constant.
+We also determine analytical approximations in the limit of a large bandwidth filter and high mechanical quality
+factor (Q = Ω
+Γ ≫ 1). In that limit, and with our chosen filter, the expressions for gY and gX can be approximated as:
+gY =
+�
+�
+�
+�2C
+�
+1 +
+G2
+64C2
+1 + G/2
+�
+,
+(S20)
+and
+gX =
+1
+gY(1 + 2/G).
+(S21)
+Using these expressions and approximating the spectral densities in the high quality limit, we calculate analytic
+expressions for the variance of each component through the residue theorem. The results are:
+VXtrans = 1
+2gX
+2,
+(S22)
+VQmech =
+1
+1 + G/2
+�1
+2 + nth
+�
+,
+(S23)
+VQmismatch = 0,
+(S24)
+VQ η = 1
+4
+�1 − η
+η
+�
+gX
+2,
+(S25)
+VQmismatchQsignal = 0,
+(S26)
+and
+VPQ = VQP = 0.
+(S27)
+Eq. S24 only occurs in the variance for Q.
+The analytic variances for P are similar, with the subscript Q (X)
+substituted for P (Y).
+
+5
+IV.
+WIGNER FUNCTIONS
+The Wigner functions used as optical input states are given by the following equations in Wigner space (r = (QP)T ):
+WCat (Q=2,P=0)(r) = exp(−r · r)
+�
+exp(−2α · α) cosh
+�
+2
+√
+2r · α
+�
++ cos
+�
+2
+√
+2r · (ϖα)
+��
+π(exp(−2α · α) + 1)
+WFock (n=1)(r) = e−r·r (2r · r − 1)
+π
+WCoherent, vacuum(r) = e− r·r
+2
+2π ,
+(S28)
+where α = (αr, αi), and we use αr = 2, and αi = 0 for the calculation of the fidelity in the main text, and ϖ =
+((0, 1), (−1, 0)) is a symplectic matrix.
+As stated in the main text, the interactions are Gaussian and we can construct a Wigner function associated with
+the noise in the system:
+Wnoise(r) = 1
+2π
+1
+�
+det (Vnoise)
+exp
+�Q2V11 − 2QPV12 + P 2V22
+2V 2
+12 − 2V11V22
+�
+,
+(S29)
+where Vij are the elements of the correlation matrix that only contains all the noise contributions. Specifically:
+Vnoise =
+�
+VQQ, optical noise + VQQ, mechanical noise
+VQnoise Pnoise
+VPnoise Qnoise
+VPP, optical noise + VPP, mechanical noise
+�
+.
+(S30)
+The transferred Wigner functions Wtransferred are found by:
+Wtransferred(r′) = (Wtarget ⊛ Wnoise) (r),
+(S31)
+which we use directly for the calculation of the transfer fidelity:
+F = 2π
+� ∞
+−∞
+Wtarget(r)Wtransferred(r)dr.
+(S32)
+REFERENCES
+[S1] W. P. Bowen and G. J. Milburn, Quantum optomechanics (CRC Press, Boca Raton London New York, 2015), ISBN
+978-0-367-57519-9.
+

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+Silicon Carbide Metasurfaces for Controlling the
+Spontaneous Emission of Embedded Color
+Centers
+Mohammed Ashahar Ahamad and Faraz Ahmed Inam
+Department of Physics, Aligarh Muslim University, Aligarh, Uttar Pradesh 202002, India
+[email protected]
+Stefania Castelletto
+School of Engineering, RMIT University, Melbourne, Victoria 3001, Australia
+[email protected]
+Abstract:
+Nanopillars fabricated in diamond or silicon-carbide (SiC) have been used to
+enhance the light harvesting or absorption or to increase the collection efficiency of embed-
+ded single photon emission in the visible or near infrared for their detection using confocal
+microscopy. While electric and magnetic dipolar resonances in SiC have been studied in
+the far-infrared, they have not been studied in the near infrared. Here we show for the first
+time that electromagnetic Mie-scattering moments within SiC metasurfaces can control the
+spontaneous emission process of point defects in the near infrared. Using SiC nanopillars
+based metasurfaces, we theoretically demonstrate a control over the spontaneous emission
+rate of embedded color-centers by using the coherent superposition of the electric dipolar
+and magnetic quadrupolar electromagnetic Mie-scattering moments of the structure. More
+than an order of magnitude emission/decay rate enhancement is obtained with the maximum
+enhancement close to 30. We also demonstrate that the relative phase of the Mie-scattering
+moments helps in controlling the emission directionality. SiC metasurfaces in the spectral
+range of color centres, from the visible to the near infrared, can be used to control the con-
+finement and directionality of their spontaneous emission, increasing the opportunities to
+study light-matter interaction and to advance quantum photonic and quantum sensing de-
+vice integration.
+Keywords: Mie-scattering moments, silicon carbide nanopillars metasurface, emission en-
+hancements, radiation directionality, color centres
+© 2022 The Author(s)
+1.
+Introduction
+Color-centers in silicon-carbide (SiC) are example of emitters that possess single photon emission [1], optical spin
+read out and control, and have been amongst the most studied for optical coherence spin control and spin-photon
+interface [2,3] due to their very long coherence time [4,5] and photo-stability. SiC is quite distinguished from the
+other material platforms as it possesses color centre with optical-spin properties combined with advanced material
+fabrication methods, metal-oxide-semiconductor functionalities [6] and nonlinear second and third order optical
+properties. Due to its wide electronic bandgap which leads to broad optical transparency, photostable color centres
+emission [7] which extend to the near infrared, CMOS compatibility [8] and availability of quantum-grade wafer-
+scale SiC on insulator, it has emerged as one of the most a promising material for integrated quantum photonic
+applications [9,10].
+In particular, SiC can host a wide range of point defects/color centers including silicon vacancy VSi (V1, V2,
+V3), divacancies VSiVC and carbon antisite vacancy pair CSiVC [11]. The VSi in SiC is a promising single photon
+source (SPS) for spin-photon photon interface in the near infrared region around 917nm [12, 13]. At present
+the main challenge in applying SiC for quantum networks is to significantly enhance the rate of single photon
+generation and collection from embedded color-centers. Photonics is mainly used to enhance the properties of
+these systems.
+So far in SiC bulk material, nanopillars have been fabricated to enhance the light harvesting or collection
+efficiency of embedded single photon emission for their detection using confocal microscopy [12,14]; while meta-
+lenses [15] are used to modify the phase front of the emitted light, achieving high focusing and large emission
+directionality of the color-centers emitting below these meta-lenses. Currently metasurfaces used to excite/enhance
+arXiv:2301.04961v1  [physics.optics]  12 Jan 2023
+
+the magnetic and electric resonances in SiC have been investigated only in the far infrared [16] and in the context
+of surface phonon polaritons studies [17]. Recently it has been shown that metamaterial/metasurface light matter
+interaction can be used to control, enhance and tune the quantum properties of bulk materials [18,19]. In particular
+all dielectric metasurfaces due to their zero absorption losses have emerged as the preferred platform compared to
+plasmonics in photo-luminescence enhancement [20]. When a dielectric structure is placed under electromagnetic
+excitation, various charge and current distributions are excited in it. These distributions results in multi-polar Mie-
+scattering resonances being excited in structures with dimensions of the order of the excitation wavelength [21]. A
+coherent superposition of these resonances leads to many interesting phenomena like, bound states in continuum
+(BIC) [22], tuning of the radiation directionality in the lateral or transverse directions [23] and tuning of the local
+optical density of states (LDOS) to achieve emission rate enhancement for emitters embedded in the metasurfaces
+[24].
+Here for the first time we study the electric dipolar and magnetic quadrupolar resonances in the near infrared
+in SiC for controlling the spontaneous emission rate of the embedded color centers in the dielectric nanopillars
+forming Mie resonators.
+In this study, using the coherent superposition of Mie-scattering resonances in SiC pillars based metasurface, we
+theoretically demonstrate that it is possible to control the spectral spontaneous emission process of the embedded
+color-centers.We first optimise the scattering efficiency of the SiC metasurface when excited by a plane wave and
+then by a dipole emitter. In the case the light source is a dipole, namely the VSi embedded in the SiC metamaterial,
+we study the effects of the metasurface based on array of nanopillars Mie resonances on the LDOS and emission
+directionality. In particular, we study the effect of the periodicity of the nanopillars array to increase the emission
+rate and maintain high directionality compared to the case of a single pillar.
+2.
+Theoretical background
+Scattering is the phenomenon of re-emission of radiation by a particle after undergoing interaction with radiation
+[25]. When a plane electromagnetic wave is incident on a particle, charge distribution and displacement currents
+J(r) = −iωε0(εr − 1)E(r) (here E(r) is the field at the position vector r, ω = 2πr is the angular frequency, εr
+and ε0 are the permittivity of the particle and surrounding medium) are excited within it. When the particle’s
+dimensions are of the order of the excitation wavelength, the excited charge and current distributions leads to the
+development of multipolar Mie-scattering modes [26, 27]. The amplitude and phase of excitation of the electric
+and magnetic resonances or multi-polar Mie-scattering moments inside the scatterer are totally governed by its
+size, shape and surrounding electromagnetic environment [28]. These multi-polar Mie resonances in the visible
+spectral range have been demonstrated experimentally in the last decade using a silicon spherical nanopartciles
+and nanodiamonds [29,30].
+The total scattering efficiency (SE) Ctotal
+sca
+is calculated by normalizing the total far field scattered power to the
+energy flux of the incident wave on the scatterer [31]. The total SE Ct
+sca is the sum of partial SE from different mul-
+tipoles: Cp
+sca, Cm
+sca, CQ
+sca and CM
+sca represents contributions from electric dipole, magnetic dipole, electric quadrupole
+and magnetic quadrupole respectively [32].
+Ctotal
+sca
+=
+Cp
+sca +Cm
+sca +CQ
+sca +CM
+sca
+(1)
+Ctotal
+sca
+=
+k4
+6πε2
+0|Einc|2
+�
+�∑
+�
+|pα|2 +
+���mα
+c
+���
+2�
++ 1
+120 ∑
+�
+�|kQe
+αβ|2 +
+�����
+kQm
+αβ
+c
+�����
+2�
+�
+�
+�
+(2)
+where, pα and mα are the electric and magnetic dipole moments with Qe
+αβ and Qm
+αβ being the corresponding
+quadrupole moments. |Einc| is the amplitude of the incident electric field, k is the wave-vector and c is the speed
+of light. They are mathematically expressed as [32]:
+ED : pα
+=
+− 1
+iω
+��
+d3rJω
+α j0(kr)+ k2
+2
+�
+d3r
+�
+3(r.Jω)rα −r2Jω
+α
+� j2(kr)
+(kr)2
+�
+(3)
+MD : mα
+=
+3
+2
+�
+d3r(r×Jω)α
+j1(kr)
+kr
+(4)
+EQ : Qe
+αβ
+=
+− 3
+iω
+��
+d3r[3(rβJω
+α +rαJω
+β )−2(r.Jω)δαβ] j1(kr)
+(kr)
++
+2k2
+�
+d3[5rαrβ(r.Jω)−(rαJβ +rβJα)r2 −r2(r.Jω)δαβ] j3(kr)
+(kr)3
+�
+(5)
+MQ : Qm
+αβ
+=
+15
+�
+d3r
+�
+rα(r×Jω)β +rβ(r×Jω)α
+� j2(kr)
+(kr)2
+(6)
+
+Fig. 1. (a) Schematic of the metasurface with a 2D periodic lattice of SiC pillars under plane-wave
+excitation at 917 nm with wave-vector along the +z-direction. (b) Schematic of the unit cell. Each
+pillar has a length L = 2 µm and diameter D = 424 nm with a dipole emitter located at the center
+of each pillar. (c) The SE of the individual multipolar Mie-scattering moments as a function of the
+lattice periodicity, P, under plane-wave excitation at 917 nm. P is varied from 450 nm to 2500 nm.
+(d) The corresponding phase of the individual multipolar Mie-scattering moments as a function of
+the lattice periodicity, P. The dotted black lines corresponds to overlapping ED and MQ resonances
+with P = 915 nm, 1095 nm, 1500 nm and 2315 nm.
+The Mie-resonances control the electromagnetic field amplitudes within the scatterer and therefore contribute
+in tuning the local electromagnetic density of states (LDOS). The LDOS due to the local electromagnetic environ-
+ment around a point dipole emitter is defined as [33]
+ρ(ω,r) = ∑
+k,σ
+| ˆd ·Ek,σ(r)|2δ(ω −ωk,σ).
+(7)
+Here, ˆd is the unit vector specifying the direction of the transition dipole moment with ω being the transition
+frequency. The summation is over all wavevectors (k) and polarizations (σ). E is the total electric field at the
+source position resulting from the superposition of the fields directly radiated by the dipole emitter embedded
+inside the scatterer with the fields reflected and scattered back from the surroundings. The LDOS govern the
+complete radiation process of a dipole emitter. Hence the Mie-scattering modes play a vital role in tuning the
+spontaneous emission process of the emitter by controlling the scattered electric field at the source point.
+The balancing of the electric and magnetic Mie-scattering moments leads to the directionality of the scattered
+radiation pattern [32]. The radiation pattern is controlled by the relative phase of the balanced electric and magnetic
+multi-polar moments [34]. When the electric and magnetic dipolar moments are balanced and in phase, |ED| =
+|MD|, arg(ED) = arg(MD), this leads to a completely forward radiation directionality, known as the Kerker
+condition [32]. When these dipolar moments are out-of-phase, |ED| = |MD|, arg(ED) = arg(MD)+π, it results in
+a completely backward directionality, known as the anti-Kerker condition [34]. When the superposition of dipolar
+
+a
+L=2um
+Unit cell
+D = 424 nm
+X
+P
+10
+！
+(c)
+ED
+！！
+MD
+Phase(rad)
+EQ
+MQ
+0
+L
+S
+T
+业
+0
+500
+1000
+1500
+2000
+2500
+1000
+1500
+2000
+P(nm)
+P(nm)Fig. 2. The 2D electric field norm within (a) a single SiC pillar and the SiC pillar metasurface with
+P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under (i) plane wave excitation with wave-vector
+along the +z-direction and electric field polarized along the +x-direction and (ii) dipole excitation
+with dipole emitter placed at the center of the SiC pillar with orientation along the x-direction.
+as well as the quadrapolar moments are balanced and are in phase, |ED + MD| = |EQ + MQ| with arg(ED +
+MD) = arg(ED + MD), the radiation pattern is highly directional along the forward direction, known as the
+generalised Kerker condition [34]. However, when these superpositions are out-of-phase, this leads to a complete
+transverse scattering [34]. The phase of the Mie-scattering moments therefore controls the far-field scattering
+radiation pattern of the scatterer. Under a point dipole emitter excitation of the structure, the far-field scattering
+pattern of the structures also influences the radiation pattern of the dipole emitter placed in the vicinity of the
+scatterer [35].
+In the following we investigate these effects in SiC nanopillars array under plane wave-excitation and under a
+single dipole excitation simulating the VSi.
+3.
+Results and discussion
+3.1.
+Scattering efficiency and decay-rate enhancement
+We have computationally optimised the SiC metasurface, shown in Fig. 1(a) with unit cell in Fig. 1(b), to achieve
+the generalised Kerker’s condition in SiC for the specific color centre of interest. The metasurface consists of a
+periodic 2D lattice of SiC pillars, each of length, L = 2µm. The electrodynamics calculations are performed using
+the commercial Comsol Multiphysics RF module. The details of the calculations are presented in the Methods
+sections. The metasurface is excited by a plane wave with wavelength, λexc, travelling along the +z-direction
+(arrow symbol in Fig. 1(a)) with the electric field polarized along the +x-direction. Under the influence of the
+plane electromagnetic wave, Mie scattering moments are excited within the SiC pillars. We first optimised the
+diameter, D, of the SiC pillars for the maxima in the SE at λexc = 917 nm corresponding to the zero phonon line
+(ZPL) of the silicon vacancy, VSi in SiC [36,37]. The optimised D value was found to be around 424 nm. We then
+study the coherent superposition of the Mie-scattering modes of the individual SiC pillars by varying the lattice
+periodicity, P. For P ≫ λexc, the structure is expected to behave as a single isolated pillar. With decreasing P, the
+interactions between the Mie-scattering modes of the individual pillars will increase. When these modes will be
+in phase, their coherent superposition will lead to a maxima for the total SE of the 2D SiC pillar lattice. Fig. 1(c)
+and (d) show the amplitude and the phase of the individual Mie-scattering moments of the SiC pillar metasurface
+as a function of P. Sharp resonance peaks are observed in the amplitude of the individual Mie-scattering moments
+
+(iD)
+(a) Single pillar
+(a)Single pillar (b)P: 2315 nm
+(b) P: 2315 nm
+X
+3
+2.5
+('n'e)
+1.5
+(c) P: 1500 nm
+(d) P: 1095 nm
+(c)P: 1500 nm
+(d)P: 1095 nm
+0.5Fig. 3. The SE of the individual excited multipolar Mie-scattering moments and the emitter’s (VSi
+color-center) relative decay rate in the SiC pillar metasurface as a function of the lattice periodicity,
+P. The dipole emitter is placed at the center of the SiC pillars with dipole orientation along the
+horizontal plane. The γ∞ is the emitter’s decay rate in the bulk SiC. (b) The schematic representation
+of the tuning of the embedded color-center’s emission with the lattice periodicity, P set to (i) off-
+resonant P1 and (ii) resonant P2 values.
