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+A sluggish random walk with subdiffusive spread
+Aniket Zodage1,2, Rosalind J. Allen3,4, Martin R. Evans4,5, Satya N. Majumdar5
+1 Department of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
+2 Department of Physics, UC San Diego, 9500 Gilman Dr. La Jolla, California 92093, USA
+3 Theoretical Microbial Ecology, Institute of Microbiology, Faculty of Biological Sciences,
+Friedrich Schiller University Jena, Buchaer Strasse 6, 07745 Jena, Germany
+4 SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, EH9 3FD and
+5 LPTMS, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, France
+(Dated: January 31, 2023)
+We study a one-dimensional sluggish random walk with space-dependent transition probabilities.
+Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases
+logarithmically with distance from the origin. This leads to a random walk which has symmetric
+transition probabilities that decrease with distance |k| from the origin as 1/|k| for large |k|. We show
+that the typical position after time t scales as t1/3 with a nontrivial scaling function for the position
+distribution which has a trough (a cusp singularity) at the origin. Therefore biased random motion
+emerges even though the transition probabilities are symmetric.
+We also compute the survival
+probability of the walker in the presence of a sink at the origin and show that it decays as t−1/3 at
+late times. Furthermore we compute the distribution of the maximum position, M(t), to the right
+of the origin up to time t, and show that it has a nontrivial scaling function. Finally we provide a
+generalisation of this model where the transition probabilities decay as 1/|k|α with α > 0.
+arXiv:2301.13077v1  [cond-mat.stat-mech]  30 Jan 2023
+
+2
+I.
+INTRODUCTION
+Slow dynamics is a common feature of many physical systems, including glasses, granular media and colloids [1, 2].
+well Slow dynamics commonly arises when the system becomes trapped for increasing periods of time in deeper and
+deeper local free energy minima in the configuration space. This phenomenon has inspired the study of simplified
+toy models known as trap models [3–7]. In these models, the many minima of the complex disordered landscape are
+represented by traps whose depths are taken to be random variables. For a single particle hopping between nearby
+traps the mean squared displacement typically grows more slowly than linearly in time, thus the particle’s motion is
+subdiffusive [8–10].
+Similar slow dynamics can also arise in an inhomogeneous landscape where the trap depth is position-dependent
+(as opposed to being random). Here, the hopping dynamics between the traps is an example of a Markov chain
+with space-dependent transition probabilities [11, 12], or in other words, an inhomogeneous random walk. For such
+systems, explicit solutions for observables beyond the simple position distribution – such as first passage probabilities
+[13–15] or extreme value statistics [16, 17] – are generally hard to obtain.
+A classic example of an inhomogenous random walk is the centrally-biased Gillis model [18–22]. In this model, a
+single particle hops on a one-dimensional lattice where the hopping probability is asymmetric in a position-dependent
+manner. Specifically, for k ̸= 0, the hopping probabilities from site k to k ± 1 are 1
+2(1 ∓ ϵ/k), while for k = 0 the
+hopping probabilities to sites ±1 are both 1/2. Because the hopping probabilities (for k ̸= 0) are asymmetric, the
+particle undergoes a biased random walk in which the parameter ϵ ∈ [−1, 1] controls the strength of the bias. For
+ϵ > 0 there is a drift towards the origin while for ϵ < 0 there is a drift away from the origin. Far away from the origin,
+where |k| ≫ 1, the bias is small and the dynamics tends towards a symmetric random walk which, in the continuum
+limit, reduces to a particle moving in a logarithmic potential U(k) → 2ϵ ln |k| [19].
+The Gillis model [18–22], and its continuous limit of particle motion in a logarithmic potential [19, 23–27], have
+aroused much interest because of their relevance to vortex dynamics, interactions between tracer particles in a driven
+fluid, cold atoms trapped in optical lattices and the nonequilibrium behaviour of systems with long-range interactions
+[28–32]. These models have the appealing feature of allowing the exact calculation of various observables going beyond
+the position distribution.
+Motivated by these works, here we consider the counterpart problem of an inhomogeneous trap model in which
+the trap depth increases logarithmically with increasing distance from the origin. The dynamics of a particle in this
+model corresponds to a symmetric random walk with space-dependent hopping probabilities that decrease inversely
+with distance from the origin. This random walk has the interesting property of being ‘sluggish’ since the particle’s
+motion slows down as it goes further away from the origin. We show that the physics of this model is quite different
+from the previously studied case of a particle in a logarithmic potential. Instead, in the continuum limit and for
+|k| ≫ 1, our model corresponds to a particle moving in a potential U(k) ∼ 1/|k| with, additionally, a space-dependent
+diffusion constant that also decays as 1/|k|. The interplay of these two features leads to an emergent bias in the
+dynamics away from the origin, even though the hopping probabilities are symmetric. The position distribution has
+a non-trivial and non- Gaussian form in which distance scales with time as t1/3 at late times. Moreover, we show
+that other observables such as the survival probability in the presence of an absorbing site and the distribution of the
+maximum displacement to one side of the origin can be computed explicitly and exhibit non-trivial scaling behaviour.
+Finally we discuss how the model can be easily generalised to higher dimensions and other space-dependent hopping
+probabilities, such as a 1/|k|α decay with exponent α > 0, without losing its solvability.
+II.
+MODEL DEFINITION AND HYDRODYNAMIC LIMIT
+As discussed, we consider an ordered array of traps arranged on a one-dimensional lattice such that the depth of
+the trap at site k is a ln(|k| + 2) (as illustrated in Fig. 1). The corresponding Arrhenius escape rate from the trap at
+site k is A(|k| + 2)−α with α = β a, where A is an overall constant and β is the inverse temperature. Without loss
+of generality we will set A = 1. We will mostly focus on the case where α = 1, although we briefly discuss α ̸= 1 in
+Section IX.
+We consider the discrete-time dynamics of a particle moving at random on this infinite one-dimensional lattice.
+The key feature of the dynamics is that, as time progresses, the particle explores sites further and further away from
+the origin in which it gets trapped to a greater and greater extent because of the increasing trap depths. Hence the
+particle is subject to diminishing transition probability for exiting the traps.
+At each integer time step t the particle’s position evolves according to the following rules (illustrated in Fig. 1).
+From a site k at time t, the particle hops to site k + 1 with probability 1/(|k| + 2), it hops to site k − 1 with equal
+probability 1/(|k|+2), or it stays at site k with the complementary probability |k|/(|k|+2). The time is then updated
+to t + 1. We note that (in contrast to the Gillis model [18–22]) the hopping probabilities are symmetric for all k;
+
+3
+FIG. 1. Schematic illustration of our model, showing the ordered arrangement of traps and the hopping probabilities 1/(|k|+2)
+for a particle to move to a nearest neighbour of site k on the lattice, as well as the probability |k|/(|k| + 2) of remaining at site
+k.
+however they differ from those of a simple random walk everywhere except at the origin, k = 0, where the hopping
+probability is 1/2 to either of k = ±1.
+Let P(k, t) denote the position distribution at time t for a particle that starts from k0 = 0 at t = 0. The distribution
+evolves via the forward master equation:
+P(k, t + 1) =
+1
+|k + 1| + 2 P(k + 1, t) +
+1
+|k − 1| + 2 P(k − 1, t) +
+|k|
+|k| + 2 P(k, t) .
+(1)
+The initial condition is P(k, 0) = δk,0 and the boundary condition is P(k, t) → 0 as |k| → ∞. The solution P(k, t) is
+symmetric around k = 0, hence we can just focus on k ≥ 0. We first write (1) in in the more suggestive form
+P(k, t + 1) − P(k, t) =
+1
+|k + 1| + 2 P(k + 1, t) +
+1
+|k − 1| + 2 P(k − 1, t) −
+2
+|k| + 2 P(k, t) .
+(2)
+For large k > 0 and large t, we can expand the right hand side (rhs) of Equation (2) as a Taylor series in k and
+replace the left hand side (lhs) by a time derivative. This gives, keeping all terms of the same order, the hydrodynamic
+equation that captures the behaviour of the system at long distance and at late time:
+∂
+∂tP(k, t) ≈ 1
+k
+� ∂2
+∂k2 P(k, t) − 2
+k
+∂
+∂k P(k, t) + 2
+k2 P(k, t)
+�
+.
+(3)
+This hydrodynamic equation can be written as a continuity equation:
+∂
+∂tP(k, t) = − ∂
+∂k j(k, t) ,
+(4)
+where the space-time dependent current density j(k, t) reads
+j(k, t) = −1
+k
+∂
+∂k P(k, t) + 1
+k2 P(k, t) .
+(5)
+We identify the first term on the rhs of (5) as a diffusive probability current and the second term as a drift away from
+the origin. The original equation (3) can also be expressed in the standard Fokker-Planck form
+∂
+∂tP(k, t) = ∂
+∂k
+�
+D(k) ∂
+∂k P(k, t) +
+� ∂
+∂k U(k)
+�
+P(k, t)
+�
+,
+(6)
+where D(k) = 1/k and U(k) = 1/k. Equation (6) shows clearly the two features of the dynamics that lead to non-
+trivial behaviour. Firstly, the effective diffusion constant D(k) = 1/k is space dependent, such that it slows down
+
+4
+the dynamics as the particle moves away from the origin. Additionally, an effective external potential U(k) = 1/k
+emerges, which is repulsive, such that it pushes the particle away from the origin. This repulsion arises from the
+microscopic dynamics: the hopping probability from site k to k + 1 is 1/(k + 2), while the reverse event (from (k + 1)
+to k) has probability 1/(k + 3) < 1/(k + 2) for any k > 0. Thus, even though the hopping probabilities out of site
+k (i.e. to (k + 1) or (k − 1)) are symmetric, the space-dependence of the hopping probability produces an outward
+bias away from the origin (symmetrically for k < 0) which leads to the drift term in the current in equation (5) in
+the hydrodynamic description.
+For large time and space we expect to see a scaling regime in which the probability distribution becomes a function
+of a combination k/tν (with ν > 0). The following argument suggests that ν takes the value 1/3. Let us assume that
+after time t, the typical value of the position k is ktyp. The number of steps N that have been taken by the particle
+will scale as N ∼ t/ktyp, since the time for one step is the typical escape time 1/ktyp. Since all steps are equal in
+distance and the hopping probability is symmetric, the position scales with the number of steps in the same way as
+for a simple random walk, ktyp ∼ N 1/2. Putting these arguments together we obtain ktyp ∼ N 1/2 ∼ (t/ktyp)1/2 which
+implies that the position of the particle scales with time as ktyp ∼ t1/3; hence ν = 1/3.
+This scaling can be confirmed by assuming the following scaling form for the probability distribution P(k, t) in the
+limit when both k and t are large, keeping the ratio k/tν (with ν > 0) fixed:
+P(k, t) →
+1
+b tν G
+� k
+b tν
+�
+,
+(7)
+where G(z) is the scaling function. We have also incorporated an adjustable constant b which can be chosen appro-
+priately. Substituting the scaling form (7) in equation (1), one readily finds that for leading order terms to be of the
+same order we must have ν = 1
+3. For convenience we will also choose b = 31/3. We will discuss the precise form of
+the scaling function G(z) in Section IV.
+The scaling k ∼ t1/3 also manifests itself in other observables, such as the survival probability and the distribution of
+the maximum position of the random walk that we study in this paper. In the next section, for clarity, we summarize
+our main results. Then, in the following sections, we discuss each result in detail.
+III.
+SUMMARY OF KEY RESULTS
+In this paper we derive exact results in the scaling limit for three observables: the position distribution, the survival
+probability and the distribution of the maximum of the random walk. For clarity, we state these results here; their
+derivations will be presented in the following sections.
+Position distribution: in the large t and large k limit, such that z = k/(3t)1/3 is fixed, the position distribution of
+the walker P(k, t) is given by
+P(k, t) →
+1
+(3t)1/3 G
+�
+k
+(3t)1/3
+�
+(8)
+where the scaling function G(z) is given by
+G(z) =
+31/3
+2 Γ(2/3) |z| e−|z|3/3 .
+(9)
+This function has a trough at z = 0 and the function is bimodal with peaks at z = ±1 (see Fig. 2).
+Survival probability: For a walker that starts from k0 > 0, the probability that the trap at k = 0 has not been
+visited by time t is equivalent to the survival probability Q(k0, t) in the presence of an absorbing site at the
+origin k = 0. This is given in the scaling limit by
+Q(k0, t) ≈ f
+�
+k0
+(3t)1/3
+�
+,
+where
+f(z) = 1 −
+1
+Γ(1/3) Γ(1/3, z3/3) ,
+(10)
+(see Fig. 3). This implies that in the long time limit the survival probability decays as t−1/3 (see equation (26)).
+We also compute the joint distribution of position and survival (equations (31) and (34)).
+
+5
+Distribution of maximum: The distribution of M, the furthest site to the right visited by the walker up to time
+t, or equivalently the deepest trap visited up to time t, is given in the scaling limit by
+P(M = L, t) →
+1
+(3t)1/3 g
+�
+L
+(3t)1/3
+�
+(11)
+where y = L/t1/3 is now the scaling variable. The scaling function g(y) (see Fig. 5) is described in equations
+(56),(58) and (59).
+IV.
+SCALING FORM OF THE POSITION DISTRIBUTION
+−3
+−2
+−1
+0
+1
+2
+3
+k/(3t)
+1
+3
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+(3t)
+1
+3P(k)
+Position Distribution
+Scaling Function G(z)
+FIG. 2. The scaling function G(z) plotted as a function of z. Symbols are obtained from Monte Carlo simulation data for the
+random walk. Starting from k0 = 0 at t = 0, the random walk is numerically evolved up to t = 20000. The symbols show the
+scaled histogram of final positions obtained from n = 150000 runs of the random walk simulation.
+We now derive equations (8) and (9) for the position distribution P(k, t) of the random walk. As discussed earlier,
+the form of equation (3) implies that the correct scaling variable involving k and t is z = k/(bt1/3), and we choose the
+arbitrary constant as b = 31/3 for later convenience. Therefore, to solve the hydrodynamic equation (3), we assume a
+scaling solution at late times and large k of the form
+P(k, t) =
+1
+(3t)1/3 G
+�
+k
+(3t)1/3
+�
+,
+(12)
+where G(z) is symmetric around z = 0, and is normalized to 1, i.e.,
+� ∞
+−∞ G(z) dz = 1, or equivalently
+� ∞
+0
+G(z) dz = 1
+2 .
+(13)
+
+6
+Substituting the scaling ansatz (12) in equation (1) and taking the scaling limit k → ∞, t → ∞ keeping z = k/(3t)1/3
+fixed, we find that G(z), for z > 0, satisfies a second order ordinary differential equation
+G′′(z) +
+�
+z2 − 2
+z
+�
+G′(z) +
+�
+z + 2
+z2
+�
+G(z) = 0 .
+(14)
+Remarkably, the general solution of this differential equation can be expressed in a simple closed form
+G(z) = c1 z e−z3/3 + c2 z e−z3/3
+� z
+0
+eu3/3 du ,
+(15)
+where c1 and c2 are arbitrary. However, the second solution (the second term in (15)) behaves, for large z, as 1/z,
+and hence is not normalisable, implying that we must have c2 = 0. The constant c1 can be fixed via the normalization
+constant
+� ∞
+0
+G(z)dz = 1/2. Using the symmetry around z = 0, the full solution for the scaling distribution (12) is
+then given by
+G(z) =
+31/3
+2 Γ(2/3) |z| e−|z|3/3 .
+(16)
+This function is plotted in Fig.
+(2) where we also plot results of Monte Carlo simulations that approach the
+scaling curve. Strikingly, in contrast to a simple random walk (where the scaling variable is z = k/(2t)1/2 and the
+corresponding scaling function is Gaussian with a peak at z = 0), G(z) has a trough at z = 0 where the solution has a
+cusp singularity. The origin of this trough can be traced back to the drift term (away from the origin) in the current
+in equation (5), that leads to a depletion of probability density near the origin at long times. Thus by creating an
+emergent current away from the origin, the sluggish dynamics that is manifested in our model keeps the particle away
+from the origin and produces two peaks (i.e. bimodality) in the probability distribution; these peaks are located at
+z = ±1 or equivalently |k| = (3t)1/3. The distribution of the depth of the trap occupied at time t also follows from
+equation (8) since the trap depth is a ln(|k| + 2).
+V.
+SURVIVAL PROBABILITY
+We now introduce a sink at the origin, such that if the random walker arrives at k = 0, it dies. We consider the
+survival probability of the walker in the presence of this absorbing site at k = 0. The ‘survival probability’ Q(k0, t)
+denotes the probability that the walker is still alive after t steps, given that it starts at site k0 at time zero. Clearly,
+Q(k0, t) is symmetric in k0, so we will consider only k0 ≥ 0, implying the walk that is defined on the positive integers.
+It is convenient to use the backward master equation for the survival probability:
+Q(k0, t + 1) =
+1
+k0 + 2 Q(k0 + 1, t) +
+1
+k0 + 2 Q(k0 − 1, t) +
+�
+1 −
+2
+k0 + 2
+�
+Q(k0, t) ,
+(17)
+for k0 ≥ 1. This equation has a simple interpretation, corresponding to the events that may occur in the first step
+of the walk. In the first step, the walker either hops from site k0 (rightwards to k0 + 1, or leftwards to k0 − 1), or it
+stays at k0. Then, starting from its position at time step 1, it has to survive a further t steps. Summing these three
+possibilities for the first step leads to equation (17), which needs to be solved for k0 ≥ 1 with the boundary conditions
+Q(k0 = 0, t) = 0
+(18)
+Q(k0 → ∞, t) = 1 .
+(19)
+The first boundary condition corresponds to the fact that if the walker starts at the absorbing site k0 = 0 it dies
+immediately. The second condition follows from the fact that if the walker starts far away from the origin, it survives
+with probability 1 as long as t is finite. In the limit of continuous time t and space k the backward equation becomes
+∂
+∂tQ(k0, t) = 1
+k0
+∂2
+∂k2
+0
+Q(k0, t) .
+(20)
+It is convenient to use a scaling approach to quickly derive the large t asymptotic behaviour of the survival prob-
+ability. We aim to solve equation (20) in the scaling limit introduced in section IV when both k0 and t are large.
+Following the discussion in section IV we expect that Q(k0, t) will satisfy a scaling form
+Q(k0, t) → f
+�
+k0
+(3t)1/3
+�
+,
+(21)
+
+7
+where f(z) is the scaling function. We now substitute the scaling form (21) in equation (20) and expand to leading
+order to obtain the following second order ordinary differential equation in z ≥ 0 for the scaling function
+f ′′(z) = −z2 f ′(z) ,
+(22)
+subject to the two boundary conditions
+f(z = 0) = 0
+and
+f(z → ∞) = 1 ,
+(23)
+which follow from Eqs. (18) and (19) respectively.
+0
+2
+4
+6
+8
+10
+12
+14
+k0/(3t)
+1
+3
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Q(k0, t)
+k0 = 25, t ∈ [1, 30000]
+k0 = 50, t ∈ [1, 30000]
+k0 = 100, t ∈ [1, 30000]
+k0 = 400, t ∈ [1, 30000]
+Scaling Function f(z)
+FIG. 3. Full curve: the scaling function f(z) plotted as a function of scaling variable z. Symbols are obtained from Monte
+Carlo simulation data for different values of k0. For a given k0, the random walk is numerically evolved over the time window
+shown in the legend. The symbols show histograms obtained from n = 10000 runs of the random walk simulation. These
+histograms are plotted against the scaling variable z = k0/(3t)1/3 . The different intervals of z for different values of k0 are
+chosen for purposes of clarity.
+The solution of equation (22) can be found trivially. Integrating (22) once gives f ′(z) = C e−z3/3. Integrating once
+more, using the boundary conditions (23), leads to the exact solution for the scaling function:
+f(z) =
+� z
+0 e−x3/3 dx
+� ∞
+0
+e−x3/3 dx = 1 −
+1
+Γ(1/3) Γ(1/3, z3/3) ,
+(24)
+where Γ(s, x) =
+� ∞
+x e−t ts−1 dt is the incomplete Gamma function. The scaling function f(z) is plotted in Fig. (3).
+It is linear for small z (t1/3 ≫ k0) and saturates at f = 1 for large z (t1/3 ≪ k0). More precisely, the scaling function
+has the asymptotic behaviours
+f(z) ≈
+�
+�
+�
+�
+�
+32/3
+Γ(1/3) z + O(z4)
+as z → 0
+1 −
+32/3
+Γ(1/3) z2 e−z3/3
+as z → ∞ .
+(25)
+
+8
+In particular, for z → 0
+Q(k0, t) ≃
+31/3
+Γ(1/3)
+k0
+t1/3 .
+(26)
+Equation (26) implies that in the limit t → ∞ the asymptotic behaviour of the survival probability is Q ∼ t−1/3. The
+exponent 1/3 is smaller than the value 1/2 that is obtained for a simple diffusive process, implying that the decay
+is slower than for simple diffusion. Again, the sluggish dynamics results in a significant difference in the dynamical
+properties, compared to those of a simple random walk.
+VI.
+JOINT SURVIVAL AND POSITION DISTRIBUTION
+Next we consider the probability Ps(k, t|k0) that a walker, starting at k0 > 0 at time t = 0, arrives at k at time t,
+having in the meantime avoided the sink at k = 0. This is the joint distribution of survival and position, with the
+subscript s in Ps(k, t|k0) denoting survival.
+For this calculation we use the forward master equation, for k > 0:
+Ps(k, t + 1|k0) =
+1
+k + 3 Ps(k + 1, t|k0) +
+1
+k + 1 Ps(k − 1, t|k0) +
+k
+k + 2 Ps(k, t|k0) ,
+(27)
+with the boundary condition Ps(0, t|k0) = 0 and the initial condition, Ps(k, t = 0|k0) = δk,k0. When summed over
+k = 1, 2, · · · , one should recover the survival probability of section V, namely
+∞
+�
+k=1
+Ps(k, t|k0) = Q(k0, t) .
+(28)
+-4
+-2
+2
+4 z
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+H(z)
+FIG. 4. The scaling function H(z), given by equation (34), plotted as a function of z.
+For simplicity, we will again work in the scaling limit where t → ∞, k → ∞ and k0 → ∞, keeping z = k/(3t)1/3
+and y = k0/(3t)1/3 fixed. We expect a scaling form
+Ps(k, t|k0) ≈
+1
+(3t)1/3 W
+�
+k
+(3t)1/3 ,
+k0
+(3t)1/3
+�
+,
+(29)
+such that when integrated over k, we recover the scaling of the survival probability survival probability Q(k0, t) in
+equation (24) with
+� ∞
+0
+W(z, y) dz = f(y) ,
+(30)
+
+9
+where f(y) is given in equation (24). Here we assume k0 ∼ O(1), so that the second argument of the scaling function
+W in equation (29) approaches zero. From the small argument behaviour of the survival probability in (26), we expect
+that W(z, y → 0) → y H(z). This leads us to the scaling ansatz, valid for any k0 ∼ O(1):
+Ps(k, t|k0) ≈
+k0
+(3t)2/3 H
+�
+k
+(3t)1/3
+�
+.
+(31)
+Substituting this scaling ansatz in equation (27), we get, to leading order in 1/t, the following ordinary differential
+equation for H(z), for any z ≥ 0 (for z < 0, this function is symmetric, hence we consider only z ≥ 0):
+H′′(z) +
+�
+z2 − 2
+z
+�
+H′(z) +
+�
+2z + 2
+z2
+�
+H(z) = 0 .
+(32)
+The scaling function H(z) should satisfy the absorbing boundary condition H(0) = 0. One more condition can be
+derived by substituting the scaling ansatz (31) in equation (28), and taking the limit y = k0/(3t)1/3 → 0. Using the
+small y behaviour of f(y) in equation (25), we obtain the following condition:
+� ∞
+0
+H(z) dz =
+32/3
+Γ(1/3) .
+(33)
+One can easily check that the normalised solution of (32) is simply
+H(z) =
+32/3
+Γ(1/3) z2e−z3/3 .
+(34)
+H(z) is plotted in Fig. (4). We note that the trough around z = 0 is quadratic in z for this calculation in the presence
+of a sink, in contrast to the linear |z| dependence for the trough in the position distribution for the calculation without
+a sink (equation (16)). The quadratic behaviour of H(z) near the origin also contrasts with the analogous result for
+the simple random walk case where linear behaviour is obtained as z → 0. This limit z → 0 gives information on the
+long time behaviour; from (31),(34) we obtain
+Ps(k, t|k0) ≃
+k0k2
+32/3Γ(1/3)
+1
+t4/3 .
+(35)
+The t−4/3 long-time behaviour of the survival probability in equation (35) contrasts with the corresponding t−3/2
+behaviour for a simple random walk.
+VII.
+DISTRIBUTION OF THE MAXIMUM OF THE RANDOM WALK
+We now remove the sink at the origin and instead consider a walker that starts at the origin (k0 = 0) and moves
+freely. We study the statistics of its maximum displacement M(t) on the positive side up to time t. This corresponds
+to the deepest trap visited to the right of the origin up to time t. Then the cumulative distribution Prob. [M(t) ≤ L]
+is just the probability that the walker, starting at the origin, does not visit the site L up to time t. Let S(k0, t) denote
+the probability that starting from k0 at t = 0, the walker does not visit L up to t. We then have
+Prob. [M(t) ≤ L] = S(0, t) .
+(36)
+To compute S(0, t), we will first solve S(k0, t) for a general starting point k0 and then set k0 = 0. The survival
+probability S(k0, t) again evolves according to the backward master equation
+S(k0, t + 1) =
+1
+|k0| + 2 S(k0 + 1, t) +
+1
+|k0| + 2 S(k0 − 1, t) +
+�
+1 −
+2
+|k0| + 2
+�
+S(k0, t) ,
+(37)
+with boundary condition
+S(L, t) = 0 ,
+(38)
+i.e. we impose a sink at site k = L. The initial condition (starting from k0 < L) is
+S(k0, 0) = 1 .
+(39)
+
+10
+Following the approach of section V, we expand in k0 to obtain the backward Fokker Planck equation:
+∂
+∂tS(k0, t) =
+1
+|k0|
+∂2
+∂k2
+0
+S(k0, t) ,
+(40)
+which is valid for k0 ≤ L, with an absorbing boundary condition S(k0 = L, t) = 0 at the sink k = L and the initial
+condition S(k0, 0) = 1 for all k0 < L.
+To solve equation (40), it is convenient to consider the Laplace transform
+�S(k0, s) =
+� ∞
+0
+S(k0, t) e−s t dt .
+(41)
+This satisfies
+∂2
+∂k2
+0
+�S(k0, s) = s|k0|�S(k0, s) − |k0| ,
+(42)
+where we used the initial condition S(k0, 0) = 1. Due to the presence of the absolute value k0 in the differential
+equation (42), we need to solve for 0 ≤ k0 ≤ L and k0 ≤ 0 separately, and then match the solution and its first
+derivative at k0 = 0.
+The general solution of (42) for 0 ≤ k0 ≤ L and k0 ≤ 0 reads
+�S(k0, s) = 1
+s + a1 Ai(s1/3k0) + b1 Bi(s1/3k0)
+for
+0 ≤ k0 ≤ L
+(43)
+�S(k0, s) = 1
+s + a2 Ai(−s1/3k0)
+for
+k0 ≤ 0 ,
+(44)
+where Ai(x) and Bi(x) are the two linearly independent solutions of the Airy differential equation U ′′(x)−xU(x) = 0.
+Since Bi(−x) diverges as x → −∞, we discarded this in the solution for k0 ≤ 0 in equation (44). The three constants
+(independent of k0) a1, a2, b1 are fixed by the continuity of �S(k0, s), the continuity of ∂k0 �S(k0, s) at k0 = 0 and
+the absorbing boundary condition �S(k0 = L, s) = 0, which yield three linear equations. These three constants can
+then be straightforwardly determined explicitly (we do not give the details here). If the walker starts at k0 = 0 (for
+simplicity), from equation (44), we just need the constant a2(s) since
+�S(0, s) = 1
+s + a2(s) Ai(0) .
+(45)
+It turns out that the expression of a2(s) is rather simple:
+a2(s) =
+1
+2π Ai(0) Ai′(0) s Bi(s1/3 L) = −
+√
+3
+s Bi(s1/3 L) ,
+(46)
+where we used Ai(0) = 3−2/3/Γ(2/3) and Ai′(0) = −3−1/3/Γ(1/3). Plugging in equation (45) then gives the exact
+Laplace transform, valid for all s:
+�S(0, s) = 1
+s
+�
+1 −
+1
+31/6 Γ(2/3)
+1
+Bi(s1/3 L)
+�
+.
+(47)
+Taking the Laplace transform of equation (36), and plugging in the result (47), we obtain the exact Laplace
+transform of the cumulative distribution of the maximum:
+� ∞
+0
+Prob. [M(t) ≤ L] e−s t dt = 1
+s
+�
+1 −
+1
+31/6 Γ(2/3)
+1
+Bi(s1/3 L)
+�
+.
+(48)
+This result can be further simplified by noting that Prob. [M(t) ≥ L] = 1 − Prob. [M(t) ≤ L]. Consequently,
+� ∞
+0
+Prob. [M(t) ≥ L] e−s t dt =
+1
+31/6 Γ(2/3)
+1
+s Bi(s1/3 L) .
+(49)
+Formally inverting this Laplace transform using the Bromwich contour and rescaling sL1/3 = λ, one sees immediately
+that for all t and L, the cumulative distribution takes the scaling form
+Prob. [M(t) ≥ L] = Y
+�
+t
+L1/3
+�
+,
+(50)
+
+11
+10−1
+100
+m/(3t)
+1
+3
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+(3t)
+1
+3P(m)
+Distribution of Maximum
+Higher End Tail of g(z)
+Lower End Tail of g(z)
+FIG. 5. Distribution of the maximum of the random walk. The two full curves denote the lower end tail of g(z), equation (59),
+and higher end tail of g(z), equation (58). The symbols are obtained from Monte Carlo simulation data for the random walk.
+Starting from k0 = 0 at t = 0, the random walk was evolved up to t = 20000. The symbols show the scaled histogram obtained
+from n = 105000 runs of the random walk simulation.
+where the scaling function Y (y) has the exact Laplace transform
+� ∞
+0
+e−λy Y (y) dy = 31/3Γ(1/3)
+2π
+1
+λBi(λ1/3) .
+(51)
+While it is difficult to invert the Laplace transform exactly, it is straightforward to extract its asymptotic behaviours,
+as shown below.
+The large y behaviour of Y (y) is controlled by the small λ expansion of (51)
+� ∞
+0
+e−λyY (y) dy ≃ 1
+λ − 31/3Γ(2/3)
+Γ(1/3)
+1
+λ2/3 + 32/3Γ2(2/3)
+Γ2(1/3)
+1
+λ1/3 + . . . ,
+(52)
+which yields the large y asymptotic expansion
+Y (y) ∼ 1 −
+31/3
+Γ(1/3)
+1
+y1/3 + 32/3Γ2(2/3)
+Γ3(1/3)
+1
+y2/3 . . . .
+(53)
+The small y behaviour of Y (y) can be obtained from the large λ asymptotic behaviour of (51)
+� ∞
+0
+e−λyY (y) dy ∼
+π1/2
+31/6Γ(2/3)λ11/12 e−2/3 λ1/2 ,
+(54)
+which can be inverted to give the small y behaviour
+Y (y) ≃
+32/3
+Γ(2/3)y1/3e−1/(9y) .
+(55)
+
+12
+Using equation (50), we can now express the probability density Prob. [M(t) = L] of the maximum of the random
+walk in a scaling form:
+Prob. [M(t) = L] = − d
+dLProb. [M(t) ≥ L] =
+1
+(3t)1/3 g
+�
+L
+(3t)1/3
+�
+,
+(56)
+where the scaling function g(z) is simply related to the scaling function Y (y) and we deduce that
+g(z) = z−4 Y ′(y)|y=1/(3z3) .
+(57)
+Using the asymptotic behavior of Y (y), we can then obtain the asymptotic tails of g(z) as
+g(z) ∼
+31/3
+Γ(2/3) z e−z3/3
+for
+z → ∞
+(58)
+g(z) ∼
+32/3
+Γ(1/3)
+�
+1 − 2.32/3Γ2(2/3)
+Γ2(1/3)
+z . . .
+�
+for
+z ≪ 1 .
+(59)
+In Fig. (5) the tails of g(z), equations (58) and (59), are compared with the numerical simulations. It is useful to
+compare equation (58) with equation (16). We see that for large z, the scaling function of the position distribution
+(16) and that of the maximum (58) have the same asymptotic tails up to an overall factor 1/2. This is similar to
+what occurs for a simple random walk, although in that case the tails are Gaussian. The small z behaviour (59) for
+the scaling function is a constant with a linear correction. The constant is consistent with the large time limit of the
+survival probability (26). The linear correction contrasts with the case of a simple random walk where the correction
+to the constant term is quadratic in the scaling variable.
+VIII.
+GENERATING FUNCTION APPROACH
+In sections IV - VII, we adopted a scaling approach to obtain long-time asymptotic results for the sluggish random
+walk problem. We now illustrate how a generating function approach may be employed to find the exact solution
+for all times. We will see that the long time limit of the solution obtained using the generating function approach
+recovers the results of the scaling approach. For the sake of brevity, we restrict ourselves to the computation of the
+survival probability.
+Consider again Q(k0, t), the survival probability for a walker starting at k0 in the presence of a sink at the origin
+k = 0. Q(k0, t) satisfies the backward master equation (17). We define a generating function with parameter λ:
+G(k0) =
+∞
+�
+t=0
+λtQ(k0, t) .
+(60)
+Substituting (60) into (17) and imposing the initial condition Q(k0, 0) = 1, we obtain
+�
+k0
+1 − λ
+λ
++ 2
+λ
+�
+G(k0) − G(k0 + 1) − G(k0 − 1) = k0 + 2
+λ
+.
+(61)
+Equation (61), in which k0 takes integer values, can be solved using Bessel functions. However it is easier to take a
+continuum limit and expand to second order in k0, to obtain
+∂2G(k0)
+∂k2
+0
+−
+�(k0 + 2)(1 − λ)
+λ
+�
+G(k0) = −(k0 + 2)
+λ
+.
+(62)
+The homogeneous version of (62) (i.e. equating the lhs to zero) has Airy functions as solutions:
+Ghom(k0) = B0Ai(C(k0 + 2)) + B1Bi(C(k0 + 2)) ,
+(63)
+where
+C =
+�1 − λ
+λ
+�1/3
+,
+(64)
+
+13
+and the constants B0 and B1 are to be fixed by the boundary conditions. We can set B1 = 0 and discard the Bi
+solution, as it diverges as k0 → ∞.