+(Fig. 1(c)). At these sharp resonances, a sharp jump in the phase of the corresponding Mie-scattering moment is
+observed (Fig. 1(d)). We will now focus on the local maxima arising due to the ED and the MQ moments (under
+dipole excitation of the structure only these two resonances were excited and were observed to have an influence
+on the dipole emitter’s decay rates). These local maxima are observed for P =915 nm, 1095 nm, 1500 nm and
+2315 nm (black dotted lines in Fig. 1(c) and (d)). At P =915 nm, the ED resonance peak is much greater than the
+MQ resonance with the phase of these two resonances being equal. For P =1095 nm, 1500 nm and 2315 nm, the
+ED and MQ moments are nearly balanced and a sharp jump is also observed in their phase. We will now examine
+the balanced superposition of these two moments at P =1095 nm, 1500 nm and 2315 nm.
+Figure 2(i) shows the normalised electric field distribution within a (a) single SiC pillar and the SiC pillar meta-
+surface with P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under plane wave excitation with wave-vector along
+the +z-direction and electric field polarized along the +x-direction. Strong confinement of the electric field within
+the SiC cylinder is observed at these P values corresponding to the balanced superposition of the ED and MQ
+resonances, with the maximum field confinement observed for P = 1500 nm (the SE was also observed to be max-
+imum at this P value). The field confinement will in-turn lead to LDOS enhancement within the SiC pillar (Eq 7).
+For a dipole emitter placed at the field maxima points, the LDOS enhancement will lead to its decay rate enhance-
+ment. Figure Fig. 2(ii) shows the normalised electric field distribution within (a) a single SiC pillar and the SiC
+pillar metasurface with P = (b) 2315 nm, (c) 1500 nm and (d) 1095 nm under dipole excitation with dipole emitter
+placed at the center of the SiC pillars with orientation along the x-direction. Large field confinement/enhancement
+which will lead to large LDOS enhancement can be observed here.
+We now study the influence of the above LDOS enhancement on the spontaneous emission rates of a dipole
+emitter, the VSi color-center embedded at the center of each SiC pillar. Figure 3(a) shows the emitter’s (VSi color-
+center) relative decay rate together with the SE of the individual Mie-scattering moments in the SiC pillar meta-
+surface as a function of the lattice periodicity, P. The decay rates of the VSi emitter in the SiC pillar metasurface,
+γ are scaled relative to its decay rates in a bulk SiC crystal, γ∞. The influence of the LDOS enhancement arising
+from the electric field confinement (Fig. 2) in tuning the emitter’s decay rate can be clearly observed here. Also, it
+can be observed that the relative decay rates ( γ
+γ∞ ) (dash-dotted red curve) only tunes with the local maxima which
+are dominated by ED and MQ resonances. These resonances corresponds to P = 1095 nm, 1500 nm and 2315
+nm, respectively. A schematic representation of the embedded dipole emitter’s radiation tuning with the SiC pillar
+lattice periodicity, P at an off-resonant (i) and resonant (ii) value is presented in Fig. 3.
+In Fig. 4, we study the SE spectral response due to all the excited Mie-scattering moments and the effect
+on the relative decay rates of a horizontally oriented (along x-direction) dipole source for the above resonant
+
+20
+(a)
+ED
+(b)
+30
+MD
+15
+EQ
+SE (a.u.)
+-MQ
+20
+8
+10
+-8
+10
+(i)
+(iD)
+500
+1000
+1500 2000
+2500
+P (nm)Fig. 4. The spectral response of the SE with the individual excited multipolar Mie-scattering mo-
+ments under horizontal dipole excitation and the emitter’s (VSi color-center with dipole orientation
+along the horizontal plane) relative decay rate in a (a) single SiC pillar; SiC pillar metastuface with
+P = (b) 2315nm, (c) 1500 nm and (d) 1095 nm, respectively.
+periodicity values (P = 1095 nm, 1500 nm and 2315 nm) of the metasurface. Here, the Mie-scattering moments
+of the metasurface are excited by the dipole source itself. For an isolated SiC pillar (Fig. 4(a)), all the studied
+Mie-scattering moments are observed to be weakly excited with no clear resonances. The relative decay rate
+(dash-dotted red curve) is observed to be around 2.7 at 917 nm. However, for all resonant P values (P = 1095
+nm, 1500 nm and 2315 nm), significant contributions are observed only from the ED and MQ moments. Their
+superposition is controlling the behaviour of the SE and the relative decay rate. The maximum relative decay rate
+enhancement is close to 30 at 917 nm for P = 1500 nm and 1095 nm. For P = 2315 nm the enhancement is about
+20. Therefore, it can be concluded that the coherent superposition of the ED and MQ Mie-scattering moments
+of the individual pillars are enhancing the decay rates of an embedded dipole emitter by more than an order of
+magnitude.
+We now study the role of the phase of the excited Mie-scattering moments (ED and MQ) of the SiC pillar
+metasurface on the far field radiation pattern of an embedded dipole emitter.
+3.2.
+Phase analysis and Radiation pattern
+Figure 5 shows a narrow range of values for both the relative amplitudes and phase of the ED and MQ moments
+at the resonant P values. At the VSi color-center’s peak emission wavelength of 917 nm, the MQ moment appears
+to be slightly larger than the ED moment. For P = 1500 nm, the ED
+MQ = 0.7 and for P = 1095 nm and 2315 nm, the
+ED
+MQ = 0.81. The corresponding phase difference between ED and MQ moments is π at 917 nm for all resonant P
+values.
+In Fig. 6(i) we show the in-phase and out-of-phase superposition of the balanced ED and MQ moments. The
+influence of this superposition on both the in-plane (X-Z plane, blue curve) and out-of-plane (Y-Z plane, red
+curve) far-field scattering patterns is shown here. The in-phase (ED + MQ) superposition results in longitudinal
+(along top and bottom directions) scattering and the out-of-phase (ED − MQ) superposition results in transverse
+scattering along the out-of-plane direction (red curve).
+We now study the influence of the ED and MQ moments superpositions on the embedded dipole emitter radi-
+ation pattern for a single nanopillar and SiC metasurface of array of interacting nanopillars. The emitter’s dipole
+orientation is along the X-direction. Fig. 6(ii) shows the far-field emission patterns of the embedded VSi color-
+
+1.5
+1.5
+Single Pillar
+ED
+P = 2315nm
+-ED
+(b)
+30
+30
+MD
+MD
+-EQ
+-EQ
+1
+SE(a.u.)
+20
+-MQ
+20.8
+8
+-MQ
+a
+.... TOTAL
+..... TOTAL
+0.5
+10
+10
+8
+8
+0
+0
+0
+900
+910
+920
+930
+940
+950
+900
+910
+930
+940
+920
+950
+入(nm)
+入(nm)
+1.5
+1.5
+= 1500 nm
+P = 1095nm
+(d)
+D
+(c)
+ED
+ED 
+30
+30
+MD
+-MD
+-EQ
+-EQ
+L
+(a.u.)
+20_8
+一MQ
+20.8
+一MQ
+....TOTAL
+..... TOTAL
+10
+10
+8
+0
+0
+0
+900
+910
+920
+930
+940
+950
+900
+910
+920
+930
+940
+950
+入(nm)
+入(nm)Fig. 5. The spectral response of the (a) relative amplitudes and (b) phase difference of the elec-
+tric dipolar (ED) and magnetic quadrapular (MQ) Mie-scattering moments excited under horizontal
+dipole excitation.
+center at 917 nm for different resonant P values. In a single pillar, the majority of the emission is observed to be
+directed towards the bottom surface and lies mainly along the longitudinal direction (bottom and top). However,
+in the SiC metasurface at the resonant P values corresponding to ED and MQ moments, the out-of-phase super-
+position of the ED and MQ moments (the phase difference between these moments was observed to be close to π
+at 917 nm in Fig. 5) directs the embedded emitter’s radiation pattern along the transverse direction, especially in
+the out-of-plane (Y-Z plane, red curve).
+4.
+Conclusion
+We studied for the first time the coherent superposition of the eletric and magnetic dipolar and quadrupolar Mie-
+scattering moments of SiC metamaterial nanopillars array in the near infrared emission (917 nm). We first de-
+termined the design of the metasurface periodicity to induce sharp resonances in the amplitude and phase of the
+Mie-scattering moments. Strong electric field confinement was observed within the SiC pillar when the periodicity
+of the lattice matched with the resonance of the ED and MQ modes. The field confinement leads to large LDOS
+and subsequently decay rate enhancement for a color-center dipole embedded at the center of the SiC pillar. Under
+a point dipole emitter excitation within SiC, it was determined that only the ED and MQ moments are contribut-
+ing to the electromagnetic scattering in the SiC nanopillars metasurface. Both these moments were observed to
+be nicely coupled and were showing collective resonance at the optimised wavelength (917 nm). The coherent
+superposition of these two moments controls the complete spontaneous emission process of the embedded color
+center. At the collective resonance point of these two moments (λ = 917 nm), we determined more than an order
+of magnitude decay rate enhancement with the maximum enhancement reaching 30. Such an enhancement has
+never been reported in dielectric neither in metal-dielectric individual nanopillars [38], thus paving the way for
+the use SiC metasurfaces to to enhance and control light extraction from quantum emitters, to study light matter
+interaction effects in integrated quantum photonics and for applications in quantum sensing. We also observed that
+by designing specific resonant structures, the coherent superposition of the ED and MQ moments can be used to
+better control the radiation/emission pattern and hence the emission directionality of the embedded dipole emitter
+compared to a single nanopillar. Specifically the embedded emitter’s radiation pattern can be more confined along
+the transverse direction, especially in the out-of-plane (Y-Z plane, red curve), thus facilitating the planar emission
+propagation. Such result is relevant for applications of SiC metasurfaces for planar integrated photonics. Our study
+can prompt further studies on SiC quantum states of light based on multi-emitters-resonators coupling enabled by
+metamaterials such as superradiance [39].
+5.
+Materials and Methods
+All the electrodynamics calculations have been performed using the commercial COMSOL Multiphysics Radio
+Frequency (RF) module. The periodic boundary conditions are applied to all the horizontal planes defining the 2D
+lattice of the SiC substrate to build an array of dielectric pillars metasurface. The Scattering boundary conditions
+are applied at the top and bottom boundaries of the computational domain. The optical constant for the SiC has
+been extracted from the experimentally reported values by Singh et.al. [40]. During the entire calculation the
+minimum meshing size was 1 nm with the maximum being λ/7.
+
+(a)
+(b)
+- Single pillar
+Phase difference (rad)
+T
+ED
+.....P: 1095 nm
+e
+arg(ED) -
+1.5
+MQ
+-P: 1500 nm
+arg (MQ)
+- P: 2315 nm
+1
+0.5
+-T
+0
+913
+915
+917
+919
+921
+913
+915
+917
+919
+921
+入 (nm)
+入(nm)Fig. 6. (i) In-plane (X-Z plane, blue curves) and out-of-plane (Y-Z plane, red curves) far-field scat-
+tering pattern corresponding to the in-phase (ED+MD) and out-of-phase (ED-MQ) superposition
+of the electric dipolar (ED) and magnetic quadrupolar (MQ) Mie-scattering moments. (ii) Farfield
+radiation patterns in-plane (X-Z plane, blue curves) and out-of-plane (Y-Z plane, red curves) of a
+dipole emitter (orientation along the X-direction, same as that of electric field in (i)) placed at the
+center of (a) single pillar; SiC pillar metastuface with P = (b) 2315nm, (c) 1500 nm and (d) 1095
+nm, respectively.
+5.1.
+Scattering efficiency calculation
+The scattering cross section is defined as the amount of power scattered by the scatterer to the amount of power
+per unit area carried by the incident wave. The SE is obtained just by dividing the scattering cross-section by
+the geometrical cross-section. Mathematically it is expressed as SE = σs/G [41]. Here σs is the scattering cross
+section and G is geometrical cross section.
+The SE is calculated semi-analytically using electric field values at each mesh point in the computational grid
+under plane-wave excitation using Comsol Multi-physics module. Using these field values and permittivity profile
+at each mesh points, current density is calculated as: Jω(r) = iωε0(εr −1)Eω(r). Here ε0 and εr are permittivity of
+free space and SiC medium, respectively. The computationally obtained values of E(r), Jω(r) and ε(r) are used to
+calculate individual multipolar Mie-scattering moments, pα, mα, Qe
+αβ and Qm
+αβ described in Eq. 2. The integration
+referred in Eq. 2 is carried on the domain of the SiC pillar.
+5.2.
+Relative decay rate calculations
+In these calculations, the SiV center is treated as a classical radiating point dipole source. In the computational
+domain, it is modelled as a point current source driven at the emission frequency, ν = c
+λ [42]. Scattering/perfectly
+matched layer (PML) boundary conditions are applied on the exterior boundaries of the computational domain.
+The total power radiated by the dipole is integrated over a closed surface enclosing the current source. The relative
+decay-rate is calculated as Γrel = γ/γ∞ = P/P∞ [42], where P∞ is the power corresponding to the point dipole’s
+emission in the bulk SiC. The permittivity of SiC is taken from [43].
+Acknowledgement The authors would like to acknowledge the financial support from the Department of Sci-
+ence and Technology (DST), India (CRG/2021/001167). The authors thank Dr Nadeem Ahmed for his help regard-
+ing the plotting of the figures and the formatting of the manuscript. MA and FAI also thank Dr Ahmed Mekawy
+for his help regarding multipole decomposition of the Mie-scattering moments.
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+

Z9E0T4oBgHgl3EQf4QJC/content/tmp_files/2301.02735v1.pdf.txt

@@ -0,0 +1,758 @@
+ 
+1 
+ 
+ 
+Designing an Improved Deep Learning-based Model for 
+COVID-19 Recognition in Chest X-ray Images: A Knowledge 
+Distillation Approach 
+AmirReza BabaAhmadi1, Sahar Khalafi2, Masoud ShariatPanahi3 , Moosa Ayati4 
+ 
+Abstract 
+Background and Objectives: COVID-19 has adversely affected humans and societies in different aspects. Numerous 
+people have perished due to inaccurate COVID-19 identification and, consequently, a lack of appropriate medical 
+treatment. Numerous solutions based on manual and automatic feature extraction techniques have been investigated 
+to address this issue by researchers worldwide. Typically, automatic feature extraction methods, particularly deep 
+learning models, necessitate a powerful hardware system to perform the necessary computations. Unfortunately, many 
+institutions and societies cannot benefit from these advancements due to the prohibitively high cost of high-quality 
+hardware equipment.  As a result, this study focused on two primary goals: first, lowering the computational costs 
+associated with running the proposed model on embedded devices, mobile devices, and conventional computers; and 
+second, improving the model's performance in comparison to previously published methods (at least performs on par 
+with state of the art models) in order to ensure its performance and accuracy for the medical recognition task. 
+Methods: This study used two neural networks to improve feature extraction from our dataset: VGG19 and 
+ResNet50V2. Both of these networks are capable of providing semantic features from the nominated dataset. 
+Streaming in a fully connected classifier layer that feeds richer features, therefore feature vectors of these networks 
+have been merged, and this action resulted in satisfactory classification results for normal and COVID-19 cases. On 
+the other hand, these two networks have many layers and require a significant amount of computation.  To this end, 
+An alternative network was considered, namely MobileNetV2, which excels at extracting semantic features while 
+requiring minimal computation on mobile and embedded devices. Knowledge distillation (KD) was used to transfer 
+knowledge from the teacher network (concatenated ResNet50V2 and VGG19) to the student network (MobileNetV2) 
+to improve MobileNetV2 performance and to achieve a robust and accurate model for the COVID-19 identification 
+task from chest X-ray images. 
+Results: Pre-trained networks were used to provide a more useful starting point for the COVID-19 detection task. 