+Then for G(k0) a particular solution to (62) is 1/(1 − λ) and the general solution to (62) is
+G(k0) = B0Ai((k0 + 2)C) +
+1
+1 − λ .
+(65)
+The boundary condition is G(0) = 0, which fixes the constant B0, and we obtain the solution to (62) as
+G(k0) =
+1
+1 − λ
+�
+1 − Ai((k0 + 2)C)
+Ai(2C)
+�
+.
+(66)
+We are interested in the long time asymptotic behaviour, which we can extract from the λ → 1 limit of (66). Since
+C → 0 as λ → 1, we require the small argument expansion of the Airy function:
+Ai(x) ≃
+1
+32/3Γ(2/3) −
+x
+31/3Γ(1/3) .
+(67)
+We then find, in the limit λ → 1,
+G(k0) ≃ 31/3 Γ(2/3)
+Γ(1/3)
+k0
+(1 − λ)2/3 .
+(68)
+Thus the leading singularity is at λ∗ = 1 and is of the form (λ∗ − λ)−2/3. Invoking the usual Tauberian theorem [33],
+this singularity gives the following large t asymptotic behaviour:
+Qt(k0) ∼
+31/3
+Γ(1/3)k0t−1/3 .
+(69)
+This matches perfectly with the small z asymptotic of the scaling behaviour in (24) upon using the small z expansion
+of f(z) in equation (25).
+IX.
+GENERALISATION TO THE CASE WHERE α ̸= 1
+Up to now, we have considered only the case where the probability of hopping to the right or left is proportional
+to 1/(|k| + 2), i.e. the exponent α = 1 in the general expression for the hopping probability, A(|k| + 2)−α. We now
+generalise to the case where α ̸= 1, i.e. the hopping probability is proportional to 1/(|k| + 2)α. In this case equation
+(3) generalises for k ≥ 0 to
+∂
+∂tP(k, t) ≈ 1
+kα
+� ∂2
+∂k2 P(k, t) − 2α
+k
+∂
+∂k P(k, t) + α(α + 1)
+k2
+P(k, t)
+�
+.
+(70)
+Equation (70) can be put into the standard form (6), where now D(k) = 1/kα and U(k) = 1/kα. One can again
+solve (70) by the scaling approach discussed earlier. For general positive α it is easy to show that the scaling variable
+becomes k/tν where ν = (2 + α)−1. Therefore the solution of (70) for P(k, t) has a scaling form
+P(k, t) = t−
+1
+α+2 G
+�
+k t−
+1
+α+2
+�
+,
+(71)
+where the scaling function G(z) is symmetric and, for positive z, satisfies the nontrivial differential equation
+G′′(z) +
+� zα+1
+α + 2 − 2α
+z
+�
+G′(z) +
+� zα
+α + 2 + α(α + 1)
+z2
+�
+G(z) = 0 ,
+(72)
+with boundary condition G(z) → 0 as z → ∞. Remarkably, this equation admits the simple solution, satisfying the
+boundary condition,
+G(z) = A zα exp
+�
+−
+zα+2
+(α + 2)2
+�
+,
+(73)
+
+14
+where the normalisation constant A is given by
+A−1 = 2(α + 2)
+α
+α+2 Γ
+�α + 1
+α + 2
+�
+.
+(74)
+Using the symmetry G(z) = G(−z), the full solution for all z can be written as
+G(z) = A |z|α exp
+�
+− |z|α+2
+(α + 2)2
+�
+.
+(75)
+When α = 0 we recover the standard Gaussian result for a simple random walk, while for α = 1 we recover the result
+(16) upon rescaling z → 31/3z. We note that for any α > 0 there is a trough, i.e. a cusp singularity, at z = 0. The
+trough at z = 0 disappears only for the case of simple diffusion (α = 0).
+Similar scaling analyses can be performed for the survival probability as well as the distribution of the maximum
+site visited to the right. We do not repeat the analysis, but just note that the scaling implies that the asymptotic
+decay of the survival probability is Q(t) ∼ t−1/(α+2) and the maximum scales as M(t) ∼ t1/(α+2) .
+X.
+CONCLUSION
+In this paper we have studied a random walk with space-dependent transition probabilities. Our study was motivated
+by trap models of slow dynamics, but in contrast to most such models, our trap depths are not random but instead
+increase logarithmically with distance k from the origin. The dynamics of a particle moving on the lattice of traps
+follows an inhomogeneous random walk which has symmetric transition probabilities that decrease with k as 1/k. Thus
+the motion of a walker slows down as it goes further and further away from the origin, a phenomenon that we term
+‘sluggish dynamics’. The sluggish dynamics causes the typical distance explored up to time t to grow subdiffusively
+as t1/3, in contrast to the standard t1/2 law for a simple random walk.
+We used a scaling approach, in which the scaling variable is k/t1/3, to compute long-time asymptotic results for
+various properties of this inhomogeneous random walk: the position distribution, the survival probability in the
+presence of a sink at the origin, the joint survival and position distribution, and the distribution of the maximum
+distance to the right. Interestingly, the position distribution has a trough (a cusp singularity) at the origin and is
+bimodal, with two peaks located at |k| = (3t)1/3. The contrasts with the usual Gaussian distribution for diffusion
+(which has a single maximum at k = 0). The bimodal distribution and the t1/3 scaling reflect the sluggish nature of
+the dynamics. The survival probability shows an asymptotic decay ∼ t−1/3 at large time, which contrasts with the
+t−1/2 decay for a simple random walk. The fact that the survival probability decays to zero as t → ∞ implies that
+the walk is recurrent in d = 1, as is the simple random walk. The distribution of the maximum of the walk up to time
+t has a nontrivial scaling function.
+We further showed how a generating function approach can be used to find exact solutions for all times. Using
+this approach to compute the survival probability in the presence of a sink at the origin, we recover our scaling
+result in the long-time limit. Application of the same generating function approach to other observables should be a
+straightforward extension.
+Finally, we generalised the model to cases where the transition probability decays as 1/|k|α with positive α. Except
+for α = 0 (simple random walk), the position distribution always shows a trough at the origin (k = 0), where it
+exhibits a singularity, behaving as |k|α. Remarkably, the scaling function for the position distribution takes on a
+simple form (equation (75)) and there is always a trough at the origin with associated singularity |z|α for α > 0.
+It is worthwhile comparing the behaviour of our sluggish random walk model with that of the Gillis model outlined
+in the introduction. In the continuum limit the Gillis model becomes diffusion in a logarithmic potential [19, 24] and
+the corresponding Fokker-Planck equation reads
+∂
+∂tP(k, t) = ∂
+∂k
+� ∂
+∂k P(k, t) +
+� ∂
+∂k U(k)
+�
+P(k, t)
+�
+,
+(76)
+where the potential U(k) = 2ϵ ln |k|. The relevant case for us is ϵ < 0 whereby the potential is repulsive and the
+particle is pushed away from the origin. In this Gillis case, the solution for the time-dependent position distribution
+has scaling form [19, 24]
+P(k, t) →
+1
+t1/2 GGill
+� k
+t1/2
+�
+(77)
+
+15
+where the scaling function, GGill(z), is given by
+GGill(z) =
+2ϵ−1/2
+Γ(1/2 − ϵ) |z|−2ϵ e−|z|2/2 .
+(78)
+This is to be compared with the scaling function G(z) (16) for the sluggish random walk model (where the scaling
+variable is z = k/(3t)1/3). As with (16), the scaling function (78) is bimodal, with peaks at z = ±(−2ϵ)1/2, and has a
+trough at the origin. However, the model exhibits diffusive scaling and is thus not sluggish. The difference between
+the sluggish random walk and diffusion in a logarithmic potential is evident when one compares the Fokker Planck
+equations (6) and (76). The key difference is the space-dependent diffusion constant D(k) = 1/k appearing in (76),
+along with the potential U(k) = 1/k . It is these features that lead to a change of the scaling variable to z = k/(3t)1/3
+and consequent sluggish behaviour.
+The sluggish random walk model and its analysis are straightforward to generalise to higher dimensions and other
+observables. For example, it would interesting to study the return probabilities and recurrence/transience transition
+in a higher dimension for general α. It would also be of interest to study the time for the walker to traverse from
+one maximum of the position distribution to the other. More generally our study has shown that inhomogenous
+space-dependent random walks can exhibit surprising properties and it remains to explore the full range of such
+behaviour.
+AZ acknowledges support of the INSPIRE fellowship from DST India and the Physics Computing Facility lab at
+UCSD. RJA was supported by the European Research Council under consolidator grant 682237 EVOSTRUC and
+by the Excellence Cluster Balance of the Microverse (EXC 2051 - Project-ID 390713860) funded by the Deutsche
+Forschungsgemeinschaft (DFG). For the purpose of open access, the author has applied a Creative Commons Attribu-
+tion (CC BY) licence to any Author Accepted Manuscript version arising from this submission. MRE thanks LPTMS
+for the award of a CNRS Visiting Professorship, during which this work was written up.
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+

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+Building Coverage Estimation with Low-resolution Remote Sensing Imagery
+Enci Liu, Chenlin Meng, Matthew Kolodner, Eun Jee Sung, Sihang Chen
+Marshall Burke, David Lobell, Stefano Ermon
+Stanford University
+{chenlin, jesslec}@cs.stanford.edu, {mkolod, ejsung, schen22, mburke, dlobell}@stanford.edu, [email protected]
+Abstract
+Building coverage statistics provide crucial insights into the
+urbanization, infrastructure, and poverty level of a region, fa-
+cilitating efforts towards alleviating poverty, building sustain-
+able cities, and allocating infrastructure investments and pub-
+lic service provision. Global mapping of buildings has been
+made more efficient with the incorporation of deep learning
+models into the pipeline. However, these models typically
+rely on high-resolution satellite imagery which are expensive
+to collect and infrequently updated. As a result, building cov-
+erage data are not updated timely especially in developing re-
+gions where the built environment is changing quickly. In this
+paper, we propose a method for estimating building coverage
+using only publicly available low-resolution satellite imagery
+that is more frequently updated. We show that having a multi-
+node quantile regression layer greatly improves the model’s
+spatial and temporal generalization. Our model achieves a co-
+efficient of determination (R2) as high as 0.968 on predicting
+building coverage in regions of different levels of develop-
+ment around the world. We demonstrate that the proposed
+model accurately predicts the building coverage from raw in-
+put images and generalizes well to unseen countries and con-
+tinents, suggesting the possibility of estimating global build-
+ing coverage using only low-resolution remote sensing data.
+Introduction
+The quantity and location of buildings provide important in-
+sight into the human activities and urban development of
+a region. Not only are building statistics themselves key
+socioeconomic indicators, they also help predict other key
+sustainable development indices, including poverty (Ayush
+et al. 2021; Uzkent, Yeh, and Ermon 2020; Yeh et al. 2020),
+population density (Huang et al. 2021), and climate out-
+comes (Chini et al. 2018). Moreover, building coverage
+statistics help policymakers and NGOs make informed deci-
+sions regarding the provision of public services, the targeting
+of humanitarian aid, and priorities for large-scale infrastruc-
+ture investments.
+The development of deep learning detection (Redmon and
+Farhadi 2018) and segmentation models (Ronneberger, Fis-
+cher, and Brox 2015; Sun et al. 2019) has allowed for more
+efficient global mapping of buildings. As a result, there has
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+been an increasing number of global settlement map datasets
+in the past decade (Esch et al. 2017; Marconcini et al. 2020;
+Sirko et al. 2021), allowing researchers to develop insights
+into the socioeconomic development of different regions.
+Nevertheless, detection- or segmentation-based methods
+typically rely on a large amount of high-resolution satel-
+lite imagery and the corresponding pixel- or instance-level
+labels for training, which are often prohibitively expensive
+and unaffordable for researchers and policymakers. More-
+over, high-resolution imagery are updated less frequently
+than low-resolution ones. In addition, running detection or
+segmentation models over high-resolution images covering
+a large area requires a large amount of compute. Due to
+these reasons, building data gathered in this way is often not
+updated timely. For example, the Microsoft Global Build-
+ing Footprints dataset1 is collected from satellite images be-
+tween 2014 and 2021. However, during the 7-year period,
+population continues to grow and new buildings are con-
+structed, especially in fast developing regions. For instance,
+from 2015 to 2020, the population increased by 44.1%
+for Malappuram and 34.2% for Abuja.2 Moreover, fine-
+grained detection and segmentation models usually gener-
+alize poorly to unseen geographies, timestamps, and image
+sources because the appearance of buildings vary widely in
+satellite images (Yuan and Cheriyadat 2014).
+Compared with its high-resolution counterpart, low-
+resolution satellite imagery (e.g. Sentinel-1 and Sentinel-
+2) are publicly available and updated every month, making
+them desirable for studying the economic and urban devel-
+opment of a region. However, prior works have not fully
+utilized low-resolution imagery. In this paper, we propose
+a cheaper and more generalizable way to update building
+statistics using only low-resolution satellite imagery from
+Sentinel-1 and Sentinel-2. Instead of detecting or segment-
+ing each building in satellite images, the proposed model
+directly predicts the building coverage from the raw input
+pixels, using input imagery from a public source that is up-
+dated nearly weekly. Specifically, we found that incorporat-
+ing a multi-node quantile regression loss helps improve the
+generalization of the model. The proposed method achieves
+a coefficient of determination (R2) as high as 0.968 in re-
+1https://github.com/microsoft/GlobalMLBuildingFootprints
+2https://worldpopulationreview.com/
+arXiv:2301.01449v1  [cs.CV]  4 Jan 2023
+
+gions from different continents and of different levels of de-
+velopment. We also conduct ablation studies and show that
+the incorporation of additional multi-spectral bands avail-
+able from the low-resolution satellite imagery and the multi-
+node quantile regression design help improve the model per-
+formance.
+Method
+In this paper, we propose a deep-learning based regression
+model that accurately predicts building coverage in low-
+resolution satellite imagery from Sentinel-1 and Sentinel-
+2. Unlike detection- or segmentation-based methods, our
+model does not require high-resolution training data or
+hand-crafted threshold and generalizes well to unseen re-
+gions. The proposed method accurately predicts the building
+coverage in the raw input low-resolution satellite image with
+the help of a quantile regression.
+Problem Definition
+Given a geographical region, we want to estimate the build-
+ing coverage within it. Due to the lack of direct statistics of
+building coverage in square meter or kilometer, we instead
+predict the number of building pixels y ∈ R, in the satellite
+image X representing the target region, assuming that the
+number of building pixels is proportional to the actual build-
+ing coverage (Figure 1(a)). We want to build a model that
+predicts y from the raw image input X.
+Multi-node Quantile Regression Model
+We observe that the distribution of building pixel counts
+across Africa and South America are heavy-tailed, with
+more than 75% of the samples having less than 20% of all
+pixels being buildings. Regular linear regression trained on
+root-mean-square error objective estimates the conditional
+mean and assumes normality of the data distribution, fail-
+ing to model non-normal asymmetric distribution accurately.
+Moreover, linear regression is not robust to outlier values,
+which characterize the building pixel count distribution for
+our task.
+To address the aforementioned problems, we adopt quan-
+tile regression (Koenker and Bassett Jr 1978), which esti-
+mates the conditional quantile (e.g., median) of the response
+variable. Quantile regression allows us to incorporate uncer-
+tainty in prediction and captures the relationship between the
+input and different quantiles of the data. As we will see in
+Section , multi-node quantile regression indeed empirically
+performed better than regular linear regression for our task.
+Specifically, we modify the ResNet18 (He et al. 2016) ar-
+chitecture to have K output channels (i.e. nodes), each cor-
+responding to a different quantile (see Figure 1). As it is ob-
+served that the median of a distribution gives the minimum
+absolute error from the ground truth (Hanley et al. 2001), at
+inference time, we collect the model predictions from only
+the 0.5 quantile node, which is expected to predict the con-
+ditional median of the response variable.
+Multi-node Quantile Loss
+For each node that represents a quantile q ∈ (0, 1), we com-
+pute an asymmetric quantile loss, or pinball loss. Depending
+on the quantile q, over- and under-estimation are penalized
+unevenly. Specifically, the node-wise pinball loss for a given
+prediction ˆy and ground truth label y is computed as follows:
+Lpinball(q, y, ˆy) =
+�q · |ˆy − y|
+if ˆy ≥ y
+(1 − q) · |ˆy − y|
+otherwise
+When q = 0.5, the pinball loss is the same as the absolute
+error.
+To compute the final loss, we take the mean of the pinball
+losses among the K output nodes as follows:
+Lquantile(y, ˆy) = 1
+K
+K
+�
+n=1
+Lpinball(qn, y, ˆy)
+where the subscript n indicates the index of the quantile in
+the list of quantiles predicted by the model. Notice that when
+K = 1 and q = 0.5, the quantile loss formula boils down to
+the regular L1 loss.
+Experiment Setup
+Before introducing the experiment setups, we define tile and
+patch in the context of this paper. We define a patch to be the
+small rasters of 50×50 pixels that we crop larger rasters into.
+A tile is a larger satellite imagery that could be cropped into
+multiple smaller patches (see Figure 2). We train and eval-
+uate the multi-node quantile regression model on patches of
+50 × 50 pixels and add post-processing steps to collect tile-
+level results.
+Training Data
+As the majority of the African continent is covered by for-
+est or desert and does not contain any buildings, we sample
+locations based on the population density so that the train-
+ing data contains sufficient tiles with buildings for the model
+to learn from. Our training set contains 15,000 input-label
+pairs. Sentinel-1 and Sentinel-2 images are used as as the
+input data and the Open Buildings dataset is used to derive
+the building coverage label, as we detail next.
+Sentinel-1 (S1)
+satellites collect radar imagery for land
+and ocean monitoring. We download the S1 satellite im-
+agery collected in 2020 from Google Earth Engine. The im-
+ages have 10m GSD and are composites that take the median
+value over the target period of time. We include band 1, the
+VV overall mean, as one of the input channels. The S1 tiles
+are cropped into 50 × 50-pixel patches. As areas with high
+built-up density are shown to result in stronger signals in
+S1 band 1 (Koppel et al. 2017), we expect that adding this
+channel to the input will improve the model’s performance.
+Sentinel-2 (S2)
+satellites are equipped with the mission of
+land monitoring. We download the 2020 composites of S2
+imagery from Google Earth Engine. We include the RGB
+channels (i.e. bands 4, 3, 2) and the near-infrared (NIR)
+channel (i.e. band 8) from S2 as input. Compared with other
+channels, NIR is useful for distinguishing the vegetation
+from the buildings (Luo et al. 2019; Pessoa et al. 2019;
+Schlosser et al. 2020). All of the four bands are available
+in 10m GSD. The S2 tiles are cropped into 50 × 50-pixel
+patches.
+
+(a) Predict the number of building pixels 𝑦 in image 𝑋
+𝑦 = 874
+Base map (left) and the binary mask 
+(right) used to derive the label 𝑦
+0.1
+0.9
+Concat
+ResNet18
+(b) Multi-node quantile regression model 
+#𝑦
+0.5
+ℒ!"#$%&'(
+𝐾 = 3
+Input 𝑿
+𝐴 = 34.96%
+Figure 1: (a) Given an input image X, we want to predict the number of building pixels y within it. We assume that y ap-
+proximate the actual building coverage. (b) Architecture of the multi-node quantile regression model. The input to the model
+includes five channels, which are Sentinel-1 band 1 (Overall Mean), Sentinel-2 band 4, 3, 2, 8 (R, G, B, NIR). The model has
+K output nodes representing different quantiles.
+Tile
+Patch
+Crop
+Summation
+50
+50
+Figure 2: A tile can be cropped into multiple 50 × 50 pix-
+els patches. The tile-level results are computed by taking
+summation of all the patch-level predictions.
+Open Buildings
+(Sirko et al. 2021) contains 516M build-
+ing footprints across 43 African countries that cover 64%
+of the continent. The building footprints were detected us-
+ing state-of-the-art segmentation model collected from high-
+resolution imagery at different timestamps. We downsample
+the original high-resolution mask (0.5m GSD) to 10m GSD
+to match the resolution of the input data. Then, we convert
+the continous-valued rasters into binary masks by threshold-
+ing at 0. The building pixel count labels are derived for the
+50 × 50-pixel patches.
+Experiment Settings
+To evaluate the generalization performance of the model, we
+consider three experiment settings that captures likely cases
+of real-world application.
+Holistic.
+In the holistic setting, we train and test on data
+points sampled across the African continent based on popu-
+lation density.
+Intra-country.
+In the intra-country setting, we train and
+test the model on samples from the same country.
+Exclusive.
+In the exclusive setting, we train the model on
+all African countries except for the one country on which we
+test our model.
+Baselines
+To evaluate the performance of the proposed method, we
+use existing settlement map products from different years
+as baselines to compare against. These off-the-shelf prod-
+ucts provide researchers and policymakers with general in-
+formation about urban shapes and boundaries, but could be
+less useful in providing up-to-date building coverage statis-
+tics, which change more frequently than the shape of urban
+area. We believe that these existing settlement map products
+serve as good baselines to compare our method against and
+provide information of them in this section.
+Gloal Urban Footprint (GUF)
+GUF was collected from
+2011 to 2012. It contains mappings of human settlements in
+the form of binary masks.
+Global Human Settlement Layer (GHSL)
+GHSL was
+collected from Sentinel-1 images in 2018. The building map
+is available as binary masks.
+World Settlement Footprint (WSF)
+WSF (Marconcini
+et al. 2020) was collected in 2015. It contains binary masks
+of global human settlements.
+Evaluation Settings
+We evaluate the model performance at both patch-level and
+tile-level and describe the evaluation metrics in this section.
+Patch-level Evaluation
+As the model is trained on
+patches, we want to evaluate the model’s performance at the
+same scale. We use two evaluation metrics for patch-level
+evaluation: mean absolute error (MAE) and Pearson’s r2 be-
+tween the predicted and the ground truth labels.
+Tile-level Evaluation
+In real-world applications, building
+coverage statistics are needed over a large geography. To re-
+flect this use case, we also evaluate the model performance at
+
+1661 (1687)
+82 (83)
+93 (87)
+665 (645)
+698 (704)
+987 (945)
+Figure 3: Examples of model predictions in Brazil, which was unseen during training. The first row is the RGB input image; the
+second row is the binary masks (where the bright yellow pixels are building pixels) from which we derive the label. The model
+prediction is in black; the ground truth labels are highlighted in green in the parentheses. Our method accurately estimated the
+results.
+Expt./Eval. settings*
+Open Buildings
+SpaceNet7
+Holistic
+✓
+Intra-country
+✓
+Exclusive
+✓
+Patch-level
+✓
+✓
+Tile-level
+✓
+Table 1: Checkmarks indicate that the corresponding dataset
+is used as validation data under the experiment (Expt.) or
+evaluation (Eval.) settings. *Different experiment settings
+require retraining the model, while different evaluation set-
+tings do not.
+tile-level using absolute error in building coverage. To com-
+pare the statistics of building coverage across baselines with
+different GSDs, we compute the percentage of building pix-
+els within a tile as a proxy for building coverage. For a tile
+R cropped into N patches at inference time, let yi be the
+number of building pixels in patch i, the building coverage
+percentage is computed as:
+Cbuilding(R) =
+�N
+i=1 yi
+Htile × Wtile
+× 100
+where Htile and Wtile denote the height and width (in pixel)
+of the tile. The absolute error between a method and the
+ground truth is computed as the absolute difference between
+the two building coverage percentages.
+Evaluation Data
+We evaluate our model on Open Buildings and SpaceNet 7
+Challenge datasets and provide the information in Table 1.
+Open Buildings
+The Open Buildings (Sirko et al. 2021)
+dataset provides the training labels. We evaluate the model
+performance on a hold-out test subset of the Open Buildings
+dataset at patch-level only. We do not use Open Buildings for
+tile-level evaluation because it contains data from different
+timestamps.
+SpaceNet 7 Challenge dataset (SpaceNet7)
+SpaceNet73
+was published in 2020 as the data for the SpaceNet 7 Multi-
+Temporal Urban Development Challenge. The dataset pro-
+vides 4km × 4km tiles and the building polygons in each
+tile. We downsample the raster to 10m GSD to match the
+resolution of the input data. We evaluate the model perfor-
+mance on SpaceNet7 at both patch-level and tile-level.
+We use the SpaceNet7 tiles for validation because the la-
+bels were collected the same year as the Sentinel-1/-2 in-
+put data. Furthermore, SpaceNet7 includes tiles from re-
+gions outside of Africa, allowing us to evaluate the model’s
+performance on unseen countries. Specifically, we evaluate
+our model on six SpaceNet7 tiles from different regions:
+Uganda, Zambia, Ghana, Peru, Brazil, and Mexico. Note
+that we do not use SpaceNet7 as the training labels be-
+cause the data is limited in quantity. These labels are human-
+generated and likely of higher quality compared to those
+from Open Buildings, which are generated by a model.
+Results
+In this section, we evaluate the performance of the proposed
+method. We first show that our model accurately predicts
+building coverage in Africa and generalizes to South Ameri-
+can regions unseen during training. We also conduct ablation
+studies demonstrating that the major design choices – non-
+RGB bands and multi-node quantile regression – are neces-
+sary for boosting model performance and generalization.
+3SpaceNet on Amazon Web Services (AWS). “Datasets.” The
+SpaceNet Catalog. Last modified October 1st, 2018. Accessed on
+November 20th, 2021. https://spacenet.ai/datasets/
+
+Africa
+South America
+Uganda
+Zambia
+Ghana
+Brazil
+Peru
+Mexico
+Method
+R2 ↑
+Tile ↓
+R2 ↑
+Tile ↓
+R2 ↑
+Tile ↓
+R2 ↑
+Tile ↓
+R2 ↑
+Tile ↓
+R2 ↑
+Tile ↓
+GUF (2012)
+0.092
+7.23
+-0.715
+0.96
+0.466
+1.59
+–
+–
+–
+–
+–
+–
+WSF (2015)
+0.286
+0.21
+-5.444
+38.68
+-0.290
+3.28
+0.579
+9.21
+0.562
+6.06
+-0.099
+18.52
+GHSL (2018)
+0.057
+6.83
+0.023
+2.46
+0.771
+0.75
+0.863
+0.97
+0.516
+10.06
+0.187
+8.12
+w/o multi-node QR
+-15.51
+33.15
+-10.41
+54.27
+-15.64
+31.68
+-0.069
+18.30
+-2.807
+38.28
+-2.480
+36.24
+w/o S1 band 1
+0.809
+1.95
+0.779
+3.74
+0.252
+3.37
+0.864
+3.91
+0.906
+2.34
+0.422
+10.96
+w/o S2 band 8
+0.772
+2.33
+0.580
+7.14
+0.804
+0.54
+0.751
+6.19
+0.836
+3.13
+0.337
+13.36
+Ours
+0.868
+1.18
+0.866
+3.21
+0.835
+2.31
+0.968
+0.83
+0.798
+6.63
+0.707
+0.36
+Table 2: Results on SpaceNet7. The bottom four rows are the proposed methods with the corresponding components removed
+and the complete version. In the table, r2 is the patch-level Pearson’s r2 and Tile is the tile-level absolute error between
+SpaceNet7 and the corresponding method. The model was trained with 15,000 samples in Africa for 1200 epochs and tested on
+the corresponding SpaceNet7 tiles.
+Expt. setting
+Train
+Test
+MAE ↓
+R2 ↑
+Holistic
+Africa
+Africa
+75.43
+0.888
+Intra-country
+Rwanda
+Rwanda
+27.60
+0.938
+Exclusive
+Africa*
+Rwanda
+42.04
+0.844
+Exclusive
+Africa*
+Uganda
+207.74
+0.568
+Excluisve
+Africa*
+Kenya
+110.67
+0.915
+Table 3: Patch-level results for different experiment settings
+on Open Buildings. All models in the table are trained with
+15,000 samples from the corresponding train regions for
+1200 epochs and tested on 1,000 samples from the corre-
+sponding test regions. *Under the Exclusive setting, the cor-
+responding test region is removed from the training set so
+that the model is tested on unseen regions.
+Accurate Prediction in Africa
+To demonstrate the effectiveness of our method, we evaluate
+it on both SpaceNet7 (see Table 2) and Open Buildings (see
+Table 3) in Africa. We define a tile as multiple patches and
+compute the tile-level results by taking summation of the
+patches within it (see Section 4.1).
+As shown in Table 2, our model achieves an R2 as high as
+0.968 at patch-level, outperforming the baselines on most of
+the African regions. Some examples of the proposed model’s
+patch-level prediction are provided in Figure 3. Furthermore,
+the proposed method yields fairly accurate building cov-
+erage estimates at tile-level, achieving a low error rate of
+0.54% on Ghana. Additionally, we observe that baselines
+like WSF and GUF give good estimates at only tile-level
+but not the other, indicating that the errors at patch-level are
+cancelled out. In contrast, the proposed method yields con-
+sistently accurate estimations at both tile- and patch-level.
+We emphasize that having good performance on both scales
+is ideal, since it gets us closer to finer-grained pixel-level
+predictions (semantic segmentation).
+Generalization to Unseen Regions
+Prior methods for generating building coverage statistics
+generalize poorly under domain shift. To evaluate the gen-
+eralizability of the proposed method, we train and test the
+multi-node quantile regression model under three exper-
+iment settings (defined in Section 4.3) – Holistic, Intra-
+country, and Exclusive. We provide the patch-level results
+on the Open Buildings dataset in Table 3.
+We see that among the three experiment settings, Intra-
+country gives the highest Pearson’s r2 of 0.971 because the
+model is tested on in-domain data and the region is small.
+We also observe that the proposed multi-node quantile re-
+gression model generalizes well to regions not seen during
+training. Speicifcally, under the Exclusive setting, the pro-
+posed method achives an r2 as high as 0.962 even with the
+test regions removed from the training set (see Table 2).
+Furthermore, the proposed model generalizes to regions
+outside of the African continent. We evaluate our method
+on SpaceNet7 tiles from South America and provide the
+results in Table 2. We observe that the proposed model
+achieves comparable or superior performance when evalu-
+ated on Brazil, Peru, and Mexico compared with the base-
+lines. This generalization to a different continent indicates
+that our model could potentially be applied to the globe for
+collecting building coverage statistics, while using only pub-
+licly available low-resolution satellite imagery.
+Ablation Studies
+In this part, we carry out ablation studies on 1) the incorpo-
+ration of S1 band 1 and S2 band 8 as input and 2) the multi-
+node quantile regression, and provide the results in Table 2.
+Multi-node Quantile Regression
+To see the effect of hav-
+ing multiple output nodes for different quantiles, we com-
+pare the performance of the multi-node quantile regression
+model with the single-node model trained on the L1 objec-
+tive. We provide the ablation study results on SpaceNet7 in
+Table 2 (see row “w/o multi-node QR”). We observe that
+the proposed multi-node model outperforms the single-node
+model by a large margin on all test regions. In addition, as
+shown in the scatter plots for ground truth VS. predicted
+building pixel counts (Figure 4), the model tends to overesti-
+mate the building coverage without the multi-node quantile
+regression. We speculate that having multiple output nodes
+improve performance because the pinball losses from the 0.1
+and 0.9 quantile nodes help bound the predictions.
+Multi-Spectral Bands
+We conduct ablation studies on the
+two non-RGB multi-spectral bands — i.e. S1 band 1 and S2
+
+w/o S2 band 8
+w/o S1 band 1
+w/o Multi-node QR
+Ours
+𝑹𝟐 = 0.751
+𝑹𝟐 = 0.864
+𝑹𝟐 = -0.069
+𝑹𝟐 = 0.968
+Ground truth building pixel count
+Predicted building pixel count
+Figure 4: Scatter plots of patch-level predicted number of
+building pixels against the ground truth from the ablation
+studies. All models are trained on Africa and tested in the
+Brazil tile. Removing the multi-node quantile regression or
+any of the non-RGB bands makes the model be prone to
+overestimation or underestimation.
+band 8 — and provide the ablation study results evaluated on
+SpaceNet7 in Table 2 . We observe that at both patch- and
+tile-level, incorporating S1 band 1 and S2 band 8 boosts the
+performance. From the scatter plots in Figure 4, we see that
+removing either S1 band 1 or S2 band 8 makes the model un-
+derestimates at patch-level. This suggests that the non-RGB
+bands provide additional information that helps correct the
+model’s tendency to under- or over-estimate.
+Discussion and Social Impact
+United Nations’ Sustainable Development Goals (SDGs)
+present an urgent call for action in all countries and collabo-
+ration between different sectors of policy-making for a more
+sustainable development (Nations 2016). However, relevant
+data for informing the decision makers and organizations are
+often lacking or infrequently collected, especially in fast de-
+veloping countries. Building coverage is an important so-
+cioeconomic indicator and also helps predict other indica-
+tors. In this paper, we develop a framework for estimat-
+ing building coverage using only free low-resolution satel-
+lite imagery from Sentinel-1 and Sentinel-2. The proposed
+multi-node quantile regression model yields fairly accurate
+estimates and generalizes well to unseen countries and con-
+tinents.
+This paper offers a cost-efficient and generalizable way
+to collect global building coverage statistics, accelerating
+the progress towards multiple SDGs. For example, build-
+ing coverage helps predict the level of economic develop-
+ment, which informs policymakers’ decisions for alleviating
+poverty (SDG 1 No Poverty), allocating infrastructure re-
+sources (SDG 9 Industry, Innovation and Infrastructure), and
+building more sustainable cities (SDG 11 Sustainable Cities
+and Communities). In addition, building coverage provides
+important information about the interaction between human
+and environment, including the monitoring of agriculture to
+reduce hunger (SDG 2 Zero Hunger) and climate measure-
+ment (SDG 13 Climate Action).
+Furthermore, the proposed method could potentially be
+applied to track changes of building coverage for different
+regions in the world, assisting existing efforts for this. For
+example, United Nations’ World Urbanization Prospects re-
+port provides estimates and projections of urban and rural
+data, including population and area, throughout the time
+(Nations 2018). The proposed method could be applied to
+derive building coverage statistics as soon as satellite images
+are updated (the low-resolution satellite imagery are updated
+on a weekly basis). As building coverage is highly corre-
+lated with or can be used to derive other values like building
+density, urban area, and population, our method could po-
+tentially aid the efforts to track urban development.
+This paper demonstrates the viability of using low-
+resolution free satellite imagery for estimating building cov-
+erage statistics over a large geography. Future research could
+explore how the model can be applied to classify urban and
+rural areas, and estimate population and poverty levels.
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+
+Appendix A: Datasets
+Sentinel-1
+provides free global Synthetic Aperture Radar
+(SAR) imagery available with 9 bands at ground sampling
+distance (GSD) of 10m. The revisit time of the Sentinel-1
+satellite is 12 days, meaning that the images are updated fre-
+quently. We include band 1, the VV overall mean, as one of
+the input channels. Each Sentinel-1 tile that we download is
+2km-by-2km and has the shape 200×200 pixels. Each tile is
+then cropped into 16 smaller patches of shape 50×50 pixels
+when fed into our model.
+Sentinel-2
+provides multi-spectral images in from the vis-
+ible to the shortwave infrared spectral range (SWIR) with the
+revisit time of 10 days. The Sentinel-2 satellite imagery con-
+tains 13 multi-spectral bands with GSDs ranging from 10 to
+60m. Besides the RGB channels, we also included the near-
+infrared (NIR) channel (i.e. band 8) as input. Like Sentinel-
+1, all the Sentinel-2 imagery are downloaded as 2km-by-
+2km tiles and cropped into 16 patches of size 50 × 50-pixel.
+Open Buildings
+contains 516M building footprints across
+43 African countries that cover 64% of the continent. The
+building footprints were detected using state-of-the-art seg-
+mentation models. The Open Buildings data is originally in
+the format of polygons labeled with three confidence inter-
+vals of buildings presence - 0.6 to 0.65, 0.65 to 0.7, and
+greater than 0.7. For our purpose, we download the Open
+Buildings data as high-resolution rasters with 0.5m GSD and
+two bands. Band 1 demonstrates the model confidence that a
+building is located in the region with the confidence interval
+preserved from the original polygon data. A band 1 value of
+zero indicates no building presence. Band 2 is a reclassifica-
+tion of the confidence scores into four buckets.
+We use band 1 to derive the binary label of building and
+non-building. Specifically, we first downsample the rasters
+to 10m GSD to match the resolution of the Sentinel-1 and
+Sentinel-2 input images. Then, we convert the continous-
+valued band 1 into a binary mask by treating all pixels with a
+non-zero value as a building pixel – i.e. a pixel that contains
+buildings. Like Sentinel-1 and Sentinel-2, the downsampled
+binary mask is cropped into 50 × 50-pixel patches. For each
+smaller patch, we use the number of building pixels as the
+labels for training and testing.
+Appendix B: Baselines
+Figure 5 shows the visualization of different benchmarks de-
+scribed below and datasets we used in experiments.
+Gloal Urban Footprint (GUF)
+is a worldwide mapping
+of human settlement patterns in the form of binary masks
+available in 12m GSD. The building footprints are derived
+from satellite images of TerraSAR-X and TanDEM-X from
+2011 to 2012.
+Global Human Settlement Layer (GHSL)
+was collected
+from backscattered information of Sentinel-1 images. The
+building map is available as binary masks at 20m GSD.
+World Settlement Footprint (WSF)
+is a binary mask of
+global human settlements available in 10m GSD. The map
+Base map
+Open Buildings (2021)
+WSF (2015)
+GHSL (2018)
+GUF (2012)
+SpaceNet (2020)
+Figure 5: Binary settlement maps in Zambia (1 = red build-
+ing pixel; 0 = white non-building pixel) from different
+sources. The years of collection for the settlement maps are
+provided in the parentheses.
+(a) Geo-locations of training samples
+(b) Per km2 cost and resolution of 
+different satellite imagery 
+Figure 6: (a) The geo-locations of our training samples, all
+of which are from Africa. (b) High-resolution satellite im-
+agery are expensive, while many lower-resolution ones are
+publicly available.
+was derived from Sentinel-1 and Landsat-8 satellite imagery
+in year 2015.
+Appendix C: Experiments
+Model Implementation
+We modify the ResNet18 architecture to incorporate multi-
+node quantile regression. Specifically, we replace the final
+fully-connected (FC) layer in ResNet18 with two FC layers,
+each followed by a ReLU activation. We set the number of
+output channels (i.e. nodes) to be K, which is the number
+of quantiles the model predicts. For all models in this pa-
+per, we use K = 3 and predict the quantiles {0.1, 0.5, 0.9}.
+Each channel corresponds to a quantile and the correspond-
+ing pinball loss is computed using the output of that partic-
+ular channel. To prevent the model from overfitting, we add
+dropout layers after each ResNet block, which we found em-
+pirically that it helps the model generalize to unseen regions.
+
+Price($/km^2)
+GSDmRegion
+Population (%)
+Building (%)
+Dhaka
+19.23
+29.68
+Kampala
+28.18
+7.28
+Athens
+-0.19
+1.55
+Boston
+4.45
+-9.42
+Table 4: Temporal change experiment results on four cities
+of different levels of development. The second and third are
+percentage change from 2016 to 2021. All results are col-
+lected from the model trained with 15,000 samples in Africa
+for 1200 epochs.
+Training Details
+The training samples’ geo-locations are shown in Figure 6
+(a). For all models in the experiment section, we train them
+on 15000 samples for 1200 epochs with a learning rate of
+0.002. The models have fully converged at the point where
+we end training.
+Temporal Experiments
+Evaluation
+To evaluate our model’s ability to track tem-
+poral changes in building coverage, we run the model on
+satellite images from 2016 and 2021.4 Specifically, we chose
+four cities with various levels of development from four dif-
+ferent continents – Dhaka, Kampala, Athens, and Boston –
+and downloaded all images covering the cities. Then, we
+compute the building coverage growth rate over the 5-year
+interval for the chosen cities using the model trained on
+15,000 African images for 1200 epochs.
+One challenge of temporal evaluation is the lack of build-
+ing coverage ground truth data. To get a sense of how well
+the proposed method tracks temporal changes in building
+coverage, we use population change as a proxy for build-
+ing coverage change. Specifically, we assume that popula-
+tion and building coverage grows together, as more people
+requires more buildings. However, we only use population
+growth rate as a rough reference for the relative level of de-
+velopments of different cities.
+Results
+Tracking changes in building coverage across
+time allows us to understand the urban development of a
+region, especially in cities or urban areas. We show that
+the proposed model can be used to track building coverage
+changes in regions of different levels of development and
+provide the temporal experiment results in Figure 4.
+4Sentinel-2 mission started in 2014.
+

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+arXiv:2301.00994v1  [quant-ph]  3 Jan 2023
+Pinhole quantum ghost imaging
+Andres Vega,1, a) Sina Saravi,1 Thomas Pertsch,1, 2 and Frank Setzpfandt1
+1)Institute of Applied Physics, Abbe Center of Photonics, Friedrich Schiller University Jena, Albert-Einstein-Str. 15, 07745 Jena,
+Germany
+2)Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-Str. 7, 07745 Jena,
+Germany
+(Dated: 4 January 2023)
+We propose a quantum ghost imaging scheme based on biphotons, that, by using a collimated pump beam of the right
+size for biphoton generation, obviates the need for lenses to achieve imaging. The scheme is found to be analogous
+to the classical pinhole camera, where we show that the equivalent to the classical pinhole size depends mainly on the
+width of the pump beam, but also on the thickness of the nonlinear crystal and the wavelengths of the biphoton.
+Quantum ghost diffraction1 and imaging2 rely on the spa-
+tial correlations of a biphoton, which can be created by the
+nonlinear process of spontaneous parametric down conversion
+(SPDC)3,4, where a pump (P) photon impinging on a crystal
+with second-order nonlinearity χ(2) is split into a pair of pho-
+tons called signal (S) and idler (I). After creation, the two
+photons are separated into two different paths and only one of
+them interacts with the object, e.g. the signal photon. Then,
+the signal photon is measured by a detector with no spatial res-
+olution, whereas another detector with spatial resolution mea-
+sures the idler photon that never interacted with the object.
+None of the detectors alone can recover a diffraction pattern
+or image of the object. Remarkably, these can be retrieved
+by correlating the two measurements5,6. This measurement
+technique has two main advantages. First, very low numbers
+of photons can be used due to the inherently better signal-to-
+noise ratio of quantum ghost imaging compared to imaging
+with classical light7,8. Additionally, ghost imaging with two-
+color biphotons can overcome limitations due to inaccessible
+wavelength ranges for illumination and detection9–11.
+To form a ghost image, usually lenses are placed in the path
+of the signal and/or idler after the crystal2,7,8 or in the pump
+beam before the crystal12. The lenses introduce a parabolic
+phase-front in either of the beam paths, which results in
+the formation of the image in the coincidence measurement.
+However, quantum ghost imaging can be also realized without
+lenses by adding the parabolic phase-front through engineer-
+ing the nonlinearity profile of the nonlinear crystal, e.g. by
+using a nonlinear photonic crystal13. Furthermore, as ghost
+imaging can be also realized with classical light using ther-
+mal light sources14–19, their inherent property of acting like
+a phase-conjugated mirror16 in a ghost imaging scheme can
+also be used for lensless ghost imaging20,21.
+In classical optics, lensless imaging can be also realized us-
+ing the principle of pinhole imaging22,23. In a pinhole camera,
+the object is located on one side of an opaque screen with a
+small pinhole, whereas the detector is on the other side. With-
+out the need of lenses, the detector captures a shadow of the
+object, which can be optimized by adapting the pinhole size22.
+Throughout this manuscript, we will refer to this shadow as an
+image although strictly no imaging is taking place. For appli-
+a)Electronic mail: [email protected]
+cations where high spatial resolution is not needed, this type
+of lensless imaging has several advantages over imaging with
+lenses, among which are a larger depth of field, a wide angular
+field of view23, and its applicability in wavelength ranges for
+which high-quality lenses are less available24,25.
+In this work, we want to show that the advantages of pin-
+hole imaging can be also harnessed in quantum ghost imag-
+ing. For ghost imaging with thermal light, a pinhole-based
+scheme has been already proposed, where the optimal lensless
+imaging condition depends on the size of the thermal source26.
+We extend this approach of lensless imaging to the quantum
+regime with entangled photons based on the setup sketched
+in Fig. 1. Here, we assume that the nonlinear crystal gener-
+ating photon pairs is illuminated by a collimated pump beam
+and, contrary to ghost imaging with thermal light, we use a
+bucket detector instead of a point detector behind the object.
+We will show, that for specific pump beam diameters, pinhole
+quantum ghost imaging can be realized and we investigate its
+properties and optimum regime of operation. To this end, we
+start by discussing the biphoton joint spatial probability (JSP)
+and the quantum ghost pattern (G) together with a numeri-
+cal example. Later, we derive a simplified analytical model
+for our imaging scheme that explains the observations of the
+numerical example and allows the connection to the classical
+pinhole camera. Using this model, we will furthermore dis-
+cuss the spatial resolution of pinhole quantum ghost imaging.
+Throughout this work, z is the propagation direction and
+we restrict our analysis, without loss of generality, to one
+transverse dimension x in position space, whose conjugate in
+momentum space is the transverse component of the wave-
+vector kx. We also assume an infinitely extended nonlinear
+crystal in the transverse direction, which ensures transverse
+Nonlinear
+crystal
+Bucket 
+detector
+Object
+Coincidence 
+circuit
+Spatially 
+resolving detector
+FIG. 1. Sketch of the considered setup.
+
+2
+phase matching kxP = kxS + kxI. We assume an undepleted
+monochromatic classical pump beam of the form EP(x,z,t) =
+� dkxP φP(kxP) exp[i(kxPx+ kzPz− ωPt)], where φP(kxP) is the
+pump spatial spectrum, the longitudinal component of the
+wave-vector is kz = [(ωn/c)2 − k2
+x]1/2 with ω/c = 2π/λ, λ
+being the wavelength in free space and ω the corresponding
+frequency. Additionally, we ignore the effects of the bound-
+aries between the crystal and its surrounding free space, which
+would cause refracted and reflected waves, and assume a sim-
+plified case with the crystal’s refractive index n = 1. We in-
+vestigate signal and idler photons at fixed frequencies of ωI
+and ωS, such that ωP = ωS + ωI. This can be achieved exper-
+imentally by placing narrow bandpass filters centered around
+these frequencies in their beam paths. Under these condi-
+tions, the biphoton quantum state after the filters will have
+the form27 |Ψ⟩ ∝
+� dkxSdkxI ψSPDC(kxS,kxI) |kxS,ωS⟩|kxI,ωI⟩,
+with ψSPDC(kxS,kxI) = φP(kxS + kxI) sinc(∆kz lz/2).
+Here,
+∆kz = kzP − kzS − kzI, lz is the thickness of the crystal, and
+|kx,ω⟩ is the single-photon state defined by the transverse
+component of the wave-vector kx and frequency ω.
+The JSP of the biphoton state at the two detectors in Fig. 1
+is28
+JSP(xS,xI) ∝
+���F−1�
+hI(kxI)h2S(kxS)
+�
+to(kxS)∗
+�
+h1S(kxS)
+×ψSPDC(kxS,kxI)
+������
+2
+,
+(1)
+where F−1 is a two-dimensional inverse Fourier transform,
+(kxS,kxI) → (xS,xI), and h are free space transfer func-
+tions with hI(kxI) = exp(ikzI zI), h1S(kxS) = exp(ikzS d) and
+h2S(kxS) = exp[ikzS(zS − d)]. Finally, ∗ denotes the convo-
+lution only in kxS as the object with transmission To(xS) is
+in the signal arm.
+The ghost pattern G(xI) for a bucket
+detector that collects all signal photons is derived from the
+JSP as G(xI) ∝
+� dxS JSP(xS,xI). We assume a pump beam
+with Gaussian spatial spectrum, whose waist is located at
+the center of the nonlinear crystal at the plane z = 0, where
+φP ∝ exp[−σ2
+P(kxS + kxI)2/2] has a flat wave front and a width
+σP in position space (see zoomed out region in Fig. 1).
+As shown in pinhole ghost imaging with a thermal source26,
+the size of the photon source has a similar role as the pin-
+hole size in classical optics, determining the optimal regime
+of imaging. This suggests that in the quantum regime, the
+same role can exist for the size of the biphoton source, which
+depends on the width of the pump beam. To examine this
+premise, we numerically calculate quantum ghost imaging in
+a setup with a nonlinear crystal with thickness lz = 3 mm,
+a pump with wavelength of λP = 350 nm, degenerate down-
+converted photons with λS = λI = 700 nm, and detectors lo-
+cated at zS = 1.2 m and zI = 1.5 m from the crystal. As ob-
+ject, we consider a double-slit with 940 µm slit separation,
+with unity transmission in each slit of 50 µm width and no
+transmission elsewhere.
+In Fig. 2(a-d) we present the en-
+suing normalized JSPs calculated using Eq. (1) for different
+sizes of the pump beam. The double-slit always results in
+two spots, whose separation is approximately five times larger
+than the slit separation. Depending on the width of the pump
+beam, they change their widths and begin to overlap and in-
+terfere. The interference is minimal in Fig. 2(c) with pump
+0
+1
+(d)
+(e)
+(a)
+(c)
+-8
+0
+8
+-8
+0
+8
+-8
+0
+8
+-8
+0
+8
+-8
+0
+8
+102
+103
+(b)
+(c)
+0
+1
+(d)
+(a)
+(b)
+FIG. 2. (a-d) JSP(xS,xI) and corresponding (e) quantum ghost pat-
+tern G(xI) of a double-slit, 940 µm slit separation and 50 µm slit
+width, located at d = 30 cm produced by a pump width σP of (a)
+58 µm, (b) 102 µm, (c) 167 µm, and (d) 800 µm.
+width σP = 167 µm. In Fig. 2(e), we show the corresponding
+ghost patterns, where the cases of (a-d) are marked. We see,
+that for a specific range of pump widths, two separate maxima
+are visible, corresponding to an image of the double-slit. We
+note, that the well-known quantum ghost diffraction pattern1
+of the object could be recovered using a large pump width,
+as in Fig. 2(d), and replacing the bucket with a point detector
+which would measure only a horizontal cut through the JSP.
+This numerical example portrays the core idea of this work.
+Other schemes12,13 have shown that a pump wave with a
+curved wave front, obtained by means of a lens placed be-
+fore the nonlinear crystal or using a photonic crystal, can also
+be used for quantum ghost imaging without lenses in the paths
+of the biphoton. Fig. 2(c) now shows, that lensless quantum
+ghost imaging can be also achieved by simply using a colli-
+mated pump beam with an optimal width. This is easily seen,
+as the Rayleigh length29 of the pump, 2πσ2
+P/λP = 50 cm, is
+much larger than the crystal thickness, lz = 3 mm.
+To find the optimal conditions for this imaging scheme,
+we derive a simplified analytical model. Here, we consider
+only one of the slits and assume it has infinitesimal width and
+is located at xS = a, which means that To(xS) = δ(xS − a).
+This object is put into Eq. (1) and in paraxial approxima-
+tion an analytical solution can be calculated (see supplemen-
+tary material), where we approximate the sinc function ap-
+pearing due to phasematching in the nonlinear crystal by
+a Gaussian30.
+We find a Gaussian ghost intensity pattern
+G(xI) ∝ exp[−(xI − x0)2/(2σ2
+G)] with a width σG and a max-
+imum located at x0, given by
+σG =
+�
+2Re
+�
+α−1
+1
+��−1/2,
+(2)
+x0 = a Re
+�
+α−1
+1 α2
+��
+Re
+�
+α−1
+1
+��−1 ,
+(3)
+where
+α1 = σ2
+P + γ lz
+π (λI − λP)+ iλIzI
+2π −
+�
+σ2
+P − γ lzλP
+π
+�
+α2,
+(4)
+α2 =
+�
+σ2
+P − γ lzλP
+π
+��
+σ2
+P + γ lz
+π (λS − λP)+ idλS
+2π
+�−1
+(5)
+
+102
+-8
+0
+83
+with γ = 0.455/4, a constant that comes from the sinc to Gaus-
+sian approximation. Due to the bucket detector that collects
+all signal photons behind the object, the equations do not de-
+pend on zS, the distance of the object to the bucket detector;
+however, the model does depend on the location of the re-
+solving detector zI. These distances remain at zS = 1.2 m and
+zI = 1.5 m throughout the manuscript. Fig. 3 shows the width
+σG of the ghost pattern and the normalized position x0/a with
+respect to the width of the pump σP and the distance of the ob-
+ject to the crystal d. We observe in Fig. 3 that, for each object
+position d, the width of the ghost pattern σG has a minimum
+at a certain pump width σP. This confirms the observations of
+the numerical example in Fig. 2 that uses d = 30 cm, where
+the optimal case for imaging, marked with a dot in Fig. 2(c),
+leads to the narrowest ghost pattern for each of the slits. In
+the rest of the cases, the pattern of each slit is too wide, result-
+ing in considerable overlap between them and hence the loss
+of visibility of the ghost pattern. If a second infinitesimal slit
+would by at xS = −a, the distance between the two maxima in
+the ghost patterns would be 2x0, therefore, x0/a represents the
+magnification. Fig. 3(b) verifies that for the numerical exam-
+ple in Fig. 2 the magnification is approximately equal to five.
+It also tells that the magnification is always negative, implying
+that the ghost image is inverted.
+Fig. 3(a) shows, that the minimum width of the ghost pat-
+tern becomes smaller as the object is placed farther from the
+crystal. This value, however, does not reach zero, the behavior
+of σG converges to approximately the orange curve in the limit
+where the object is very distant from the crystal, d → ∞. Here,
+Eq. (2) can be reduced to σ2
+G = σ2
+0 + σ−2
+0
+[zIλI/(4π)]2. The
+width of the ghost pattern of an infinitesimal slit object with
+the spatially resolving detector placed right after the crystal,
+σ0 = σG(zI = 0), is
+σ0 =
+�1
+2σ2
+P + γ
+� λI
+λS
+��λPlz
+2π
+��1/2
+,
+(6)
+an expression that depends only on the parameters of the
+biphoton source. For a fixed value of zIλI, the ghost pattern
+width σG has a minimum value, namely σmin
+G
+=
+√
+2σ0, when
+σ2
+0 = zIλI/(4π). Remarkably, this result is equivalent to the
+optimal pinhole size σpinhole in a classical pinhole camera that
+creates the smallest point image of a slit upon spatially inco-
+herent illumination22, σ2
+pinhole ∝ λz. Hence, σ0 can be consid-
+ered the pinhole size of quantum ghost imaging. It does not
+only depend on the width of the pump but also on the thickness
+of the crystal and the biphoton wavelengths. However, the de-
+viation of the equivalent pinhole size σ0 from the pump width
+σP due to the biphoton wavelength is small as for a pump with
+negligible diffraction inside the crystal σ2
+P ≫ λPlz/(2π).
+Next, we analyze the magnification to complement the anal-
+ogy.
+The magnification of the classical pinhole camera is
+given by geometric optics as −z/d with z the distance of the
+detector to the pinhole and d the distance of the object to the
+pinhole. A similar relation can be found from the analytical
+model of the proposed ghost imaging scheme using Eq. (3),
+under the conditions that σ2
+P ≫ (γ lz/π)(max(λS,λI) − λP)
+and {dλS/(2π), zIλI/(2π)} ≫ σ2
+P. The first condition en-
+sures, that signal, idler and pump waves have negligible
+(a)
+(b)
+0.4
+1
+1.6
+2.2
+2.8
+0
+200
+400
+-45
+-30
+-15
+0
+0
+200
+400
+d=3cm
+↓
+d=5cm
+d=10cm
+d=30cm
+d→∞
+FIG. 3. (a) Width σG and (b) magnification of the position x0/a of
+the ghost pattern of a infinitesimal slit at xS = a with respect to the
+pump width σP at various distances d of the slit to the crystal. The
+circle marks the parameters of the numerical example in Fig. 2(c).
+diffraction inside the nonlinear crystal. In this case, σP de-
+fines the size of the generated signal and idler beams inside the
+crystal, and hence their Rayleigh lengths are also much larger
+than the crystal thickness lz. The second condition states that
+both the object and the resolving detector are in the far-field
+of the biphoton source. Under these conditions, following the
+steps detailed in the supplementary material, we find the mag-
+nification to be x0/a ≈ −(zI/d)(λI/λS). This is similar to
+the geometrical magnification of the classical pinhole camera
+but including again the biphoton wavelengths. From this ex-
+pression, the magnification of the numerical example in Fig. 2
+can be quickly found −(1.5m/30cm)(700nm/700nm) = −5,
+see the supplementary material for an example with non-
+degenerate wavelengths. Noteworthy, our imaging scheme is
+not limited to the far-field domain. We only take the approxi-
+mations to find the simple expression for the magnification to
+build the analogy with the classical pinhole camera.
+We further harness the analytical model of Eqs. (2) and (3)
+to derive the transverse resolution of pinhole quantum imag-
+ing. In the following, we use Rayleigh’s resolution criterion
+that is defined as the minimum distance between two point-
+like objects to distinguish them from one another29. A smaller
+ghost pattern width σG of a point-like object results in a bet-
+ter distinction between neighboring objects, as was demon-
+strated in Fig. 2. At the same time, a larger magnification
+|x0/a| would results in a better distinction between two neigh-
+boring objects, as their images are further apart. This suggests
+that, to optimally tell two objects apart, not only the ghost pat-
+tern width has to be taken into account but also the magnifi-
+cation. To test this hypothesis, we begin by taking the derived
+Gaussian ghost pattern from an infinitesimal slit A at xS = a.
+For the sake of simplicity we normalize its maximum to one,
+resulting in the ghost pattern GA = exp[−(xI − x0)2/(2σ2
+G)],
+where σG is given by Eq. (2) and x0 by Eq. (3). If we in-
+clude another similar slit B but at xS = −a, the resulting ghost
+pattern is symmetric with respect to xI = 0 and is given ap-
+proximately by G ≈ GA + GB, assuming negligible interfer-
+ence. The two slits can be distinguished in this pattern when
+its visibility is above a certain threshold, here heuristically
+chosen to be 0.4.
+That means, the intensity at xI = 0 be-
+tween the two maxima should be smaller than 0.4 of the maxi-
+mal intensity at xI = ±x0, which gives the threshold condition
+G(xI = 0)th ≈ GA(0)+GB(0) = 2GA(0) = 0.4. Using this, the
+
+4
+0
+100
+200
+300
+1.3
+1.6
+0.1
+0.4
+0.7
+1
+d=3cm
+d=5cm
+d=10cm
+d=30cm
+�
+0
+1
+0
+1
+-10
+0
+10
+(��
+�� �
+���
+� 
+
+d=1m
+100
+101
+102
+10-1
+100
+101
+d
+
+∞
+d=3cm
+d=30cm
+d=1m
+FIG. 4. (a) Resolution R with respect to the pump width σP. The
+green line connects the minima of curves of a wider range of object
+distances d. (c) Number of spatial modes N with respect to the crystal
+thickness lz at various d. Numerically, JSP (top) and G (bottom) of
+a (b) double-slit and a (d) complex object (magnified transmission in
+yellow). Circles in (a) and (c) mark the parameters of (b) and (d).
+expression for GA, and Eqs. (2) and (3), we find the transverse
+spatial resolution R corresponding to the minimum resolvable
+distance 2a between two identical infinitesimal slits to be
+R = 2
+�
+−ln[G(0)th/2]Re
+�
+α−1
+1
+��1/2���Re
+�
+α−1
+1 α2
+����
+−1
+.
+(7)
+Fig. 4(a) displays its dependence on the pump width σP at dif-
+ferent object distances d. The thick green line connects the
+minima of several curves over a wider range of d to portray
+the tendency. One could naively expect from the width of
+the ghost pattern σG in Fig. 3(a) that the resolution R could
+improve as the object is farther from the crystal since the min-
+imum width becomes smaller. However, Fig. 4(a) tells the
+opposite, as R is enhanced by the large magnification at small
+object distances d. This is confirmed numerically by consid-
+ering again the case depicted in Fig. 2(b), that uses a pump
+width of 102 µm and d = 30cm, and changing the position of
+the object to d = 10cm. The resulting JSP and ghost pattern G
+are shown in Fig. 4(b). Compared to Fig. 2(b), the visibility is
+increased due to the larger magnification. This originates from
+the fact that, for a fixed object distance d, the minimum ghost
+pattern width σG does not coincide with the largest magnifi-
+cation x0/a. Hence, the optimal pump width σP that results in
+the best resolution is not necessarily when σG is minimized.
+This is only true for larger values of d where x0/a is almost
+independent of σP, as in the numerical example of Fig. 2.
+Noteworthy, the green tendency line in Fig. 4(a) hints that
+the closer the object is to the crystal and the smaller the pump
+width, the better the resolution R. However, the smaller the
+pump width, the broader its spatial spectrum becomes, in such
+case the used paraxial approximation does not hold. A non-
+paraxial formulation is a matter of future research. Addition-
+ally, R will depend linearly on the object position d for large
+values of d. This is because in such case, the ghost pattern
+width σG is nearly independent of d, see Fig. 3(a), and the
+magnification is |x0/a| ∝ 1/d, as already explained.
+In addition to the resolution, it is also of great interest to
+describe the extent of the signal illumination onto the object,
+which limits the object size that can be imaged. This is finite
+and can be found from a projection of the JSP at the object po-
+sition onto the signal axis (see the analytical expression in the
+supplementary material). This illumination size σS increases
+with the position of the object d and decreases with the crystal
+thickness lz. Importantly, the ratio between the illumination
+size and the resolution tells the maximum number of identical
+infinitesimal slits that can be resolved inside the illuminated
+region of the object, N ≡ σS/R, i.e. describes the number of
+independent spatial modes in the object illumination31. Its de-
+pendence on the crystal thickness lz is displayed in Fig. 4(c)
+for various d, each at a pump width σP that minimizes the res-
+olution as described by the green line in Fig. 4(a). To increase
+the number of spatial modes N, a thinner crystal can be used,
+similar to other quantum imaging schemes2,32, as it allows a
+larger range of transverse wave-vectors33. Also, the object
+could be put farther away from the crystal. Here, the increase
+of N with d has a limit due to the linear dependence of both
+σS and R on d for very distant objects. Finally, we found that
+photon-pairs with non-degenerate wavelengths show a minor
+improvement in the resolution and number of modes, see the
+supplementary material for some examples.
+Lastly, we sum up the main features of the proposed setup
+with a ghost image of a more complicated object with vari-
+ous shapes and transmissions, see Fig. 4(d). This object is
+placed at d = 1 m and we use a pump width σP = 258 µm that
+optimizes the ghost image resolution to R = 1.5 mm, allows
+N = 10 spatial modes and has a magnification x0/a = −1.2.
+Moreover, unlike the setup with a pseudo-thermal source26,
+we see in the JSP of this example that an integrating bucket
+detector is necessary to show the whole illuminated section of
+the object in the ghost pattern, a point detector would not be
+able to detect signals from all objects as it would just “take a
+horizontal thin slice” of the JSP.
+To conclude, we proposed a quantum ghost imaging
+scheme without lenses in the biphoton arms by means of a
+collimated pump beam with an optimal size. This imaging
+scheme is best suited for applications where lenses for the
+biphoton wavelengths are less available and a high transverse
+resolution is not required. We demonstrated that the proposed
+scheme is analogous to the classical pinhole camera where the
+biphoton source plays the role of the pinhole and derived its
+spatial resolution and number of spatial modes.
+See the supplementary material for further details of the an-
+alytical model and examples with non-degenerate photons.
+
+5
+ACKNOWLEDGMENTS
+We thank E. Santos and V. Gili for their insightful com-
+ments. This work was supported by the Thuringian Ministry
+for Economy, Science, and Digital Society; the European So-
+cial Funds and the European Funds for Regional Develop-
+ment (2017 FGR 0067, 2017 IZN 0012); the German Federal
+Ministry of Education and Research (FKZ 13N14877, FKZ
+03ZZ0434) and the Deutsche Forschungsgemeinschaft (DFG,
+German Research Foundation, project ID 407070005).
+DATA AVAILABILITY
+The data that supports the findings of this study are avail-
+able within the article and its supplementary material.
+This article may be downloaded for personal use only.
+Any other use requires prior permission of the author and
+AIP Publishing.
+This article appeared in A. Vega et al.,
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+

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+GAN-Based Content Generation of Maps for
+Strategy Games
+Vasco Nunes
+Instituto Superior T´ecnico
+University of Lisbon
+Lisbon, Portugal
+[email protected]
+Jo˜ao Dias
+Faculty of Science and Technology
+University of Algarve and CCMAR and INESC-ID
+Faro, Portugal
+[email protected]
+Pedro A. Santos
+Instituto Superior T´ecnico / INESC-ID
+University of Lisbon
+Lisbon, Portugal
+[email protected]
+Keynotes—Heightmap; Procedural Content Generation; Genera-
+tive Adversarial Network
+Abstract—Maps are a very important component of strategy
+games, and a time-consuming task if done by hand. Maps
+generated by traditional PCG techniques such as Perlin noise or
+tile-based PCG techniques look unnatural and unappealing, thus
+not providing the best user experience for the players. However
+it is possible to have a generator that can create realistic and
+natural images of maps, given that it is trained how to do so. We
+propose a model for the generation of maps based on Generative
+Adversarial Networks (GAN). In our implementation we tested
+out different variants of GAN-based networks on a dataset
+of heightmaps. We conducted extensive empirical evaluation to
+determine the advantages and properties of each approach. The
+results obtained are promising, showing that it is indeed possible
+to generate realistic looking maps using this type of approach.
+I. INTRODUCTION
+In maps for strategy games, the map’s visual characteristics
+play a very important role in the player’s experience. When we
+talk about visual characteristics, we usually refer to the map’s
+outline and level of detail. Complex elements like peninsulas,
+mountain ranges or islands provide more tactical information,
+improving the player’s decisions in a strategic way. Therefore,
+these details improve the way the information contained in the
+map is assessed, in order for the player to make decisions in
+a strategic way. A game which uses the same kind of map
+numerous times with no variety can cause players to become
+bored after replaying the game a few times.
+One of the ways to generate maps for strategy games is
+by using Procedural Content Generation (PCG) 1 techniques.
+The most common traditional approach for initial Heightmap
+generation is the Perlin Noise [1], followed by complementing
+techniques such as Hydraulic erosion [2]. Unfortunately the
+maps generated look a bit unnatural and unappealing. More
+so, most of the methods generated by PCG suffer from some
+kind of uncontrollability. The ideal scenario would be to have
+a generator that learned to create realistic images of maps and
+Published in the Proceedings of GAME ON 2022; Cite as:
+Nunes, V., Dias, J., Santos, Pedro A.: GAN-Based Content Generation of
+Maps for Strategy Games. Proceedings of GAME-ON’2022, pg 20-31, ISBN
+978-9-492859-22-8
+1Creation of game content algorithmically with limited or indirect user
+input
+with the complex elements (peninsulas or mountain ranges)
+appearing next to each other2.
+Recent advances in Deep Neural Networks, and in particular
+GANs (Generative Adversarial Networks) highlight the po-
+tential for a new approach for the automatic generation of
+maps. GAN is a framework proposed by Ian J. Goodfellow
+et al. [3] that is trained to generate data (images in most
+cases) with the same characteristics of a given training set. For
+example, a GAN trained on images of human faces would be
+able to generate realistic samples that are authentic to human
+observers.
+The framework consists of two networks competing against
+each other, thus the term adversarial: a generative network,
+Generator G, which creates fake data from a random distri-
+bution pz (usually normal or uniform) and a discriminative
+network, Discriminator D, which, by giving it some data x,
+estimates whether x came from real data distribution pdata or
+from the generator’s distribution pg.
+A real world analogy would be the job of an art counterfeiter
+and a cop. The cop (D) learns to detect false paintings while
+the counterfeiter (G) improves on producing perfectly fake
+paintings indistinguishable from real ones.
+The generation of images using GANs reached great success in
+recently years. Recent applications using this network include
+creation of realistic faces [4], pose guided person image
+generation [5] or transforming images from one domain to
+another [6].
+Taking this into account, the research problem addressed in
+this work is to explore several GAN’s techniques in order to
+generate realistic and appealing maps for strategy games.
+By “realistic”, we mean that maps should be perceived in
+a similar way to natural land formations. By “appealing”,
+we mean that maps should have suitable characteristics and
+elements discussed above, that allow players to have a more
+challenging and interesting experience.
+Taking into account the research problem, if we have a
+proper and balanced heightmap’s dataset of natural landscapes,
+we believe that this type of technique can be successfully
+used to generate maps that resemble the original dataset,
+2From an interview conducted for this research with Andy Gainey, game-
+play programmer at Paradox Development Studio
+arXiv:2301.02874v1  [cs.LG]  7 Jan 2023
+
+Fig. 1. Generative Adversarial Network Model architecture.
+which will have the type of characteristics existing in natural
+landscapes, such as peninsulas and mountain ranges. Starting
+from a baseline GAN architecture, we will slowly improve its
+structure in order to achieve our goal, taking into account the
+proper evaluation of the models.