+Additionally, a 5-fold cross-validation technique was used on both the teacher and student networks to evaluate the 
+ 
+1 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran (email: [email protected]) 
+2 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran 
+3 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 
+4 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 
+
+ 
+2 
+ 
+ 
+proposed method's performance.  Finally, the proposed model achieved 98.8% accuracy in detecting infectious and 
+normal cases. 
+Conclusion: The study results demonstrate the proposed method's superior performance. With the student model 
+achieving acceptable accuracy and F1-score using cross-validation technique, it can be concluded that this network is 
+well-suited for conventional computers, embedded systems, and clinical experts' cell phones. 
+Keywords: COVID-19, Deep Learning, Medical Image Analysis, Knowledge Distillation, Chest X-ray images, 
+Teacher-Student Model 
+ 
+ 
+1. Background and Objectives 
+COVID-19 has been a significant threat to human life and health in recent years. It has delivered a detrimental effect 
+on healthcare systems worldwide. Unfortunately, COVID-19 is highly contagious and transmissible. As a result, it is 
+inevitable to develop prediction systems capable of rapidly diagnosing it and averting its adverse effects. Numerous 
+scientists have conducted multiple studies to develop human-level prediction and diagnosis systems to aid 
+communities in combating this disease. 
+These methods are typically effective at detecting COVID-19 using CT and chest X-ray images. However, the fatal 
+flaw is that automatic feature extraction techniques, particularly deep learning models, require a significant amount of 
+computation to perform a specific task. Numerous hospitals and institutions are unable to procure expensive hardware 
+systems capable of performing these algorithms. 
+Additionally, in the absence of medical experts, the health of the population in many developing countries may be 
+jeopardized. In some developing countries, the scarcity of human experts capable of identifying medical diseases from 
+medical images is a stark reality. As a result, an effort is made to offer an alternate solution to remedy this issue in 
+this study.  
+This paper focuses on two primary objectives and charts the course accordingly: first, a fast algorithm capable of being 
+deployed in conventional computers and embedded in mobile devices and tablets, and other embedded systems 
+requires development. The second objective, which prompted developing a novel deep learning model, was to deliver 
+more accurate and reliable results than previously published methods to ensure the algorithm's performance for 
+medical diagnosis. 
+Recognizing COVID-19 is the first step toward treatment. The main objective of this paper is an attempt to address 
+this step. COVID-19 is typically accompanied by symptoms including coughing, colds, and shortness of breath, among 
+others. According to the WHO, respiratory problems are the fatal symptoms of COVID-19 and can be identified using 
+CT scans or chest X-rays obtained in hospitals and medical clinics. 
+
+ 
+3 
+ 
+ 
+Deep learning and its associated techniques have been widely applied to computer vision tasks such as medical image 
+analysis in recent years. Examples include tumor detection [1], diabetic retinopathy [2], among others. 
+Considering that medical image analysis is one of the most exciting applications of computer vision and deep learning, 
+automatic feature extractor algorithms can be critical in diagnosing diseases at levels comparable to human experts. 
+Due to the critical nature of the initial diagnosis, researchers have identified this step's accuracy as a significant 
+challenge. The following section examines related works and studies in order to create a solid framework for our 
+research. In [3], a model based on stacked convolutional neural networks with multiple pre-trained networks was 
+proposed. In [4], a residual-layer-based neural network with a modified kernel was used to prognosticate the presence 
+of COVID-19 in normal and pneumonia chest X-ray images. 
+[5] proposes a model based on an ensemble learning classifier to address the imbalance dataset issue. Another paper 
+[6] introduced a combination of LSTM5 (for detection) and a convolutional neural network (for recognition). This 
+method produced a 99.4% accuracy metric. [7] describes the development of a model using LSTM and GAN 
+architectures. This method does not require the addition of a network for feature extraction for the binary classification 
+task. [8] investigated the presence of COVID-19 in medical images using a deep neural network based on a CNN with 
+additional components in different layers. 
+In [9], CNN and SVM6 were combined to analyze COVID-19 CXR images. [10] proposes a new framework based on 
+the use of DarkNet53 to perform chest X-ray images. [11] conducted a comparative study to determine which model 
+performs the best among pre-trained deep learning architectures. Fifteen models were examined in order to determine 
+the optimal architecture. According to this study, VGG19 was the most accurate pre-trained model. Another 
+comparative study [12] classified COVID-19 from normal cases and pneumonia using novel pre-trained architectures 
+such as DenseNet 201, Inception V3, Resnet50, and a few other networks. [13] introduced a new model based on 
+GAN and CNN. Multiple classifiers were used to perform the classification task in [13]. Furthermore, this paper 
+developed a COVID-19 segmentation model using a unique dataset.  
+[14] compared pre-trained architectures and then modified those networks to reduce trainable parameters and enable 
+faster algorithm training. [15] examined a variety of pre-trained architectures, including Alex-Net, VGG16, and the 
+ResNet family. Moreover, this article concentrated on evaluating freezing layers and other configurations used in 
+transfer learning methods. A study was conducted in [16] to determine the best candidate for COVID-19 recognition. 
+Based on this research, the outstanding architecture for COVID-19 detection is achieved through the Inception 
+Network. Another comparative study was conducted in [17] using DenseNet121, ResNet50, VGG16, and VGG19. 
+In [18], a binary classification task has been performed. Fine-tuning was performed on the final layer of the pre-trained 
+SqueezeNet, DenseNet121, ResNet50, and ResNet18 networks. [19] depicts a voting-based procedure for evaluating 
+ 
+5 Long Short-Term Memory (LSTM) 
+6 Support Vector Machine (SVM) 
+
+ 
+4 
+ 
+ 
+VGG16, InceptionV3, and ResNet50 predictions. The final layer (output) compares the performance of all of the 
+networks mentioned previously in a simultaneous manner and selects the best result. 
+In [20], an assessment of the effect of preprocessing on the results of CNN-based networks is made. [21] describes 
+the development of a customized network dubbed COVID-Net for CXR and chest X-ray medical images. Another 
+study [22] compared COVID-19, bacterial pneumonia, and viral-pneumonia classification results using state-of-the-
+art algorithms and CNN architecture. In another study [23], PCA7 was used to increase the efficiency of feature 
+extraction. YOLO8 Networks with additional convolution layers involving modified kernels were used in [24] to detect 
+COVID-19 from chest X-ray images. 
+[25] examined the performance of the Efficient-Net family in detecting COVID-19 and pneumonia in normal cases. 
+In [26], a network, termed CoroNet, was introduced based on the Xception network to perform multi-class 
+classification from normal category images, including pneumonia-viral, pneumonia-bacterial, and COVID-19. 
+This paper is structured as follows: The second section provides an overview of previous publications on the task. The 
+third section describes the knowledge distillation method and explains why it was incorporated into the algorithm. 
+Section four discusses the dataset that was used to conduct this research. Section five describes the training phase and 
+the evaluation metrics. Finally, section six presents the proposed model's evaluation results, and section seven 
+summarizes our paper's findings. 
+ 
+2. Methods 
+Nowadays, Deep Neural Networks9 have revolutionized the field of computer vision. Their applications have been 
+extensively investigated in a variety of fields, including self-driving cars [27], medical image analysis [28][29], and 
+agriculture [30], among others. CNNs10 have established themselves as the most effective tools for automatic feature 
+extraction in computer vision and NLP11, speech processing, and video classification tasks. 
+This paper demonstrates how CNN-based neural networks can improve semantic feature extraction for binary 
+classification tasks involving COVID-19 and normal cases. DenseNet [31], ResNet [32], VGG [33], Xception [34], 
+and Mobile-NetV2 [35] are some of the most powerful pre-trained networks. VGG19 is an extension of VGG16; it 
+features sixteen convolutional layers and three fully connected layers. Five MaxPool layers are used, and the final 
+layer is a SoftMax layer. ResNet50V2 is an enhanced version of ResNet50 that outperforms previous versions such 
+as ResNet101. 
+ 
+7 Principal Component Analysis (PCA) 
+8 You Only Look Once (YOLO) 
+9 Deep Neural Networks (DNN) 
+10 Convolutional Neural Networks (CNN) 
+11 Natural Language Processing (NLP) 
+
+ 
+5 
+ 
+ 
+ResNet50V2 has been equipped with several new connections between different blocks, allowing this network to 
+achieve a high level of accuracy in the ImageNet competition. MobileNetV2 is a convolutional network designed with 
+depth-wise convolution layers to improve accuracy while lowering computational costs. This reduction is caused by 
+decreasing trainable parameters. Due to the use of inverted residual blocks, this network can be embedded in mobile 
+phones, tablets, and other conventional embedded devices. 
+Since both ResNet50V2 and VGG19 generate the same size output layer (feature vector), their feature vectors were 
+combined to produce a richer semantic feature set for the specified task. Afterward, a CNN layer was added to this 
+architecture (kernel size 1 and 1024 filters). The network's output was then flattened and streamed into the fully 
+connected layers. Notably, no activation function in the final CNN layer was used.  The classification layer was 
+constructed using 64 neurons in fully connected layers, with a dropout rate of 0.5. Another fully connected layer has 
+been added to this layer, the final layer, whose neurons count is equal to the size of the problem's classes. 
+The entire architecture is termed the teacher network, with MobileNetV2 serving as the student network in a broader 
+context referred to as the teacher-student model or knowledge distillation framework. The structure of the teacher 
+model is depicted in Figure 1. The tensors' feature size for both ResNet50V2 and VGG19 is 10*10*2048, and the size 
+of the concatenated feature is 10*10*4096. 
+ 
+ 
+Figure 1. Teacher Model Architecture 
+ 
+
+Teacher Model Architecture
+ResNet50V2
+10*10*2048
+CNNlayer
+Filteri024
+Dropout(50%)
+VGG19
+Kernel(1,l)
+10*10+2048
+10*10*4096
+n
+Classifier
+Merged Features 
+6 
+ 
+ 
+ 
+ 
+3. Knowledge Distillation 
+Hinton et al. pioneered  Knowledge Distillation12 in [36]. KD is a process that involves training a smaller network to 
+imitate the behavior of a more extensive network.  The purpose of designing a complex network as a teacher is to 
+learn more sophisticated features and deliver better results. However, we typically want to run our network on a 
+standard computer or embedded device. 
+Due to the limitation of memory size and computational cost, frequent issues arise. As a result, a solution is required 
+to address these issues. A weighted average (mean) is necessary to distill knowledge from teacher to student. Cross-
+Entropy with soft targets is the initial objective function. Through the softmax function in the smaller network, this 
+objective function is calculated based on high temperature. A more significant architecture (network) must be used to 
+generate soft targets. Cross-Entropy with valid labels is the second objective function. This function is calculated 
+using the softmax output from the student model by setting the temperature to zero. 
+The teacher network and the student network begin receiving training data in parallel. The teacher model contains a 
+softmax with temperature in its output.  By contrast, the student model generates two different outputs. The first output 
+is softmax with temperature, while the second output contains standard softmax. The student model is intended to 
+produce softened probabilities (the output of the teacher model). The following formula is used to calculate the loss 
+of knowledge distillation:  
+2
+( . )
+(1
+) (
+. )
+KD
+s
+L
+KL p q T
+L W x
+
+
+=
++
+−
+     
+(1) 
+Where p and q denote the probabilities generated by student and teacher networks in a specific temperature (T), 
+respectively, and KL denotes the Kullback-Leibler divergence, which measures the level of distinction between two 
+probabilistic distributions. The Cross-Entropy of the student model with T=1 is (LWs.x). According to [36],  and T  
+are hyperparameters where the greater the value of  , the better the learning experience for the student model. 
+Back-propagation must be performed only in the student network during the distillation phase to add a significant 
+element to this description since the teacher has already tuned its parameters. The teacher's knowledge is then 
+transferred to the student model throughout the distillation procedure. Notably, the student model can be trained at a 
+faster rate than the teacher model. For more details regarding the distillation procedure, please refer to [37]. The 
+procedure for knowledge distillation is depicted in Figure 1. 
+ 
+12 Knowledge Distillation (KD) 
+
+ 
+7 
+ 
+ 
+ 
+Figure 2.  Knowledge Distillation for COVID-19 detection 
+4.  Dataset 
+Two public datasets were used to train the proposed deep learning model to build the required dataset. First, a public 
+dataset available at (https://github.com/ieee8023/covid-chestxray-dataset) was used for positive samples of COVID-
+19. Afterward, the dataset available at (https://www.kaggle.com/c/rsna-pneumonia-detection-challenge) was used to 
+collect negative samples (normal cases).  
+Following the two datasets being merged, 118 COVID-19 cases and 8851 normal cases were established. It became 
+clear that an unbalanced dataset was created due to the number of positive cases (COVID-19) being significantly 
+lower than the number of normal chest X-ray medical images. As a result, the issue was mitigated through the use of 
+sampling techniques. The central concept is to select an equal number of items from each category for the binary 
+classification task. The oversampling method was used to increase COVID-19 (positive cases) samples to ensure that 
+both positive and negative classes had an equal number of samples. 
+The number of positive cases increased to 8851 following the oversampling technique, while the number of negative 
+(normal) samples remained unchanged. It should be noted that no images of pneumonia were used in this study. 
+Pneumonia is classified into several different classes, including SARS, Streptococcus, ARDS, and Pneumocystis. In 
+this respect, treating all of these categories as a single class was deemed as impractical. This cannot be very clear in 
+terms of interpreting recognition task results as distinct pneumonia types require a unique type of treatment. As a 
+result, developing a pneumonia-type classifier was deferred to a later date. Figure 3  illustrates a selection of patients 
+with COVID-19 and normal images from the dataset. 
+
+Generating soft targets
+Teacher
+Network(ResNet50V2/VGG19)
+Chest X-ray Dataset
+Backpropagation
+Loss
+Student
+Trainingthe student
+Network(MobileNetV2) 
+8 
+ 
+ 
+ 
+ 
+Figure 3(a): Healthy person  
+ 
+Figure 3(b): Patient with COVID-19 
+4.1 Data Augmentation 
+Before data augmentation, the images were normalized to avoid issues with vanishing and exploding gradients. After 
+that, the image was resized to 224x224. The model was then enhanced with data augmentation techniques to make it 
+more responsive to variations within the medical images. During data augmentation, it was assumed that variations in 
+the images did not affect the label's (ground truth) definition. Random rotation was the only data augmentation 
+
+ 
+9 
+ 
+ 
+approach considered for the dataset, between (0-200). This can prevent overfitting and also helps the learning curve 
+converge more quickly. 
+5. Training Phase 
+The training phase consumed approximately 80% of the dataset, with the remaining 20% used for the test phase. The 
+K-fold Cross-Validation technique was used to ensure the accuracy of the performance evaluation. The loss function 
+was binary Cross-Entropy, and the optimizer was Adam (with a learning rate=1e-5). The number of folds was 5 (k=5). 
+The batch size was 32, and the epoch number was set to 120. 
+5.1 Evaluation Metrics 
+Several specific criteria need to be established for evaluating the model's performance on the test dataset. True Positive 
+refers to correctly identifying COVID-19 positives among both positive and normal cases. In the conceptualization, 
+true negativity entails accurately identifying normal cases. False Positive refers to the practice of misdiagnosing 
+COVID-19 cases as normal. False-negative prediction is misclassifying normal cases as COVID-19 cases. 
+Precision is defined as the ratio of True Positives over the sum of True Positives and False Positives. 
+ 
+TP
+Precision
+TP
+FP
+=
++
+                  (1) 
+ 
+Sensitivity is the ratio of True Positives over to the sum of True Positives and False Negatives. 
+ 
+(
+)
+TP
+Sensitivity recall
+TP
+FN
+=
++
+ 
+(2) 
+ 
+Specificity is the ratio of True Negatives over the sum of False Positives and True Negatives. 
+ 
+TN
+Specificity
+TN
+FP
+=
++
+ 
+(3) 
+ 
+
+ 
+10 
+ 
+ 
+Finally, three additional criteria were used to measure the performance of the proposed model on the study's dataset 
+(F1 Score and Balanced-Accuracy in addition to conventional accuracy).  