+II. RELATED WORK
+In this section we describe different types of GANs that
+appeared in recent works and that we will use.
+A. DCGAN
+In the work of Alec Radford et al. [7] the authors managed
+to consolidate the junction of GAN and Convolutional Neu-
+ral Network (CNN) frameworks, after several unsuccessfully
+attempts to do so in the past years.
+CNN are a subset of neural networks most commonly applied
+to image and video recognition or computer vision problems.
+They were largely inspired by the visual cortex, small regions
+of cells sensitive to specific regions of the visual field [8].
+These networks are mostly composed by convolutional layers,
+that are responsible for applying the convolution3 operation
+of the input using a filter, or kernel, and sending the result,
+also known as feature map to the next layer. If this kernel is
+designed to detect a specific type of feature on the input, then
+filtering it across the whole image would allow the kernel to
+detect that feature anywhere in the image independently of
+the feature’s location. By having this translation invariance
+characteristic and a shared-weight architecture, CNN are also
+known as Shift Invariant Artificial Neural Networks [9].
+3In image processing, refers to the process of adding each pixel of the
+image to its local neighbors, weighted by a kernel.
+Their methodology consisted in adopting and modifying three
+demonstrated changes to CNN architectures:
+1) Replacing the pooling layers of the CNN baseline ar-
+chitecture for convolutional layers with stride 2 in order
+for G and D to learn its own spatial upsampling and
+downsampling respectively;
+2) Elimination of fully connected layers for convolutional
+layers. To make the most use of these convolution layers,
+they added another layer at the beginning of G that takes
+the input vector z and reshapes it into a 4-dimensional
+tensor. They also added another layer at the end of D that
+flattens the image to a single value output;
+3) Applying Batch Normalization to layers in order to stabi-
+lize learning by normalizing the input of a layer to have
+zero mean and unit variance, for each minibatch.
+B. Progressively Growing GANs
+Generation of high-resolution images is a difficult task since it
+is easier to discriminate between the fake and real images. In
+the work of Karras et al. [4], the authors proposed a training
+methodology that consists on starting with low-resolution
+images, and then progressively increasing the resolution by
+adding layers to both the discriminator and generator (which
+are mirror images of each other and always grow in syn-
+chrony). In other words, the authors are slicing a bigger
+complex problem into smaller ones and slowly increasing the
+complexity to prevent the training from becoming unstable.
+The incremental addition of the layers allows the models to
+effectively learn coarse-level detail and later learn even finer
+detail, both on the generator’s and discriminator’s side.
+The insertion of layers cannot be done directly due to sudden
+shocks to the already well-trained layers. Instead they phase in
+the layer. This operation consists on using a skip connection4
+to connect the new block to the input of D or output of G
+using a weighted sum with the existing input or output layer,
+that represents the influence of the new block. It is controlled
+by a parameter α that starts at a very small value and increases
+linearly to 1 over the process of training. In other words, we
+can think of this operation as the layers slowly being inserted
+during the training phase.
+In terms of results, this network was capable of generating high
+resolution images of 1024 × 1024, creating a high-resolution
+version of the CELEBA dataset [10].
+C. Wasserstein GAN
+Martin Arjovsky et al. [11] propose a GAN that has an
+alternate way of training so that the generator model better
+approximates the distribution of data of a given dataset. More
+so, they present an alternate loss function in which D is always
+giving enough information for the G to improve himself, even
+if D has reached its optimality.
+The discriminator D is replaced by a critic C that instead of
+classifying an image as fake or real (in the interval [0,1]),
+4Skip connections are connections of outputs from early layers to later
+layers through addition or concatenation
+
+Latent space
+Real example
+Fake example
+Updatemodel
+UpdatemodelI
+Probability of
+fake/realit scores the fakeness or realness of an image (in the interval
+]−∞,+∞[). This score is also known as Wasserstein estimate.
+The critic is looking to estimate the Wasserstein distance
+between the dataset sample distribution and the generated
+images distribution, which corresponds to the distance between
+the average critic score on real and the average critic score on
+fake images. Thus the network’s objective function can be
+summarized as follows:
+• C objective function is the difference between the average
+critic score on fake images and the average critic score on
+real images;
+• G objective function is the average critic score on fake
+images.
+Both the networks are trying to maximize these objective
+functions. Therefore, for G, a larger score of the fake images
+will result in a higher output for the G, encouraging C to
+output higher scores for the fake images. For C, a larger
+score for real images results in a lower value for the model,
+penalizing it, thus the encouragement for the critic to score
+lower scores for the real images.
+D. VAE + GAN
+The idea of combining the power of Variational Autoencoders
+(VAE) and GAN was the basis for the work of Larsen et
+al. [12]. The motivation behind their work was to leverage
+learned representations to better measure similarities in the
+data distribution.
+Autoencoders are another subset of neural networks used to
+generate images with good results. They learn a representation
+of a given data [13]. They are commonly used for face
+recognition or acquiring semantic meanings of words. The
+general idea behind this neural network is that they learn
+to copy the input to the output through a latent space that
+compresses the input maintaining only the most relevant and
+important information. The decoder then decompresses the
+information retained in the latent space ( also known as code)
+leading to a very similar copy of the output.
+VAE are a subset of autoencoders specialized in content
+generation. They inherit the architecture from the traditional
+autoencoders but instead of mapping the input to a fixed latent
+space, the VAE maps the input onto a latent distribution,
+which allows us to take random samples from the latent
+space. These samples are then decoded using the decoder
+segment to generate outputs very similar to the inputs used
+to train the encoders. Instead of having a fixed vector as the
+latent space, the vector is replaced by two separate vectors,
+that represent the mean and the standard deviation of the
+distribution, respectively.
+Typically, VAE uses element-wise similarity in their recon-
+struction error, however, Larsen et al. [12] propose using the
+GAN discriminator to measure the sample similarity. In other
+words, they use the GAN’s discriminator as a way to measure
+the difference between the VAE’s output and the original
+image.
+The authors’ proposed architecture is comprised of a single
+model that simultaneously learns to encode, generate and
+compare samples from the dataset. The GAN’s generator
+coincides with the VAE’s decoder. The authors define the loss
+of the model as:
+L = LV AE + LGAN
+(1)
+where LGAN is the Binary cross entropy loss and LV AE is
+defined as:
+LV AE = Lprior + LDisl
+llike
+(2)
+where Lprior is a prior regularization term, the Kullback-
+Leibler divergence and LDisl
+llike is the expected log likelihood
+(reconstruction error) expressed in the GAN discriminator:
+LDisl
+llike = −Ez∼Enc(x)[log p(Disl(x)|z)]
+(3)
+with Disl denoting a hidden representation of the lth layer of
+the Discriminator.
+III. DATASET CREATION
+In order to apply the GAN-based techniques, it is necessary to
+have a group of examples of what the system is supposed to
+generate. Therefore, we began by building a proper dataset of
+natural landscapes which had the characteristics of the already
+existing land formations of the planet Earth, such as peninsulas
+and mountain ranges.
+Data gathering
+When looking for a proper dataset we had the idea that no data
+could resemble more the characteristics discussed in Section I
+than data from the real world, in which the terrain had already
+the desired complex elements and characteristics. Taking this
+idea into account, we used a public dataset 5 with nearly
+global coverage of the planet Earth generated by a satellite
+radar topography mission. The dataset consists of several
+Digital Elevation Model (DEM) files created by a Ground
+Data Processing System supercomputer with a 3 arc-second
+sample spacing. We grouped the different DEM files together
+and exported them into a Tagged Image File Format using
+a program called Global Mapper, resulting in a Heightmap
+image with 43200 × 18000 resolution, as shown on Fig 2.
+Fig. 2. Heightmap of the planet Earth after joining the DEM files together
+5https://www2.jpl.nasa.gov/srtm/cbanddataproducts.html
+
+Preprocessing
+As one can see in Fig 2 most regions of the Earth are a
+bit too dark making the terrain not very noticeable. This is
+not ideal for the techniques to be used since the altitude data
+should be more evenly distributed. In other words, we needed
+a higher range of pixel values so that their difference would
+be better distributed between 0 and 255, instead of the original
+linear mapping between altitudes and pixel values. So, using
+the same program, we changed the image’s brightness level so
+that lower altitude regions could have higher pixel greyscale
+values.
+We proceeded to crop the high resolution image into 1024 ×
+1024 images with a sliding window of 512 pixels, which gave
+in total 2822 images.
+Dataset Augmentation / Removal of unwanted images
+Due to the lacking of enough images on the dataset for a
+problem with a moderate level of complexity such as the one
+approached here, we had to perform dataset augmentation in
+order to increase the number of images. The procedure is
+described below:
+1) To the 43200 × 18000 image, we applied different sets of
+image processing techniques: rotation between 0 and 180
+degrees, and horizontal / vertical flipping. The rest of the
+image would then be filled with the pixel value 255.
+2) As done in the preprocessing phase, the 43200×18000 was
+cropped into 1024 × 1024 images with a sliding window
+of 512.
+3) Resulting images that had little to no continental land or
+were cut due to the image processing techniques, were
+removed. In other words, images that had 95% of the pixel
+values ≤ 25, or had the pixel value of 255 were removed.
+This procedure was repeated 15 times, ending up with 12640
+images of 1024 × 1024 resolution, which we found more
+than reasonable for the problem. Unfortunately, Working with
+1024×1024-sized images would result in our models having a
+high number of parameters thus requiring more computational
+resources, such as memory, which we had not at our disposal.
+Therefore, we had to downsize the dataset to a 128 × 128
+resolution. The image rescaling was done using a nearest
+neighbor interpolation filter.
+IV. GANS’ ARCHITECTURE AND TRAINING
+Instead of just focusing on one type of GAN, we decided to
+explore several models in order to decide which one would be
+the most appropriate for the problem, making a comparison in
+terms of quality of the images generated, training efficiency
+and ease in training and convergence. We added tables which
+detail each model’s architecture in the Appendix.
+A. DCGAN
+The Deep Convolutional Generative Adversarial Network
+(DCGAN) architecture was the first to be tested, to serve as
+an initial baseline model and a foundation for the other mod-
+els. We followed the majority of the architectural guidelines
+explained in [7] such as:
+• Using strided convolutions on D and fractional-strided
+(transposed) convolutions on G, allowing the network its
+own spatial downsampling and upsampling;
+• Batch Normalization layers in both networks, which helps
+the network on its learning process;
+• Removal of fully connected layers and replacing them with
+the convolutional layers except in the beginning of G and
+in the end of D;
+• Leaky ReLU activation function for all layers of D, since it
+is more effective for models generating images with higher
+resolution;
+However, instead of applying the Rectified Linear Unit (ReLU)
+activation function to all layers of G, we instead used the
+Leaky ReLU, since the latter activation function has been
+proven to be more effective than the ReLU [14].
+Fig. 3 and Table 4 depicts the model of the GAN generator
+and discriminator, respectively.
+Fig. 3. DCGAN’s Generator Model
+Fig. 4. DCGAN’s Discriminator Model
+All of the convolutional and transposed convolution layers had
+a kernel size of 5, SAME padding and a stride of 2, except for
+the first convolution layer of D which had stride of 1. Those
+layers’ weights were initialized using a zero mean normal
+distribution with standard deviation 0.02. The values used for
+
+32
+64
+128
+1024
+256
+1
+128
+128
+16
+32
+64
+8
+100
+8
+16
+32
+64
+128
+128
+Convolution layer followed by Batch Normalization Layer,
+LeakyReLu layer and Dropout Layer
+Reshape1
+1
+32
+64
+128
+256
+128
+128
+64
+32
+16
+8
+>output
+8
+16
+32
+128
+128
+64
+Convolution layer followed by a LeakyReLu layer
+Convolution layer followed by Batch Normalization Layer
+and LeakyReLu layer
+Flattenthe Batch Normalization, Leaky ReLU layers and the kernel
+size on both networks were the same as the ones used in [7].
+It is well-known that GANs suffer from the vanishing gradient
+problem, where an optimal D ( one that is really good at
+classifying the images as real and fake), doesn’t provide
+enough information for the G to improve itself [11], [15].
+Regarding the DCGAN’s training phase, in order to battle the
+GAN’s vanishing gradient problem we decided to add three
+hindering methods that would help in achieving this goal.
+These hindering methods, have the purpose of preventing D
+from rapidly reaching a perfect discriminator scenario [16],
+[17], [15].
+• Adding a unique noise vector to each sample of the mini-
+batch of m real samples from pdata and the minibatch of
+m fake samples;
+• Applying One-sided Label Smoothing with β = 0.2.
+• Adding a Dropout layer after each Leaky ReLU layer on D.
+For these layers we chose to set 50% of the input’s values
+to 0;
+We came up with three expressions for the noise factor.
+The first one illustrates an increasing noise factor over the
+number of epochs:
+noise factor = epoch × 0.5
+epochs
+(4)
+In the second and third one, the noise factor increases till
+half the number of epochs and then decreases.
+noise factor =
+� ep×0.5
+half ep
+if ep ≤ half ep
+0.5 − ep×0.5
+ep
+otherwise
+(5)
+noise factor =
+�
+e×0.5
+h ep
+if e ≤ h ep
+0.5 − (e−h ep)×0.5
+h ep
+otherwise
+(6)
+In (5) the noise drops to half after half of the epochs have
+passed (e.g. 0.3 - 0.4 - 0.5 - 0.25 - ...) while in (6) the noise
+ascends and descends at a same rate (e.g 0.3 - 0.4 - 0.5 -
+0.4 - 0.3 - ...). We wanted to test what would happen if we
+hindered D till half of the training phase and then ease it either
+by suddenly decreasing the noise factor to half at the middle
+of the training phase or having a similar rate for increasing
+and decreasing the noise factor.
+B. WGAN
+The second tested architecture was the Wasserstein Generative
+Adversarial Network (WGAN) discussed in Section II-C. With
+this model we were expecting to work on some of the problems
+that came with the DCGAN model such as the failure to
+converge and the vanishing gradient problems.
+The structure of the generator from the DCGAN model and
+the WGAN model is exactly the same. The same goes for the
+discriminator from DCGAN model and the critic from WGAN
+model except we’re using a linear activation function as the
+last layer on the critic instead of the sigmoid, since the score
+for realness or fakeness of an image doesn’t have a limit value.
+There are some things to be taken in consideration about the
+training phase:
+• As discussed in section II-C, during each epoch, C is trained
+n critic times more than G;
+• It was not to possible to implement the Wasserstein distance
+with the formulation presented in [11] using Keras’ API
+methods because Keras doesn’t allow a sum of losses from
+independent batches. We used the fact that maximizing the
+Wasserstein distance is equivalent to increasing the distance
+between the Wasserstein score for positive examples and the
+score for negative examples. One simple way of achieving
+this is by returning positive estimates for real examples and
+negative estimates for fake examples6.
+• When updating the weights, we clip them to stay in an
+interval between a constant −c and c. This clipping is neces-
+sary to ensure the critic’s approximation to the Wasserstein
+distance, as explained in [11].
+C. ProgGAN
+For a third architecture we decided to apply the principles of
+the WGAN model and the idea of generating high-resolution
+images discussed in Section II-B together to create a GAN
+based on the Progressive Growing Generative Adversarial Net-
+work (ProgGAN). We wanted to verify if this technique would
+have interesting results in terms of training time optimization.
+We also wanted to see if it would generate images with higher
+resolution while maintaining a good quality.
+Fig. 5. ProgGAN’s Generator Model
+Fig. 5 and 6 depict the models of the ProgGAN’s generator
+and critic, respectively.
+The upsampling layers use an upscaling factor of 2 for
+both dimensions and a nearest neighbor interpolation. The
+downsampling layers perform an average pooling operation,
+using a downscaling factor of 2 for both dimensions and
+a stride of 2. During training, the phasing in from giving
+full weight to the 64 × 64 model (at the beginning of the
+training phase) to full weight to the 128 × 128 model (end
+of the training phase) is done through a weighted sum layer
+controlling how much to weight the input from the 64 × 64
+6https://machinelearningmastery.com/how-to-implement-wasserstein-loss-
+for-generative-adversarial-networks/
+
+64
+64
+64
+128
+128
+1
+64
+64
+64
+128
+128
+128
+100
+64
+64
+64
+128
+128
+128
+Transposed convolution layer followed by Batch Normalization Layer
+and LeakyReLu layer
+-64x64network
+Upsampling
+128x128 network
+ReshapeFig. 6. ProgGAN’s Critic Model
+and the 128 × 128 models. It uses a parameter α that grows
+linearly over the training phase.
+Each of the models was trained separately, in this order: 64×64
+model → growth model → 128×128 model. We treated each
+of the models as being a WGAN.
+D. VAE + WGAN
+As a final contribution, we wanted to combine the VAE and
+WGAN together. The idea was to train a VAE to encode and
+decode the images of our dataset of heightmaps. We hoped that
+using a generator that was already trained to create features
+from our dataset, such as in the VAE, would accelerate the
+WGAN’s training by increasing the efficiency in training time.
+Fig. 7 depicts the model of the encoder.
+Fig. 7. VAE + WGAN’s Encoder Model
+The decoder and the discriminator followed the same structure
+as the WGAN’s generator and critic. The only difference is
+that the last layer of the decoder uses a sigmoid activation
+function instead of the hyperbolic tangent used in WGAN’s
+generator.
+In terms of training, we trained the VAE and WGAN sepa-
+rately. We started with the VAE, by just training the model for
+a fixed number of epochs. Then we took the decoder model
+of the VAE and used it as our WGAN’s generator. However,
+instead of normalizing our dataset between −1 and 1 for both
+the networks, we normalized it between 0 and 1 since the last
+activation function of the decoder is a sigmoid, which ranges
+between the latter values. For some of the experiments, instead
+of generating the z vectors from N(µ, σ2) we generated
+them using the vectors µ and σ learned from the VAE. With
+this latter idea, we wanted to test if the decoder would be able
+to generate images with closer resemblance to the Heightmap
+dataset, since the vectors µ and σ contained features of that
+dataset.
+V. RESULTS
+All experiments were done in a desktop workstation with
+architecture x86 64, CPU AMD Ryzen 5 2600X Six-Core
+processor and one GeForce RTX 2080 Ti graphic cards with
+11 GB RAM. For the implementation and training of the
+networks we used Keras7, a deep learning API running on top
+of Tensorflow8. It provides abstractions and building blocks
+for developing and solving machine learning solutions.
+A. DCGAN
+In terms of results, in the four experiments we performed,
+the networks were compiled using the Adam optimizer with
+learning rate 0.0002 and β19 = 0.5. Both values were chosen
+as suggested in [7]. The networks were trained for 1000
+Epochs. A brief description of every experiment made is
+described below:
+• E1: None of the discriminator hindering methods were used;
+• E2: All the discriminator hindering methods were imple-
+mented; Equation (4) for the noise vector;
+• E3: All the discriminator hindering methods were imple-
+mented; Equation (5) for the noise vector;
+• E4: All the discriminator hindering methods were imple-
+mented; Equation (6) for the noise vector;
+In Fig. 9 one can observe the losses of D and G throughout the
+epochs for E1, and we can see the GAN’s vanishing gradient
+problem appearing.
+As we can observe, D’s loss rapidly converges to 0, while G’s
+loss keeps growing unsteadily. Early during the training phase,
+D reaches its optimal state, which is perfectly discriminating
+between the real and fake images. Thus, G is unable to
+improve, leading to a loss growth.
+For E2, E3 and E4, despite the fact that the hindering methods
+above mentioned did in fact help battle the GAN’s vanishing
+gradient problem, the results obtained have a poor quality.
+In E1, we can see that the model entered in a mode collapse,
+since the images generated have similar shapes, with some
+of them even having a blurry artifact. In E2 however, the
+quality of the images decayed, with the resulting images
+presenting some kind of artifacts in the form of a squared
+pattern. Experiments E3 and E4 yielded better results in terms
+of quality. Despite this results, all the images represented in
+Fig. 10 were classified as fake by the experiment’s respective
+discriminators.
+7https://keras.io/, the version used was 2.3.1
+8https://www.tensorflow.org/, the version used was 1.14.0
+9Exponential decay for the running average of the gradient
+
+1
+64
+64
+128
+128
+128
+128
+128
+128
+64
+64
+Output
+64
+128
+128
+128
+Convolution layer followed by Batch Normalization Layer
+and LeakyReLu layer
+-64x64network
+Downsampling
+- 128 x 128 network
+Flatten1
+16
+32
+64
+128
+1
+128
+U
+512
+64
+32
+16
+8
+>1024
+8
+16
+a
+512
+32
+128
+64
+Convolution layer followed by Batch Normalization Layer
+andReLulayer
+Flatten
+Dense layerFig. 8. VAE + WGAN architecture. Train the VAE (1) to learn the representation of the Heightmap dataset. Then use the sample vector z created from the
+distribution’s mean and standard deviation to train the WGAN (2). The VAE’s decoder will be the generator of the WGAN.
+Fig. 9. D and G Binary Cross Entropy loss of the Experiment 1
+Fig. 10. Some of the images generated by DCGAN’s generator after training
+for 1000 epochs, in the several experiments.
+B. WGAN
+The network was compiled using the RMSProp optimizer with
+learning rate of 0.0005, as in [11]. In terms of results, we
+first ran some tests to see which value we should use as a
+constraint for clipping the weights values from this list of
+values: 0.01, 0.02, 0.05, 0.1, 0.15, 0.2. We ended up choosing
+0.1 as the constraint since it gave the best results in terms of
+the Wasserstein estimate.
+We ran a single but extensive experiment E5 where we trained
+the network for 5000 epochs with a training time of 35 hours
+and 27 minutes. Fig. 11 depicts the Wasserstein estimates of
+E5. The blue and yellow lines, which represent the Wasserstein
+estimate for the critic real and fake images, respectively, are
+distancing from each other. We can also verify this by looking
+at the other graphic, where the red line, which represents
+the difference between those two estimates, keeps growing
+troughout the training epochs.
+Fig. 11.
+Left: Wasserstein estimate for C real and fake images and G
+generated images during the training phase; Right: Difference between the
+C Wasserstein estimate of the real images and the fake images
+Fig. 12 depicts some of the images generated during the last
+epochs. The generator was able to reproduce realistic images
+with a relative good quality and level of detail. Some complex
+structures are also present such as peninsulas, mountain ranges
+and groups of islands. Fig. 13 depicts a 3D representation of
+a heightmap generated in E5, where we can see a peninsula
+and some small islands next to it.
+Given the results we achieved with the previous experiment,
+we decided to run another experiment in order to check what
+type of results another WGAN, with a more complex structure
+
+Experiment1
+Experiment 2
+Experiment3
+Experiment41e7
+criticandgeneratorestimate
+le7
+criticdifferencewassersteinestimate
+crit_real
+4-
+crit_fake
+gen
+8
+2 -
+6
+estimate
+0
+wasser
+-2
+2 
+-4-
+-0
+0
+1000
+2000
+3000
+4000
+5000
+0
+1000
+2000
+3000
+4000
+5000
+epoch
+epoch2
+z
+Decoder/
+Update model
+Generator
+L
+G(z)
+G(z)
+Decoder /
+Encoder
+Generator
+G(z)
+D(x)
+D(x)
+Update model
+Heightmap
+Datasetdiscriminatorloss
+generatorloss
+6
+0.6
+0.5 
+7
+0.4 
+6.
+loss
+0.3
+4.
+0.2
+3
+0.1
+2
+0.0
+0
+200
+400
+600
+800
+1000
+200
+400
+600
+800
+1000
+epoch
+epochFig. 12. Images generated by WGAN’s generator during the last 5 training
+epochs
+Fig. 13. 3D representation of a heightmap generated in E5
+and higher number of neurons than the previous one, could
+generate. So, in experiment E6 we trained the 128 × 128
+model presented in subsection IV-C for 5000 epochs, which
+took 217 hours of training time. Fig. 14 shows some of the
+images generated through some training epochs of E6. Using
+a WGAN with a more complex structure proved not to be so
+efficient, given that the images between experiments E5 and
+E6 have similar quality.
+Fig. 14. Images generated by the 128 × 128 model’s generator during the
+training of E6
+C. ProgGAN
+Taking into account the network structure and training algo-
+rithm previously discussed, we made one experiment regarding
+the ProgGAN model. To be able to verify the efficiency in
+training time we had to run an experiment E7 with similar
+amount of epochs to E6. Therefore, we trained the model for
+4950 epochs ( 1650 epochs for each of the networks ).
+Table I depict the training time for E6 and E7. As we can
+see, the ProgGAN model took less 57 hours to train than the
+WGAN model.
+TABLE I
+TRAINING TIME FOR EACH EXPERIMENT
+E6
+E7
+64 × 64
+25h 3min
+growth
+74h 9min
+128 × 128
+217h
+71h 10min
+TOTAL
+217h
+170h 23 min
+Fig 15, 16 and 17 depict the Wasserstein estimate of both
+generators and critics for the 64 × 64, growth and 128 × 128
+models, respectively.
+Fig. 15.
+Left: Wasserstein estimate for C real and fake images and G
+generated images during the training phase of the 64 × 64 model; Right:
+Difference between the C Wasserstein estimate of the real images and the
+fake images
+Fig. 16.
+Left: Wasserstein estimate for C real and fake images and G
+generated images during the training phase of the growth model; Right:
+Difference between the C Wasserstein estimate of the real images and the
+fake images
+Despite the figures showing an erratic and unstable movement
+of the Wasserstein estimate during the training phase, we can
+see that in all of the models, the estimates of the real and fake
+images were moving away from each other, hence the line of
+the difference of the Wasserstein estimate on the critic going
+up.
+Fig. 18 shows some of the images generated through some
+training epochs of E7. We can see that the quality of the
+images from the 64 × 64 model to the growth model and
+to the 128 × 128 didn’t deteriorate. We can even say that
+
+Epocho
+Epoch1250
+Epoch2500
+Epoch3750
+Epoch5000critic and generatorwasserstein estimate
+criticdifferencewassersteinestimate
+350000
+crit_real
+70000
+crit_fake
+300000
+gen
+60000
+imate
+250000
+50000
+estll
+200000
+40000
+150000
+30000
+100000
+20000-
+50000
+10000
+0
+250
+500
+750
+1000
+1250
+1500
+0
+250
+500
+750
+1000
+1250
+1500
+epoch
+epoch1e7
+criticandgeneratorwassersteinestimate
+le7
+criticdifferencewassersteinestimate
+2
+crit_real
+crit_fake
+gen
+2.5
+1
+wassersteinestimate
+2.0
+1.5
+1.0 
+-1
+0.5
+-2.
+0.0
+0
+250
+500
+750
+1000
+1250
+1500
+0
+250
+500
+750
+1000
+1250
+1500
+epoch
+epochFig. 17.
+Left: Wasserstein estimate for C real and fake images and G
+generated images during the training phase of the 128 × 128 model; Right:
+Difference between the C Wasserstein estimate of the real images and the
+fake images
+the quality improved, since there is more variety of complex
+structures present in the end of the training phase such as bays,
+peninsulas, mountain ranges near the shore and islands.
+Fig. 18. Images generated by the ProgGAN’s generator during the training
+epochs for the several models’ networks in E7
+D. VAE + WGAN
+We did several experiments with the VAE + WGAN architec-
+ture, to analyze factors such as:
+• Number of epochs needed to train the VAE in order for it to
+recreate the dataset images with a reasonable level of detail;
+• Generation of the random z vector in order to be given as
+input to WGAN’s generator;
+• Comparison between training a WGAN with the decoder
+having already learned features from the Heightmap dataset
+and a WGAN without a pre-trained decoder.
+Taking into account these factors the experiments are described
+as follow:
+• E8: VAE trained for 1000 epochs. Decoder with the weights
+already trained used as the WGAN’s generator. WGAN
+trained for 1000 more epochs, using a normal distribution
+with µ = 0 and σ = 1 for generating z;
+• E9: VAE trained for 150 epochs. Decoder with the weights
+already trained used as the WGAN’s generator. WGAN
+trained for 1000 more epochs, using a normal distribution
+with µ = 0 and σ = 1 for generating z;
+• E10: Decoder, with the weights already trained in Experi-
+ment 2, used as the WGAN’s generator. WGAN trained for
+1000 more epochs, using the vectors µ and σ learned from
+the VAE, for generating z.
+• E11: WGAN trained for 1150 epochs to make a comparison
+between the models.
+The training time for each experiment is denoted in Table
+II. As we can see, the difference between E9 or E10, which
+corresponds to training VAE for 150 epochs and training
+the WGAN for 1000 epochs and E11 which corresponds to
+training the WGAN for 1150 epochs isn’t that big, saving at
+maximum 20 minutes with the latter experiment. Fig. 23 below
+depicts images, for the several experiments, generated by the
+WGAN after training.
+TABLE II
+TRAINING TIME FOR EACH EXPERIMENT
+Experiment
+VAE
+WGAN
+Total
+E8
+8h 42 min
+7h 28 min
+16h 8 min
+E9
+1h 18 min
+7h 30 min
+8h 48 min
+E10
+1h 18 min
+7h 20 min
+8h 38 min
+E11
+8h 24 min
+8h 24 min
+Regarding E8, the loss and estimates are shown in Fig. 19. The
+VAE’s training went well as expected, with its loss rapidly
+decreasing in the initial epochs, but stagnating around 7250
+towards the end of the training. The WGAN’s training also
+went smoothly, despite some moments between the epochs
+800 and 1000 where all the estimates suddenly got close to
+each other.
+Fig. 19.
+E8: Top: VAE’ loss during the training phase; Left: Wasserstein
+estimate for C real and fake images and decoder’s generated images during
+the training phase; Right: Difference between the C Wasserstein estimate of
+the real images and the fake images
+Since the stagnation of VAE’s loss happened early in its
+training phase we thought we could have trained it for a lower
+amount of epochs and still be able to generate images with
+good quality. Taking that into account, in E9, as shown in
+Fig. 20, we trained the VAE for only 150 epochs. In this
+experiment, the WGAN training went really smoothly, as can
+be seen by the Wasserstein estimate graphic and the images
+generated. However, in this experiment, there were images,
+
+1e8
+criticandgeneratorwassersteinestimate
+1e8
+criticdifferencewassersteinestimate
+1.5
+crit real
+crit_fake
+2.5
+1.0
+gen
+2.0
+wassersteinestimate
+0.5
+1.5
+0.0
+1.0
+-0.5
+0.5
+-1.0 -
+0.0
+0
+250
+500
+750
+1000
+1250
+1500
+0
+250
+500
+750
+1000
+1250
+1500
+epoch
+epoch64 x 64
+Growth
+128 x 128
+Epocho
+Epoch825
+Epoch1650VAEloSS
+7800
+7700
+7600
+SSO
+7500
+7400
+7300
+7200
+0
+200
+400
+600
+800
+1000
+Epoch
+1e7
+criticandgeneratorestimate
+le7
+criticdifferencewassersteinestimate
+1.0
+crit_real
+crit_fake
+2.0
+dec
+0.5
+1.5
+estimate
+0.0
+1.0
+0.5
+0.5 
+1.0 -
+0.0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+epoch
+epochgenerated throughout the training process, that presented some
+strange artifacts as seen in Fig. 24. Even after the training
+was complete, some images with those kind of artifacts were
+generated.
+Fig. 20.
+E9: Top: VAE’ loss during the training phase; Left: Wasserstein
+estimate for C real and fake images and decoder’s generated images during
+the training phase; Right: Difference between the C Wasserstein estimate of
+the real images and the fake images
+In E10 we took the weights of the decoder trained in the
+VAE of the previous experiment and trained the WGAN for
+1000 epochs again. However, we are using the vectors µ and
+σ learned from the VAE to generate our vectors z, since
+they contain features of the Heightmap dataset. Fig. 21 and
+23 depict the results of E10. The graphic of the Wasserstein
+estimate is not what we expected it to be. From epoch 400, the
+images generated by the decoder managed to fool the critic
+into thinking they were becoming more real throughout the
+remaining epochs, given that the Wasserstein estimate of the
+fake images went up after that epoch.
+Fig. 21.
+E10: Left: Wasserstein estimate for C real and fake images and
+decoder’s generated images during the training phase; Right: Difference
+between the C Wasserstein estimate of the real images and the fake images
+As the last experiment, we wanted to just train the WGAN for
+1150 epochs, in order to make a proper comparison between
+the other experiments. Fig. 22 and 23 depict the results.
+Discussion
+The results obtained with the DCGAN were unsatisfying, as
+we were already expecting, due to several problems with this
+architecture already mentioned in the literature. The WGAN
+Fig. 22.
+E11: Left: Wasserstein estimate for C real and fake images and
+decoder’s generated images during the training phase; Right: Difference
+between the C Wasserstein estimate of the real images and the fake images
+Fig. 23. Images generated by WGAN after training, in the several experiments
+Fig. 24. Images with some strange artifacts generated by WGAN during the
+epochs for E10
+proved to be a good option since throughout the whole training
+phase the generator always had enough information provided
+by the critic to improve itself, thus ending up generating
+heightmaps with good quality. With the ProgGAN we saw
+that we could achieve the same results as in the WGAN with
+less training time. Curiously, the Wasserstein estimates did not
+follow the expected pattern. With the VAE + WGAN model,
+although it seemed to be a promising model in paper, we didn’t
+get quite the results we expected.
+VI. CONCLUSIONS
+In this work, our purpose was to explore the GANs’ capabili-
+ties to generate images with high resolution and quality to try
+to create maps that look realistic and appealing for players, in
+
+VAEloSS
+7700
+7600
+7400
+7300
+20
+40
+60
+80
+100
+120
+140
+Epoch
+1e7
+criticandgeneratorestimate
+1e7
+critic difference wasserstein estimate
+crit_real
+crit_fake
+2.5
+1.0
+dec
+2.0
+0.5
+1.5 
+0.0
+1.0
+0.5
+0.5 
+1.0 
+1.5 
+0.0 
+200
+400
+600
+008
+1000
+0
+200
+400
+600
+800
+1000
+epoch
+epochle6
+criticandgeneratorestimate
+1e6
+criticdifferencewassersteinestimate
+2.0
+crit real
+3.5
+1.5
+crit_fake
+dec
+3.0
+1.0
+2.5
+0.5
+ate
+2.0
+0.0
+estim
+1.5
+0.5
+1.0
+1.0
+-1.5
+0.5
+2.0
+0.0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+epoch
+epoch1e6
+criticandgeneratorestimate
+1e6
+criticdifferencewassersteinestimate
+2.00
+crit_real
+crit_fake
+1.0
+dec
+1.75
+1.50
+0.5
+1.25
+imate
+1.00
+ISe
+0.0
+0.75
+0.50
+-0.5
+0.25
+-1.0
+0.00
+0
+200
+400
+600
+800
+1000
+1200
+0
+200
+400
+600
+800
+1000
+1200
+epoch
+epochE8
+E9
+E10
+E11a visual and even strategic way. That could not be done with
+the traditional approaches of PCG. Instead of just focusing on
+one model of the current state of the art of GANs we decided
+to explore several ones and test if they could indeed be used
+to generate such maps. We ended up exploring four models:
+DCGAN, WGAN, ProgGAN and VAE + WGAN, each one
+with its upsides and downsides. The DCGAN and VAE +
+WGAN models provided results with poor quality while the
+WGAN and ProgGAN models provided the best results, with
+the latter one being more efficient in terms of training time.