+2*
+*
+1
+(
+)
+precision recall
+F
+Score
+precision
+recall
+−
+=
++
+ 
+(4) 
+ 
+2
+specificity
+sensitivity
+Balanced Accuracy
++
+=
+   (5) 
+ 
+TP
+TN
+Accuracy
+Positive
+Negative
++
+=
++
+ 
+(6) 
+ 
+6. Results and Discussion 
+In this section, we present the results of our method applied to the dataset mentioned above. The images which have 
+been used for the test have not been seen before by the algorithm. Therefore, these results are approvable that the 
+proposed algorithm can perform very well on unseen medical images. Table 1 presents the results of the training 
+teacher model. It is conspicuous that the teacher's performance is enough to detect chest X-ray images with good 
+accuracy and F1-score. Table 2 shows the results of MobileNetV2 (student model). Its results are satisfactory for the 
+classification task. Despite presenting good results of the student network, we decided to improve its classification 
+ability via the knowledge distillation approach. Table 3 is presenting the results of the student network after 
+knowledge distillation is completed. Eventually, the student network performs  on par with the existing methods in 
+the literature and sometimes achieves better accuracy and F1-score in comparison with previous publications. 
+ 
+Table 1: Evaluation results of the Teacher Network (ResNet50V2/VGG19) 
+Fold No.  
+Acc 
+Precision 
+Specificity 
+Recall(sensitivity) 
+F1 score 
+Balanced- 
+Acc 
+1 
+0.978 
+0.978 
+0.978 
+0.978 
+0.978 
+0.978 
+2 
+0.989 
+0.978 
+1.000 
+1.000 
+0.988 
+1.000 
+3 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+4 
+0.989 
+0.978 
+1.000 
+1.000 
+0.988 
+1.000 
+5 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+Average 
+0.992 
+0.987 
+0.996 
+0.996 
+0.991 
+0.996 
+ 
+
+ 
+11 
+ 
+ 
+ 
+ 
+Table 2: Evaluation results of the Student Network (MobileNetV2) Before Knowledge-Distillation 
+Fold No.  
+Acc 
+Precision 
+Specificity 
+Recall(sensitivity) 
+F1 score 
+Balanced- 
+Acc 
+1 
+0.987 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+2 
+0.974 
+0.976 
+0.976 
+0.976 
+0.972 
+0.976 
+3 
+0.974 
+0.976 
+0.976 
+0.976 
+0.972 
+0.976 
+4 
+0.987 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+5 
+0.987 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+Average 
+0.980 
+0.977 
+0.990 
+0.990 
+0.984 
+0.990 
+ 
+Table 3: Evaluation results of the Student Network (MobileNetV2) After Knowledge-Distillation 
+Fold No.  
+Acc 
+Precision 
+Specificity 
+Recall(sensitivity) 
+F1 score 
+Balanced- 
+Acc 
+1 
+0.974 
+0.976 
+0.976 
+0.976 
+0.972 
+0.976 
+2 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+3 
+0.978 
+0.978 
+1.000 
+1.000 
+0.992 
+1.000 
+4 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+1.000 
+5 
+0.989 
+0.978 
+1.000 
+1.000 
+0.988 
+1.000 
+Average 
+0.988 
+0.987 
+0.996 
+0.996 
+0.991 
+0.996 
+%Improvement 
+0.8 % 
+1.0 % 
+0.6 % 
+0.6 % 
+0.7 % 
+0.6 % 
+ 
+ 
+Table 4: Number of total parameters in each architecture 
+Teacher Network (ResNet50V2/VGG19) 
+49,222,390 
+Student Network (MobileNetV2) 
+2,334,966 
+Number of Parameters Reduction (%) 
+-95.3  %  
+ 
+ 
+It is noteworthy that the student network has outperformed its previous version(without KD) in terms of evaluation 
+metrics performance, and this is because knowledge distillation improves MobileNetV2's performance to some extent. 
+Meanwhile, KD aids the network in mitigating common neural network forgetting problems. Thus, the student model's 
+performance demonstrates that it can be used for medical recognition tasks in embedding systems while requiring 
+minimal computation, owing to the use of depthwise convolutional layers and knowledge distillation. Table 4 shows 
+the number of parameters in each architecture. It’s conspicuous that not only KD can improve student model’s 
+performance, but also it reduces  the number of parameters about 95.3% while maintainging perfromance. Therfore, 
+student model can be an eligible candidate for COVID-19 recognition task. 
+
+ 
+12 
+ 
+ 
+ 
+7. Conclusion 
+In this paper, a novel method was developed for identifying COVID-19 medical chest X-ray images. Due to 
+encountering an unbalanced dataset, this issue was resolved using oversampling and data augmentation techniques. In 
+evaluating the algorithm, the fivefold cross-validation method was used to ensure the proposed model's performance. 
+After performing knowledge distillation, an accuracy and F1-score of 98.8% and 99.1% were achieved respectively. 
+The proposed method, we believe, is an excellent choice for the COVID-19 recognition task. However, adding more 
+diverse datasets from different countries will help improve the algorithm. For future works on medical image datasets, 
+incremental learning techniques, self-supervised deep learning methods, vision transformer architectures, knowledge 
+distillation under adversarial attacks are proposed. 
+ 
+Declaration 
+Conflict of interest 
+The corresponding author declares that there are no conflicts of interest on behalf of all authors. 
+ 
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+

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+arXiv:2301.13711v1  [astro-ph.HE]  31 Jan 2023
+MNRAS 000, 1–14 (2022)
+Preprint 1 February 2023
+Compiled using MNRAS LATEX style file v3.0
+Wideband Study of the Brightest Black Hole X-ray Binary 4U 1543−47 in
+the 2021 Outburst: Signature of Disk-Wind Regulated Accretion
+Geethu Prabhakar1⋆, Samir Mandal1†, Bhuvana G. R.2, and Anuj Nandi3
+1 Department of Earth and Space Sciences, Indian Institute of Space Science and Technology (IIST), Trivandrum - 695547, India
+2 Department of Physics, Dayananda Sagar University, Bengaluru - 560068, India
+3 Space Astronomy Group, ISITE Campus, U R Rao Satellite Centre, Bengaluru - 560037, India
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+A comprehensive wideband spectral analysis of the brightest black hole X-ray binary 4U 1543 − 47 during its 2021 outburst
+is carried out for the first time using NICER, NuSTAR, and AstroSat observations by phenomenological and reflection mod-
+elling. The source attains a super-Eddington peak luminosity and remains in the soft state, with a small fraction (< 3%) of
+the inverse-Comptonized photons. The spectral modelling reveals a steep photon index (Γ ∼ 2 − 2.6) and relatively high inner
+disk temperature (Tin ∼ 0.9 − 1.27 keV). The line-of-sight column density varies between (0.45 − 0.54)×1022 cm−2. Reflection
+modelling using the RELXILL model suggests that 4U 1543 − 47 is a low-inclination system (θ ∼ 32◦ − 40◦). The accretion
+disk is highly ionized (log ξ > 3) and has super solar abundance (3.6−10 AFe,⊙) over the entire period of study. We detected a
+prominent dynamic absorption feature between ∼ 8 − 11 keV in the spectra throughout the outburst. This detection is the first
+of its kind for X-ray binaries. We infer that the absorption of the primary X-ray photons by the highly ionized, fast-moving
+disk-winds can produce the observed absorption feature. The phenomenological spectral modelling also shows the presence of
+a neutral absorption feature ∼ 7.1 −7.4 keV, and both ionized and neutral absorption components follow each other with a delay
+of a typical viscous timescale of 10 − 15 days.
+Key words: accretion, accretion disc - black hole physics - X-rays: binaries - stars: individual: 4U 1543 − 47
+1 INTRODUCTION
+X-ray spectroscopy of black hole X-ray binaries (BH-XRBs)
+holds the key to unveil the geometry of the system and the
+dynamics of the accretion process. The spectrum of BH-XRBs
+mainly consists of a hard powerlaw and a soft thermal compo-
+nent. The soft component, which is a multi-temperature black-
+body, is assumed to be originated from an optically thick, ge-
+ometrically thin accretion disk (Shakura & Sunyaev 1973). The
+hard powerlaw component is generally believed to be emitted
+from an optically thin, hot electron cloud called ‘corona’ by the
+Comptonization of the soft disk photons (Sunyaev & Titarchuk
+1980, 1985; Zdziarski et al. 1994; Chakrabarti & Titarchuk 1995;
+Poutanen & Coppi 1998; Chakrabarti & Mandal 2006; Iyer et al.
+2015; Poutanen et al. 2018). The relative strength of these com-
+ponents leads to different states in outbursting BH-XRBs. In the
+Low/Hard State (LHS), the non-thermal component dominates and
+in the High/Soft State (HSS), the disk emission dominates. There
+are short-lived intermediate states also, namely, the Hard Interme-
+diate State (HIMS) and Soft Intermediate State (SIMS), lying be-
+tween LHS and HSS (Homan et al. 2001; Homan & Belloni 2005;
+Remillard & McClintock 2006; Nandi et al. 2012; Sreehari et al.
+2019; Aneesha et al. 2019; Bhuvana et al. 2021; Prabhakar et al.
+2022). A typical outburst starts with the LHS and proceeds through
+⋆ [email protected]
+† [email protected]
+intermediate states to HSS then back to LHS again and finally reach
+quiescence. However, it does not always have to go through all the
+states mentioned above (Debnath et al. 2015; Radhika et al. 2016;
+García et al. 2019; Baby et al. 2020, 2021; Prabhakar et al. 2022).
+The advent of high resolution spectroscopy reveals the presence
+of reflection features in the spectra of many BH-XRBs. Irrespective
+of the geometry of the corona, it is believed that the photons upscat-
+terd by the corona, the primary photons interact with the disc mate-
+rial and a part of which produces the reflection features (Basko et al.
+1974). The reprocessed X-ray spectrum consists of fluorescent line
+emission from various elements, a soft thermal continuum and a
+Compton hump peaked at ∼ 20 − 30 keV. The most prominent fea-
+ture among the fluorescent emission lines is the iron K-edge at ∼ 7.1
+keV (Ueda et al. 1998) and Kα line at ∼ 6 − 7 keV (White et al.
+1986; Barret & Olive 2002; Di Salvo et al. 2005). This is because
+the fluorescence yield increases with the atomic number (Burhop
+1952). For a distant observer, these reflection features appeared to
+be diluted/broadened and distorted (asymmetric) due to relativistic
+effects of the strong gravity region in the close vicinity of the BH
+(Fabian et al. 1989, 2000). Spectral modelling using relativistic re-
+flection models can address the effect of blurring of the spectral fea-
+tures and helps to probe the physics of the strong gravity region at
+the inner disk. The accretion disk characteristics, such as the ion-
+ization of the disk material, the iron abundance, inclination of the
+system, spin of the BH etc., can also be obtained from reflection
+modelling. The line broadening can also be due to Comptonization
+in a highly ionized, optically thick cloud, and the resultant feature
+© 2022 The Authors
+
+2
+Prabhakar et al.
+is broad and symmetric. However, this mechanism is important for
+high inclination systems only (Petrucci et al. 2001).
+The Fe−K band (5 − 8 keV) is the energy range where most of
+the emission/absorption features appear. The first observational ev-
+idence of the Fe−K absorption lines was provided by Ueda et al.
+(1998) with ASCA in the spectra of galactic superluminal BH source
+GRO J1655 − 40. Kotani et al. (2000) and Lee et al. (2002) also de-
+tected similar features in the superluminal jet source GRS 1915+105
+with ASCA. Later, it is revealed that the absorption features are
+very common in the spectra of BH-XRBs (Shidatsu et al. 2013;
+King et al. 2014; Xu et al. 2018). Photon interaction with neutral
+and static material produces sharp fluorescent lines at their corre-
+sponding transition energy levels. In case of ionized absorbers, there
+would be an increase in the transition line energy compared to their
+neutral ones. The absorption lines from highly ionized ions give an
+insight into the highly ionized plasma around the compact object.
+The process of accretion in XRBs is usually accompanied with
+outflows and/or jets (Fender et al. 1999, 2004, 2010; Miller et al.
+2012, 2013; Radhika & Nandi 2014; Radhika et al. 2016). The per-
+sistent jets are present in the LHS of the system, and it gets turned
+off in HSS. Accretion disk-wind is generally observed in the disk-
+dominated HSS, though it can exist in other spectral states as well
+(Lee et al. 2002; Miller et al. 2008; Neilsen & Lee 2009; Neilsen
+2013). The disk-winds carry a sufficient amount of matter which
+suppresses the launch of radio jets (Neilsen & Lee 2009) in HSS.
+The disk-wind can also be highly ionized and their presence can be
+inferred by the blue-shifted absorption features in the X-ray spec-
+trum (Ebisawa 1997; Kotani et al. 1997). In general, it seems that
+the absorption lines are absent in the LHS, which is still a matter
+of debate. Neilsen & Lee (2009) suggests that the wind gets pho-
+toionized completely in LHS, and the medium becomes transparent;
+this could be a possible reason for the absence of absorption lines in
+the spectra. Usually, the disk-winds are observed in high inclination
+systems (Ponti et al. 2012). Such systems may show intensity ‘dips’
+in their X-ray spectra, for e.g., GRS 1915 + 105, 4U 1630 − 47, H
+1743 − 322, MAXI J1305 − 704, GRO J1655 − 40 (Leahy 1997;
+Kuulkers et al. 1998; Shidatsu et al. 2013). The dips are believed to
+be caused by obscuring material associated with the accretion disk
+(Frank et al. 1987) and are visible for highly inclined systems with
+inclination angle 60◦ ≲ θ ≲ 80◦ (Frank et al. 1987). The disk-winds
+play a major role in regulating the accretion scenario of BH-XRBs.
+For example, Muñoz-Darias et al. (2016) showed how winds control
+the violent outburst of V404 Cygni by diminishing a significant frac-
+tion of the outer disk. Disk-wind studies in BH-XRBs can provide
+great insights into the physical mechanisms involved in the accretion
+process.
+4U 1543 − 47 is a BH-XRB, discovered by Uhuru satellite in
+1971 (Matilsky et al. 1972). Since the discovery, it has undergone
+five outbursts; the first four are in an interval of ∼ 10 years, in
+1984 (Kitamoto et al. 1984), 1992 (Harmon et al. 1992) and 2002
+(Park et al. 2004). After a gap of 19 years, the fifth outburst hap-
+pened in 2021 (Negoro et al. 2021a), which marks the source as
+the brightest BH-XRB with a peak X-ray intensity of 11 Crab in
+2 − 4 keV with MAXI/GSC (Negoro et al. 2021b). The 2002 out-
+burst was also brighter (4 Crab in 2 − 12 keV), while the previ-
+ous three outbursts have comparable intensities (Park et al. 2004).
+Its optical counterpart, IL Lupi, was discovered by Pedersen (1983).
+The central engine is a dynamically confirmed BH with a mass of
+9.4 ± 1.0 M⊙, and the companion is an A2V star of mass 2.45 ±
+0.15 M⊙ (Russell et al. 2006). It is located at RA = 15h47m8s.27,
+Dec = −47◦40
+′10
+′′.8 (J2000) (Park et al. 2004) at a distance of
+7.5 ± 0.5 kpc (Jonker & Nelemans 2004). Orosz (2003) estimated
+the orbital inclination of the system as 20.7◦ ± 1.5◦. There were
+multiple attempts to estimate the spin (a∗, dimensionless spin pa-
+rameter) of the BH in 4U 1543 − 47 using RXTE observations of
+the 2002 outburst. Shafee et al. (2006) estimated a spin of ∼ 0.75 −
+0.85 using continuum-fitting of RXTE data. Miller et al. (2009) and
+Morningstar & Miller (2014) estimated the spin value as 0.3 ± 0.1
+and 0.43+0.22
+−0.31 respectively using relativistic disk reflection and disk
+continuum modelling. These three estimations are based on a BH
+mass of 9.4 ± 1.0 M⊙ and a distance of 7.5 ± 0.5 kpc. Shafee et al.
+(2006) and Morningstar & Miller (2014) used the binary inclination
+(θ) of 20.7 ± 1.5 degree, while Miller et al. (2009) used a θ of 32+3
+−4
+degree for the spin estimation. Dong et al. (2020) reported a spin of
+0.67+0.15
+−0.08 and θ of 36.3+5.3
+−3.4 degree by reflection modelling of RXTE
+data using the model RELXILL.
+The Giant Metrewave Radio Telescope (GMRT) detected radio
+flares from the source in 2002 outburst (Park et al. 2004). Multi-
+ple flaring occasions are reported at different phases of the outburst.
+Russell et al. (2020) reported the presence of a compact jet in the
+SIMS of the 2002 outburst of 4U 1543 − 47 using multiwavelength
+observations (X-ray, optical, IR, and radio). Since the system has
+a low inclination, the jet angle and axis of rotation may coincide.
+Russell et al. (2020) tested the chances of jet contribution to the lu-
+minosity of the system and renounced that possibility.
+Until now, there is no study in literature based on the 2021 out-
+burst of 4U 1543 − 47. We aim for a detailed analysis of the wide-
+band spectral characteristics of the 2021 outburst using three dif-
+ferent instrument data from NICER (Neutron star Interior Compo-
+sition ExploreR), NuSTAR (Nuclear Spectroscopic Telescope Array)
+and AstroSat during outburst decay. The evolution of spectral pa-
+rameters is investigated using phenomenological and reflection mod-
+elling. Even though the reflection modelling of RXTE data of 2002
+outburst (Miller et al. 2009; Morningstar & Miller 2014; Dong et al.
+2020) unveil the fundamental quantities of the system like a∗ and
+θ, data from much better spectral resolution instruments like NuS-
+TAR (Harrison et al. 2013) are highly promising. It can also provide
+outburst specific quantities like the iron abundance and ionization
+of the accretion disk. We report the presence of strong and dynamic
+absorption features in the 2021 outburst spectra, which has not been
+observed in any previous outbursts of 4U 1543 − 47. We examine
+these features quantitatively using phenomenological modelling of
+NuSTAR data.
+This paper is structured as follows: The observations and the data
+reduction procedure are discussed in §2. The evolution of the out-
+burst lightcurve and hardness ratio are examined in §3. The spectral
+modelling and parameter evolution are presented in §4. Phenomeno-
+logical and reflection modelling of different epochs are discussed in
+§4.1 and §4.2, respectively. The detailed study of the absorption fea-
+tures in the spectra of 4U 1543 − 47 is carried out in §4.3. We dis-
+cussed our overall findings in §5. Finally, we summarise the results
+in §6 and then conclude.
+2 OBSERVATIONS AND DATA REDUCTION
+We perform the present study based on the 2021 outburst of 4U
+1543 − 47 using NuSTAR, NICER and AstroSat observations over
+a period from 17 June 2021 (MJD 59382) to 14 September 2021
+(MJD 59471). We considered all the NuSTAR and AstroSat obser-
+vations in this period and used the NICER observations which are
+simultaneous with NuSTAR. The list of observations considered for
+this study is given in Table 1. There are a total of 16 epochs of ob-
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+3
+Table 1. The list of observations of the source 4U 1543 − 47 considered for the study. There are 16 epochs consisting of ten NuSTAR and six AstroSat
+observations. Seven NuSTAR epochs have simultaneous NICER coverage also.
+Epoch
+Obs. ID (MJD)
+Remarks
+NuSTAR
+NICER
+AstroSat
+1
+80702317002 (59382.42)
+4655060101 (59382.44)
+2
+80702317004 (59389.47)
+4655060201 (59389.47)
+3
+T04_018T01_9000004494 (59396.04)
+Offset
+4
+80702317006 (59396.18)
+4655060301 (59396.19)
+5
+80702317008 (59403.02)
+4655060401 (59403.04)
+6
+T04_021T01_9000004526 (59405.36)
+Offset
+7
+T04_030T01_9000004588 (59421.19)
+Pointed
+8
+90702326002 (59421.67)
+9
+90702326004 (59428.18)
+4202230143 (59428.13)
+10
+T04_035T01_9000004622 (59430.59)
+Pointed
+11
+90702326006 (59450.19)
+12
+90702326008 (59455.55)
+13
+T04_046T01_9000004680 (59457.06)
+Pointed
+14
+T04_051T01_9000004686 (59461.05)
+Pointed
+15
+90702326010 (59465.67)
+4202230166 (59466.07)
+16
+90702326012 (59471.51)
+4202230171 (59471.43)
+servations consisting of ten NuSTAR and six AstroSat observations.
+Seven NuSTAR epochs have simultaneous NICER coverage also.
+2.1 NuSTAR Data Reduction
+NuSTAR (Harrison et al. 2013) has observed the source several
+times in the 2021 outburst. NuSTAR is devoid of pile-up issues
+and moreover, its good energy resolution in the energy cover-
+age (3 − 79 keV) makes it suitable for the study of enormously
+bright sources like 4U 1543 − 47. NuSTAR consists two focal
+plane module telescopes (FPMA and FPMB), both are operat-
+ing in 3 − 78 keV band. The NuSTAR data for the 2021 out-
+burst is reduced using HEASOFT v.6.29, NUSTARDAS pipeline
+v.2.1.1 and CALDB v.20211115. For extremely bright sources,
+we set statusexpr="STATUS==b0000xxx00xxxx000"1 and set
+saamode to strict and tentacle to yes. A circular region of
+radius 35 pixels centered on the brightest pixel is extracted as the
+source region and as the background region, we also choose a 35
+pixel circular region away from this. These files are used for gener-
+ating science products such as the spectrum, background, lightcurve,
+Auxiliary Response File (ARF) and Response Matrix File (RMF),
+through the NUPRODUCTS task, independently for both FPMA and
+FPMB. The spectra are grouped with a minimum of 50 counts per
+bin without any systematics.
+2.2 NICER Data Reduction
+The
+X-ray
+Timing
+Instrument
+(XTI)
+onboard
+NICER
+(Gendreau et al. 2016) operates in 0.2 − 12 keV band. NICER
+has observed the source 4U 1543 − 47 in almost every day during
+the 2021 outburst. We analyse NICER data of the source between
+MJD 59382 and MJD 59471 which is simultaneous with the
+NuSTAR observations (Table 1). The data is reduced using the
+tool NICERDAS2 in HEASOFT v.6.29 with the 20210707 caldb
+version. There are 56 focal plane modules (FPMs) of NICER/XTI.
+1 https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/
+2 https://heasarc.gsfc.nasa.gov/docs/nicer/nicer_analysis.html
+We excluded FPM-14 and 34 in addition to the non-functional
+FPMs (FPM-11, 20, 22, and 60) due to increased noise levels.
+Since 4U 1543 − 47 is extremely bright at the beginning of the
+outburst, the initial epochs (till ∼ MJD 59425) are affected by
+telemetry saturation. For such observations, a lower number of
+FPMs were kept active by the instrument team and we considered
+only the active detectors in the data reduction. Level-2 standard
+calibration and filtering are done using nicerl2 task and applied
+barycenter corrections through barycorr with refframe="ICRS".
+Spectra is generated using XSELECT (V2.4m). Lightcurve of
+the NICER observation on MJD 59428.18 shows a flaring in the
+high energy band; therefore, the corresponding GTIs are excluded
+from the extraction. The ARF and RMF files are generated for
+each observation based on the number of active detectors. The
+task nibackgen3C503 (Remillard et al. 2021) is used for creating
+background files. Finally, the source spectra are grouped with 25
+photons per bin and a systematic uncertainty of 1 % is added to the
+spectra.
+2.3 AstroSat Data Reduction
+The Soft X-ray Telescope (SXT) and Large Area X-ray Propor-
+tional Counter (LAXPC) on-board AstroSat (Yadav et al. 2016;
+Agrawal et al. 2017) together observes the astronomical sources in
+wideband energy range (0.3−80 keV). AstroSat has observed the
+2021 outburst of 4U 1543-47 during 6 different epochs. The first
+two of these observations are carried out with an offset of 40′ since
+the source was too bright to have pointed observation (Garg et al.
+2021). We obtain Level-1 LAXPC and Level-2 SXT data of all six
+observations available at data archive hosted by ISSDC4.
+LAXPC consists of three identical proportional counts namely
+LAXPC10, LAXPC20 and LAXPC30. However, for our analysis, we
+have used data from LAXPC20 alone because of its steady gain (see
+also Bhuvana et al. 2021; Baby et al. 2021; Bhuvana et al. 2022;
+Prabhakar et al. 2022). To extract the Level-2 LAXPC data file i.e.,
+source spectrum, lightcurve, RMF and background spectrum and
+3 https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer_bkg_est_tools.html
+4 https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp
+MNRAS 000, 1–14 (2022)
+
+4
+Prabhakar et al.
+0
+5
+10
+0
+25
+50
+75
+100
+125
+150
+175
+0
+20
+(a)
+NICER
+NuSTAR
+AstroSat
+0.0
+0.5
+0
+25
+50
+75
+100
+125
+150
+175
+0.0
+0.2
+(b)
+0
+25
+50
+75
+100
+125
+150
+175
+Time (days since MJD 59370)
+0.0
+0.2
+0.4
+(c)
+Flux (photons/sec/cm2)
+Ratio (4-10/2-4 keV)
+10-20 keV
+2-10 keV
+Flux (Crab)
+Figure 1. MAXI/GSC daily average lightcurve in the energy bands (a) 2 − 10 keV and (b) 10 − 20 keV with flux in units of photons/sec/cm2 and Crab in the left
+and right Y-axes respectively. The hardness ratio (c) is defined by the flux in 4 − 10 keV to 2 − 4 keV. The NICER, NuSTAR and AstroSat observations for this
+study are marked using lines with the colours cyan, blue and red respectively.
+lightcurve, we make use of latest version of single routine LAXPC
+software LaxpcSoftversion3.4.35 (Antia et al. 2022). Level-2
+files are extracted from a single event and the top layer of LAXPC
+unit to avoid the instrument effects at high energy. While the soft-
+ware generated LAXPC response files are used for pointed obser-
+vations, a 40′ offset LAXPC response file provided by the instru-
+ment is used for the offset observations (see also Baby et al. 2020;
+Katoch et al. 2021).
+SXT has observed the source in Photon Counting (PC) mode dur-
+ing all the epochs. The orbit-wise SXT cleaned Level-2 event files
+are merged to get single event file for each observation using event
+merger python routine6 based on Julia v 1.1.1. The merged event file
+is then loaded into XSELECT, where we select single-pixel events
+by applying grade 0 filter to avoid optical data leakage (Singh et al.
+2021; Prabhakar et al. 2022). From the XSELECT images, we find
+that the first two offset observations have count rate < 40 counts s−1
+and hence the corresponding spectra wouldn’t have pileup issues.
+We select a circular region of radius 10′ in the image to extract the
+source spectrum and lightcurve files. In all the pointed observations
+(see Table 1), we find the central region of the image to be very
+bright which could cause a pile-up effect. Therefore, source files are
+extracted from an annular region of the outer radius of 15′ and inner
+radius of 2′ for these observations. The standard SXT background
+spectrum and instrument response file provided by the instrument
+team7 are used. ARF for the selected region is obtained from python-
+based tool sxtarfmodule provided by the SXT team. Extracted SXT
+and LAXPC spectra are grouped to have 30 counts per bin in the first
+5 http://www.tifr.res.in/~astrosat_laxpc/LaxpcSoft.html
+6 https://www.tifr.res.in/~astrosat_sxt/dataanalysis.html
+7 https://www.tifr.res.in/~astrosat_sxt/dataanalysis.html
+two observations and 20 counts per bin in the rest of the observations
+based on the source brightness. A systematics of 2% (Sreehari et al.
+2019; Athulya et al. 2022) is applied for both SXT and LAXPC spec-
+tra.
+3 OUTBURST PROFILE AND HARDNESS RATIO
+During the 2021 outburst of 4U 1543 − 47, the flux reached the
+peak value within a few days of the commencement of the out-
+burst. The outburst is monitored by multiple X-ray instruments. The
+MAXI/GSC8 daily lightcurve of the source is generated for two dif-
+ferent energy bands, 2 − 10 keV and 10 − 20 keV and are plotted in
+Fig. 1. The MJD 59370 (05 June 2021) is defined as day 0 through-
+out the study and according to this, the outburst continues over ∼175
+days. The lightcurve reveals that the source is extremely luminous in
+low energies with a very high count rate (Fig. 1a), while the contri-
+bution to the luminosity in the high energy band (Fig. 1b) is an order
+of magnitude lower. The highest value of flux in 2 − 10 keV band is
+32.67 photons/sec/cm2 (∼ 10 Crab) on day 9, whereas the same in
+10 − 20 keV is just 0.19 photons/sec/cm2 (∼ 0.5 Crab). The source
+flux reached 11.65 Crab in 2 − 4 keV, which is the highest value ob-
+served among the BH-XRBs. We define hardness ratio (HR) as the
+ratio of flux in 4 − 10 keV to 2 − 4 keV since beyond 10 keV the
+contribution is significantly low. The HR evolution (Fig. 1c) shows
+that the source mostly remains in the soft state during the outburst.
+The NICER, NuSTAR and AstroSat observations used in this study
+are marked by cyan, blue and red dashed lines, respectively. There
+is no simultaneous broadband observation in the rising phase of the
+outburst.
+8 http://maxi.riken.jp/mxondem/
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+5
+10
+5
+0.8
+1
+1.2
+Ratio
+Energy (keV)
+Figure 2. The ratios (model to data) of the spectral fitting of NuSTAR obser-
+vations (Table 1) using tbabs(diskbb+powerlaw) model. The colours black,
+red, green, blue, cyan, pink, magenta, orange, yellow and grey represent the
+NuSTAR epochs in ascending order (Table 1). A strong absorption feature ex-
+ists between ∼ 8 − 11 keV for all the epochs. The absorption depth increases
+up to Epoch 9 (pink in colour) and then decreases as the outburst progresses.
+The figure is zoomed around the absorption feature for better clarity.
+4 SPECTRAL MODELLING AND RESULTS
+We studied the spectral properties of the 2021 outburst of 4U 1543−
+47 from MJD 59382 (17 June 2021) to MJD 59471 (14 September
+2021). All the three instruments, NICER, NuSTAR and AstroSat have
+good coverage over this period. Table 1 summarises the list of obser-
+vations used in this work. In total, there are 16 epochs, comprising
+ten NuSTAR and six AstroSat (SXT-LAXPC) observations. In addi-
+tion, there are seven NICER observations which are simultaneous
+with that of NuSTAR and we used these pairs for NICER-NuSTAR
+wideband spectral analysis. We carried out phenomenological and
+reflection modelling of each NuSTAR observations and extended that
+to wideband NICER-NuSTAR and AstroSat observations.
+4.1 Phenomenological Spectral Modelling
+We used HEASOFT v.6.29 and XSPEC V12.12.0 package for the
+spectral modelling of NICER, NuSTAR and AstroSat data. We have
+done spectral modelling of NICER-NuSTAR and AstroSat data to
+see the nature of the broad-band spectrum. The NICER data below
+0.8 keV shows large residuals; therefore, we used 0.8 − 10 keV for
+NICER spectra and 4 − 60 keV for NuSTAR spectra. Spectra from
+both FPMA and FPMB telescopes of NuSTAR give similar results.
+We present only FPMA spectra throughout the study. We used 0.5−7
+keV for SXT and 3 − 30 keV for LAXPC as significant data is not
+available beyond this range.
+To accommodate the interstellar absorption, we used the tbabs
+model which uses an equivalent hydrogen column density nH
+through the solar abundance table provided by Wilms et al.
+(2000). We initially modelled the NuSTAR observations with
+tbabs(diskbb+powerlaw) model. Here, diskbb (Mitsuda et al. 1984)
+represents the multicolor blackbody spectrum from the accretion
+disc and powerlaw employs the inverse Comptonization of the soft
+blackbody photons. We detected a broad absorption feature at ∼
+8−11 keV in all epochs. The ratio of data to model of all the NuSTAR
+observations are shown in Fig. 2. The different NuSTAR epochs are
+shown in black, red, green, blue, cyan, pink, magenta, orange, yel-
+low and grey colours in ascending order. It shows that the depth of
+absorption feature starts with a low value (black in colour) and then
+keeps on increasing as the outburst progress, reaching the maximum
+on Epoch 9 (pink in colour). Finally, the absorption depth decreases
+towards the end (grey in colour) of our study. We found a similar
+absorption feature in the AstroSat/LAXPC data as well.
+We used a partial covering fraction absorption model pcfabs9 in
+XSPEC to check if this strong absorption feature at ∼ 8−11 keV can
+be due to an intervening absorber, but it did not improve the fitting
+and the low energy residuals were high. We also tried several other
+models, for example, the thermal Comptonization model nthcomp
+(Zdziarski et al. 1996) or thcomp (Zdziarski et al. 2020) in-place of
+powerlaw, and diskbb was replaced by kerrbb (Li et al. 2005). Var-
+ious combinations of these models, fit to the data also showed the
+presence of the absorption feature at ∼ 8 − 11 keV. Model combina-
+tion with kerrbb as the seed photon source did not provide a good
+fit; moreover, it failed to constrain the BH mass and spin. So, we
+prefer to use diskbb model in combination with thcomp which is an
+improved version of nthcomp. The thcomp is a convolution model
+which allows a variable fraction (parameter cov_ f rac) of seed pho-
+ton to Comptonize both up-scattering and down-scattering. Other
+parameters are the photon index (Γ) and electron temperature (kTe)
+of the corona.