+Despite the poor results of the VAE + WGAN model, we
+think that this last model is one that should be further studied
+because its concept is promising.
+Given the server memory limitations we were only able to
+work with images of 128 × 128 resolution while the ideal
+scenario would be to work with an higher resolution such as
+1024×1024. Further individual studies of each of these models
+or the ability to be able to expand these networks to images
+with higher resolution, given the proper hardware, are some
+examples of future work that could be explored and studied
+regarding this topic.
+ACKNOWLEDGMENTS
+This work was supported by national funds through FCT,
+Fundac¸˜ao para a Ciˆencia e a Tecnologia, under projects
+UIDB/04326/2020, UIDB/50021/2020, UIDP/04326/2020 and
+LA/P/0101/2020.
+REFERENCES
+[1]
+Ken Perlin. “An Image Synthesizer”. In: New York,
+NY, USA: Association for Computing Machinery, 1985,
+pp. 287–296.
+[2]
+Xing Mei, Philippe Decaudin, and Bao Gang Hu.
+“Fast hydraulic erosion simulation and visualization
+on GPU”. In: Proceedings - Pacific Conference on
+Computer Graphics and Applications. 2007.
+[3]
+Ian Goodfellow et al. “Generative adversarial net-
+works”. In: arXiv (2014).
+[4]
+Tero Karras et al. “Progressive growing of GANs for
+improved quality, stability, and variation”. In: 6th In-
+ternational Conference on Learning Representations,
+ICLR 2018 - Conference Track Proceedings. 2018.
+[5]
+Mehdi Mirza and Simon Osindero. “Conditional Gen-
+erative Adversarial Nets”. In: arXiv (2014).
+[6]
+Jun Yan Zhu et al. “Unpaired Image-to-Image Transla-
+tion Using Cycle-Consistent Adversarial Networks”. In:
+2017, pp. 2242–2251.
+[7]
+Alec Radford, Luke Metz, and Soumith Chintala. “Un-
+supervised representation learning with deep convo-
+lutional generative adversarial networks”. In: arXiv
+(2016).
+[8]
+Ian Goodfellow, Yoshua Bengio, and Aaron Courville.
+Learning deep architectures for AI. MIT Press, 2016.
+[9]
+Wei Zhang et al. “Parallel distributed processing model
+with local space-invariant interconnections and its opti-
+cal architecture”. In: Applied Optics (1990).
+[10]
+Ziwei Liu, Ping Luo, and Xiaogang Wang andXiaoou
+Tang. “Deep Learning Face Attributes in the Wild”. In:
+Proceedings of International Conference on Computer
+Vision (ICCV). 2015.
+[11]
+Martin Arjovsky, Soumith Chintala, and L´eon Bottou.
+“Wasserstein GAN”. In: arXiv (2017).
+[12]
+Anders Boesen Lindbo Larsen et al. “Autoencoding
+beyond pixels using a learned similarity metric”. In:
+arXiv (2016).
+[13]
+Yoshua Bengio. Learning deep architectures for AI.
+Hanover, MA, USA: Now Publishers Inc., 2009.
+[14]
+Andrew Gordon Wilson, Been Kim, and William Her-
+lands. “Proceedings of NIPS 2016 Workshop on Inter-
+pretable Machine Learning for Complex Systems”. In:
+arXiv (2016).
+[15]
+Martin Arjovsky and L´eon Bottou. “Towards Princi-
+pled Methods for Training Generative Adversarial Net-
+works”. In: arXiv (2017).
+[16]
+Tim Salimans et al. “Improved techniques for training
+GANs”. In: arXiv (2016).
+[17]
+Phillip Isola et al. “Image-to-Image Translation with
+Conditional Adversarial Networks”. In: arXiv (2018).
+APPENDIX A
+In this appendix we show the detailed architecture of the
+implemented models and the hyper-parameters used, to allow
+for reproducibility of the results. For the DCGAN network,
+we used the Adam Optimizer with learning rate 0.0002 and
+β1 = 0.5. For the WGAN and ProgGAN network, we used the
+RMSProp optimizer with learning rate 0.0005. For the VAE +
+WGAN model we trained the VAE and then the whole model,
+using the RMSProp optimizer with learning rate 0.0003 and
+0.0005, respectively.
+TABLE III
+GENERATOR STRUCTURE OF THE DCGAN MODEL USED
+Layer name
+Act.
+Input shape
+Output shape
+Dense
+-
+100 × 1 × 1
+65536 × 1 × 1
+Reshape
+-
+65536 × 1 × 1
+1024 × 8 × 8
+Deconv
+-
+1024 × 8 × 8
+256 × 16 × 16
+BatchNorm
+-
+256 × 16 × 16
+256 × 16 × 16
+LeakyReLU
+LeakyReLU
+256 × 16 × 16
+256 × 16 × 16
+Dropout
+-
+256 × 16 × 16
+256 × 16 × 16
+Deconv
+-
+256 × 16 × 16
+128 × 32 × 32
+BatchNorm
+-
+128 × 32 × 32
+128 × 32 × 32
+LeakyReLU
+LeakyReLU
+128 × 32 × 32
+128 × 32 × 32
+Dropout
+-
+128 × 32 × 32
+128 × 32 × 32
+Deconv
+-
+128 × 32 × 32
+64 × 64 × 64
+BatchNorm
+-
+64 × 64 × 64
+64 × 64 × 64
+LeakyReLU
+LeakyReLU
+64 × 64 × 64
+64 × 64 × 64
+Dropout
+-
+64 × 64 × 64
+64 × 64 × 64
+Deconv
+-
+64 × 64 × 64
+32 × 128 × 128
+BatchNorm
+-
+32 × 128 × 128
+32 × 128 × 128
+LeakyReLU
+LeakyReLU
+32 × 128 × 128
+32 × 128 × 128
+Dropout
+-
+32 × 128 × 128
+32 × 128 × 128
+Deconv
+tanh
+32 × 128 × 128
+1 × 128 × 128
+
+TABLE IV
+DISCRIMINATOR STRUCTURE OF THE DCGAN MODEL USED
+Layer name
+Act.
+Input shape
+Output shape
+Conv
+-
+1 × 128 × 128
+1 × 128 × 128
+LeakyReLU
+LeakyReLU
+1 × 128 × 128
+1 × 128 × 128
+Conv
+-
+1 × 128 × 128
+32 × 64 × 64
+BatchNorm
+-
+32 × 64 × 64
+32 × 64 × 64
+LeakyReLU
+LeakyReLU
+32 × 64 × 64
+32 × 64 × 64
+Conv
+-
+32 × 64 × 64
+64 × 32 × 32
+BatchNorm
+-
+64 × 32 × 32
+64 × 32 × 32
+LeakyReLU
+LeakyReLU
+64 × 32 × 32
+64 × 32 × 32
+Conv
+-
+64 × 32 × 32
+128 × 16 × 16
+BatchNorm
+-
+128 × 16 × 16
+128 × 16 × 16
+LeakyReLU
+LeakyReLU
+128 × 16 × 16
+128 × 16 × 16
+Conv
+-
+128 × 16 × 16
+256 × 8 × 8
+BatchNorm
+-
+256 × 8 × 8
+256 × 8 × 8
+LeakyReLU
+LeakyReLU
+256 × 8 × 8
+256 × 8 × 8
+Flatten
+-
+256 × 8 × 8
+16384 × 1 × 1
+Dense
+sigmoid
+16384 × 1 × 1
+1 × 1 × 1
+TABLE V
+BLOCKS OF LAYERS USED IN THE PROGGAN NETWORK’S STRUCTURES
+Layer name
+Act.
+Output shape
+Deconv
+-
+128 × 64 × 64
+BatchNorm
+-
+128 × 64 × 64
+LeakyReLU
+LeakyReLU
+128 × 64 × 64
+DECONV 1
+Deconv
+-
+128 × 64 × 64
+BatchNorm
+-
+128 × 64 × 64
+LeakyReLU
+LeakyReLU
+128 × 64 × 64
+UpSampling
+-
+128 × 128 × 128
+Deconv
+-
+64 × 128 × 128
+BatchNorm
+-
+64 × 128 × 128
+DECONV 2
+LeakyReLU
+LeakyReLU
+64 × 128 × 128
+Deconv
+-
+64 × 128 × 128
+BatchNorm
+-
+64 × 128 × 128
+LeakyReLU
+LeakyReLU
+64 × 128 × 128
+Conv
+-
+64 × 128 × 128
+BatchNorm
+-
+64 × 128 × 128
+LeakyReLU
+LeakyReLU
+64 × 128 × 128
+CONV 1
+Conv
+-
+128 × 128 × 128
+BatchNorm
+-
+128 × 128 × 128
+LeakyReLU
+LeakyReLU
+128 × 128 × 128
+Downsample
+-
+128 × 64 × 64
+Conv
+-
+128 × 64 × 64
+BatchNorm
+-
+128 × 64 × 64
+CONV 2
+LeakyReLU
+LeakyReLU
+128 × 64 × 64
+Conv
+-
+128 × 64 × 64
+BatchNorm
+-
+128 × 64 × 64
+LeakyReLU
+LeakyReLU
+128 × 64 × 64
+TABLE VI
+GENERATOR STRUCTURE OF THE 64 × 64 NETWORK
+Layer name
+Act.
+Input shape
+Output shape
+Dense
+-
+100 × 1 × 1
+262144 × 1 × 1
+Reshape
+-
+262144 × 1 × 1
+64 × 64 × 64
+DECONV 1
+Deconv
+Tanh
+128 × 64 × 64
+1 × 64 × 64
+TABLE VII
+CRITIC STRUCTURE OF THE 64 × 64 NETWORK
+Layer name
+Act.
+Input shape
+Output shape
+Conv
+-
+1 × 64 × 64
+128 × 64 × 64
+LeakyReLU
+LeakyReLU
+128 × 64 × 64
+128 × 64 × 64
+CONV 2
+Flatten
+-
+128 × 64 × 64
+524288 × 1 × 1
+Dense
+linear
+524288 × 1 × 1
+1 × 1 × 1
+TABLE VIII
+GENERATOR STRUCTURE OF THE 128 × 128 NETWORK
+Layer name
+Act.
+Input shape
+Output shape
+Dense
+-
+100 × 1 × 1
+262144 × 1 × 1
+Reshape
+-
+262144 × 1 × 1
+64 × 64 × 64
+DECONV 1
+DECONV 2
+Deconv
+Tanh
+64 × 128 × 128
+1 × 128 × 128
+TABLE IX
+CRITIC STRUCTURE OF THE 128 × 128 NETWORK
+Layer name
+Act.
+Input shape
+Output shape
+Conv
+-
+1 × 128 × 128
+64 × 128 × 128
+LeakyReLU
+LeakyReLU
+64 × 128 × 128
+64 × 128 × 128
+CONV 1
+CONV 2
+Flatten
+-
+128 × 64 × 64
+524288 × 1 × 1
+Dense
+linear
+524288 × 1 × 1
+1 × 1 × 1
+TABLE X
+ENCODER STRUCTURE OF THE VAE + WGAN MODEL USED
+Layer name
+Act.
+Input shape
+Output shape
+Conv
+-
+1 × 128 × 128
+16 × 64 × 64
+BatchNorm
+-
+16 × 64 × 64
+16 × 64 × 64
+ReLU
+ReLU
+16 × 64 × 64
+16 × 64 × 64
+Conv
+-
+16 × 64 × 64
+32 × 32 × 32
+BatchNorm
+-
+32 × 32 × 32
+32 × 32 × 32
+ReLU
+ReLU
+32 × 32 × 32
+32 × 32 × 32
+Conv
+-
+32 × 32 × 32
+64 × 16 × 16
+BatchNorm
+-
+64 × 16 × 16
+64 × 16 × 16
+ReLU
+ReLU
+64 × 16 × 16
+64 × 16 × 16
+Conv
+-
+64 × 16 × 16
+128 × 8 × 8
+BatchNorm
+-
+128 × 8 × 8
+128 × 8 × 8
+ReLU
+ReLU
+128 × 8 × 8
+128 × 8 × 8
+Flatten
+-
+128 × 8 × 8
+8192 × 1 × 1
+Dense
+-
+8192 × 1 × 1
+1024 × 1 × 1
+BatchNorm
+-
+1024 × 1 × 1
+1024 × 1 × 1
+ReLU
+ReLU
+1024 × 1 × 1
+1024 × 1 × 1
+µ (Dense)
+-
+1024 × 1 × 1
+512 × 1 × 1
+σ (Dense)
+-
+1024 × 1 × 1
+512 × 1 × 1
+

BNAyT4oBgHgl3EQfd_jC/content/tmp_files/2301.00314v1.pdf.txt

@@ -0,0 +1,2042 @@
+Edited: In the Proc. of the 26th International Conference on Pattern Recognition (ICPR’22) Montreal, Canada, Aug. 21-25, 2022.
+Causal Deep Learning
+M. Alex O. Vasilescu
+[email protected]
+IPAM, University of California, Los Angeles
+Tensor Vision Technologies, Los Angeles
+Abstract—We derive a set of causal deep neural networks whose
+architectures are a consequence of tensor (multilinear) factor
+analysis. Forward causal questions are addressed with a neural
+network architecture composed of causal capsules and a tensor
+transformer. The former estimate a set of latent variables that rep-
+resent the causal factors, and the latter governs their interaction.
+Causal capsules and tensor transformers may be implemented
+using shallow autoencoders, but for a scalable architecture we
+employ block algebra and derive a deep neural network composed
+of a hierarchy of autoencoders. An interleaved kernel hierarchy
+pre-processes the data resulting in a hierarchy of kernel ten-
+sor factor models. Inverse causal questions are addressed with
+a neural network that implements multilinear projection and
+estimates the causes of effects. As an alternative to aggressive
+bottleneck dimension reduction or regularized regression that may
+camouflage an inherently underdetermined inverse problem, we
+prescribe modeling different aspects of the mechanism of data
+formation with piecewise tensor models whose multilinear projec-
+tions are well-defined and produce multiple candidate solutions.
+Our forward and inverse neural network architectures are suitable
+for asynchronous parallel computation.
+Index Terms—causality, factor analysis, tensor decomposition
+transformer, Hebb learning, neural networks
+I. INTRODUCTION
+Building upon prior representation learning efforts aimed at
+disentangling the causal factors of data variation [28][8][92]
+[72][71]1, we derive a set of causal deep neural networks
+that are a consequence of tensor (multilinear) factor analysis.
+Tensor factor analysis is a transparent framework for modeling
+a hypothesized multi-causal mechanisms of data formation,
+computing invariant causal representations, and estimating
+the effects of interventions [103][100][108][105]. The validity
+and strength of causal explanations depend on causal model
+specifications in conjunction with experimental designs for
+acquiring suitable training data [81].
+Unlike conventional statistics and machine learning which
+model observed data distributions and make predictions about
+one variable co-observed with another, or perform time series
+forecasting, causal inference is a hypothesis-driven process,
+as opposed to a data-driven process, that models the mecha-
+nism of data formation and estimates the effects of interven-
+tions [78][46][89][103][99]. Inverse causal “inference” estimates
+the causes of effects given an estimated forward model and
+constraints on the solution set [32][100][109].
+1Representation learning has been performed with deep neural net-
+works [8][63][84] composed of architectural modules, such as Restricted
+Boltzmann Machines (RBMs) [37][72], spike-and-slab RBMs [28][71], autoen-
+coders [61][9][70], or encoder-decoders [79][15][50], and have been trained
+in a supervised [64][22][71], unsupervised [37][8][28][61][79], and semi-
+supervised manner [73]. Deep neural networks have been employed in life-
+critical application areas, such as medical diagnosis [54][68][94], and face
+recognition [91][42][90][20].
+Fig. 1: A forward causal neural network is composed of
+a set of causal capsules and a tensor transformer. Causal
+capsules estimate the causal factor representations Um, and a
+tensor transformer T governs their interaction. Causal capsules
+and tensor transformers can be implemented with a set of
+shallow autoencoders and tensor autoencoders, respectively. For
+a scalable architecture, each autoencoder is replaced with a
+deep neural network composed of a part based hierarchy of
+autoencoders (Fig. 3f). The above neural network depicts the M-
+mode SVD [106][105], a parallel multilinear rank decomposition.
+(In practice, images are vectorized and centered.)
+Causal Inference Versus Regression
+Neural networks and tensor factorization methods may be causal
+in nature and perform causal inference, or simply perform
+regression from which no causal conclusions are drawn. For
+causal inference, hypothesis-driven experimental design for
+generating training data [81], and model specifications (Fig. 2)
+trump algorithmic design and analysis.2
+2Tensor causal factor analysis have been employed in the analysis and
+recognition of facial identities [108][103], facial expressions[44], human motion
+signatures [97][24][41], and 3D sound [33]. It has been employed in the
+transfer of facial expressions [111], the rendering of textures suitable for
+arbitrary geometries, views and illuminations [107], etc.Tensor factor analysis
+has also been employed in psychometrics [95][34][17][10][60], econometrics
+[52],[69], chemometrics
+[13], and other fields. Tensor regression has been
+employed to estimate missing data [21] and to perform dimensionality reduction
+[115][112][59] [14][49][40][7] by taking advantage of the row, column and
+fiber redundancies. Recently, tensor regression has been employed in machine
+learning to reduce neural network parameters. Network parameters are organized
+into “data tensors”, and dimensionally reduced [62][74][56][55][76].
+arXiv:2301.00314v1  [cs.LG]  1 Jan 2023
+
+Causal Capsules:
+Interventions
+I, Illums.
+Mode-m Matrix C omputation, Um
+^ I Views
+Ip People
+二
+Interventions
+p
+Pixels
+m = D ×U+··*m- Ut-*+U++·** U
+matrixize
+X,[v]
+Xi,[L]
+X
+invariant
+Xi
+p,[P]
+invariant
+IP
+code
+xi
+code
+2（电（
+x2
+X2
+ 
+ll;
+[P]
+V
+IP
+/ [P]
+1X
+Nx
+XNpIx
+AT
+il
+Training Initialization
+Tensor Transformer:
+vec (Ri ini)
+Core Tensor Computation, T
+V3
+V4
+d
+0
+23
+P2
+Rt
+I, Illums.
+1.14P2
+di
+Y111
+d.
+Iy Views
+d.
+1
+ipivit
+d.
+Y211
+d2
+ tensorize
+1 p
+People
+ripivit
+matrixize
+13
+3'4P
+[x
+14
+N,P
+'2P
+'3P
+4P
+1s
+15vP
+5'2P
+T
+Ix Pixels
+YiplviL
+d.x
+R.A. Causal Neural Networks
+Causal neural networks are composed of causal capsules and
+tensor transformers (Fig. 1). Causal capsules estimate the latent
+variables that represent the causal factors of data formation.
+A tensor transformer governs the interaction of the latent
+variables. Causal capsules may be implemented as shallow
+Hebb autoencoders, which perform principal component analysis
+(PCA) [87][83][82][1][75] (see supplemental Sec VI-A) when
+the neurons are linear with non-deterministic activation [4][48].
+The tensor transformer may be implemented as a tensor
+autoencoder, a shallow autoencoder whose code is the tensor
+product of the latent variables.
+Causal deep neural networks are composed of stacking Hebb and
+tensor autoencoders. Each causal capsule or tensor transformer
+in a shallow causal neural network is replaced by mathematically
+equivalent deep architectures composed either of a part-based
+hierarchy of Hebb autoencoders or of a part-based hierarchy
+of tensor autoencoders. An interleaved hierarchy of kernel
+functions [86] serves as a pre-processor that warps the data
+manifold for optimal tensor factor analysis.3 The resulting deep
+causal neural network models the mechanism of data formation
+with a hierarchy of tensor factor models [103][102][99, Sec
+4.4] (see Supplemental Section VI-C).
+Inverse causal neural networks implement the multilinear
+projection algorithm to estimate the causes of effects [109][100].
+A neural network that addresses an underdetermined inverse
+problem is characterized by a wide hidden layer. Dimensionality
+reduction removes noise and nuisance variables [38], [93], and
+has the added benefit of reducing the widths of hidden layers.
+However, aggressive bottleneck dimensionality reduction may
+camouflage an inherently ill-posed problem. Alternatively or in
+addition to dimensionality reduction and regularized regression,
+we prescribe modeling different aspects of the data formation
+process with piecewise tensor (multilinear) models (mixture of
+experts) whose projections are well-defined [104]. Candidate
+solutions are gated to yield a unique solution.
+II. FORWARD CAUSAL QUESTION: “WHAT IF?”
+Forward causal inference is a hypothesis-driven process that
+addresses the “what if” question. Causal hypotheses drive both
+the experimental design for generating training data and the
+causal model specification.
+Training Data: For modeling the unit level effects of causes,
+the training data is generated by combinatorially varying
+each causal factor while holding the other factors fixed. The
+best causal evidence comes from randomized experimental
+studies. When physical, or statistical experiments for generating
+training data are unethical or infeasible, experiments may be
+approximated with carefully designed observational studies [81],
+such as natural experiments [2][16][47]. The certainty of causal
+conclusions are dependent on the type of evidence employed.4
+Models: Within the tensor mathematical framework (Supple-
+mental Section VI-B) a “data tensor,” D ∈ CI0×I1···×Im···×IM,
+3There have been a number of related transformer architectures engineered
+and empirically tested with success [29], [113], [66].
+4Datasheets for datasets, as proposed by Gebru et al. [31], may help facilitate
+the approximation of experimental studies.
+(a)
+(b)
+Fig. 2: Same data, same algorithm, but two different model
+specifications (problem setups) result in two semantically
+different decompositions. (a) Causal Inference: The M-mode
+SVD (Algorithm 1) factorizes a “data tensor” of vectorized
+observations into a set of latent variables that represent the
+causal factors. (b) Regression: The M-mode SVD factorizes a
+“data tensor” composed of images as a “data matrix” into the
+image column and row space as well as the normalized PCA
+coefficients. (Images represent their vectorized versions except
+in Fig. 2b.)
+Algorithm 1 M-mode SVD (parallel computation)[106], [105]
+Input D ∈ CI0×···×IM, dimensions R0, R1 . . . Rm . . . RM
+1. Initialize Um := I or random matrix, 0 ≤ m ≤ M
+2. Iterate until convergence
+For m := 0, . . . , M,
+• X := D ×0 UT
+0 × · · · ×m-1 UT
+m-1 ×m+1 UT
+m+1 · · · × UT
+M
+• Set Um to the ˜Rm leading left-singular vectors of the
+SVD of X[m] or SVD of [X[m]XT
+[m]]. a, b
+3. Set Z := D ×0 UT
+0 · · · ×m UT
+m · · · ×M UT
+M := X × UT
+M
+c
+Output mode matrices U0, U1, . . . , UM and core tensor Z.
+aThe computation of Um in the SVD X[m] = UmΣVmT can be performed
+efficiently, depending on which dimension of X[m] is smaller, by decomposing
+either X[m]X[m]T = UmΣ2UmT (note that VmT = Σ+UmTX[m]) or
+by decomposing X[m]TX[m] = VmΣ2VmT and then computing Um =
+X[m]VmΣ+.
+b For a neural network implementation, the SVD of X[m] is replaced with
+a Hebb autoencoder that sequentially computes the orthonormal columns of
+Um/Vn by performing gradient descent or stochastic gradient descent [12][80].
+In Fig. 1, the autoencoders learn the columns in Vm. Matrix Vm,r contains the
+first r columns; vm,r is column r.
+Hebb Autoencoder
+�
+��
+�
+For r := 1 . . . Rm.
+Iterate until convergence
+∆vm,r(t+1)=η
+�
+X[m] − Vm,r(t)VT
+m,r(t)X[m]
+�
+XT
+[m]vm,r(t)
+�
+��
+�
+code
+ˆvm,r(t+1)=
+�
+vm,r(t) + ∆vm,r(t+1)
+�
+∥vm,r(t) + ∆vm,r(t+1)∥
+cThe columns in Z[0] may be computed by initializing the code of an
+autoencoder to (UM · · · ⊗ Um · · · ⊗ U0), where ⊗ is the Kronecker product.
+In Fig. 1, the columns of the extended core T are computed by initializing the
+code of the autoencoder with (UM · · · ⊗ Um · · · ⊗ U1).
+
+people variance
+person signature, p
+X
+★peoplevariance
+Up
+Views
+lew
+People
+variance
+M-mode SVD
+viewvariance
+Pixels
+illumination variance
+illuminations
+Uv
+x
+UL
+illumination
+view
+representation.IT
+illuminationvariance
+representation,vT Images
+Ri
+Normalized PCA
+Images
+CoefficientMatrix
+U
+R
+IT
+M-mode SVD
+Rxc
+Pixels
+Z
+U
+Pixels
+xr
+Rxc
+Column
+Row
+Space
+Space
+Ixr
+XC
+IxcCausal Deep Learning
+ICPR 2022, Montreal, Canada
+Algorithm 2 Kernel Tensor Factor Analysis [99, Sec 4.4][108]
+Kernel Multilinear Independent Component Analysis (K-MICA) and Kernel Principal Component Analysis (K-MPCA).
+Input the data tensor D ∈ CI0×···×IM , where mode m = 0 is the measurement mode, and the desired ranks are ˜R1, . . . , ˜RM.
+Initialize Cm = I or random matrix, ∀ 0 ≤ m ≤ M
+Iterate until convergence.
+1) For m := 1, . . . , M
+a) Set Xm := D ×1 C+
+1 · · · ×m−1 C+
+m−1 ×m+1 C+
+m+1 · · · ×M C+
+M.
+b) Compute the elements of the mode-m covariance matrix, for j, k := 1, . . . , Im:
+[X[m]X[m]
+T]jk:=
+I1
+�
+i1=1
+...
+Im−1
+�
+im-1=1
+Im+1
+�
+im+1=1
+...
+IM
+�
+iM=1
+K(xi1...im-1 j im+1...iM, xi1...im-1 k im+1...iM).
+(1)
+c) a
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+For K-MPCA:
+Set Cm := U to the left matrix of the SVD, of [X[m]X[m]
+T] = UmΣ2UT
+m from (1)
+Truncate to ˜Rm columns Um ∈ CIm× ˜
+Rm.
+For K-MICA:
+Set Cm := UmW−1
+m . The additional invertible matrix Wm may be computed based on negentropy,
+mutual information, or higher-order cumulants [108].
+The initial SVD of [X[m]X[m]] T (1) truncates the subspace to ˜Rm.
+2) Set B := XM ×M C+
+M. For K-MPCA, C+
+M = CT
+M.
+Output the converged extended core tensor T ∈ CI0× ˜
+R1×···× ˜
+RM and causal factor mode matrices C1, . . . , CM.
+aEvery SVD step may be computed by gradient descent and replaced with a Hebb autoencoder-decoder. See Algorithm 1 footnotes a and b. See Fig. 3 for a
+scalable neural network implementation.
+Linear kernel:
+K(u, v) = uTv = u · v
+Polynomial kernel of degree d:
+K(u, v) = (uTv)d
+Polynomial kernel up to degree d:
+K(u, v) = (uTv + 1)d
+Sigmoidal kernel:
+K(u, v) = tanh(α uTv + β)
+Gaussian (radial basis function (RBF)) kernel:
+K(u, v) = exp
+�
+− ∥u−v∥2
+2σ2
+�
+TABLE I: Common kernel functions. Kernel
+functions are symmetric, positive semi-definite
+functions corresponding to symmetric, positive
+semi-definite Gram matrices. The linear kernel
+does not modify or warp the feature space.
+contains a collection of vectorized5 and centered observations,
+di1...im...iM ∈ CI0 that are the result of M causal factors. Causal
+factor m (1 ≤ m ≤ M) takes one of Im values that are
+indexed by im, 1 ≤ im ≤ Im. An observation and a data tensor
+are modeled by a multilinear equation with multimode latent
+variables:
+D = T ×1 U1 · · · × Um ×M UM + E,
+(2)
+di1,...,iM = T ×1 (ˆu
+T
+i1 + ϵ
+T
+i1) · · · ×M (ˆu
+T
+iM + ϵ
+T
+iM) + ξi1,...,iM,
+where T is the extended core that contains the basis vectors and
+governs the interaction between the latent variables ˆuT
+im (row
+i of Um) that represent the causal factors of data formation,
+ϵim ∈ N(0, Σm) are disturbances with Gaussian distribution,
+and ξi1,...,iM is a Gaussian measurement error.
+Minimizing the cost function
+L=∥D − T ×1 U1... ×m Um... ×M UM∥ +
+M
+�
+m=1
+λm∥UmU
+T
+m − I∥
+is equivalent to maximum likelihood estimation [27] of the
+causal factor parameters, assuming the data was generated by
+the model with additive Gaussian noise. The optimal mode
+matrices Um are computed by employing a set of M alternating
+least squares optimizations.
+Lm
+=
+∥Xm − T ×m Um∥ + λm∥U
+T
+m × Um − I∥, where
+5 It is preferable to vectorize an image and treat it as a single observation
+rather than as a collection of independent column/row observations. Most
+assertions found in highly cited publications in favor of treating an image as a
+“data matrix” or “tensor” do not stand up to analytical scrutiny [99, App. A].
+Xm:=D ×1 ... ×m-1 U
+T
+m-1 ×m+1 U
+T
+m+1... ×M U
+T
+M - parallel
+(3)
+= Xm(t−1) ×n U
+T
+n(t)Un(t−1), ∀n ̸= m, - asynchronous (4)
+= (Xm-1 ×m-1 U
+T
+m-1) ×m Um = T ×m Um
+- sequential
+(5)
+The M-mode SVD [105] (Algorithm 1) minimizes M alternat-
+ing least squares in closed form by employing M different
+SVDs. It is suitable for parallel computation, but can be
+performed asynchronously or sequentially by employing (4)
+and (5), respectively.6 The core tensor T is computed by
+multiplying the data tensor with the inverse mode matrices,
+T = D ×1 UT
+1 · · · ×m UT
+m · · · ×M UT
+M, or more efficiently as
+T = Xm × UT
+m.
+A. Kernel Tensor Factor Analysis:
+When data D are a combination of non-linear independent causal
+factors C,
+D = B ×1 φ(C1) · · · × φ(Cm) · · · ×M φ(CM) + E
+(6)
+Cm = UmW−1+ Em,
+kernel
+multilinear
+independent
+component
+analysis
+(K-
+MICA) [99, Ch 4.4] models the mechanism of data formation. K-
+MICA employs the “kernel trick” [85][110] as a pre-processing
+step which makes the data suitable for multilinear independent
+component analysis [108] (Algorithm 2), where the additional
+rotation matrix Wm may be computed based on negentropy,
+mutual information, or higher-order cumulants. K-MICA is
+6A sequential computation of the M-mode SVD is known in the literature
+as a tensor ring [116]. A single-time iteration is known as a tensor train [77].
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+Fig. 3: Deep neural network. Subfigures (a-e) depict (7–10) [99, pg.38-40]. (a) The mode matrix computation Um is a constrained
+cluster-based PCA that is rewritten in terms of block SVDs. Matrixizing may be viewed as a concatenation of “cluster” data.
+The matrix W transforms the basis matrix U(n)
+0
+such that the causal factor representation Um is the same regardless of cluster
+membership. (b) Mode matrix Um computation using a single autoencoder-decoder. (c) Mode matrix computation as a hierarchy
+of autoencoder-decoders, (d) Mode matrix computation written as a deep learning model (e) Concurrent-autoencoders; i.e.,
+constrained cluster-based autoencoders. (f) Forward causal model with a set of capsules implemented by deep neural networks.
+For parallel computation, we break the chain links and shuttle causal information between capsules. (g) Each capsule in (f) may
+be replaced with a part-based deep neural network by permuting the rows in DT
+[m], and segmenting them based on adaptive
+subdivision [96]. The capsules are efficiently trained with a part-based hierarchy of autoencoders (Fig. 4).
+a tensor generalization of the kernel PCA [86] and kernel
+ICA [3][114].
+To accomplish this analysis, recall that the computation
+of covariance matrix D[m]D[m]
+T involves inner products
+dT
+i1...im-1 j im+1...iMdi2...im-1 k im+1...iM between pairs of data points
+in the data tensor D associated with causal factor mode m, for
+m = 1, . . . , M (Step 2.2 in Algorithm 1). We replace the inner
+products with a generalized distance measure between images,
+K(di1...im−1 j im+1...iM, di2...im−1 k im+1...iM), where K(·, ·) is a
+suitable kernel function (Table I) that corresponds to an inner
+product in some expanded feature space. This generalization
+naturally leads us to a Kernel Multilinear PCA (K-MPCA)
+Algorithm, where the covariance computation is replaced by
+[D[m]D[m]
+T]jk :=
+I1
+�
+i1=1
+· · ·
+Im−1
+�
+im−1=1
+Im+1
+�
+im+1=1
+· · ·
+IM
+�
+iM=1
+K(di1...im−1 j im+1...iM, di1...im−1 k im+1...iM).
+When a causal factor is a combination of multiple independent
+sources that are causal in nature, we employ a rotation matrix
+W to identify them. The rotation matrix is computed by
+
+Initialization:
+Cm = D × U+..·Xm- Ut- *m++ U+1°.· U parallel computation
+Iv Views
+= m(t-1) xn Ut(t)Ur(t-1), Vn±m
+asynchronous computation
+= (n-1 *m- Ut-) *m Um = T *m Um sequential computation
+Ip
+D
+People
+extended core,T
+matrixize 
+Tx.U.=α
+T×,U,→s
+T×,U=m
+T×,U, =
+matrixize
+matrixize
+M
+M
+T
+T
+TM
+[zlix
+Xi;
+Xi
+Xi
+invariant
+x.
+invariant
+invariant
+xt
+xi
+shared
+shared
+shared
+.code l
+x
+ code
+.codel:
+x2
+12
+x2
+w
+W.:
+IwM
+xt
+21
+21
+U.