+The observed absorption feature has a symmetrical profile and in-
+clusion of a Gaussian absorption model gabs10 fits the absorption
+feature well. The parameters of gabs component are line energy
+(line E, Eg), line width (σ) and line depth (strength). However, if
+we use a smeared absorption edge model, smedge (Ebisawa et al.
+1994) in XSPEC to compensate for the absorption feature, it re-
+sults in an abnormally high value of absorption width due to the
+asymmetric nature of the model. In addition, the NuSTAR data show
+a weak presence of a Fe Kα absorption edge. Therefore, we also
+used an edge component to improve the fit residual for NuSTAR.
+The model parameters for the edge component are the threshold
+energy of the absorption edge (edge E, Ee) and the correspond-
+ing absorption depth (D). Therefore, the final model for NICER-
+NuSTAR data is tbabs(thcomp × diskbb)edge × gabs (model M1,
+hereafter). In contrast, the absorption edge feature was not visible
+in the AstroSat/LAXPC data, possibly due to the low spectral resolu-
+tion of LAXPC. Therefore, the model M1 for AstroSat data becomes
+tbabs(thcomp×diskbb)×gabs. However, the AstroSat data of Epoch
+3 & 6 show the presence of a weak Fe Kα emission line feature in
+the residual, and we include a gauss model component for these two
+epochs of AstroSat data.
+All the seven NICER-NuSTAR simultaneous pairs and six AstroSat
+observations are fitted with the model M1. The Fig. 3a and Fig. 3b
+show the wideband spectra of NICER-NuSTAR (Epoch 2 & 9) and
+AstroSat (Epoch 3 & 7) respectively modelled using M1. In each
+case, the spectrum in the red colour is relatively softer than that of
+the black colour; therefore, both figures illustrate moderate spectral
+changes during the outburst decay.
+The goodness of the fit is determined using χ2 statistics. The re-
+duced χ2 (χ2
+red) varies between 0.9 − 1.3. All the parameters es-
+timated from the wideband phenomenological modelling are pre-
+sented in Table 2. The parameter uncertainties are calculated within
+the 90% confidence range. Note that the NuSTAR observations on
+Epoch 8, 11 & 12 (Table 1) are not included here as no simultane-
+ous NICER observations available. The nH is left free and it varies
+between (0.45 − 0.54) × 1022 cm−2. We aim to find out the evolution
+9 https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSmodelPcfabs.html
+10 https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node246.html
+MNRAS 000, 1–14 (2022)
+
+6
+Prabhakar et al.
+10−4
+10−3
+0.01
+0.1
+1
+10
+Photons cm−2 s−1 keV−1
+1
+10
+2
+5
+20
+50
+1
+1.5
+Ratio
+Energy (keV)
+(a)
+Epoch 2
+Epoch 9
+10−3
+0.01
+0.1
+1
+10
+Photons cm−2 s−1 keV−1
+1
+10
+0.5
+2
+5
+20
+1
+1.5
+Ratio
+Energy (keV)
+(b)
+Epoch 3
+Epoch 7
+Figure 3. (a) Simultaneous NICER-NuSTAR pair on Epoch 2 (black in colour) and Epoch 9 (red in colour) fitted with the model tbabs(thcomp × diskbb)edge ×
+gabs. Epoch 2 spectrum is harder compared to that of Epoch 9. (b) The AstroSat spectra on Epoch 3 (black) and 7 (red) respectively modelled using
+tbabs(thcomp × diskbb) × gabs. We include an additional gauss component for AstroSat (Epoch 3) data. Both instruments show spectral changes during
+the outburst.
+Table 2. Wideband NICER-NuSTAR simultaneous pairs and AstroSat observations (highlighted with grey colour) using the model tbabs(thcomp×diskbb)edge×
+gabs and tbabs(thcomp×diskbb)×gabs respectively. The error values represent 90% confidence interval. The NuSTAR data on Epoch 8, 11 & 12 are not included
+here as no simultaneous NICER observations available. The bolometric (0.5 − 100 keV) observed flux and estimated luminosity for each epoch are also shown.
+Epoch
+nH
+Model
+χ2
+red
+Fbol
+Lbol
+diskbb
+thcomp
+edge or gauss∗
+gabs
+(×1022
+Tin
+norm
+Γ
+kTe
+cov_frac
+line E
+D or σ
+line E
+σ
+strength
+(×10−8
+cm−2)
+(keV)
+(×103)
+(keV)
+(×10−2)
+(keV)
+(keV)
+(keV)
+(keV)
+(keV)
+erg cm−2 s−1)
+(LEdd)
+1
+0.470+0.002
+−0.002 1.272+0.002
+−0.002 7.96+0.05
+−0.05 2.44+0.06
+−0.05
+20f
+2.3+0.2
+−0.2
+7.16+0.05
+−0.05 0.04+0.01
+−0.01
+10.0+0.1
+−0.1
+1.60+0.09
+−0.09 0.84+0.08
+−0.08 0.90
+27.64+0.04
+−0.04
+1.52+0.14
+−0.14
+2
+0.458+0.002
+−0.002 1.159+0.002
+−0.002 6.07+0.04
+−0.04
+2.0+0.1
+−0.1
+11.6+3.9
+−1.9
+0.9+0.2
+−0.1
+7.19+0.07
+−0.07 0.03+0.01
+−0.01
+9.9+0.1
+−0.1
+1.65+0.07
+−0.08
+1.2+0.1
+−0.1
+1.00
+17.22+0.02
+−0.03
+0.95+0.09
+−0.09
+3
+0.52+0.02
+−0.02
+0.99+0.01
+−0.01
+15.4+1.5
+−1.4
+1.84+0.14
+−0.12
+20f
+0.39+0.07
+−0.06 6.77+0.15
+−0.15
+0.6f
+7.35+0.34
+−0.39
+2f
+1.73+0.47
+−0.47 1.17
+19.5+0.1
+−0.1
+1.07+0.001
+−0.001
+4
+0.461+0.002
+−0.002 1.094+0.002
+−0.002 5.82+0.05
+−0.05
+2.2+0.1
+−0.1
+20f
+0.3+0.1
+−0.1
+7.27+0.07
+−0.07 0.05+0.01
+−0.01
+10.4+0.1
+−0.1
+1.94+0.08
+−0.07
+2.2+0.2
+−0.1
+0.95
+12.38+0.02
+−0.03
+0.68+0.06
+−0.06
+5
+0.458+0.002
+−0.002 1.050+0.002
+−0.002 5.77+0.04
+−0.04
+2.3+0.2
+−0.2
+20f
+0.2+0.1
+−0.1
+7.28+0.05
+−0.05 0.06+0.01
+−0.01
+10.7+0.1
+−0.1
+1.97+0.08
+−0.07
+2.7+0.2
+−0.2
+1.02
+10.10+0.01
+−0.02
+0.56+0.05
+−0.05
+6
+0.50+0.02
+−0.02
+0.97+0.02
+−0.02
+12.1+1.5
+−1.4
+2.3f
+20f
+0.3+0.02
+−0.02
+6.77+0.18
+−0.18
+0.6f
+9.05+0.51
+−0.55
+2.5f
+2.27+0.66
+−0.68 1.27
+13.9+0.06
+−0.05
+0.76+0.001
+−0.001
+7
+0.54+0.01
+−0.01
+0.98+0.01
+−0.01
+7.64+0.33
+−0.33 2.26+0.06
+−0.05
+20f
+0.2f
+-
+-
+10.7+0.2
+−0.2
+1.46+0.24
+−0.18 1.56+0.25
+−0.33 1.29
+9.36+0.21
+−0.20
+0.51+0.004
+−0.004
+9
+0.451+0.003
+−0.003 0.966+0.002
+−0.002 5.60+0.05
+−0.05
+2.6+0.5
+−0.4
+20f
+0.1+0.2
+−0.1
+7.44+0.08
+−0.08 0.08+0.02
+−0.02
+10.5+0.2
+−0.1
+2.02+0.09
+−0.09
+3.3+0.3
+−0.2
+1.05
+6.88+0.02
+−0.02
+0.38+0.04
+−0.04
+10
+0.53+0.01
+−0.01
+0.96+0.01
+−0.01
+5.63+0.35
+−0.31 1.92+0.30
+−0.25
+20f
+0.11+0.05
+−0.03
+-
+-
+10.1+0.2
+−0.2
+1.56+0.20
+−0.18 2.13+0.39
+−0.32 1.28
+7.59+0.04
+−0.03
+0.40+0.001
+−0.001
+13
+0.52+0.01
+−0.01
+0.91+0.01
+−0.01
+7.48+0.24
+−0.23 1.77+0.03
+−0.03
+20f
+0.12+0.05
+−0.05
+-
+-
+9.42+0.27
+−0.27 0.75+0.30
+−0.28 0.47+0.13
+−0.11 1.10
+6.59+0.06
+−0.07
+0.36+0.002
+−0.003
+14
+0.48+0.01
+−0.01
+0.91+0.01
+−0.01
+7.92+0.58
+−0.53 1.94+0.02
+−0.02
+20f
+0.4f
+-
+-
+10.7+0.24
+−0.21 1.34+0.18
+−0.17 1.47+0.18
+−0.18 1.31
+6.22+1.89
+−6.56
+0.34+0.08
+−0.29
+15
+0.465+0.004
+−0.010 0.904+0.030
+−0.004 6.10+0.09
+−0.69 2.09+0.04
+−0.04
+20f
+1.6+0.1
+−0.1
+7.31+0.08
+−0.14 0.10+0.05
+−0.03
+9.6+0.3
+−0.3
+2.1+1.2
+−0.4
+1.1+1.1
+−0.3
+1.27
+5.66+0.02
+−0.02
+0.31+0.03
+−0.03
+16
+0.471+0.002
+−0.002 0.897+0.002
+−0.001 6.22+0.05
+−0.05 2.04+0.04
+−0.06
+20f
+1.6+0.1
+−0.1
+7.35+0.06
+−0.03 0.12+0.02
+−0.03
+9.4+0.4
+−0.3
+1.5+0.4
+−0.3
+0.5+0.1
+−0.1
+1.00
+5.39+0.01
+−0.01
+0.30+0.03
+−0.03
+∗ edge is used for NICER-NuSTAR pairs whereas gauss component is used only for Epoch 3 & 6 of AstroSat data.
+f Frozen parameters
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+7
+20
+40
+60
+80
+100
+1.0
+1.2
+Tin (keV)
+(a)
+20
+40
+60
+80
+100
+2.0
+2.5
+3.0
+Γ
+(b)
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+0.25
+0.50
+τ0
+(c)
+Figure 4. Evolution of (a) Tin and (b) Γ from simultaneous NICER-NuSTAR
+pairs (green in colour) and AstroSat observations (red in colour) fitted using
+the model M1. We calculate the (c) optical depth (τ0) of the absorber from
+gabs components for each epoch. The Γ of Epoch 13 marked with a blue
+circle carries a special signature that is discussed in §5.
+of various parameters with the progress of the outburst. Fig. 4 gives
+the variation of the inner disk temperature Tin, photon index Γ and
+optical depth (τ0) with time. Here, points in green and red colour
+represent the parameter value estimated from NICER-NuSTAR and
+AstroSat spectral modelling respectively.
+We find that the inner disk temperature (Fig. 4a and Table 2)
+monotonically decreases throughout the outburst decay. The evolu-
+tion of the diskbb norm (Table 2) estimated using NICER-NuSTAR
+decreases till Epoch 9 and a reverse trend is observed for later
+epochs. The diskbb norm from the AstroSat data also shows a sim-
+ilar pattern though AstroSat values are higher than that of NICER-
+NuSTAR. The variation of photon index Γ is presented in Fig. 4b. The
+value of Γ estimated from AstroSat data (red square) differs from that
+of NICER-NuSTAR pairs (green square). This can be due to the non-
+availability of the high energy contribution (beyond 30 keV) in the
+LAXPC data. From NICER-NuSTAR fitting, Γ varies between 2−2.6,
+and it shows spectral softening till Epoch 9. We wanted to estimate
+the electron temperature (kTe) of the corona using the thcomp model.
+But the broadband spectral fitting could not constrain the value of
+kTe, except for Epoch 2, for which we obtained kTe = 11.6+3.9
+−1.9 keV
+(Table 2). For all the remaining epochs, we freeze kTe at 20 keV.
+Only a tiny fraction, cov_frac < 3 % (Table 2), of the soft photons
+Comptonized in the corona. It gradually decreases till Epoch 10 and
+increases afterwards. This behaviour is consistent with the spectral
+softening trend shown in Fig. 4b.
+The broad absorption feature at ∼ 8−11 keV in the spectrum is
+well represented by the gabs model. The strength shows an increas-
+ing trend and reaches the maximum in Epoch 9 and declines beyond
+that. We calculate the optical depth (τ0) associated with the gabs
+component using gabs strength and σ as τ0 = strength/σ
+√
+2π. The
+evolution of τ0 is shown in Fig. 4c, which shows that the absorption
+optical depth increases and reaches a maximum on Epoch 9 and then
+decreases. The dynamic behaviour of the absorption strength seems
+10−4
+10−3
+10−2
+10−1
+100
+101
+Photons cm−2 s−1 keV−1
+1.00
+1.25
+(a)
+χ2
+red =2.46
+tbabs(diskbb+relxill)
+0.75
+1.00
+1.25
+(b)
+χ2
+red =1.60
+tbabs(diskbb+relxill)gabs
+0.8
+1.0
+1.2
+(c)
+χ2
+red =1.63
+tbabs(diskbb+relxillCp)gabs
+0.75
+1.00
+1.25
+(d)
+χ2
+red =1.79
+tbabs(diskbb+relxilllpCp)gabs
+100
+101
+Energy (keV)
+1.0
+1.2
+(e)
+χ2
+red =0.91
+tbabs(diskbb+relxilllp)gabs
+Ratio
+Figure 5. Unfolded spectrum (top panel) of simultaneous NICER-NuSTAR
+pair on Epoch 2 and ratio of the model to the data using various re-
+flection models (a) tbabs(diskbb+relxill), (b) tbabs(diskbb+relxill)gabs,
+(c) tbabs(diskbb+relxillCp)gabs, (d) tbabs(diskbb+relxilllpCp)gabs and (e)
+tbabs(diskbb+relxilllp)gabs. The model and the value of χ2
+red are mentioned
+at the top left and bottom left corners, respectively, in each case.
+interesting. We attempt to characterize the strong and dynamic ab-
+sorption features in §4.3. The evolution of the edge component is
+listed in Table 2. We discuss a possible connection between edge
+and gabs components in §5.
+We estimate the observed bolometric flux (Fbol) in 0.5 − 100
+keV with uncertainty in 90% confidence interval from the wide-
+band simultaneous spectral data, which is also shown in Table 2.
+Corresponding bolometric luminosity (Lbol) of the source is also
+calculated by assuming the distance to the source as 7.5 ± 0.5 kpc
+(Jonker & Nelemans 2004). The Eddington luminosity of the source
+is LEdd = 1.22±1×1039 ergs s−1 with an assumed BH mass of 9.4±1
+M⊙ (Russell et al. 2006). It can be seen from Table 2 that the lumi-
+nosity of the source exceeds the LEdd at the peak (Epoch 1 is close
+to the peak) of the outburst, and the luminosity decreases gradually
+with the decay of the outburst.
+MNRAS 000, 1–14 (2022)
+
+8
+Prabhakar et al.
+4.2 Spectral Modelling for Reflection Studies
+To understand the reflection features in the spectra of 4U 1543 − 47,
+we use the relativistic reflection model RELXILL11v1.4.3. Dif-
+ferent flavours of the RELXILL model are tried. The unfolded
+NICER-NuSTAR spectra and data-to-model ratios using various re-
+flection models are shown in Fig. 5 for Epoch 2. We started with
+the model tbabs(diskbb+relxill) which gives a χ2
+red of 2.46. The
+data-to-model ratio (Fig. 5a) shows that the absorption feature at
+∼ 8 − 11 keV in the spectrum cannot be fitted by the reflection
+model. We added the absorption model gabs with this, and the model
+tbabs(diskbb+relxill)gabs (Fig. 5b) improves the residual but still
+the χ2
+red for this combination is 1.6. Then, we replaced the model
+relxill with relxillCp (Fig. 5c) where a thermal Comptonizing con-
+tinuum is assumed for the illuminating flux. This combination has
+χ2
+red = 1.63, which is also unacceptable. Then, we used relxilllpCp
+(Fig. 5d) as the reflection model, where a lamp-post (lp) geome-
+try is assumed for the corona. In all the ‘lp’ flavours of the RELX-
+ILL model, the inner disk is illuminated by a point-like corona sit-
+uated at a height ‘h’ from the disk surface on the axis of rotation.
+The modelling with relxilllpCp also resulted in a large residual, with
+χ2
+red = 1.79. We replaced relxilllpCp with relxilllp (Fig. 5e) where
+the illuminating flux is modelled as a powerlaw with a high-energy
+cutoff just like the relxill. For this trial, we got a reasonable fit with
+χ2
+red = 0.91, and we decided to proceed with the model combina-
+tion, tbabs(diskbb+relxilllp)gabs (model M2, hereafter) as the final
+model to study the reflection features in the spectra. Note that, no ad-
+ditional edge or gauss component is required in the reflection mod-
+elling.
+Using the model M2, we did the spectral fitting of the simultane-
+ous NICER-NuSTAR pairs and AstroSat observations. Here, nH is a
+free parameter. Inner and outer disk radii are frozen at the innermost
+stable circular orbit RISCO and 400 rg (where rg ≡ GM/c2, the gravi-
+tational radius of the BH), respectively. The powerlaw cutoff energy,
+Ecut, is fixed at 60 keV since it is hitting the upper limit. All other
+parameters of relxilllp are kept free; the lamp-post height h (in units
+of rg), inclination angle of the system θ (in degree), Γ of the incident
+radiation, ionization parameter log ξ (erg cm s−1), iron abundance
+(AFe) of the accretion disk in terms of the solar abundance AFe,⊙, and
+the reflection fraction Rf. Here, Rf is defined as the ratio of the pri-
+mary photon flux illuminating the disk to that reach the observer at
+infinity (Dauser et al. 2016). We wish to estimate the spin parameter,
+a∗, of the system, but it is found to hit the upper limit for almost all
+the epochs. Based on previous studies (Morningstar & Miller 2014;
+Dong et al. 2020), we freeze a∗ = 0.4 for this study.
+The estimated reflection model (M2) parameters are listed in Ta-
+ble 3. The errors represent 90% confidence interval. The evolution
+of Tin follows the same trend as that observed in the phenomeno-
+logical spectral modelling (Table 2). The value of Γ varies between
+2 − 3.3 and appears slightly steeper than that of phenomenologi-
+cal modelling (Table 2). It may be due to the additional low energy
+contribution from the reflection component over the diskbb compo-
+nent. However, reflection modelling also shows spectral softening
+till Epoch 9. The NICER-NuSTAR results suggest that 4U 1543 − 47
+is a low inclination system with θ varies between ∼ 32◦ − 40◦. We
+could not constrain the inclination angle from AstroSat data and
+freeze it to 40◦.
+The evolution of few important model parameters (h, log ξ
+and Rf ) are shown in Fig. 6 for better presentation. Simultane-
+ous NICER-NuSTAR pairs and AstroSat observations are marked in
+11 http://www.sternwarte.uni-erlangen.de/~dauser/research/relxill/index.html
+20
+40
+60
+80
+100
+0
+50
+100
+h
+(a)
+20
+40
+60
+80
+100
+2
+4
+log ξ
+(b)
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+1
+5
+10
+Rf
+(c)
+Figure 6. Evolution of (a) h, (b) log ξ and (c) Rf from reflection modelling
+of simultaneous NICER-NuSTAR pairs (green in colour) and AstroSat obser-
+vations (red in colour) using the model tbabs(diskbb + relxilllp)gabs. See
+text for details.
+green and red colour, respectively. We could not estimate the un-
+certainty of h in most of the epochs. The evolution of h (Fig. 6a)
+indicates that the primary source is moving away from the BH till
+Epoch 6 and then gradually coming closer to the central object. The
+ionization structure of the disk is established through the parameter
+log ξ. Its value gradually increases (Fig. 6b) and reaches the maxi-
+mum around Epoch 10 and then gradually decreases. The high value
+of log ξ (>3) suggests a highly ionized disc material throughout the
+outburst. We could estimate iron abundance AFe for Epoch 1, 2, 4
+and 5, and it hits the upper limit during the rest of the epochs. Our
+study reveals an overabundance (3.6−10 AFe,⊙) of iron in the disk.
+The reflection fraction Rf is estimated well at the first three and
+last two epochs only. Fig. 6c suggests that the fraction of primary
+photons reaching the disk increases till Epoch 9, and it decreases af-
+terwards. The gabs strength shows a similar behaviour found in the
+phenomenological modelling. We have discussed more on this result
+in §4.3.
+Very recently, the RELXILL model (version v2.2) has undergone
+some modifications by considering the effect of returning radiation
+in the calculation of reflected flux. Particularly in relxilllpCp, where
+the effects of returning radiation, the density profile and ionization
+gradient of the disk and the velocity of the primary source are also
+included. However, the velocity of the primary source and the effects
+of returning radiation are the new parameters added to the relxilllp
+model. We applied this modified relxilllp model to the broadband
+NICER-NuSTAR data but could not constrain the source velocity.
+Also, we tried to estimate the parameters using the v2.2 flavour of
+relxilllpCp. We fitted all the wideband NICER-NuSTAR observations
+using the model tbabs(diskbb+relxilllpCp)gabs. But, we could not
+constrain most of the parameters since the number of free parameters
+is very large. Therefore, we need to essentially freeze all the new
+parameters introduced in the updated version, and RELXILL v2.2
+does not bring any improvement in the result.
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+9
+Table 3. Reflection modelling of NICER-NuSTAR simultaneous pairs and AstroSat observations (highlighted with grey colour) using the model
+tbabs(diskbb+relxilllp)gabs. The error values represent 90% confidence interval. The NuSTAR data on Epoch 8, 11 & 12 are not included here as no si-
+multaneous NICER observations available.
+Epoch
+nH
+Model
+χ2
+red
+diskbb
+relxilllp
+gabs
+(×1022
+Tin
+norm
+h
+θ
+Γ
+log ξ
+AFe
+Rf
+norm
+line E
+σ
+strength
+cm−2)
+(keV)
+(×103)
+(GM/c2)
+(deg)
+(erg cm s−1)
+(AFe,⊙)
+(×10−3)
+(keV)
+(keV)
+(keV)
+1
+0.49+0.01
+−0.01
+1.276+0.001
+−0.001 6.31+0.03
+−0.03
+25.50a
+32.7+3.4
+−2.9
+2.81+0.03
+−0.03
+3.58+0.09
+−0.07
+8.5+1.2
+−1.3
+1.4+0.1
+−0.2
+282.7+84.3
+−46.1
+9.9+0.1
+−0.1
+2.05+0.1
+−0.03
+1.1+0.1
+−0.1
+0.80
+2
+0.70+0.04
+−0.05
+1.151+0.006
+−0.005
+6.3+0.2
+−0.2
+9.4a
+36.3+1.6
+−1.8
+2.94+0.04
+−0.08
+3.65+0.05
+−0.08
+7.7+1.2
+−1.1
+3.5+0.9
+−0.7
+200.4+47.8
+−53.9
+9.8+0.1
+−0.1
+1.94+0.09
+−0.09
+1.3+0.2
+−0.2
+0.91
+3
+0.46+0.02
+−0.01
+1.01+0.006
+−0.006
+11.7+0.76
+−0.44
+30f
+40f
+3.0f
+4.7b
+9.37b
+−0.40 5.59+1.85
+−3.48 0.32+0.08
+−0.08
+9.37+0.27
+−0.13 0.75+0.37
+−0.64 0.48+0.03
+−0.19 1.37
+4
+0.683+0.004
+−0.03
+1.081+0.001
+−0.001 6.14+0.008
+−0.04
+30.5+24.3
+−2.0
+35.7+2.2
+−2.7
+3.23+0.01
+−0.06
+4.23+0.05
+−0.04
+5.8+0.8
+−0.7
+10.0a
+41.1+20.3
+−2.0
+10.04+0.04
+−0.07 1.92+0.04
+−0.06 1.99+0.2
+−0.08 0.83
+5
+0.67+0.01
+−0.03
+1.044+0.001
+−0.001 5.83+0.01
+−0.04 84.0+56.6
+−18.3 39.6+2.6
+−3.1 3.385+0.006
+−0.05
+4.7b
+7.1+0.6
+−0.7
+10.0a
+33.6+3.7
+−2.5
+10.33+0.03
+−0.03 2.06+0.03
+−0.05 2.76+0.02
+−0.06 0.83
+6
+0.49+0.01
+−0.02
+0.99+0.07
+−0.01
+9.23+1.46
+−0.19
+100f
+40f
+3f
+4.30+0.21
+−0.21
+10b
+7.18a
+7.11+1.95
+−4.1
+10.0+0.13
+−0.14 1.30+0.13
+−0.13 1.39+0.28
+−0.11 1.30
+7
+0.53+0.03
+−0.01
+0.99+0.003
+−0.003
+6.94+0.16
+−0.27
+8.26a
+40f
+2.46+0.36
+−0.25
+3.70+0.75
+−0.68
+10b
+9.99a
+5.75+0.55
+−0.69
+10.5+0.16
+−0.14 1.45+0.20
+−0.17 1.65+0.30
+−0.25 1.44
+9
+0.496+0.01
+−0.007 0.958+0.001
+−0.001 5.85+0.04
+−0.04
+40.29a
+40f
+3.17+0.1
+−0.06
+4.7b
+10b
+10f
+4.0+2.3
+−1.0
+10.11+0.08
+−0.08 1.86+0.06
+−0.06
+3.1+0.2
+−0.2
+1.10
+10
+0.44+0.03
+−0.01
+0.98+0.004
+−0.004
+4.95+0.10
+−0.90
+70f
+40f
+2.87+0.33
+−0.24
+4.7b
+10b
+8.10a
+1.33+0.28
+−0.12
+10.2+0.13
+−0.12 1.78+0.16
+−0.19 2.76+0.36
+−0.38 1.19
+13
+0.47+0.01
+−0.01
+0.94+0.002
+−0.003
+5.90+0.08
+−0.09
+44.3a
+40f
+2.25f
+3.64+0.09
+−0.18
+3.56+0.56
+−0.46
+8.10a
+1.11+0.07
+−0.16
+9.22+0.21
+−0.18 0.84+0.20
+−0.21 0.74+0.10
+−0.13 1.11
+14
+0.48+0.01
+−0.01
+0.91+0.002
+−0.002
+7.95+0.21
+−0.12
+6.69a
+40f
+2.61+0.08
+−0.19
+3.96+0.07
+−0.26
+10b
+10.6a
+0.14+0.01
+−0.01
+10.1+0.13
+−0.13 1.03+0.12
+−0.13 1.25+0.14
+−0.14 1.24
+15
+0.475+0.004
+−0.003 0.906+0.001
+−0.001 5.91+0.04
+−0.04
+33.74a
+38.0+8.5
+−4.4
+2.07+0.06
+−0.06
+2.7+0.1
+−0.3
+10.0b
+0.9+0.2
+−0.2
+4.9+1.5
+−0.7
+9.1+0.1
+−0.1
+1.5+0.1
+−0.2
+1.0+0.1
+−0.1
+1.12
+16
+0.482+0.004
+−0.004 0.904+0.001
+−0.001 5.63+0.05
+−0.05
+50.01a
+36.7+9.4
+−5.5
+2.05+0.05
+−0.05
+3.7+0.1
+−0.2
+10.0b
+0.9+0.2
+−0.1
+4.2+0.8
+−0.6
+8.7+0.1
+−0.1
+1.49+0.08
+−0.08
+1.2+0.2
+−0.2
+0.87
+a Parameter uncertainty can’t be estimated.
+b Parameter hits the boundary.
+f Frozen parameters.
+4.3 Absorption Features in the Spectra of 4U 1543−47
+The wideband spectral analysis of 4U 1543 − 47 reveals the pres-
+ence of a very strong absorption feature (Fig. 2, Table 2, Table 3)
+whose strength changes throughout the outburst. We use gabs model
+to characterize the absorption feature in phenomenological and re-
+flection modelling. The gabs strength estimated from both methods
+follow the same trend; getting more stronger as the outburst pro-
+gresses and reaches the maximum value on Epoch 9, then declines
+gradually.
+In general, the absorption features in the spectrum can be due to
+multiple reasons like the presence of obscuring cloud in the line-
+of-sight, occultation due to the companion star, strong accretion
+disk-wind and/or the stellar wind from the companion (Miller et al.
+2008; Szostek & Zdziarski 2008; Koljonen & Tomsick 2020). We
+have discarded the chances of absorption due to obscuring cloud in
+the line-of-sight by fitting the data with the partial covering fraction
+model pcfabs and found no improvement in the fitting. If the ab-
+sorption feature is produced by the occultation or stellar wind of the
+binary companion, the features must show some orbital variations.
+Precise diagnostic of the orbital variations provide significant insight
+into the understanding of the nature and origin of the absorption fea-
+tures. Since 4U 1543−47 is a low inclination system (θ ∼ 32◦ −40◦),
+the expected orbital variation of the absorption feature, if any, will be
+weak. Therefore, we avoid using multi-instrument data to check the
+orbital variation of the absorption feature, as the differences in the
+estimated parameters between instruments may screw up the varia-
+tion. We use only the NuSTAR observations for this purpose.
+We extracted the spectrum from different patches of GTIs of each
+NuSTAR observation epoch. Since the GTI-patches have low expo-
+sure time, we grouped the spectrum with only 30 counts per bin.
+Patches with an exposure time less than 500 seconds are merged to-
+gether before extracting the spectrum. We did a simultaneous joint
+fitting of all the GTI-patches under each epoch using the model M1.
+In the joint fitting, all parameters are tied between the patches ex-
+cept gabs strength. The line-of-sight column density, nH, is frozen
+at 0.45 × 1022 cm−2 found from broadband NICER-NuSTAR spec-
+tral modelling. In Fig. 7, we plot the simultaneous joint fitting of the
+spectra for Epoch 9, which has 7 patches of GTIs. The black, red,
+green, blue, cyan, magenta and yellow colours represent them in the
+ascending order of time.
+We also fitted all the 10 epochs (Table 1) of NuSTAR observations
+using the models M1 and M2 discussed before. We calculated the
+gabs strength (Si) for each NuSTAR epoch using both models. The
+evolution of Si is shown in Fig. 8a. The colours black, red, green,
+blue, cyan, pink, magenta, orange, yellow and grey indicate the NuS-
+TAR epochs in chronological order. The gabs strength of NuSTAR
+data is showing the same trend of wideband spectral data; reach-
+ing maximum on Epoch 9 (pink in colour). The absorption strength
+MNRAS 000, 1–14 (2022)
+
+10
+Prabhakar et al.
+10−5
+10−4
+10−3
+0.01
+0.1
+1
+Photons cm−2 s−1 keV−1
+10
+5
+20
+1
+1.2
+Ratio
+Energy (keV)
+Figure 7. Folded spectra of different patches in NuSTAR observation of
+Epoch 9 using the model tbabs(thcomp × diskbb)edge × gabs. The param-
+eters, except the gabs strength, are tied between the patches. See text for
+details.
+0
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+1
+2
+(a)
+0
+100
+200
+300
+Phase (degree)
+−0.5
+0.0
+0.5
+1.0
+Residual Strength (keV)
+(b)
+Strength (keV)
+Figure 8. (a) Evolution of gabs strength (Si) estimated from NuSTAR us-
+ing the models tbabs(thcomp × diskbb)edge × gabs (square symbol) and
+tbabs(diskbb + relxilllp) × gabs (star symbol). (b) Residual strength (Sp -
+Si) for different patches in each NuSTAR epoch with the orbital phase. The
+colours black, red, green, blue, cyan, pink, magenta, orange, yellow and grey
+represent the NuSTAR epochs in chronological order. See text for details.
+represented by the square symbol is estimated using the model M1,
+whereas the same using M2 are denoted by the star symbol. We no-
+tice that the value estimated using M2 are marginally higher than the
+same from M1. This can be the effect of an additional edge compo-
+nent used in the M1 model. Similarly, we estimated the gabs strength
+(Sp) corresponding to each GTI-patch for a given epoch from the
+patch-spectra modelling using M1. The residual strength (Sp - Si) is
+measured for each patches inside an epoch and are plotted against
+the orbital phase in Fig. 8b. The binary orbital period (P) of 4U
+1543 − 47 is 26.79377 ± 0.00007 hours (Orosz et al. 1998; Orosz
+2003). The orbital position of each NuSTAR patch has been identi-
+fied based on the start time (MJD 59382.42) of Epoch 1 as the refer-
+ence time. For Epoch 12 (data in orange colour in Fig. 8a), the value
+of Si is unusually low, and the estimated Sp from the patch-spectra
+modelling of this epoch is not reliable. Therefore, we ignore Epoch
+12 from Fig. 8b. The residual varies within ±0.5 keV (except one
+patch), and we see only a marginal variation (within uncertainties)
+in strength within an orbit. This implies that the orbital position of
+the BH and the companion is not responsible for the dynamic nature
+of the absorption features. In fact, we do not expect such behaviour
+for a low inclination system.