+U,=U2 ...=Um=IL[P′
+People
+invariant
+invariant
+Pixels
+code
+code
+xi
+invariant
+:
+invariant
+Xl
+ll;
+code
+People
+code
+ui
+Pixels
+xi
+X1
+:
+invariant
+:
+code
+ui
+People
+lli
+Pixels
+x1
+p
+p
+xi
+:
+:
+X
+People
+ul;
+Pixels
+xi
+:
+:
+Xsy
+P
+People
+X1
+xr
+Pixels
+:
+:
+ui
+u
+People
+r
+p
+p
+P
+:
+:
+Xi.
+People
+X1
+xr
+Pixels
+:
+OR
+:
+X
+u1
+u;
+People
+Xt
+"p
+Xr
+Pixels
+:
+.
+AT
+11Sequential SVDs
+Constrained cluster-based SVD
+Di
+Rotate with W.
+SVD
+SVD
+(P)
+Ip
+DOT
+V(1)
+U(OT
+V(0)
+V(1)
+W.,Z.
+UT
+V(1)
+W)
+Z.
+UT
+[P] 
+[P] 
+[P]
+[P] 
+[P]
+[P]
+[P] 
+[P]
+[P]
+Z[P]  [P]
+[P] 
+[P]  [P]
+/
+V(2)
+U(2)T
+V(2)
+V2)
+V(2)
+W(2)
+[P] 
+[P]
+[P]
+[P] 
+[P]
+[P] 
+[P] 
+[P] 
+...
+Ip
+V(Np)
+V(Wp)
+Ix
+WWp)
+[P] 
+[P] 
+[P] 
+[P] 
+[P] 
+[P]
+[P] 
+[P] 
+[P] 上
+)T
+[P]
+People
+invariant
+code
+dr
+Pixels
+d2
+d2
+(N,)T
+P
+People
+Pixels
+aNpIgdr
+dr
+d2
+d.
+invariant
+()T
+(1)
+()
+VP
+VP
+code
+U;
+M
+(2)
+(1)
+u;
+u;
+la
+d.
+n
+W
+P
+di
+(Np)
+(VP)
+u;
+d2
+u
+(WP)T
+()
+(VP)
+d
+Vp
+W
+u;
+d.
+tix
+Adi
+invariant
+di
+code
+d2
+(12
+()T
+(1)
+YP
+d
+: u;
+W
+上
+P
+d2
+(NVp)T
+(Np)
+(2
+P
+VP
+VP
+IV
+u;dr
+invariant
+code
+d2
+T
+()T
+()(D)
+-
+P
+dt
+u;
++
+dt
+dt
+d2
+(WP)T (WP)T
+(VP)(VP)
+d2
+d
+ATCausal Deep Learning
+ICPR 2022, Montreal, Canada
+employing either mutual information, negentropy, or higher-
+order cumulants [23][6][45][5]. A Kernel Multilinear ICA (K-
+MICA) Algorithm is a kernel generalization of the multilinear
+independent component analysis (MICA) algorithm [108].
+Algorithm 2 simultaneously specifies both K-MPCA and K-
+MICA algorithms. A scalable tensor factor analysis represents
+an observation as a hierarchy of parts and wholes [103][102].
+III. NEURAL NETWORK ARCHITECTURE
+Causal neural networks (Fig. 1) parallel the functionality and
+composition of tensor factor analysis models. Causal neural
+networks are composed of a set of causal capsules and a tensor
+transformer. The capsules compute the latent variables, Um, that
+represent the causal factors. The tensor transformer, T , encodes
+the interaction between the causal factors.
+Tensor factor analysis models are transformed into causal
+neural networks by using Hebb autoencoders and tensor autoen-
+coders as building blocks. The M-mode SVD (Algorithm 1)
+is transformed into a causal neural network (Fig. 1) by
+replacing every SVD step with gradient descent optimization,
+which is outsourced to a Hebb autoencoder (Supplemental
+Section VI-A). For effectiveness, we employ stochastic gradient
+descent [12][80]. The extended core tensor T[0] is computed by
+defining and employing a tensor autoencoder, an autoencoder
+whose code is initialized to the tensor product of the causal
+factor representations, D[0] = T[0](UiM ⊗· · ·⊗Uim · · ·⊗Ui1)T.
+To address a set of arbitrarily non-linear causal factors, each
+autoencoder employs kernel activation functions (Table I).
+A. Causal Deep Networks and Scalable Tensor Factor Analysis:
+For a scalable architecture, we leverage the properties of
+block algebra. Shallow autoencoders are replaced with either a
+mathematically equivalent deep neural network that requires end-
+to-end training, a part-based hierarchy of autoencoders [103],
+or a set of concurrent autoencoders (Fig. 3).
+For example, the orthonormal subspace of a data matrix, D ∈
+CI0×I1 that has I0 measurements and I1 observations may be
+computed by recursively subdividing the data and analyzing the
+data blocks,
+D =
+�
+DA
+DB
+�
+=
+�
+UASAVT
+A
+UBSBVT
+B
+�
+=
+�
+UA
+0
+0
+UB
+� �
+SAVT
+A
+SBVT
+B
+�
+�
+��
+�
+SVD
+=
+=
+�
+UA
+0
+0
+UB
+�
+WΣV
+T =
+�
+UAWA
+UBWB
+�
+ΣV
+T = UΣV
+T,
+where W is a rotation matrix that transforms the basis matrices,
+UA and UB, spanning the data blocks, DA and DB, such that their
+observations have the same representations. This approach can
+be applied bottom up to a recursively partioned data matrix. The
+above equalities provide mathematical justification for greedy
+layer training of deep neural networks [9](Fig. 3a-e).
+Computing the mode matrices Um of a tensor model may be
+viewed as equivalent to computing a set of mutually constrained,
+cluster-based PCAs [99, pg.38-40] (Fig. 3a). When dealing with
+data that can be separated into clusters, the standard machine
+learning approach is to compute a separate PCA. When data
+from different clusters are generated by the same underlying
+process (e.g., facial images of the same people under different
+(a)
+(b)
+Fig. 4: (a) Causal capsules may be implemented with a part-
+based hierarchy of autoencoders. The dataset is permuted
+by P, filtered and segmented by Hm that is mode depen-
+dent. Subdivision into parts can be determined with adaptive
+subdivision [96].(b) Implementing the capsules with a part-
+based hierarchy of autoencoders is equivalent to performing
+Incremental M-mode Block SVD[103, Sec IV][98].
+viewing conditions), the underlying data can be concatenated
+in the measurement mode and the common causal factor can
+be modeled by one PCA.
+Thus, we define a constrained, cluster-based PCA as the
+computation of a set of PCA basis vectors that are rotated such
+that the latent representation is constrained to be the invariant
+of the cluster membership.
+In the context of our multifactor data analysis, we define a cluster
+as a set of observations for which all factors are fixed but one.
+For every tensor mode, there are Nm = I1I2 . . . Im−1Im+1 . . . IM
+possible clusters and the data in each cluster varies with the same
+causal mode. The constrained, cluster-based PCA concatenates
+the clusters in the measurement mode and analyzes the data
+with a linear model, such as PCA or ICA [5], [23], [26].
+To see this, let Di1...im−1im+1...iM ∈ CI0×1×1···×1×Im×1···×1
+denote a subtensor of D that is obtained by fixing all causal
+factor modes but mode m and mode 0 (the measurement
+mode). Matrixizing this subtensor in the measurement mode
+we obtain Di1...im−1im+1...iM [0] ∈ CI0×Im. This data matrix
+comprises a cluster of data obtained by varying causal factor
+m, to which one can traditionally apply PCA. Since there are
+Nm = I1I2 . . . Im−1Im+1 . . . IM possible clusters that share the
+same underlying space associated with factor m, the data can
+be concatenated and PCA performed in order to extract the
+same representation for factor m regardless of the cluster. Now,
+consider the MPCA computation of mode matrix Um (Fig. 3a),
+which can be written in terms of matrixized subtensors as
+Dm =
+�
+�������
+D1...11...1[m]
+T
+.
+.
+.
+DI1...11...1[m]
+T
+..
+.
+DI1...Im−1Im+1...IM [m]
+T
+�
+�������
+T
+= UmΣmVm
+T.
+(7)
+This
+is
+equivalent
+to
+computing
+a
+set
+of
+Nm
+=
+I1I2 . . . Im−1Im+1 . . . IM cluster-based PCAs concurrently by
+combining them into a single statistical model and representing
+the underlying causal factor m common to the clusters. Thus,
+rather than computing a separate linear PCA model for each
+
+People
+xi
+.
+x:
+People
+ul.
+x
+x
+ui,
+People
+ul,
+ul.
+People
+ul;
+xi
+xi
+Xix
+People
+H
+Pixels
+X
+People
+Pixels
+xi
+1
+People
+xi
+Pixels
+People
+Pixels
+xiIncremental
+7M-mode Block SVD
+U1
+TFig. 5: An inverse
+causal neural network
+is
+an
+inverted
+for-
+ward
+model
+where
+the operations are per-
+formed in reverse or-
+der [100][109].
+cluster, MPCA concatenates the clusters into a single statistical
+model and computes a representation (coefficient vector) for
+mode m that is invariant relative to the other causal factor modes
+1, ..., (m − 1), (m + 1), ..., M. Thus, MPCA is a multilinear,
+constrained, cluster-based PCA. To clarify the relationship, let
+us number each of the matrices Di1...im−1im+1...iM [m] = D(n)
+m
+with a parenthetical superscript 1 ≤ n = 1 + �M
+k=1,k̸=m(in −
+1) �k−1
+l=1,l̸=m Il ≤ Nm.
+Let each of the cluster SVDs be D(n)
+m
+= U(n)
+m Σ(n)
+m V(n)
+m
+T, and
+D[m]=
+�
+U(1)
+m Σ(1)
+m
+. . . U(NM)
+m
+Σ(Nm)
+m
+�
+�
+��
+�
+SVD
+diag([V(1)
+m
+. . . V(Nm)
+m
+])
+T
+=
+UmΣmW
+T
+m diag([ V(1)
+m
+. . .
+V(Nm)
+m
+])
+T,
+(8)
+=
+UmΣm[ V(1)
+m W(1)
+m
+. . .
+V(Nm)
+m
+W(Nm)
+m
+]
+T
+(9)
+=
+UmΣmVm
+T,
+(10)
+where diag(·) denotes a diagonal matrix whose elements are
+each of the elements of its vector argument. The mode matrix
+V(nm)
+m
+is the measurement matrix U(nm)
+0
+(U(nm)
+x
+when the
+measurements are image pixels) that contains the eigenvectors
+spanning the observed data in cluster nm, 1 ≤ nm ≤ Nm. MPCA
+can be thought as computing a rotation matrix, Wm, that contains
+a set of blocks W(n)
+m
+along the diagonal that transform the PCA
+cluster eigenvectors V(nm)
+m
+such that the mode matrix Um is
+the same regardless of cluster membership (8–10)(Fig 3). The
+constrained “cluster”-based PCAs may also be implemented
+with a set of concurrent “cluster”-based PCAs.
+Causal factors of object wholes may be computed efficiently
+from their parts, by applying a permutation matrix P and
+creating part-based data clusters with a segmentation filter Hm,
+where D
+×
+T
+m HmP ⇔ HmPD[m]
+T, but leaving prior analysis
+intact (Fig. 3g). A deep neural network can be efficiently
+trained with a hierarchy of part-based autoencoders (Fig. 4). A
+computation that employs a part-based hierarchy of autoencoders
+parallels the Incremental M-mode Block SVD [103, Sec.
+IV][102][98] (Supplemental VI-C).
+A data tensor is recursively subdivided into data blocks,
+analyzed in a bottom-up fashion, and the results merged as
+one moves through the hierarchy. The computational cost is
+the cost of training one autoencoder, O(T), times O(logNM),
+the total number of autoencoders trained for each factor matrix,
+O(TlogNm). If the causal neural network is trained sequentially,
+the training cost for one-time iteration is O(MTlog ¯N), where
+¯N is the average number of clusters across the M modes.
+IV. INVERSE CAUSAL QUESTION: “WHY?”
+Inverse causal inference addresses the “why” question and
+estimates the causes of effects given an estimated forward causal
+model and a set of constraints that reduce the solution set and
+render the problem well-posed [32][100][109].
+Multilinear tensor factor analysis constrains causal factor
+representations to be unitary vectors. Multilinear projec-
+tion [109][100] relies on this constraint and performs multiple
+regularized regressions. One or more unlabeled test observations
+that are not part of the training data set are simultaneously
+projected into the causal factor spaces
+T +x ×
+T
+x dtest = R
+(M-mode SVD or CP)
+≈ r1... ◦ rm... ◦ rM, and ∥rm∥ = 1.
+An autoencoder-decoder neural network architecture that imple-
+ments a multilinear projection architecture (Fig. 5) is an inverted
+(upside down) forward neural network architecture that reverses
+the operation order of the forward model.
+Neural architectures addressing underdetermined inverse prob-
+lems are characterized by hidden layers that are wider than the
+input layer; i.e., the dimensionality of vec(R) is larger than
+the number of measurements in d. Dimensionality reduction
+reduces noise, and the width of the hidden layers [38], [93].
+Adding sparsity, non-negativity constraints, etc., can further
+reduce the solution set. Alternatively or in addition, one can
+determine a set of candidate solutions by modeling different
+aspects of the mechanism of data formation as piecewise tensor
+(multilinear) factor models, such that each of their inverses
+is well-posed. A single multilinear projection [109][100] is
+replaced with multiple multilinear projections that are well-
+posed. Vasilescu and Terzopoulos [104][99, Ch.7] rewrote the
+forward multilinear model in terms of multiple piecewise linear
+models that were employed to perform multiple well-posed
+linear projections and produced multiple candidate solutions.
+V. CONCLUSION
+We derive a set of causal deep neural networks that are a
+consequence of tensor factor analysis.7 Causal deep neural
+networks encode hypothesized mechanisms of data formation
+as a part-based hierarchy of kernel tensor models, where
+“A causes B” means “the effect of A is B”, a measurable
+and experimentally repeatable quantity [39]. The causal deep
+architectures are composed of causal capsules and tensor
+transformers.
+The former estimate the causal factor representations, whose
+interaction are governed by the latter. Inverse causal questions
+estimate the causes of effects and implement the multilinear
+projection. For an underdetermined inverse problem, as an
+alternative to aggressive “bottleneck” dimensionality reduction,
+the mechanism of data formation is modeled as piecewise tensor
+(multilinear) models, and inverse causal inference performs
+multiple well-posed multilinear projections that result in multiple
+candidate solutions, which are gated to yield a unique solution.
+7“Every theoretical physicist that is any good knows six or seven different
+theoretical representations for exactly the same physics. He knows that they
+are all equivalent, but he keeps them all in his head hoping that they will give
+him different ideas.” - Richard Feynman [30].
+
+Compute the representation, vec(R) 
+1211
+[x]
+[x
+'Iplvl
+Factorize R. into latent variabfes:
+Tensorize code vec(R) → R
+CP or M-mode SVD( R.)
+R
+[PCausal Deep Learning
+ICPR 2022, Montreal, Canada
+VI. CAUSAL DEEP LEARNING
+(SUPPLEMENTAL DOCUMENT)
+We denote scalars by lower case italic letters (a, b, ...), vectors
+by bold lower case letters (a, b, ...), matrices by bold uppercase
+letters (A, B, ...), and higher-order tensors by bold uppercase
+calligraphic letters (A, B, ...). Index upper bounds are denoted
+by italic uppercase letters (i.e., 1 ≤ a ≤ A or 1 ≤ i ≤ I). The
+zero matrix is denoted by 0, and the identity matrix is denoted
+by I. The TensorFaces paper [105] is a gentle introduction to
+tensor factor analysis, [58] is a great survey of tensor methods
+and references [99], [25], [13] provide an in depth treatment of
+tensor factor analysis.
+A. PCA computation with a Hebb autoencoder
+A Hebb autoencoder-decoder minimizes the least squares
+function,
+l
+=
+I
+�
+i=1
+∥di − Bci∥ + λ∥B
+TB − I∥,
+(11)
+and learns a set of weights, bi0,r, that are identical to the
+elements of the PCA basis matrix [19, p. 58], B ∈ CI0×R,
+when employing non-deterministic linear neurons. The weights
+are computed sequentially by training on a set of observations
+di ∈ CI0 with I0 measurements (Fig. 6). The autoencoder is
+implemented with a cascade of Hebb neurons[36].
+The contribution of each neuron, c1, . . . , cr, is sequentially
+computed, subtracted from a centered training data set, and
+the difference is driven through the next Hebb neuron, cr+1 [87],
+[83], [82], [1], [75].
+The weights of a Hebb neuron, cr, are updated by
+∆br(t + 1) = η
+�
+d −
+r
+�
+ir=1
+bir(t)cir(t)
+�
+cr(t)
+(12)
+= η
+�
+d −
+r
+�
+ir=1
+bir(t)b
+T
+ir(t)d
+�
+d
+Tbr(t),
+br(t + 1)
+= (br(t) + ∆br(t + 1))
+∥br(t) + ∆br(t + 1)∥
+where d ∈ CI0 is a vectorized centered observation with I0
+measurements, 0 ≤ η ≤ 2/∥B∥2 = σmax,B is the learning rate, br
+are the autoencoder weights of the r neuron, cr is the activation,
+Fig. 6: Autoencoder-decoder architecture and Principal Compo-
+nent Analysis. (All images have been vectorized, but they are
+displayed as a grid of numbers. )
+Fig. 7: Matrixizing a 3rd order tensor, A.
+and t is the time iteration. Back-propagation[64], [65] performs
+PCA gradient descent [19, p. 58][51]. An autoencoder may be
+trained and the weights updated with a data batch, D,
+∆bi(t + 1)
+=
+(D − Br(t)B
+T
+r (t)D) D
+Tbr(t)
+=
+�
+DD
+T −
+r
+�
+ir=1
+bir(t)b
+T
+ir(t)DD
+T
+�
+br(t).
+Computational speed-ups are achieved with stochastic gradient
+descent [12][80].
+B. Relevant Tensor Algebra
+Briefly, the natural generalization of matrices (i.e., linear
+operators defined over a vector space), tensors define multilinear
+operators over a set of vector spaces. A “data tensor” denotes
+an M-way data array.
+Definition 1 (Tensor): Tensors are multilinear mappings over a
+set of vector spaces, CIm, 1 ≤ m ≤ M, to a range vector space
+CI0:
+A :
+�
+CI1 × CI2 × · · · × CIM�
+�→ CI0.
+(13)
+The order of tensor A ∈ CI0×I1×···×IM is M + 1. An element
+of A is denoted as Ai0i1...im...iM or ai0i1...im...iM, where 1 ≤
+im ≤ Im.
+The mode-m vectors of an M-order tensor A ∈ CI0×I1×···×IM
+are the Im-dimensional vectors obtained from A by varying index
+im while keeping the other indices fixed. In tensor terminology,
+column vectors are the mode-0 vectors and row vectors as mode-
+1 vectors. The mode-m vectors of a tensor are also known as
+fibers. The mode-m vectors are the column vectors of matrix
+A[m] that results from matrixizing (a.k.a. flattening) the tensor
+A.
+Definition 2 (Mode-m Matrixizing): The mode-m matrixizing
+of tensor A ∈ CI0×I1×...IM is defined as the matrix A[m] ∈
+
+n-p
+n-p
+b1
+b
+b11
+d
+C1
+b21
+big
+d.
+bil
+b,
+C
+bIR
+DR
+D2R
+CR
+d
+1
+-μ B
+=p
+B
+p介
+C7
+C1
+d
+b1
+bR
+u
+CR
+Rlo
+loAlgorithm 3 M-mode SVD algorithm.[105]
+Input the data tensor D ∈ CI0×···×IM.
+1) For m := 0, . . . , M,
+Let Um be the left orthonormal matrix of [UmSmVT
+m] :=
+svd(D[m])a
+2) Set Z := D ×0 U0
+T ×1 U1
+T · · · ×m Um
+T... ×M UM
+T.
+Output mode matrices U0, U1, ..., UM, and the core tensor Z.
+aThe computation of Um in the SVD D[m] = UmΣVmT can be performed
+efficiently, depending on which dimension of D[m] is smaller, by decomposing
+either D[m]D[m]T = UmΣ2UmT (note that VmT = Σ+UmTD[m]) or
+by decomposing D[m]TD[m] = VmΣ2VmT and then computing Um =
+D[m]VmΣ+.
+CIm×(I0...Im−1Im+1...IM). As the parenthetical ordering indicates,
+the mode-m column vectors are arranged by sweeping all the
+other mode indices through their ranges, with smaller mode
+indexes varying more rapidly than larger ones; thus,
+[A[m]]jk= ai1...im...iM,
+where
+(14)
+j = im
+and
+k = 1 +
+M
+�
+n=0
+n̸=m
+(in − 1)
+n−1
+�
+l=0
+l̸=m
+Il.
+A generalization of the product of two matrices is the product
+of a tensor and a matrix [25], [18].
+Definition 3 (Mode-m Product, ×m): The mode-m product
+of a tensor A ∈ CI1×I2×···×Im×···×IM and a matrix B ∈
+CJm×Im, denoted by A ×m B, is a tensor of dimensionality
+CI1×···×Im−1×Jm×Im+1×···×IM whose entries are computed by
+[A ×mB]i1...im−1jmim+1...iM
+=
+�
+im
+ai1...im−1imim+1...iMbjmim,
+C = A ×m B.
+matrixize
+tensorize
+C[m] = BA[m].
+The M-mode SVD, Algorithm 3 proposed by Vasilescu and
+Terzopoulos [105] is a “generalization” of the conventional
+matrix (i.e., 2-mode) SVD which may be written in tensor
+notation as
+D = U0SU1
+T
+⇔
+D = S ×0 U0 ×1 U1
+The M-mode SVD orthogonalizes the M spaces and decom-
+poses a tensor as the mode-m product, denoted ×m , of M-
+orthonormal mode matrices, and a core tensor Z
+D = Z ×0 U0 · · · ×m Um · · · ×M UM.
+(15)
+D[m] = UmZ[m] (UM · · · ⊗ Um+1 ⊗m-1 U · · · ⊗ U0)
+T, (16)
+vec(D) = (UM · · · ⊗ Um+1 ⊗ Um-1 · · · ⊗ U0) vec(Z).
+(17)
+The latter two equations express the decomposition in matrix
+form and in terms of vec operators.
+C. Compositional Hierarchical Block TensorFaces
+Training Data: In our experiments, we employed gray-level
+facial training images rendered from 3D scans of 100 subjects.
+The scans were recorded using a CyberwareTM 3030PS laser
+scanner and are part of the 3D morphable faces database
+created at the University of Freiburg [11]. Each subject was
+combinatoriall y imaged in Maya from 15 different viewpoints
+(θ = −60◦ to +60◦ in 10◦ steps on the horizontal plane,
+φ = 0◦) with 15 different illuminations ( θ = −35◦ to +35◦
+in 5◦ increments on a plane inclined at φ = 45◦).
+Data Preprocessing: Facial images were warped to an average
+face template by a piecewise affine transformation given a set
+of facial landmarks obtained by employing Dlib software [57],
+[53], [88], [67], [35]. Illumination was normalized with an
+adaptive contrast histogram equalization algorithm, but rather
+than performing contrast correction on the entire image, subtiles
+of the image were contrast normalized, and tiling artifacts
+were eliminated through interpolation. Histogram clipping was
+employed to avoid over-saturated regions.
+Experiments:We ran five experiments with five facial part-based
+hierarchies from which a person representation was computed,
+Fig. 8. Each image, d ∈ RI0×1, was convolved with a Gaussian
+and a Laplacian filter bank {Hs∥s = 1...S} that contained five
+filters, S = 5. The filtered images, d ×0 Hs, resulted in five
+facial part hierarchies composed of (i) independent pixel parts
+(ii) parts segmented from different layers of a Gaussian pyramid
+that were equally or (iii) unequally weighed, (iv) parts were
+segmented from a Laplacian pyramid that were equally or (v)
+unequally weighed.
+The composite person signature was computed for every test
+image by employing the multilinear projection algorithm [101],
+[109], and signatures were compared with a nearest neighbor
+classifier.
+To validate the effectiveness of our system on real-world images,
+we report results on “LFW” dataset (LFW) [43]. This dataset
+contains 13,233 facial images of 5,749 people. The photos are
+unconstrained (i.e., “in the wild”), and include variation due
+to pose, illumination, expression, and occlusion. The dataset
+consists of 10 train/test splits of the data. We report the mean
+accuracy and standard deviation across all splits in Table 9.
+Fig. 8(b-c) depicts the experimental ROC curves. We follow
+the supervised “Unrestricted, labeled outside data” framework.
+Results: While we cannot celebrate closing the gap on human
+performance, our results are promising. DeepFace, a CNN model,
+improved the prior art verification rates on LFW from 70% to
+97.35%, by training on 4.4M images of 200 × 200 pixels from
+4, 030 people, the same order of magnitude as the number of
+people in the LFW database.
+We trained on less than one percent (1%) of the 4.4M total
+images used to train DeepFace. Images were rendered from
+3D scans of 100 subjects with an the intraocular distance of
+approximately 20 pixels and with a facial region captured
+by 10, 414 pixels (image size ≈ 100 × 100 pixels). We have
+currently achieved verification rates just shy of 80% on LFW.
+Summary: Compositional Hierarchical Block TensorFaces
+models cause-and-effect as a hierarchical block tensor inter-
+action between intrinsic and extrinsic causal factors of data
+formation [103][98].
+A data tensor expressed as a part-based a hierarchy is a unified
+tensor model of wholes and parts. The resulting causal factor
+representations are interpretable, hierarchical, and statistically
+invariant to all other causal factors. While we have not closed
+the gap on human performance, we report encouraging face
+
+(b)
+(c)
+Fig. 8: Compositional Hierarchical Block TensorFaces learns a hierarchy of features, and reesents each person as a part-based
+compositional representation. Figure depicts the training data factorization, D = T H ×L UL ×V UV ×P UP, where an observation is
+represented as d(p, v, l) = T H ×L lT ×V vT ×P pT and TH spans the hierarchical causal factor variance. (b) ROC curves for the
+University of Freiburg 3D Morphable Faces dataset. (c) ROC curves for the LFW dataset. The average accuracies are listed next
+to each method, along with the area under the curve (AUC). Parts refers to using compositional hierarchical Block TensorFaces
+models to separately analyze facial parts. Gaussian, Laplacian refers to using compositional hierarchical Block TensorFaces on a
+Gaussian/Laplacian data pyramid.
+Training
+Dataset
+Test
+Dataset
+PCA
+TensorFaces
+Compositional Hierarchical Block TensorFaces
+Pixels
+Gaussian
+Pyramid
+Weighted
+Gaussian
+Pyramid
+Laplacian
+Pyramid
+Weighted
+Laplacian
+Pyramid
+Freiburg
+Freiburg
+65.23%
+71.64%
+90.50%
+88.17%
+94.17%
+90.96%
+93.98%
+Freiburg
+LFW
+(grey level images)
+69.23%
+±1.51
+66.25%
+±1.60
+72.72%
+±2.14
+76.72%
+±1.65
+77.85%
+±1.83
+77.58%
+±1.45
+78.93%
+±1.77
+Fig. 9: Empirical results reported for Freiburg and Labeled Faces in the Wild (LFW) using PCA, TensorFaces and Compositional
+Hierarchical Block TensorFaces representations. Pixels denotes independent facial part analysis Gaussian/Laplacian use a multi
+resolution pyramid to analyze facial features at different scales. Weighted denotes a weighted composite signature.
+Freiburg Experiment:
+Train on Freiburg: 6 views (±60◦,±30◦,±5◦); 6 illuminations (±60◦,±30◦,±5◦), 45 people
+Test on Freiburg:
+9 views (±50◦, ±40◦, ±20◦, ±10◦, 0◦), 9 illums (±50◦, ±40◦, ±20◦, ±10◦, 0◦), 45 different people
+Labeled Faces in the Wild (LFW) Experiment:
+Models were trained on approximately half of one percent (0.5% < 1%) of the 4.4M images used to train DeepFace.
+Train on Freiburg:
+15 views (±60◦,±50◦, ±40◦,±30◦, ±20◦, ±10◦,±5◦, 0◦), 15 illuminations (±60◦,±50◦, ±40◦,±30◦, ±20◦, ±10◦,±5◦, 0◦), 100
+people
+Test on LFW: We report the mean accuracy and standard deviation across standard literature partitions [43], following the
+Unrestricted, labeled outside data supervised protocol.
+
+people variance weights
+-person signature, pT
+Up,1
+Up,s
+P,nose
+P,face
+people
+WS
+Upx
+Xp
+viewpoint variance.
+full face @ layer|6 (lowest resolution)
+XI
+illumination variance weights
+viewpoint variance weights
+Xy
+varianc
+ illuminatibn representation, IT
+vjewpoint representation, vt
+parts @ layer 5
+illumination
+Humination
+U
+ewp
+UVX
+subparts @ layer 1 (highest resolution)0.9
+0.8
+0.7
+true positive rate (recall)
+0.6
+0.5
+0.4
+0.3
+Parts+Gaussian+Weighted (94.17%) (AUC=0.9801)
+Parts+Laplacian+Weighted (93.98%) (AUC=0.9799)
+0.2
+Parts+Laplacian (90.96%) (AUC=0.9654)
+Parts (90.50%) (AUC=0.9577)
+Parts+Gaussian (88.17%) (AUC=0.9484)
+0.1
+TensorFaces (71.64%) (AUC=0.7920)
+PCA (65.23%) (AUC=0.7140)
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+false positive rate0.9
+0.8
+0.7
+true positive rate (recall)
+0.6
+0.5
+0.4
+0.3
+Parts+Laplacian+Weighted (78.93%) (AUC=0.8669)
+Parts+Gaussian+Weighted (77.85%) (AUC=0.8600)
+0.2
+Parts+Laplacian (77.58%) (AUC=0.8575)
+Parts+Gaussian (76.72%) (AUC=0.8502)
+Parts (72.72%) (AUC=0.8138)
+0.1
+PCA (69.23%) (AUC=0.7735)
+TensorFaces (66.25%) (AUC=0.7257)
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+false positive rateverification results on two test data sets–the Freiburg, and the
+Labeled Faces in the Wild datasets by training on a very small
+set of synthetic images. We have currently achieved verification
+rates just shy of eighty percent on LFW by employing synthetic
+images from 100 people, 15 viewpoints and 15 illuminations,
+for a total that constitutes less than one percent (1%) of the
+total images employed by DeepFace. CNN verification rates
+improved the 70% prior art to 97.35% only when they employed
+4.4M images from 4, 030 people, the same order of magnitude
+as the number of people in the LFW database.
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C9E4T4oBgHgl3EQf5w7z/content/tmp_files/2301.05327v1.pdf.txt

@@ -0,0 +1,637 @@
+Blind Judgement:
+Agent-Based Supreme Court Modelling With GPT
+Sil Hamilton
+McGill University
+[email protected]
+Abstract
+We present a novel Transformer-based multi-agent system
+for simulating the judicial rulings of the 2010-2016 Supreme
+Court of the United States. We train nine separate models
+with the respective authored opinions of each supreme justice
+active ca. 2015 and test the resulting system on 96 real-world
+cases. We find our system predicts the decisions of the real-
+world Supreme Court with better-than-random accuracy. We
+further find a correlation between model accuracy with re-
+spect to individual justices and their alignment between legal
+conservatism & liberalism. Our methods and results hold sig-
+nificance for researchers interested in using language mod-
+els to simulate politically-charged discourse between multi-
+ple agents.
+Introduction
+Recent and ongoing political turmoil in the United States
+has magnified the actions of the federal Supreme Court in
+the public eye. The Court has taken to overturning judicial
+precedent in recent years, with the number of such deci-
+sions in the last six years reaching over twice the number
+of overturns between 2010 to 20151. The weakening rule of
+stare decisis has encouraged judicial researchers to develop
+holistic models of Supreme Court behaviour to better pre-
+dict and account for future trends (Blake 2019; Allcorn and
+Stein 2021).
+Accurate models of Supreme Court behaviour are rare de-
+spite this focus. The best performing models only reach ac-
+curacy levels of ≈ 70% on out-of-distribution cases (Katz,
+Bommarito, and Blackman 2017). Models achieving even
+this middling accuracy are complex in their architecture,
+generally consisting of a mix of SVM and logistic regression
+models. This complexity is necessitated by the variables in-
+volved.
+Confounding variables discussed in the literature in-
+clude little agreed-upon theories regarding the legal doc-
+trines practiced by individual justices (Jr, Curry, and Mar-
+shall 2011) and their rarely-documented social realities
+(Kromphardt 2017; Peterson, Giallouri, and Menounou
+2021). While the in-court behaviour of the justices is well
+documented, exogenous factors have an equal impact on
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+12010-2015: 8 overturns, 2016-2022: 22+ overturns.
+BREYER
+GINSBURG
+KAGAN
+SOTOMAYOR
+ALITO
+KENNEDY
+ROBERTS
+SCALIA
+THOMAS
+BREYER
+GINSBURG
+KAGAN
+SOTOMAYOR
+ALITO
+KENNEDY
+ROBERTS
+SCALIA
+THOMAS
+1.0000 0.4515 0.4521 0.3441 -0.1069 0.0599 -0.1066-0.2252-0.2390
+0.4515 1.0000 0.5825 0.5426 -0.2388-0.0527-0.2074-0.0908-0.2774
+0.4521 0.5825 1.0000 0.4556 -0.2212 0.0529 -0.1857-0.0253-0.2274
+0.3441 0.5426 0.4556 1.0000 -0.2198-0.0256-0.1201-0.1524-0.2240
+-0.1069-0.2388-0.2212-0.2198 1.0000 0.1095 0.4693 0.2998 0.4692
+0.0599 -0.0527 0.0529 -0.0256 0.1095 1.0000 0.0361 0.0688 -0.0166
+-0.1066-0.2074-0.1857-0.1201 0.4693 0.0361 1.0000 0.4630 0.3800
+-0.2252-0.0908-0.0253-0.1524 0.2998 0.0688 0.4630 1.0000 0.4462
+-0.2390-0.2774-0.2274-0.2240 0.4692 -0.0166 0.3800 0.4462 1.0000
+Figure 1: Correlation matrix of justices voting on 290 cases
+between 2010 and 2016. Note the clustering of justices nom-
+inated by Democrat and Republican presidents.
+case decision-making. A model capable of both cognitive
+and social reasoning would therefore benefit justice be-
+haviour modelling. To this end, we investigate whether re-
+cent advances in social simulation with language models can
+promote simple and effective models of Supreme Court be-
+haviour.