+The X-ray luminosity of the source at the peak (see Epoch 1 in
+Table 2) of the outburst is extremely high, and it may irradiate (see
+Lasota (2001) for a review) the outer accretion disk and the compan-
+ion star. If the irradiation affects the companion star, either a fresh
+accretion of matter starts at the hot spot or enhances the stellar wind
+in the companion star. The former may produce multiple trigger-
+ing in the same outburst event, which has been observed, for exam-
+ple, in GX 339 − 4 (Aneesha et al. 2019). If the latter happens, the
+highly ionized wind material may absorb the X-ray emission from
+the primary to produce the broad absorption feature. The compan-
+ion of 4U 1543 − 47 is an A2V type star with a mass Mc=2.45 M⊙
+(Russell et al. 2006) and radius Rc=2.84 R⊙ (Orosz et al. 1998). The
+escape velocity (ve) of the stellar material from the surface of the
+companion is calculated as 573 km s−1. The binary separation (a)
+between the BH and the companion is estimated as 7.18 ×1011 cm
+by considering a BH mass, MBH=9.4 M⊙ (Russell et al. 2006) using
+the relation P2/a3 = 4π2/ G(MBH + Mc), where G is the Gravitational
+constant. We observe that the stellar wind takes only a few hours to
+reach the primary. The column density of stellar wind and the ion-
+ization state should reduce with the decrease of the X-ray luminosity
+of the primary. Therefore, we expect that the strength of the broad
+absorption feature should reduce along with the progress of the out-
+burst. Instead, we observe that the absorption strength enhances and
+becomes strongest during Epoch 9. Moreover, the estimated stellar
+wind speed is not sufficient to blue shift the highest ionized lines
+of Fe XXVI, to produce the observed absorption feature. Therefore,
+the stellar wind has no role in the dynamic absorption features in the
+spectra of 4U 1543 − 47.
+The irradiation of the outer accretion disk enhances the accre-
+tion rate; therefore, the outburst source stays in the high luminosity
+state for a longer duration (King 1998; Lasota 2001; Aneesha et al.
+2019; Aneesha & Mandal 2020). This is possibly causing 2021 out-
+burst of 4U 1543 − 47 to decline very slowly (over ∼ 175 days,
+see Fig. 1). The super Eddington peak luminosity (Epoch 1 in Ta-
+ble 2) of the source can launch strong disk-wind (e.g., King et al.
+(2015); Muñoz-Darias et al. (2018)). The presence of the accre-
+tion disk-wind is more prominent in the soft state of X-ray bi-
+naries (Miller et al. 2008; Neilsen & Lee 2009; Ponti et al. 2012),
+though disk-winds are not exclusively confined to soft spectral state
+(Lee et al. 2002). Spectral analysis of 4U 1543 − 47 (Table 2) sug-
+gests that the source was in the HSS during our study. Also, we no-
+tice spectral softening happens till Epoch 9 (∼ day 60) and beyond
+which spectra gradually become harder (Table 2 & Table 3). If the
+disk-ionized winds are responsible for the absorption features, then
+the strength of the features would be maximum when the source is
+softer. Surprisingly the strength of the absorption feature is maxi-
+mum on ∼ day 60 as per our analysis (Fig. 8a), and it is keeps-on
+decreasing further. The optical depth (τ0) evolution (Fig. 4c) also
+suggests that the absorption column is maximally populated on ∼
+day 60.
+The transition energy of the most ionized line with the highest
+absorption yield, i.e., Fe XXV and Fe XXVI, are 6.68 keV and 6.97
+MNRAS 000, 1–14 (2022)
+
+Disk-wind regulated accretion in 4U 1543−47
+11
+keV respectively (provided by XSTAR line finding list12). Assuming
+the absorption feature (with line E ∼ 10 keV) in the NuSTAR spectra
+is produced due to the absorption of the accretion disk photons by
+the highly ionized blue shifted disk-wind, the estimated wind speed
+is reaching 30% of the speed of light to blue shift the Fe XXVI line
+energy to 10 keV. Such a fast disk-wind has never been observed in
+X-ray binary systems. In fact, highly ionized wind (say, Fe XXVI)
+is never detected (Ponti et al. 2012) in BH-XRB systems with low
+inclination angle; for example, GX 339 − 4, XTE J1817 − 330, 4U
+1957 + 115, XTE J1650 − 500, GRS 1758 − 258 etc. Therefore,
+this detection is the first of its kind for X-ray binaries. However,
+mildly relativistic disk-wind is not uncommon in quasars and AGNs
+(Reeves et al. 2009; Tombesi et al. 2015; Hagino et al. 2017). The
+other difficulty is the width (σ) of the absorption feature, which is
+as broad as 2 keV on Epoch 9 (Table 2). Known line-broadening
+processes due to turbulence or scattering will face serious challenges
+in explaining the line width if it is from a single line. Instead, it is
+more likely that the broad feature can be produced by combining
+multiple lines of various ionization states of iron.
+The phenomenological spectral fitting of NuSTAR data with
+model M1 reveals the presence of neutral Fe K−α absorption edge
+and the broad ionized absorption features. We calculate the equiv-
+alent width (EW), which is a measure of the strength of an ab-
+sorption line, of both absorption features to find if there exists any
+connection between these two components. The EW is defined as
+(Arumugasamy et al. 2018),
+EW =
+� ∞
+0
+[1 − F(E)] dE.
+(1)
+The energy dependent function F(E) for the gabs component is
+given by,
+F(E) = exp(−τ);
+τ = τ0 exp
+�
+−(E − Eg)2/2σ2�
+,
+(2)
+where τ0, Eg and σ are the optical depth, line energy and line width
+respectively. Similarly, F(E) corresponds to the edge component in
+XSPEC is given by,
+I(E) =
+
+1
+if E ≤ Ee
+exp[−D (E/Ee)−3]
+if E ≥ Ee,
+(3)
+where Ee and D are the threshold energy and absorption depth, re-
+spectively.
+The evolution of EW calculated based on NuSTAR phenomeno-
+logical modelling is shown in Fig. 9 for both edge (red in colour)
+and gabs (blue in colour) components. The gabs EW increases till
+Epoch 9 and then gradually decline, except for Epoch 12 (Fig. 8a)
+& 13 (Table 2). The implication of this result and the connection
+between both components (gabs and edge) are discussed in §5.
+5 DISCUSSION
+The wideband spectral modelling of NICER-NuSTAR and AstroSat
+data reveals that the inner disc temperature Tin is highest (1.27 keV)
+on Epoch 1, and it keeps on decreasing during the decay of the out-
+burst (Fig. 4a). The estimated diskbb norm (Table 2, Table 3) sug-
+gests a marginal inward movement of the inner disk radius rin since
+diskbb norm ∝ r2
+in. However, the decrease in rin could not prevent
+the drop in Tin due to the gradual decline in ˙M as Tin ∝ ˙M1/4 r−3/4
+in
+12 https://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html
+20
+40
+60
+80
+100
+Time (days since MJD 59370)
+0.1
+0.5
+1
+2
+3
+EW (keV)
+A
+B C
+D
+E
+Figure 9. Evolution of equivalent width of gabs (blue in colour) and edge
+(red in colour) components from the NuSTAR epochs. The vertical lines (A-
+E) are used to explain the figure in the description.
+(Frank et al. 2002). The extreme luminosity in the inner disk may
+slow down the accretion of matter to the BH. On the other hand,
+the high accretion disk luminosity (Table 2) can irradiate the outer
+accretion disk, enhancing the accretion of matter. If most of the ac-
+creted matter is released as the disk-wind, the amount of matter ac-
+tually transfers to the inner disk for falling onto the BH is much less.
+Therefore, the gradual decline of Tin is due to the reduction of effec-
+tive infall of matter onto the BH through the inner disk. The source
+luminosity is completely soft-photons-dominated due to very little
+fractional Comptonization (cov_frac in Table 2), low corona tem-
+perature, and steeper photon index Γ (Fig. 4b). Therefore, the source
+was in the high/soft spectral state during our study, and it was the
+softest on Epoch 9.
+The important parameters for reflection modelling are shown in
+Table 3 and in Fig. 6. The lamp-post height comes closer to the
+central object as the source becomes softer. The reflection fraction
+(Rf) increases and hits the boundary when the source is softest.
+If the value of ionization parameter, log ξ ≳ 3, the fluorescence
+yield of the highly ionized Fe line (more ionized than Fe XXIII)
+increases (Matt et al. 1993). The high value of log ξ (Table 3) ob-
+tained from reflection modelling suggests a highly ionized accretion
+disk throughout the outburst. Combining all the factors like extreme
+luminosity, high log ξ, and an overabundance of Fe (parameter AFe
+in Table 3) refer to a highly ionized disk-wind having a significant
+yield of Fe XXV, Fe XXVI etc.
+An important characteristic of this source is the presence of a
+broad, symmetric and dynamic absorption feature in the spectrum
+∼ 8−11 keV. We presented various possibilities regarding the origin
+of this absorption feature in §4.3. Finally, we concluded that the fast
+moving ionized disk-wind could absorb the primary X-ray photons
+and produce this broad absorption feature. The phenomenological
+modelling shows the presence of the neutral Fe Kα absorption (edge)
+at ∼ 7.1 − 7.4 keV, originating from the outer part of the accretion
+disk. The initial steep rise of the EW of the neutral component (red
+star in Fig. 9) indicates the enhancement of disk matter due to irra-
+diation of the outer disk. Due to the availability of more matter, the
+radiation pressure of the highly luminous inner part could release
+more ionized matter. The source enters to the HSS, and the disk-
+wind gradually becomes very active till Epoch 9 (possibly Epoch
+MNRAS 000, 1–14 (2022)
+
+12
+Prabhakar et al.
+10 also) where the gabs EW is maximum (marked A in Fig. 9) and
+the disk spectrum is the softest (Fig. 4b). Therefore, the evolution
+of the ionized EW (blue square in Fig. 9) followed the neutral com-
+ponent till day 60 (Epoch 9). After that, EW of the ionized compo-
+nent declines gradually (AB in Fig. 9) because the disk luminosity
+has reduced significantly, and a good fraction of inner disk matter
+has already been lost in the form of wind. The neutral component
+remains unaffected as it takes a viscous timescale to propagate the
+same to the outer disk. The NuSTAR data on Epoch 12 reveals a sud-
+den drop of the strength (orange points in Fig. 8a) and EW (marked
+‘C’ in Fig. 9) of gabs component. We also notice the same signature
+in the AstroSat data on Epoch 13, observed after 2 days of the NuS-
+TAR’s Epoch 12 observation; the gabs strength (Table 2) on Epoch
+13 is smaller by a few factor compared to the nearby observations.
+We identify this sudden drop of ionized EW can be due to the evacu-
+ation of the inner disk. The huge central luminosity may slow down
+the accretion onto the central object, and most of the accreted mat-
+ter is released through disk-wind. Once the disk luminosity reduces,
+there is a sudden infall of matter onto the BH, leading to an evacua-
+tion of the inner disk.
+If this interpretation is correct, we expect a relatively harder spec-
+trum due to a significant drop of soft photon flux during Epochs 12
+& 13. To characterise this, we calculate the observed flux in 0.5 − 7
+keV (soft) and 7−20 keV (hard) band for NuSTAR and AstroSat data.
+The AstroSat soft and hard fluxes on Epoch 10 are 7.45 × 10−8 erg
+cm−2 s−1 and 1.23 × 10−9 erg cm−2 s−1 respectively. The drop of As-
+troSat soft flux on the next observation (Epoch 13) is 14%, whereas
+the hard flux increases by 5%. This resulted in a relatively harder
+spectral index (marked by a blue circle in Fig. 4b) on Epoch 13. Sim-
+ilarly, the NuSTAR soft and hard fluxes on Epoch 11 are 6.48 × 10−8
+erg cm−2 s−1 and 1.1 × 10−9 erg cm−2 s−1 respectively. The drop of
+NuSTAR soft flux on Epoch 12 is 10%, whereas the hard flux in-
+creases by a factor of 2. Therefore, the suddenly enhanced accretion
+at the inner disk resulted in these dramatic changes in the EW and
+spectral properties. However, the inner accretion disk recovers over
+the next 10 days (marked CD) due to the transfer of matter from
+the outer disk, and the ionized component returns back to a gradual
+declination. Interestingly, the neutral component (or the outer disk)
+follows the same trend as the ionized component, namely the de-
+cline and refilling signature (red stars between BE), with a delay of
+the typical viscous timescale of 10 − 15 days.
+6 SUMMARY AND CONCLUSION
+We study the wideband spectral properties of the 2021 outburst of
+4U 1543 − 47. The MAXI/GSC lightcurve (Fig. 1) shows that the
+outburst rises over 9 days followed by a slow decay over ∼ 175 days.
+We use multi-instruments data (NICER, NuSTAR and AstroSat) for
+simultaneous broadband spectral study over a period of 100 days
+from MJD 59370. We have performed the spectral study using the
+phenomenological model M1 and reflection model M2. The major
+findings from our study are summarized below:
+• The source generally remains very bright during this outburst
+with a super Eddington peak luminosity on Epoch 1 (Table 2).
+• The source was in the HSS during our study, with a steep pho-
+ton index (Fig. 4b) due to a very small fraction (< 3%) of inverse-
+Comptonized photons and low corona temperature.
+• The reflection modelling reveals that the inclination of the sys-
+tem is between 32◦−40◦.
+• The extreme luminosity, high ionization (log ξ > 3) and over-
+abundance of iron (3.6−10 AFe,⊙) indicate the presence of disk-wind
+with a significant yield of highly ionized iron species.
+• Presence of a broad, dynamic absorption feature at ∼ 8 − 11
+keV is observed throughout our study. This detection is the first of
+its kind for X-ray binaries. We propose that this feature is due to
+the absorption of the accretion disk photons by the highly ionized,
+blue shifted disk-wind. The strength of the ionized absorption fea-
+ture (Table 2 & Table 3) increases between Epoch 1 to Epoch 9 as the
+disk-wind column density is expected to increase with the spectral
+softening of the source.
+• The observed line energy of the absorption feature suggests an
+estimated wind speed of nearly 30% of the speed of light to blue
+shift the most ionized line with the highest absorption yield like Fe
+XXVI. Hence it would become the first X-ray binary source to show
+a highly relativistic disk-wind.
+• The initial steep rise of the neutral component EW (red star
+in Fig. 9) is an indication of the enhancement of disk matter due
+to irradiation of the outer disk. It enhances the accretion rate and
+hence the source remains in the high luminosity state and decays
+very slowly.
+• The evolution of EW (Fig. 9) of the neutral absorption compo-
+nent (edge) and the same of the ionized component (gabs), follow
+each other with a delay of the typical viscous timescale of 10 − 15
+days.
+• An evacuation of the inner accretion disk is observed during
+Epoch 12 − 13. This event leaves a signature of the drop in the soft
+photon flux and an enhancement of hard flux. Therefore, the spec-
+trum becomes relatively harder (blue circle in Fig. 4b).
+Finally, this study suggests that accretion dynamics of 4U 1543 −
+47 during 2021 outburst is regulated by the disk-wind.
+ACKNOWLEDGEMENTS
+The authors wish to thank the anonymous reviewer for the insightful
+suggestions which significantly improved the quality of the publi-
+cation. This work uses data from the NICER and NuSTAR mission
+by the National Aeronautics and Space Administration. This work
+also has used data from the AstroSat mission of the ISRO archived
+at the Indian Space Science Data Centre (ISSDC). The work has
+been performed utilizing the calibration databases, and auxiliary
+analysis tools developed, maintained and distributed by AstroSat-
+SXT team with members from various institutions in India and
+abroad. The High Energy Astrophysics Science Archive Research
+Center (HEASARC), which provides the software and NASA’s As-
+trophysics Data System Bibliographic Services are also acknowl-
+edged. BGR acknowledges the financial support of ISRO under As-
+troSat archival data utilization program Sanction order No. DS-2B-
+13013(2)/13/2019-Sec.2. AN thanks GH, SAG, DD, PDMSA, and
+Director, URSC for the support to carry out this research.
+DATA AVAILABILITY
+The
+data
+from
+NICER
+and
+NuSTAR
+underly-
+ing
+this
+article
+are
+available
+in
+HEASARC,
+at
+https://heasarc.gsfc.nasa.gov/docs/archive.html.
+AstroSat
+data
+archive
+is
+available
+at
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