+Background
+In this section we describe the rationale behind our project.
+Judicial Modelling
+Three major theories of judicial behaviour generally inform
+the design of Supreme Court models: the legal theory, the at-
+titudinal theory, and the strategic theory (Jr, Curry, and Mar-
+shall 2011). The legal theory suggests justices are bound by
+constitutional precedent. The attitudinal theory instead ar-
+gues justices account for policy preference first, precedent
+second. Between the two lies the strategic theory, which
+says justices vote according to a mix of precedent and pref-
+erence.
+arXiv:2301.05327v1  [cs.CL]  12 Jan 2023
+
+As we show in Figure 1, decision correlations between
+justices active between 2010 and 2016 (hereafter referred
+to as the Roberts IV court) indicate the strategic theory is
+most accurate to reality. While justices will invoke prece-
+dent when writing their rationales, evidence suggests jus-
+tices remain somewhat beholden to the political alignment
+of their nominator. We note, however, that the correlations
+are only medium in their strength. This indicates accurate
+models should account for precedent, but not exclusively.
+Integrating one of these three theories into a Supreme
+Court model requires choosing how to best cast the influ-
+ence of precedence and preference as variables. Given this
+conversion can itself result in significant drawbacks via un-
+foreseen factors, we instead choose a simulative tool which
+allows to us to make fewer assumptions as to the most cor-
+rect theory of judicial behaviour: language models.
+Simulation
+Large Language Models (LLMs) are adept at simulating
+complex social phenomena. Recent research has demon-
+strated their ability to predict populated social media plat-
+forms (Park et al. 2022), the distribution of votes for presi-
+dential candidates in the 2012-2020 American elections (Ar-
+gyle et al. 2022), and the general sentiment of news arti-
+cles reporting COVID-19 in the early stages of the pan-
+demic (Hamilton and Piper 2022). These developments
+show model bias is valuable for those in the social sciences
+given bias is derived from the underlying distributions of
+their training material.
+Prior simulation research benefits from new techniques
+for eliciting cognitive activity from LLMs nominally de-
+signed for next-token prediction. These include chain of
+thought reasoning (Wei et al. 2022), discretely-structured
+prompts (Liu et al. 2021), and fine-tuning (Drori et al. 2022).
+These techniques have the model draw on internal biases to
+make predictions, allowing researchers to embed fewer as-
+sumptions into their simulative models. LLMs are alluring
+given judicial modelling necessitates a system capable of
+both social and cognitive reasoning.
+Agent-Based Modelling
+The process by which the Court arrives at a decision is nomi-
+nally rational (Jr, Curry, and Marshall 2011). While predilec-
+tions are known to influence vote outcomes, justices are ex-
+pected to justify dissenting decisions in written documents
+called opinions. Opinions are typically one to five pages in
+length within which the justice (or their aid) lays out their
+argument in a manner similar to an essay. For our simula-
+tion task, we assume justices record their rationale honestly
+and so treat the opinions as our primary target of prediction,
+meaning any model we train will be predicting opinions.
+Because opinions are long documents (i.e. longer than the
+1024 token-long context window GPT-2 is trained for), hav-
+ing one model produce multiple opinions in the same run is
+untenable. We turn to agent-based modelling for a solution.
+Whether consolidating multiple generative LLMs into a
+single architecture is beneficial has been heretofore un-
+derstudied, with the only significant prior experiment with
+LLMs showing promise (Betz 2022). However, given the
+Case Syllabus
+...
+Judge 1
+Judge n
+...
+Opinion
+Opinion
+...
+Vote
+Vote
+Majority Opinion
+Figure 2: Flow of our multi-agent system.
+success of the Mixture of Experts (MoE) method in ma-
+chine translation (NLLB Team 2022), we argue further ex-
+ploration of similar techniques for simulation tasks is war-
+ranted. While social simulation experiments suggest a sin-
+gle language model is capable of producing a wide range
+of opinions, training multiple models separately prevents
+cross-contamination when studying multiple data sources.
+Method
+We present the general design of our architecture in three
+parts: data collection, system architecture, and measures.
+Dataset
+We source data from two datasets for this experiment. The
+first corpus is the Supreme Court Database (SCDB) released
+by researchers at Washington University in St. Louis, which
+provides variables for 9,095 cases decided between 1946
+and 2021 (Spaeth et al. 2014).
+We supplement the SCDB with all written opinions from
+all slips provided on the Supreme Court website.2 Extracting
+the opinions from the PDF documents with an optical char-
+acter recognition (OCR) utility leaves us with 145MiB of
+text written between 2003 and 2022. We then associate each
+opinion with the justice and case from which it originated.
+System Architecture
+We choose to simulate the Roberts IV court (2010-2016)
+given this period outlasts all other Supreme Court iterations
+in recent history.3
+2Found at https://www.supremecourt.gov/opinions/slipopinion
+3See http://scdb.wustl.edu/documentation.php?var=naturalCourt
+
+Design
+Our multi-agent system is composed of nine full-
+sized GPT-2 models (Radford et al. 2019). We present the
+system architecture in Figure 2. At a high level, our system
+receives the topic of a case being brought before the court
+and passes it along to nine justice models. The system then
+receives back nine opinions and corresponding decisions of
+whether to approve the appellant. The system totals the re-
+sults and returns the majority vote.4
+Prompt
+We train each justice model with a discrete
+prompt structured like a Python dictionary:
+{
+’issue’: ’Lorem ipsum...’,
+’topic’: ’Lorem ipsum...’,
+’opinion’: ’Lorem ipsum...’
+’decision’: ’Lorem ipsum...’
+}
+The issue value corresponds to the issueArea variable pro-
+vided by SCDB.5 The topic value is a short description of
+what the appellant is bringing before the court. We extract
+this information from the syllabus of each opinion slip and
+summarize it with GPT-3 Davinci (Brown et al. 2020). The
+opinion value is the corresponding rationale the justice pro-
+duces when formulating their decision value, here a categor-
+ical variable signalling (dis)approval. We provide an exam-
+ple in the appendix.
+Training
+All models are trained for a total 30 epochs at a
+learning rate of 2e−4 with the Adam optimizer (Kingma and
+Ba 2014). This training process is conducted in two steps:
+1. Construct ≤ 1000 token prompts of the above style for all
+cases in which the Roberts IV court came to a unanimous
+decision. This model serves as the base for all further
+trained models.
+2. Collect all prompts generated in step 1 for each of the
+opinions (2003-2016) written by each justice active dur-
+ing Roberts IV. We thereby collect nine training sets and
+further train the model generated in step 1 with each sep-
+arately.
+Average model loss after both steps is 1.5, indicating there
+remains significant room for improvement.
+Measures
+We assess the performance of our multi-agent system on 96
+test cases withheld from the training set with two measures:
+accuracy and a novel measure for judicial ideological align-
+ment.
+Accuracy
+We measure accuracy with a receiver’s operat-
+ing characteristic curve (ROC) together with Cohen’s κ to
+account for a slight distribution bias in our test set.
+Alignment
+Justices are understood as being more or less
+in favour of overturning precedent. We capture this align-
+ment by taking the Pearson coefficient (r) between model
+accuracy and the frequency with which the respective justice
+4We provide our code at [withheld from review copy]
+5See http://scdb.wustl.edu/documentation.php?var=issueArea
+Justice
+Accuracy
+κ
+Samuel Alito
+65%
+0.30
+Ruth Bader Ginsburg
+62%
+0.21
+Clarence Thomas
+59%
+0.18
+Stephen Breyer
+58%
+0.16
+John Roberts
+57%
+0.13
+Elena Kagan
+56%
+0.12
+Anthony Kennedy
+54%
+0.09
+Sonia Sotomayor
+51%
+0.00
+Antonin Scalia
+50%
+-0.03
+Table 1: Model accuracy by justice. Note the wide variation
+in accuracy between justices.
+voted against precedent-altering decisions between 2003
+and 2016. Our measure is intended to capture where a justice
+is aligned between conservative (e.g. textualism, formalism,
+originalism) or liberal (e.g. legal realism) frameworks of ju-
+dicial decision making (Post and Siegel 2006).
+Results
+All results are reported with a minimum confidence rate of
+80% and are controlled for training material size and topic.
+Generations are run with a temperature of 0.5 and a maxi-
+mum length of 1000 tokens.
+Accuracy
+Our system achieves an aggregate accuracy of 60% (κ ≈
+0.18) on 96 test cases. While less predictive than the state of
+the art, our model nonetheless achieves better-than-random
+performance despite having been trained solely on opinions.
+We find a wide variation in the accuracy of each simulated
+justice when examining system performance more closely.
+As shown in Table 1, model accuracy varies between 65%
+and 50% despite having controlled for training data volume
+and case outcome.
+Alignment
+We measure a moderate correlation (r ≈ 0.56) between sim-
+ulated justice accuracy and the frequency with which each
+respective justice did not agree with the Court overruling
+or re-interpreting precedent. This result suggests our system
+achieves better accuracy with justices who are less likely to
+overturn precedent. We discuss the implications of this result
+below.
+Validation
+We train a single agent model to ensure having many agents
+provides non-negligible benefits. We fine-tune this single
+agent with the majority opinions of all cases decided on by
+the Roberts IV court. Testing this single agent on the test
+set results in an overall accuracy of 54% (κ = 0.08). The
+predicted decisions differs from the original test set with
+a Cohen’s d of ≈ −0.86 versus d ≈ 0.19 for our multi-
+agent model, increasing the population overlap from 68.5%
+to 92.4%.
+
+We implement software controls to ensure program out-
+put validity given training to a low loss does not guarantee
+the model produces both the opinion and decision variables.
+We therefore rerun each case until all models have returned
+a valid result. Once having processed all 96 cases, we sam-
+ple agent-produced opinions belonging to half to ensure co-
+herency.
+Discussion
+In this section we discuss two major consequences of our
+research.
+Precedent Hallucination
+GPT-2 is not an expert on legal
+precedent, nor should one expect it to be when the only for-
+mal source of legal information ingested by the model dur-
+ing pre-training were some seven thousand pages from Find-
+Law, a website principally known for tort law (Clark 2022).6
+This becomes evident when surveying model output. While
+the model will occasionally reference real laws, these cita-
+tions prove to be happenstance as GPT-2 will confuse details
+and thus render the references meaningless.
+That the model generates its own precedent when arguing
+over a case is an example of hallucination, a well known
+property of language models (Rohrbach et al. 2018). Be-
+cause causal language models are only tasked with predict-
+ing the next most likely token given some prior sequence,
+they are not given incentive to withhold factually incorrect
+statements—the model will say whatever is necessary to re-
+turn the number of tokens requested in a cogent manner.
+Our justice models will hallucinate precedence when pro-
+ducing opinions. They produce this pretend precedence im-
+plicitly by citing it throughout the argumentation process.
+That our models achieve greater-than-random decision accu-
+racy in voting outcomes despite not producing legally valid
+arguments suggests Supreme Court decisions may not al-
+ways rest on legally coherent rationales.
+Alignment Correlation
+The correlation between model
+accuracy and judicial alignment indicates conservative jus-
+tices are more predictable given their general unwillingness
+to overturn precedent. Considering the model hallucinates
+precedent, this correlation suggests conservative justices are
+conservative for ideological rather than rational reasons.
+We find this result surprising given conservative justices
+often make it a point to rationalize their unwillingness to
+overturn precedent with legal justifications. Common for-
+malist theories of this sort include both originalism and
+textualism, doctrines practiced by conservative members of
+the current court (Esbeck 2011). Our results suggest these
+decision-making patterns are less grounded in rational logic
+than anticipated given they are partially captured in a model
+not familiar with common law.
+Conclusion
+The aim of our project was to produce a multi-agent sys-
+tem capable of predicting Supreme Court decision-making
+with little to no prior theory-based assumptions of judicial
+6https://www.findlaw.com/
+behaviour. Given our resulting model achieves better-than-
+random accuracy despite having been trained only on opin-
+ion matter, we argue our process serves as an example for
+researchers seeking to develop simulative experiments with
+language models.
+Limitations
+As should be expected of any project promoting the creative
+output of AI, we make note of the biased material used in
+the production of large language models like GPT-2. While
+we contend this culturally-derived bias is beneficial for re-
+searchers using foundation models in the the social sciences,
+we nonetheless ensure our model does not cause unwanted
+harm. As such, we clearly mark all samples as having been
+generated and refrain from releasing large collections of
+generated material to the public.
+Next Steps
+We propose the following next steps after having demon-
+strated the basic viability of our architecture.
+Larger Model
+Can we improve system accuracy with
+larger language models? Recent research suggests language
+models develop emergent cognitive features when scaled
+above 6.7 billion parameters, narrowing future possible can-
+didates to the likes of GPT-NeoX-20B and T5X (Dettmers
+et al. 2022; Black et al. 2022; Roberts et al. 2022).
+Larger Training Corpus
+Another avenue for increas-
+ing system accuracy involves fine-training GPT-2 with the
+whole corpus of American law as captured by proceedings
+and opinions written in lower courts. The principle of stare
+decisis means the practice of common law is a social ven-
+ture, suggesting language models would do well in predict-
+ing precedent-dependent cases if prepared.
+Improved Prompting
+Research indicates language mod-
+els can avoid the long tail of token probabilities by repet-
+itively querying the model (Portelli et al. 2022; Kim et al.
+2020). Integrating repetitive prompting strategies into the
+opinion-generating schema is a promising avenue for im-
+provement. Another avenue would be to assess how rein-
+forcement learning from human feedback (RLHF) models
+like InstructGPT simulate court proceedings (Ouyang et al.
+2022).
+Investigating Future Cases
+How would the Roberts IV
+court fare with cases brought before the court after 2016?
+Would their court overturn precedent at the rate the post-
+2016 Supreme Court has? Questions of this caliber would be
+made approachable with a more accurate Roberts IV system.
+Acknowledgements
+I thank Prof. Andrew Piper of McGill University and Prof.
+Kristen Thomasen of UBC for their invaluable advice during
+the research process. I furthermore thank the reviewers and
+workshop committee members for their recommendations.
+
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+Appendix
+We demonstrate our system with a real-world case fed to the
+Ruth Bader Ginsburg model. We first provide our prompt
+followed by the model’s response. Note the model correctly
+predicts Ginsburg’s real-world decision (to deny the appel-
+lant) despite the ”California Childhood Video Game Act”
+not existing.
+
+Prompt
+Issue:
+‘First Amendment’,
+Topic:
+‘The California law restricting the sale or rental of
+violent video games to minors was challenged by the video-
+game and software industries. The Federal District Court
+concluded that the Act violated the First Amendment and
+permanently enjoined its enforcement. The Ninth Circuit af-
+firmed.’,
+Appellant is seeking a:
+‘certiorari’,
+Model Response
+Opinion:
+‘The California Childhood Video Game Act is
+on its face both overbroad and vague. It prohibits the sale
+or rental of “any device or medium designed for or capable
+of reproducing visual images of decapitation, blood, or dis-
+memberment of a human body,” in the presence of a child
+under the age of 13. It also prohibits the importation into
+the State of California of “any video game system, com-
+puter game software, or computer program that contains a
+killing or violent content.” The Act specifically identifies
+video games that have been determined by the State to be
+“violent,” and any person who “possesses, or has under his
+control,” one of those video games is subject to the Act’s
+restrictions. The Act falls outside the First Amendment be-
+cause it interferes with the creative expression and acquisi-
+tive impulse that go to make up the quintessential activity of
+a child’s life.
+The majority points out that the Act does not regulate
+the mere possession of these games, but seeks to regulate
+the medium in which they are played. It reasons that the
+State may regulate what the child sees and does not see by
+preventing the child from becoming engrossed in the game
+world and thereby “sustaining or supporting the action or
+activity.” But the majority does not ask whether the regula-
+tion of what the child sees and does not see is content-based.
+It neither considers nor concludes whether the regulation is
+narrowly tailored to the interests it seeks to protect. And it
+does not consider the adverse impact that its rule may have
+on legitimate play. The majority says that the regulation is
+content-neutral because it does not regulate “the medium
+in which the [killing or violent content] is embodied,” or
+the “context in which the [killing or violent content] is pre-
+sented.” But to the extent that the regulation of medium and
+context are related, the majority’s rule will be unworkable.
+The child who might otherwise play The Last of Us might
+otherwise play The Path, instead, because the former game
+has fewer graphic content and the latter is rated M for Ma-
+ture. And the child who might otherwise enjoy the violent
+video game The Walking Dead might otherwise have played
+might instead have played The Secret of Monkey Island,
+which has no violence or blood and is rated A. The child
+who might otherwise have played Grand Theft Auto might
+instead have played The Legend of Zelda, which has no vi-
+olence or blood and is rated E. And so on. The variety of
+content that the child may choose from makes it virtually
+impossible to determine with any precision what the reg-
+ulation of what the child sees and does not see will look
+like. The only thing that the majority seems to be able to
+say with any precision is that the regulation will likely con-
+tain some content-based exceptions. But as the San Fran-
+cisco ordinance demonstrates, those exceptions are exceed-
+ingly rare. And when they do appear, they almost always
+involve content the State has determined is not suitable for
+children.
+I fear that the majority’s rule will produce a generation of
+Americans—white, middle-class, suburban—who have no
+interest in, and no capacity to, engage in the kinds of play
+the First Amendment protects. And I fear that this Court,
+which prides itself on its willingness to protect the most vul-
+nerable members of our society from harm, will be complicit
+in that generation’s continued marginalization. I respectfully
+dissent.’,
+Decision:
+‘deny’
+

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+Task-specific Scene Structure Representations
+Jisu Shin*, Seunghyun Shin*and Hae-Gon Jeon†
+AI Graduate School, GIST, South Korea
+{jsshin98, seunghyuns98}@gm.gist.ac.kr, [email protected]
+Abstract
+Understanding the informative structures of scenes is essen-
+tial for low-level vision tasks. Unfortunately, it is difficult to
+obtain a concrete visual definition of the informative struc-
+tures because influences of visual features are task-specific.
+In this paper, we propose a single general neural network ar-
+chitecture for extracting task-specific structure guidance for
+scenes. To do this, we first analyze traditional spectral cluster-
+ing methods, which computes a set of eigenvectors to model a
+segmented graph forming small compact structures on image
+domains. We then unfold the traditional graph-partitioning
+problem into a learnable network, named Scene Structure
+Guidance Network (SSGNet), to represent the task-specific
+informative structures. The SSGNet yields a set of coeffi-
+cients of eigenvectors that produces explicit feature repre-
+sentations of image structures. In addition, our SSGNet is
+light-weight (∼ 55K parameters), and can be used as a plug-
+and-play module for off-the-shelf architectures. We optimize
+the SSGNet without any supervision by proposing two novel
+training losses that enforce task-specific scene structure gen-
+eration during training. Our main contribution is to show that
+such a simple network can achieve state-of-the-art results for
+several low-level vision applications including joint upsam-
+pling and image denoising. We also demonstrate that our SS-
+GNet generalizes well on unseen datasets, compared to ex-
+isting methods which use structural embedding frameworks.
+Our source codes are available at https://github.com/jsshin98/
+SSGNet.
+1
+Introduction
+Methods for estimating scene structures have attracted wide
+research attention for the past several decades. As an exam-
+ple, texture representations based on image edges have been
+extensively studied with impressive performance on low-
+level vision tasks, i.e. image denoising (Tomasi and Man-
+duchi 1998), deblurring (Krishnan and Fergus 2009; Levin
+et al. 2007), super-resolution (Tai et al. 2010) and inpaint-
+ing (Nazeri et al. 2019; Yang, Qi, and Shi 2020; Guo, Yang,
+and Huang 2021). Another aspect of scene structures in-
+volves inferring robust object boundaries to quantify uncer-
+tainty and refine initial predictions in visual perception tasks
+*These authors contributed equally.
+†Corresponding author
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+SSGNet
+Baseline 
+Networks
+Scene
+Structure for 
+Depth Upsampling
+LR Depth
+Noisy Image
+Scene
+Structure for 
+Image Denoising
+HR Depth
+Ours
+Baseline
+HR Depth
+Denoised Image
+Denoised Image
+RMSE : 25.72cm
+RMSE : 25.15cm
+PSNR : 34.00dB
+PSNR : 30.51dB
+Figure 1: Our SSGNet is a lightweight architecture and can
+be applied as a plug-and-play module to improve the perfor-
+mance of baseline networks for low-level vision tasks.
+including joint filtering (He, Sun, and Tang 2012; Guo et al.
+2018; Li et al. 2016) and depth completion (Eldesokey et al.
+2020). Clearly, the goodness of scene structures depends on
+the target applications, and is defined by either training data
+or objective functions.
+More recent approaches of extracting informative scene
+structures have focused on capturing task-specific features
+from various learning frameworks. One interesting work for
+joint filtering in (de Lutio et al. 2022) builds graph nodes on
+learned features from a guidance image to encode semantic
+information, and represents scene structures by segmenting
+the graph edges based on objective functions. However, they
+have heavy computational burdens and are not implemented
+as an end-to-end architecture. To formulate an end-to-end ar-
+chitecture, edge priors, directly obtained from conventional
+edge detection (Irwin et al. 1968; Canny 1986), are used as
+a guide. Typically, image edges (or gradients) represent high
+frequency features and can be forced to generate fine details
+in the prediction results (Fang, Li, and Zeng 2020). Never-
+theless, the question of how to effectively exploit structure
+guidance information remains unanswered. Tremendous ef-
+forts have been made to only generate a single purpose scene
+structure with each different architecture.
+In this paper, we propose a Scene Structure Guidance Net-
+work (SSGNet), a single general neural network architecture
+for extracting task-specific structural features of scenes. Our
+SSGNet is lightweight in both size and computation, and
+is a plug-and-play module that can be applied to any base-
+line low-level vision architectures. The SSGNet computes
+a set of parameterized eigenvector maps, whose combina-
+tion is selectively determined in favor of the target domain.
+arXiv:2301.00555v1  [cs.CV]  2 Jan 2023
+
+To achieve this, we introduce two effective losses: (1) Eigen
+loss, motivated by the traditional graph partitioning prob-
+lem (Shi and Malik 2000), forms a basis set of scene struc-
+tures based on weight graphs on an image grid. (2) Spatial
+loss enforces the sparsity of each eigenvector for diverse rep-
+resentations of scene structures. We note that, without any
+supervision, our SSGNet can successfully learn to gener-
+ate task-specific and informative structural information as
+shown in Fig.1. To demonstrate the wide applicability of our
+SSGNet, we conduct extensive experiments on several low-
+level vision applications, including joint upsampling and im-
+age denoising, and achieve state-of-the-art results, even in
+cross-dataset generalization.
+2
+Related Work
+Our work is closely related to scene structure embedding for
+low-level vision tasks.
+2.1
+Low-level vision tasks
+The goal of low-level vision tasks such as denoising, super-
+resolution, deblurring and inpainting is to recover a sharp
+latent image from an input image that has been degraded by
+the inherent limitations of the acquisition systems (i.e. sen-
+sor size, depth of field or light efficiency). In the past decade,
+there have been significant improvements in low-level vision
+tasks, and recently deep learning-based techniques have es-
+pecially proven to be powerful systems.
+With the help of inductive bias (Cohen and Shashua
+2017), convolutional neural networks (CNNs) with a pixel-
+wise photo consistency loss (Li et al. 2016; Zhang and
+Sabuncu 2018; Zhong et al. 2021) are adopted. To mitigate
+the issue on inter-pixel consistency on CNNs, generative ad-
+versarial networks (GANs) (Goodfellow et al. 2014; Zhu
+et al. 2017; Karras, Laine, and Aila 2019; Liu et al. 2021a;
+Wang et al. 2021)-based methods are proposed to produce
+visually pleasing results with perceptual losses (Johnson,
+Alahi, and Fei-Fei 2016; Fuoli, Van Gool, and Timofte 2021;
+Suvorov et al. 2022) based on high-level semantic features.
+Nowadays, a vision transformer (ViT) (Dosovitskiy et al.
+2021; Liu et al. 2021b; Caron et al. 2021; Chen et al. 2021)
+has been used to capture both local and global image infor-
+mation by leveraging the ability to model long-range con-
+text.
+Such approaches have shown good progress with struc-
+tural details. For regularization, adding robust penalties
+to objective functions (Tibshirani 1996; Xu et al. 2010;
+Loshchilov and Hutter 2019; de Lutio et al. 2022) suppresses
+high-frequency components, and hence the results usually
+provide a smooth plausible reconstruction. However, those
+constraints often suffer from severe overfitting to noisy la-
+bels and are sensitive to hyperparameters, which leads to a
+lack of model generality.
+2.2
+Structural information
+Extensive studies on low-level vision have verified the feasi-
+bility and necessity of the image prior including image edges
+and gradients. One of the representative works involves joint
+image filters which leverage a guidance image as a prior
+and transfer its structural details to a target image for edge-
+preserved smoothing (Tomasi and Manduchi 1998; He, Sun,
+and Tang 2012; Zhang et al. 2014).
+Such structure information can be defined in practice,
+depending on the tasks. Both super-resolution (Pickup,
+Roberts, and Zisserman 2003; Sun, Xu, and Shum 2008;
+Xie, Feris, and Sun 2015; Fang, Li, and Zeng 2020) and im-
+age denoising (Liu et al. 2020), which utilize a patch simi-
+larity, generate gradient maps to reconstruct high frequency
+details or suppress image noises. Works in (Gu et al. 2017;
+Jin et al. 2020) infer object boundaries to refine initial pre-
+dictions in visual perception tasks, including depth estima-
+tion/completion. Also, image inpainting (Nazeri et al. 2019;
+Yang, Qi, and Shi 2020; Guo, Yang, and Huang 2021; Cao
+and Fu 2021), filling in missing parts of corrupted scenes,
+adopt edge maps from traditional method like Canny edge
+detector (Canny 1986) to hallucinate their own scene struc-
+tures.
+In spite of promising results from the state-of-the-art
+methods learning meaningful details for each task, they re-
+quire a high modeling capacity with numerous parameters
+and ground-truth structure maps for training. In contrast,
+our SSGNet, a very small network generating scene struc-
+tures without any supervision, has advantages for various
+low-level vision tasks, simply by embedding as an additional
+module.
+3
+Methodology
+Motivated by spectral graph theory (Shi and Malik 2000;
+Levin, Rav-Acha, and Lischinski 2008; Levin, Lischinski,
+and Weiss 2007), a set of basis represents scene configu-
+rations as a linear combination of the basis. Such parame-
+terization provides a restrictive solution space to accommo-
+date semantic entities like textures and object boundaries.
+Following the works in (Tang and Tan 2019; Bloesch et al.
+2018), we begin with an introduction to spectral methods,
+and then parameterize scene structures which can be used as
+guidance for various vision tasks.
+3.1
+Motivation
+Let us set a weighted undirected graph G = (V, E) in an
+arbitrary feature space with a set of nodes V, and a set of
+edges E, whose weight can be represented as an N × N
+non-negative adjacency matrix W = {w(i, j) : (i, j) ∈ E}
+where i, j denote graph nodes. The Laplacian matrix L of
+this graph is then obtained by L = D − W, where D is a
+diagonal matrix with the row-wise sum of W on its diagonal.
+Since the Laplacian matrix is a positive semidefinite matrix,
+for every N dimensional vector y from the matrix Y which
+consists of a set of vectors, it holds that
+yT Ly =
+�
+(i,j)∈E
+w(i, j){y(i) − y(j)}2 ≥ 0.
+(1)
+To minimize the Eq.(1), the indicator vector y should take
+similar values for nodes i and j. When the adjacent value
+w(i, j) is high, the two nodes are more tightly coupled.
+Spectral graph theory in (Fiedler 1973; Shi and Malik
+2000) proves that the eigenvectors of the graph Laplacian
+
+ℒ𝑒𝑖𝑔𝑒𝑛
+ℒ𝑠𝑝𝑎𝑡𝑖𝑎𝑙
+Image
+I ∈ ℝ𝐻×𝑊×3
+Affinity Matrix
+W ∈ ℝ(𝐻𝑊)×(𝐻𝑊)
+Eigenvectors
+Y ∈ ℝ𝐻×𝑊×3
+Attention
+Layer
+Baseline 
+Networks
+⋯
+Skip Connection
+CONV
+BLK
+CONV
+BLK
+CONV
+BLK
+CONV
+BLK
+CONV
+BLK
+CONV
+BLK
+CONV
+BLK 3⨯3 Conv.-LN-GeLU
+CONV
+BLK 3⨯3 Conv.-LN-Softmax CONV
+BLK 3⨯3 Conv.-LeakyReLU
+Figure 2: An overview of SSGNet. LN, GeLU, and
+LeakyReLU denote the layer normalization, GeLU activa-
+tion, and LeakyReLU activation, respectively. The eigenvec-
+tors are integrated via the attention layer, and then embedded
+to any baseline network.
+yield minimum-energy graph partitions, and each smallest
+eigenvector, like the indicator vector y, partitions the graph
+into soft-segments based on its adjacent matrix.
+In the image domain, a reference pixel and its similarity to
+neighboring pixels can be interpreted as a node and edges in
+a graph (Boykov, Veksler, and Zabih 2001), respectively. In
+general, affinity is defined by appearance similarities (i.e. the
+absolute of intensity differences). With this motivation, im-
+ages can be decomposed into soft image clusters from a pre-
+computed affinity matrix. In addition, scene configurations
+in images can be described as a set of eigenvectors whose
+smallest eigenvalues indicate connected components on the
+affinity matrix.
+3.2
+Scene Structure Guidance Network
+In this work, our goal is to train the proposed network, SS-
+GNet, without any supervision because it is infeasible to
+define a unique objective function for a task-specific struc-
+ture guidance. To accomplish this, we devise a learnable and
+parametric way of efficiently representing scene structures.
+Given single color images I ∈ Rh×w×3, our SSGNet σ
+yields a set of eigenvectors Y ∈ Rh×w×n, where n denotes
+the number of eigenvectors and is empirically set to 3:
+Y = σ(I).
+(2)
+As illustrated in Fig.2, SSGNet takes a simple encoder-
+decoder architecture (∼ 55K), consisting of two 3×3 con-
+volutional layers and three 3×3 deconvolutional layers with
+layer normalizations (Ba, Kiros, and Hinton 2016) and gelu
+activations (Hendrycks and Gimpel 2016) after each layer
+except for the last softmax layer. The output of our SSGNet
+is associated with learnable weights that will be finetuned in
+accordance with an objective function of target applications.
+To optimize SSGNet in an unsupervised manner, we de-
+fine a loss function Lssg which is a linear combination of
+two loss terms as follows:
+Eigen Loss
+The main objective of SSGNet is to obtain a
+set of smallest eigenvectors Y of the graph Laplacian L, in-
+spired by the spectral graph theory (Fiedler 1973; Shi and
+Malik 2000; Levin, Rav-Acha, and Lischinski 2008).
+To generate the graph Laplacian L, we trace back all the
+way down to some traditional similarity matrix methods.
+Since an image is segmented based on a constructed affin-
+ity matrix in spectral graph theory, the form of the matrix
+depends on the pixel-level similarity encoding (Levin, Rav-
+Acha, and Lischinski 2008; Levin, Lischinski, and Weiss
+2007; Chen, Li, and Tang 2013). In this work, we adopt the
+sparse KNN-matting matrix (Chen, Li, and Tang 2013). To
+be specific, we first collect nonlocal neighborhoods j of a
+pixel i by the k-nearest neighbor algorithm (KNN) (Cover
+and Hart 1967). Then, we define the feature vector ϕ(i) at a
+given pixel i as follows:
+ϕ(i) = (r, g, b, dx, dy)i,
+(3)
+where (r, g, b) denotes each color channel, and (dx, dy) is
+a weighted spatial coordinate for the x- and y-axes. We fol-
+low the KNN kernel function KNN(i) to construct the sparse
+affinity matrix W based on feature vectors ϕ:
+W(i, j) =
+�
+1− ∥ ϕ(i) − ϕ(j) ∥,
+j ∈ KNN(i)
+0,
+otherwise,
+(4)
+where j ∈ KNN(i) are the k-nearest neighbors of i based on
+the distance defined by ϕ. Using the sparse KNN-matting
+matrix, we can take account of both spatial distance and
+color information with less computational cost than a tra-
+ditional similarity matrix. The graph Laplacian L is finally
+obtained by L = D − W as the same manner, described
+in Sec.3.1.
+We can finally obtain a set of eigenvectors Y by minimiz-
+ing the quadratic form of L, Leigen, as below:
+Leigen =
+�
+k
+YT
+k LYk.
+(5)
+However, we observe that SSGNet sometimes produces
+undesirable results during the training phase because of the
+degenerate case, where the rank of Y may be lower, and
+needs an additional loss term to regularize it.
+Spatial Loss
+Since our SSGNet uses a softmax function
+in the last layer to prevent the eigenvectors from converging
+to zero vectors, we only need to handle the degenerate case,
+where all eigenvectors have the same value. Our spatial loss
+Lspatial considers the sparsity of each eigenvector to enforce
+diverse representations of scene structure, defined as below:
+Lspatial =
+�
+k
+(|Yk|γ + |1 − Yk|γ) − 1,
+(6)
+where | · | indicates an absolute value, and the hyperparam-
+eter γ is set to 0.9 in our implementation. We can intuitively
+figure out that Lspatial has a minimum value when Yk is ei-
+ther 0 or 1 for each pixel. With the Lspatial and the softmax
+operation together, we show that if a pixel of one eigenvector
+converges near to 1, the pixel of other eigenvectors should
+go to 0. This makes each pixel across the eigenvectors have
+different value due to the sparsity penalty, which produces
+diverse feature representations of image structures.
+In total, the final loss function for SSGNet is defined as:
+Lssg = Leigen + λLspatial
+(7)
+where λ is the hyper-parameter, and is empirically set to 40.
+Our SSGNet is pretrained on a single dataset and can
+be embedded in various baseline networks after passing
+
+(b) Eigenvectors
+𝜆=0
+𝜆=40
+𝜆=1000
+0.4
+0.0
+(a) Input Image
+Undesirable
+Seams
+Same Eigenvectors
+1.0
+Figure 3: Visualization of the sets of eigenvectors according
+to λ = 0, 40, and 1000.
+through an additional single convolution layer which acts as
+an attention module. In favor of the target domain on each
+task, this layer produces adaptive structural information of
+input scenes by linearly combining the set of eigenvectors.
+3.3
+Analysis
+To the best of our knowledge, the SSGNet is the first to un-
+fold the eigen-decomposition problem into a learnable net-
+work. To validate its effectiveness, we provide a series of
+analyses on SSGNet.
+First, we analyze our loss function in Eq.(7) by tuning
+the hyper-parameter λ used as a balancing term between
+Lspatial and Leigen. In our experiment, the best performance
+is obtained with λ = 40. In Fig.3, we show the visualization
+results for three different λ values, including λ = 0, 40, and
+1000. When λ is set to 0, Lspatial is not forced enough to
+give a sparsity penalty across eigenvectors, which leads to
+the degenerate case. Otherwise, if λ is set to 1000, the im-
+age is not well-segmented because the overwhelming major-
+ity of Lspatial causes undesirable seams on the image. From
+this, we can see that the absence of either one leads to an
+undesirable situation, which emphasizes the role of each of
+the two terms in our loss functions.
+Next, we demonstrate that our SSGNet yields task-
+specific structural guidance features. As we highlighted, the
+SSGNet can be embedded in baseline networks. When the
+pretrained SSGNet is attached to baseline networks, the net-
+work parameters on SSGNet are finetuned to produce guid-
+ance features suitable for each task as the training proceeds.
+In Fig.4, we visualize how the eigenvectors from SSGNet
+change at each iteration during finetuning, including joint
+depth upsampling (Dong et al. 2022) and single image de-
+noising (Zhang et al. 2022).
+The joint depth upsampling needs accurate object bound-
+aries as a prior (Li et al. 2014). For obvious reasons, an ob-
+jective function in the joint depth upsampling encourages a
+greater focus on reconstructing object boundaries. As shown
+in Fig.4(a), our SSGNet generates attentive features on them
+during fine-tuning. In addition, for image denoising, it is es-
+sential to preserve fine detailed textures. In Fig.4(b), with
+the meaningful scene structures from our SSGNet, the plau-
+sible result is inferred as well. We claim that it is possible for
+Denoised Image
+Noise Image
+Initial
+Guidance
+(a) Depth Upsampling
+LR Depth map
+Initial
+Guidance
+HR Depth map
+Intermediate
+Guidance
+Finetuned
+Guidance
+Intermediate
+Guidance
+Finetuned
+Guidance
+(b) Image Denoising
+Figure 4: Examples of task-specific scene structures: in-
+tial, intermediate and final results from SSGNet for (a) joint
+depth upsampling and (b) image denoising.
+(a) Depth Upsampling
+HR Depth
+LR Depth
+Scene 
+Structure
+RGB Image
+Encoder
+Decoder
+Encoder
+Baseline
+Ours
+(b) Image Denoising
+Noisy Image
+Concatenation
+C
+Baseline
+Scene 
+Structure
+Denoised Image
+Ours
+Encoder
+Decoder
+C
+SSGNet
+SSGNet
+Figure 5: Illustrations of SSGNet for low-level vision tasks.
+The yellow colored networks indicate our SSGNet that out-
+puts informative task-specific structure guidances.
+our SSGNet to capture informative and task-specific struc-
+tures through gradient updates from backpropagation (Le-
+Cun et al. 1989). We will describe the experimental details
+and SSGNet���s quantitative benefits on each task in Sec.4.
+3.4
+Training Scheme
+We implement the proposed framework using a public Py-
+torch (Paszke et al. 2019), and utilize the Adam (Kingma
+and Ba 2014) optimizer with β1 = 0.9 and, β2 = 0.999.
+The learning rate and the batch size are set to 0.0001 and
+4 on SSGNet, respectively. We train the proposed frame-
+work on images with a 256×256 resolution. Since the pro-
+posed framework consists of fully convolutional layers, im-
+ages with higher resolutions than that used in the training
+phase are available in inference. The training on SSGNet
+took about 40 hours on two NVIDIA Tesla v100 GPUs.
+4
+Experiments
+We conduct a variety of experiments on low-level vi-
+sion tasks, including self-supervised joint depth upsam-
+pling (Sec.4.1) and unsupervised single image denoising
+(Sec.4.2), to demonstrate the effectiveness of our SSGNet.
+Moreover, we provide an extensive ablation study (Sec.4.3)
+to precisely describe the effects of each component in SS-
+GNet. Note that the higher resolution version of experimen-
+tal results is reported in our supplementary material.
+Baselines with SSGNet
+In this section, our goal is to val-
+idate a wide applicability of SSGNet. To do this, we incor-
+porate SSGNet into existing CNN architectures for the joint
+depth upsampling and the unsupervised image denoising by
+simply embedding scene structures from ours to the models.
+
+Supervised
+Self-Supervised
+Dataset
+Scale
+DKN
+FDKN
+FDSR
+P2P
+MMSR
+Ours
+2005
+×4
+RMSE
+1.103
+0.964
+0.886
+1.288
+0.708
+0.612
+MAE
+0.275
+0.222
+0.211
+0.273
+0.239
+0.188
+×8
+RMSE
+1.182
+1.629
+1.043
+1.177
+1.043
+0.830
+MAE
+0.288
+0.339
+0.333
+0.280
+0.319
+0.245
+2006
+×4
+RMSE
+1.623
+1.337
+1.198
+2.604
+0.555
+0.504
+MAE
+0.297
+0.222
+0.198
+0.413
+0.232
+0.201
+×8
+RMSE
+1.790
+1.883
+1.170
+2.684
+0.723
+0.648
+MAE
+0.307
+0.305
+0.267
+0.300
+0.261
+0.225
+2014
+×4
+RMSE
+2.878
+2.593
+3.217
+4.019
+1.953
+1.819
+MAE
+0.739
+0.659
+0.595
+0.822
+0.573
+0.451
+×8
+RMSE
+3.642
+3.510
+3.606
+3.894
+2.765
+2.714
+MAE
+0.775
+0.871
+0.885
+0.920
+0.785
+0.675
+Table 1: Quantitative results on joint depth upsampling
+tasks. The best and the second best results are marked as
+bold and underlined, respectively. (unit:cm)
+DKN
+FDKN
+FDSR
+Ours
+MMSR
+Baseline
+DKN
+FDKN
+FDSR
+Ours
+MMSR
+Baseline
+Guidance
+Source
+GT
+Scene  
+Structure
+0.0
+1.0
+(a) Predicted Depth Map 
+(b) Predicted Error Map 
+Figure 6: Comparison results on the joint depth upsampling
+with a resolution factor of 8 on the Middlebury 2005 dataset.
+We visualize the predictions and their corresponding error
+maps of competitive methods and ours.
+Prior to the evaluations, we train our SSGNet on a well-
+known NYUv2 dataset (Silberman and Fergus 2011), con-
+sisting of 1,000 training images and 449 test images. With
+the pre-trained weight of SSGNet, we embed it to the base-
+line networks and finetune on each task. As mentioned
+above, we do not need any supervision for training SSGNet.
+We note that NYUv2 dataset is not used for evaluations, to
+validate the zero-shot generalization across various datasets.
+4.1
+Joint Depth Upsampling
+Joint depth upsampling leverages the explicit structure de-
+tail of the input image as a guidance and transfers it to the
+target low-resolution depth map for enhancing spatial res-
+olution. With this application, we demonstrate the synergy
+of the structure details from clean input images and the pro-
+posed learnable scene structure from SSGNet.
+For this experiment, we choose MMSR (Dong et al. 2022)
+as a baseline depth upsampling network. MMSR introduces
+a mutual modulation strategy with the cross-domain adap-
+tive filtering and adopts a cycle consistency loss to train the
+model in a fully self-supervised manner. Instead of directly
+using the input image as the guidance, we employ the struc-
+ture guidance from the pretrained SSGNet in Fig.5(a), and
+follow the training scheme of MMSR for fair comparisons
+such that all the supervised methods are trained on NYUv2
+Kodak
+BSD300
+BSD68
+Method
+σ = 25
+σ = 50
+σ = 25
+σ = 50
+σ = 25
+σ = 50
+BM3D
+PSNR
+31.88
+28.64
+30.47
+27.14
+28.55
+25.59
+SSIM
+0.869
+0.772
+0.863
+0.745
+0.782
+0.670
+N2V
+PSNR
+31.63
+28.57
+30.72
+27.60
+27.64
+25.46
+SSIM
+0.869
+0.776
+0.874
+0.775
+0.781
+0.681
+Nr2n
+PSNR
+31.96
+28.73
+29.57
+26.18
+N/A
+N/A
+SSIM
+0.869
+0.770
+0.815
+0.684
+N/A
+N/A
+DBSN
+PSNR
+32.07
+28.81
+31.12
+27.87
+28.81
+25.95
+SSIM
+0.875
+0.783
+0.881
+0.782
+0.818
+0.703
+N2N
+PSNR
+32.39
+29.23
+31.39
+28.17
+29.15
+26.23
+SSIM
+0.886
+0.803
+0.889
+0.799
+0.831
+0.725
+IDR
+PSNR
+32.36
+29.27
+31.48
+28.25
+29.20
+26.25
+SSIM
+0.884
+0.803
+0.890
+0.802
+0.835
+0.726
+Ours
+PSNR
+32.39
+29.34
+31.52
+28.33
+29.25
+26.36
+SSIM
+0.885
+0.806
+0.891
+0.805
+0.835
+0.731
+Table 2: Quantitative results on single image denoising.
+DBSN
+Noisy(𝜎 = 50)
+GT
+Ours
+IDR(baseline)
+N2N
+BSD300: 2092
+PSNR
+14.69dB
+Kodak: 1
+DBSN
+GT
+Ours
+IDR(baseline)
+N2N
+PSNR
+28.43dB
+28.69dB
+28.49dB
+31.87dB
+15.91dB
+29.68dB
+31.29dB
+25.73dB
+31.43dB
+Noisy(𝜎 = 50)
+Figure 7: Examples of the single image denoising. For
+the noisy level σ = 50, we visualize the results from
+IDR+SSGNet as well as the state-of-the-art methods.
+dataset.
+We also follow the evaluation protocol described in (Dong
+et al. 2022) to quantitatively measure the root mean
+square error (RMSE) and the mean absolute error (MAE).
+To be specific, we use the Middlebury stereo dataset
+2005 (Scharstein and Pal 2007), 2006 (Hirschmuller and
+Scharstein 2007), and 2014 (Scharstein et al. 2014)1, and
+augment them, which provides 40, 72, and 308 image-depth
+pairs, respectively, using a public code2.
+We compare with various state-of-the-art models, in-
+cluding supervised, DKN (Kim, Ponce, and Ham 2021),
+FDKN (Kim, Ponce, and Ham 2021) and FDSR (He et al.
+2021), and self-supervised manners, P2P (Lutio et al. 2019)
+and MMSR (Dong et al. 2022). As shown in Tab.1, MMSR
+with our SSGNet embedded achieves the best performance
+in almost datasets over the comparison methods. Our SS-
+GNet brings the performance gain over the second best
+method is about 10.4% and 11.8% with respect to RMSE
+and MAE, respectively. It is also noticeable that the scene
+structure contributes to reducing the errors in the star-like
+object boundary and the inside surface, visualized in Fig.6.
+We highlight that the result demonstrates the strong general-
+ization capabilities of our SSGNet on unseen data again.
+1Since Middlebury 2003 provides neither depth maps nor cam-
+era parameters, we could not use it in this evaluation.
+2Downloaded from https://rb.gy/bxyqgi
+
+Task
+Depth Upsampling
+Denoising
+RMSE
+MAE
+PSNR
+SSIM
+Ours
+0.83
+0.245
+29.34
+0.806
+Hyper-parameter λ
+0.001
+0.84
+0.246
+29.17
+0.801
+0.1
+0.83
+0.245
+29.15
+0.800
+1
+0.82
+0.244
+29.13
+0.800
+100
+0.81
+0.244
+29.16
+0.801
+1000
+0.84
+0.248
+29.26
+0.803
+# of eigenvectors
+2
+0.84
+0.272
+29.15
+0.800
+5
+0.81
+0.242
+29.16
+0.800
+7
+0.82
+0.242
+29.17
+0.800
+10
+0.82
+0.244
+29.27
+0.804
+Canny Edge
+ψ = {0.6, 0.9, 1.4}
+0.90
+0.298
+24.86
+0.510
+ψ = {1.0, 2.0, 3.0}
+0.90
+0.282
+24.37
+0.489
+Table 3: Ablation study for the effects of each component of
+SSGNet. We use the Middlebury 2005 with ×8 for the joint
+depth upsampling, and the Kodak with a noise level σ=50
+for the single image denoising.
+4.2
+Image Denoising
+We treat single image denoising to check the effectiveness of
+our SSGNet if the scene structure in the input image is cor-
+rupted by noise. For this experiment, we use IDR (Zhang
+et al. 2022) as a baseline image denoising network. IDR
+suppresses the image noise in a self-supervised manner by
+proposing an iterative data refinement scheme. The key of
+IDR is to reduce a data bias between synthetic-real noisy
+images and ideal noisy-clean images. To embed the scene
+structure to IDR, we simply concatenate it from our pre-
+trained SSGNet with the noisy input image in Fig.5(b). As
+the rounds go on iteratively, our SSGNet focuses more on
+texture information of input scenes by ignoring the image
+noise, as already displayed in Fig.4.
+To validate the applicability to the image denoising task
+as well, we compare our results with various state-of-
+the-art self-supervised models, including BM3D (M¨akinen,
+Azzari, and Foi 2019) N2V (Krull, Buchholz, and Jug
+2019), Nr2n (Moran et al. 2020) DBSN (Wu et al. 2020),
+N2N (Lehtinen et al. 2018), and IDR (Zhang et al. 2022).
+For the evaluation, we strictly follow the experimental setup
+in (Zhang et al. 2022). We quantitatively measure PSNR and
+SSIM on Kodak (Kodak 1993), BSD300 (Movahedi and El-
+der 2010) and BSD68 (Martin et al. 2001) datasets for the
+zero-shot generalization. The models are trained on Gaus-
+sian noise with the continuous noise level σ = [0, 50] and
+tested on σ = 25 and 50.
+As shown in Tab.2, IDR with our SSGNet embedded
+achieves the best performance among all the competitive
+methods regardless of the noise levels. We emphasize that
+the performance gain by our SSGNet is about 0.58dB on av-
+erage. Considering the performance difference between the
+second and the third best methods is about 0.26dB achieved
+by the paradigm shift from a statistical reasoning of im-
+age restoration (Lehtinen et al. 2018) to the iterative refine-
+ment (Zhang et al. 2022), SSGNet makes meaningful contri-
+bution. Fig.7 shows some example results. With the power-
+ful capability of IDR on the noise suppression, our SSGNet
+preserves the scene texture of the objects well.
+Guidance
+LR
+𝜆=0.001
+𝜆=0.1
+𝜆=1
+𝜆=100
+𝜆=1000
+Canny2
+EV=10
+EV=7
+EV=5
+EV=2
+Canny1
+Ours
+(a) Joint Depth Upsampling
+GT
+𝜆=0.001
+𝜆=0.1
+𝜆=1
+𝜆=100
+𝜆=1000
+Canny2
+Canny1
+EV=10
+EV=7
+EV=5
+EV=2
+Noise
+(𝜎 = 50)
+Ours
+PSNR
+26.66dB
+26.69dB
+26.68dB
+26.68dB
+26.73dB
+23.78dB
+26.92dB
+24.35dB
+26.65dB
+26.71dB
+26.69dB
+26.68dB
+16.16dB
+(b) Image Denoising
+Figure 8: Qualitative comparison for different settings of SS-
+GNet. EV denotes the number of eigenvectors, and Canny1
+and Canny2 mean edge threshold settings such as ψ = {0.6,
+0.9, 1.4} and ψ = {1.0, 2.0, 3.0}, respectively. For the joint
+depth upsampling, we display the reconstruction results and
+the error maps, together.
+4.3
+Ablation Study
+An extensive ablation study is conducted to examine the ef-
+fect of each component on SSGNet: the hyper-parameter
+λ in our loss function and the number of eigenvectors. We
+additionally test alternative scene structures computed from
+Canny Edge (Canny 1986) with different thresholds. For this
+ablation study, we measure RMSE and MAE on the Middle-
+bury 2005 dataset for the joint depth upsampling (×8), and
+PSNR and SSIM on the Kodak dataset for the single im-
+age denoising (σ = 50), whose results and examples are
+reported in Tab.3 and Fig.8, respectively.
+Choice of Hyper-parameter λ
+Since our loss function re-
+quires the selection of a hyper-parameter λ, it is important to
+study the sensitivity of the performances to the choice of λ.
+We carry out this experiment for six different values: 0.001,
+0.1, 1, 100 and 1000 as well as 40 in our setting.
+As a result, SSGNet’s performance is insensitive to the
+choice of λ. In the joint depth upsampling, the performance
+difference according to λ is very marginal in that RMSE
+and MAE are at most 0.02cm and 0.001cm off the optimal
+values. In contrast, the performance gain for the image de-
+noising when using λ = 40 is relatively large. Compared
+to λ = 1000 which shows the second best performance, the
+improvement from 0.08dB in PSNR when using λ = 40
+brings more benefits for the comparisons with the state-of-
+the-art methods. In total, we find the optimal trade-off be-
+
+tween these two tasks.
+The Number of Eigenvectors
+The number of eigenvec-
+tors to represent scene structures is closely related to the
+number of learnable parameters in SSGNet. It is impor-
+tant for us to determine the optimal trade-off parameter in
+consideration of both the minimum number and the perfor-
+mances on these two tasks.
+We investigate the performances of SSGNet with two,
+five, seven and ten as well as three eigenvectors. Interest-
+ingly, we observe the similar phenomenon just as above. The
+performance degradation on the joint depth upsampling is
+very small (about 0.02cm in RMSE and 0.003 in MAE), and
+the performance gain by 0.07dB in PSNR over the second
+best value on the image denoising is achieved. For the same
+reason, we set the number of eigenvectors to 3.
+Comparison with Hand-crafted Structure Prior
+Hand-
+crafted edge detection is widely used for representing scene
+structures, even in recent models for low-level vision i.e. in-
+painting (Guo, Yang, and Huang 2021; Dong, Cao, and
+Fu 2022) and super-resolution (Nazeri, Thasarathan, and
+Ebrahimi 2019). We employ one of the representative hand-
+crafted edge map detection methods, Canny edge.
+For fair comparison, we generate a set of edge maps with
+various thresholds for image gradients ψ, and embed it into
+the baseline networks. We note that the edge maps are used
+as input of our attention layer for the networks to selec-
+tively choose informative scene structures during the train-
+ing phase. Here, we set two types of thresholds ψ to {0.6,
+1.5, 2.4} and {1.0, 2.0, 3.0} that we manually find the best
+settings to configure image textures and object boundaries.
+As shown in Tab.3, the interesting fact is that the perfor-
+mance drop of the joint depth upsampling is not huge when
+using the hand-crafted edge maps (within 0.07cm in RMSE
+and 0.037cm in MAE). On the other hand, there is the large
+performance gap between ours and the Canny edge maps
+(about 4dB in PSNR and 0.3 in SSIM).
+Two possible reasons why the Canny edge fails to gen-
+erate task-specific scene representations are: (1) The edge
+maps are not affected by back-propagation in training phase.
+(2) Based on the experimental results for the image denois-
+ing, the Canny edge is sensitive to image noise, which may
+corrupt the estimated scene structures and eventually not
+work as a prior. On the other hand, as displayed in Fig.8, our
+SSGNet returns the sharpest images, enabling the contents
+to be read. We thus argue that this experiment demonstrates
+the efficacy of our learnable structure guidance.
+5
+Conclusion
+In this paper, we present a single general network for repre-
+senting task-specific scene structures. We cast the problem
+of the acquisition of informative scene structures as a tradi-
+tional graph partitioning problem on the image domain, and
+solve it using a lightweight CNN framework without any
+supervision, Scene Structure Guidance Network (SSGNet).
+Our SSGNet computes coefficients of a set of eigenvectors,
+enabling to efficiently produce diverse feature representa-
+tions of a scene with a small number of learnable parameters.
+With our proposed two loss terms, the eigen loss and the spa-
+tial loss, SSGNet is first initialized to parameterize the scene
+Scale
+×2
+×3
+×4
+PSNR
+SSIM
+PSNR
+SSIM
+PSNR
+SSIM
+SeaNet
+38.08
+0.9609
+34.55
+0.9282
+32.33
+0.8981
+SeaNet+
+38.15
+0.9611
+34.65
+0.9290
+32.44
+0.8981
+Ours
+38.18
+0.9612
+34.68
+0.9290
+32.51
+0.8983
+Scale
+KITTI
+NYU
+RMSE
+MAE
+RMSE
+MAE
+pNCNN
+1013.08
+251.53
+0.058
+0.144
+Ours
+1009.51
+256.06
+0.056
+0.138
+Table 4: Additional experiments on Set5 (Bevilacqua et al.
+2012) for image super-resolution with ×2, ×3 and ×4, and
+NYUv2 (Silberman et al. 2012) and KITTI (Uhrig et al.
+2017) datasets for unguided depth completion. (unit:cm)
+Baseline
+Ours
+Scene
+Structure
+GT
+Sparse Point
+Figure 9: Comparison results on unguided depth completion
+on the NYUv2 dataset with pNCNN. Even no a guidance
+image, our SSGNet establishes the scene structure well.
+structures. The SSGNet is then embedded into the baseline
+networks and the parameters are fine-tuned to learn task-
+specific guidance features as the training proceeds. Lastly,
+we show the promising performance gains for both the joint
+depth upsampling and image denoising, even with the good
+cross-dataset generalization capability.
+Discussion
+Although our SSGNet achieves the state-of-
+the-art results for the tasks with the simple embedding ap-
+proach across the baseline networks, there are still rooms
+for improvements.
+To suggest our future directions, we conduct additional
+small experiments for image super-resolution and unguided
+depth completion tasks which is a dense depth prediction
+from a sparse input depth without any guidance image. In
+these experiments, the super-resolution only use downsam-
+pled input images to extract scene structures, and the depth
+completion relies on the pretrained weight of SSGNet to rep-
+resent scene configurations. We choose SeaNet (Fang, Li,
+and Zeng 2020), a CNN architecture equipped with a sep-
+arate scene texture estimation branch, and pNCNN (Eldes-
+okey et al. 2020), a lightweight probabilistic CNN (∼ 670K)
+to refine initial dense depth predictions based on a statistical
+uncertainty measure, for the image super-resolution and the
+unguided depth completion, respectively.
+Tab.4 reports that we obtain the performance gains with
+our SSGNet over the baseline models. Particularly, the syn-
+ergy between our SSGNet and pNCNN is noticeable in
+Fig.9. Unfortunately, the baseline models with our SSGNet
+do not reach the quality of huge size models, a ViT-based
+image super-resolution (Liang et al. 2021) and a GAN-based
+unguided depth completion (Lu et al. 2020).
+One of the future directions is to devise the best incorpo-
+ration scheme of our SSGNet in that their structures are too
+tricky to intuitively embed it. Another is that a joint multi-
+modality training from heterogeneous data is expected to
+represent more informative scene structures and to extend
+the applicability of SSGNet.
+
+Acknowledgements
+This research was partially supported by ’Project for Sci-
+ence and Technology Opens the Future of the Region’ pro-
+gram through the INNOPOLIS FOUNDATION funded by
+Ministry of Science and ICT (Project Number: 2022-DD-
+UP-0312), GIST-MIT Research Collaboration funded by the
+GIST, the Ministry of Trade, Industry and Energy (MOTIE)
+and Korea Institute for Advancement of Technology (KIAT)
+through the International Cooperative R&D program in part
+(P0019797), the National Research Foundation of Korea
+(NRF) (No.2020R1C1C1012635) grant funded by the Ko-
+rea government (MSIT), Vehicles AI Convergence Research
+& Development Program through the National IT Industry
+Promotion Agency of Korea (NIPA) funded by the Ministry
+of Science and ICT (No.S1602-20-1001), and the Institute
+of Information & communications Technology Planning &
+Evaluation (IITP) grant funded by the Korea government
+(MSIT) (No.2019-0-01842, Artificial Intelligence Graduate
+School Program (GIST), No.2021-0-02068, Artificial Intel-
+ligence Innovation Hub)
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+

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+A Novel Waveform Design for OFDM-Based
+Joint Sensing and Communication System
+Yi Geng
+Cictmobile, China
+[email protected]
+Abstract—The dominating waveform in 5G is orthogonal
+frequency division multiplexing (OFDM). OFDM will remain
+a promising waveform candidate for joint communication and
+sensing (JCAS) in 6G since OFDM can provide excellent
+data transmission capability and accurate sensing information.
+This paper proposes a novel OFDM-based diagonal waveform
+structure and corresponding signal processing algorithm. This
+approach allocates the sensing signals along the diagonal of the
+time-frequency resource block. Therefore, the sensing signals in a
+linear structure span both the frequency and time domains. The
+range and velocity of the object can be estimated simultaneously
+by applying 1D-discrete Fourier transform (DFT) to the diagonal
+sensing signals. Compared to the conventional 2D-DFT OFDM
+radar algorithm, the computational complexity of the proposed
+algorithm is low. In addition, the sensing overhead can be sub-
+stantially reduced. The performance of the proposed waveform
+is evaluated using simulation and analysis of results.
+Index Terms—OFDM, 6G, JCAS, radar, waveform, DFT, IDFT
+I. INTRODUCTION
+Sixth generation (6G) will not be only about communica-
+tion. Joint communication and sensing (JCAS), which will en-
+able a myriad of new use cases, is a significant area of interest
+within 6G research. To achieve a favorable trade-off between
+communication and sensing, the waveforms for 6G need to
+be designed for simultaneous communication and sensing
+[1]. Orthogonal frequency division multiplexing (OFDM) is a
+promising waveform candidate for JCAS systems. Compared
+to other JCAS waveform candidates, e.g., frequency modu-
+lated continuous wave (FMCW), OFDM waveform naturally
+supports MIMO processing and has excellent communication
+performance [2]. In terms of sensing performance, OFDM is
+well-suited for range and velocity estimation. For example,
+an OFDM-based periodogram algorithm can obtain range-
+velocity estimation by applying 2D-discrete Fourier transform
+(DFT) to signals in modulation symbol domain [3] [4]. On
+the other hand, OFDM waveform avoids extra hardware com-
+plexity and costs compared to dual-waveform systems (e.g.,
+time-multiplexing of OFDM and FMCW) [3].
+The paper is structured as follows. Section II gives an
+example of how to design the sensing signal structure ac-
+cording to the requirements of a traffic monitoring scenario.
+Section III provides an overview of the periodogram algorithm.
+In Section IV, we propose a novel diagonal waveform structure
+and corresponding signal processing algorithm. Section V
+concludes the paper.
+TABLE I: Exemplary KPIs for traffic monitoring
+Requirement
+Symbol
+Value
+Range resolution
+∆R
+0.4 m
+Velocity resolution
+∆v
+0.2 m/s
+Maximum detection range
+Rmax
+150 m
+Maximum detection velocity
+vmax
+90 m/s
+TABLE II: OFDM system parameters under the condition of
+satisfying the KPIs in Table I
+System parameter
+Symbol
+Value
+Carrier frequency
+fc
+28 GHz
+Bandwidth
+B
+400 MHz
+Subcarrier spacing
+SCS (∆f)
+120 KHz
+Total subcarriers
+Nc
+3360
+OFDM symbol duration
+Tsym
+8.92 µs
+OFDM slot duration
+Ts
+0.125 ms
+Time-domain duration
+Tb
+240Ts (30 ms)
+Total symbols in Tb
+Nsym
+3360
+Comb size in frequency domain
+Cf
+7∆f
+Comb size in time domain
+Ct
+7Tsym
+Sensing signals in frequency domain
+Nf
+480
+Sensing signals in time domain
+Nt
+480
+Sensing signals along diagonal
+N
+480
+II. WAVEFORM DESIGN FOR JCAS SYSTEM
+The waveform design of an OFDM-based JCAS system
+depends on the key performance indicators (KPIs) requested
+by the sensing applications, such as range resolution ∆R,
+velocity resolution ∆v, maximum detection range Rmax, and
+maximum detection velocity vmax. For example, an OFDM-
+based JCAS system at 28 GHz carrier frequency (fc) with
+120 kHz subcarrier spacing (SCS), which is deployed for traf-
+fic monitoring and communication simultaneously, is designed
+to meet the KPIs tabulated in Table I. To realize such a system,
+the OFDM system parameters that will be used in this paper
+are given in Table II. A waveform structure to meet the KPIs in
+Table I is shown in Fig. 1(a). To reduce the sensing overhead,
+the sensing subcarriers are assigned in a comb structure. Nf
+sensing signals are uniformly distributed at an interval of comb
+size Cf = 7∆f within the band B. Note that the Nf sensing
+signals occupy the full bandwidth B. Hence no deterioration
+of range resolution occurs [5]. The range resolution of this
+waveform ∆R is given by
+∆R =
+c
+2B ,
+(1)
+arXiv:2301.03347v1  [cs.IT]  9 Jan 2023
+
+(a) A comb structure of sensing signals
+(b) Consecutively transmitted blocks of sensing signals
+Fig. 1: A comb structure of sensing signals to meet the KPIs
+in Table I.
+Fig. 2: Tx and Rx scheme of OFDM-based JCAS system.
+where c is the speed of light.
+The maximum unambiguous detection range of this inter-
+leaved scheme Rmax is reduced by a factor Nc
+Nf compared to
+the maximum unambiguous range with a classical contiguous
+subcarrier allocation. Rmax is given by
+Rmax =
+cNf
+2∆fNc
+.
+(2)
+Similarly, in the time domain, Nt sensing signals are uni-
+formly distributed at an interval of comb size Ct = 7Tsym within
+240 OFDM slots (30 ms). The sensing signals are transmitted
+at symbol 2 and symbol 9 of each slot. The velocity resolution
+of this waveform ∆v is thus
+∆v =
+c
+2fcTb
+.
+(3)
+The maximum unambiguous velocity vmax can be given by
+vmax = c∆fNt
+2fcNsym
+.
+(4)
+According to (1)-(4), a sensing system with the parameters
+in Table II would be suitable to support the KPIs listed in
+Table I. One sensing block illustrated in Fig. 1(a) can be used
+to derive one range-velocity estimate. The sensing blocks can
+be transmitted consecutively to track the objects with an update
+rate of 33.3 Hz, as illustrated in Fig. 1(b).
+III. OFDM RANGE-DOPPLER PROCESSING IN THE
+MODULATION SYMBOL DOMAIN
+To date, various OFDM-based radar algorithms have been
+developed. In this section, a periodogram algorithm in the
+“modulation symbol” domain [4] is presented. Fig. 2 illus-
+trates the transmitting and receiving block diagram of an
+OFDM-based JCAS system using the algorithm in the cited
+work [4] and using the waveform shown in Fig. 1. At Tx,
+sensing signals are converted from serial to parallel. Each
+parallel symbol stream modulates a 120 kHz subcarrier. Over
+30 ms, a 480×480 modulation symbol matrix DTx(m, n) is
+processed by Inverse Fast Fourier Transform (IFFT). Each
+row of DTx(m, n) represents a vector carrying Doppler in-
+formation obtained from a sensing subcarrier. The indices of
+sensing subcarriers m in the frequency domain range from
+0 to Nf − 1. Each column of DTx(m, n) represents a vector
+carrying range information obtained from a sensing symbol.
+The indices of sensing symbols n range from 0 to Nt − 1.
+After IFFT processing, CP insertion, and digital-to-analog
+conversion, the matrix DTx(m, n) is transmitted in the air.
+The objects in the detection range affect the propagation of
+DTx(m, n). Therefore, the received modulation symbol matrix
+DRx(m, n) is the combination of DTx(m, n) and information
+of the objects. The user data carried by the sensing signals are
+eliminated by element-wise division between DRx(m, n) and
+DTx(m, n), yielding
+D(m, n) = DRx(m, n)
+DTx(m, n) = yR(m) ⊗ yD(n),
+(5)
+where
+yR(m) = exp(−j4π∆fRm
+c
+), m = 0, · · · , Nf − 1
+(6)
+yD(n) = exp(j4πTsymfcvn
+c
+), n = 0, · · · , Nt − 1
+(7)
+where D(m, n) is a Nf × Nt matrix, the operator ⊗ denotes
+dyadic product, yR(m) and yD(n) are Nf×1 vector and 1×Nt
+vector, respectively. R is the range between the JCAS antenna
+and the object, v is the radial velocity of the object.
+The linear phase shifts of yR(m) and yD(n) carry the range
+and Doppler information of the object. When a D(m, n)
+is extracted from a sensing block, inverse discrete Fourier
+transform (IDFT) and DFT are performed in the frequency
+domain and time domain, respectively, to derive the range R
+as well as the velocity v of the object [6] [7],
+Yr(p) = IDFT(yR(m)) = 1
+Nf
+Nf−1
+�
+m=0
+yR(m)exp(j2πmp
+Nf
+)
+= 1
+Nf
+Nf−1
+�
+m=0
+exp(−j4π∆fRm
+c
+)exp(j2πmp
+Nf
+),
+p = 0, · · · , Nf − 1
+(8)
+
+3359
+.....
+Subcarrier No.
+21
+400 MHz
+14
+0
+Symbol 2
+Symbol 9
+Symbol 2
+Symbol 9
+Slot 0
+Slot 1
+Slot 239
+Sensing signal symbol
+Tb = 30 ms
+ Communication symbolBlock 0
+Block 1
+Block 2
+B = 400 MHz
+t
+Tp = 30 ms
+toTx
+△f
+dTx(m,n)
+Subcarriers
+CP
+S/P
+IFFT
+insertion
+to RF Txi
+= 1/△f
+Sensing symbols
+Rx
+dRx (m,n)
+CP
+from RF Rx
+FFT
+P/S
+removal
+Sensing symbolsFig. 3: Sensing signal processing of a matrix D(m, n) with
+application of 2D-DFT.
+Yv(q) = DFT(yD(n)) =
+Nt−1
+�
+n=0
+yD(n)exp(−j2πnq
+Nt
+)
+=
+Nt−1
+�
+n=0
+exp(j4πTsymfcvn
+c
+)exp(−j2πnq
+Nt
+),
+q = 0, · · · , Nt − 1
+(9)
+By performing IDFT and DFT, the phase shifts of yR(m)
+and yD(n) are transformed from the time-frequency domain to
+spectral peaks in the delay domain and Doppler domain [8].
+Range R can be calculated by the IDFT bin index p in the
+delay domain where a peak occurs. Similarly, velocity v can
+be obtained by the DFT bin index q in the Doppler domain
+where a peak occurs. The range and velocity can be calculated
+by
+R = cppeak
+2∆fNf
+,
+(10)
+v =
+cqpeak
+2fcTsymNt
+,
+(11)
+where ppeak is the bin index at peak location in the delay
+domain, qpeak is the bin index at peak location in the Doppler
+domain.
+Matrix D(m, n) in (5) can be rewritten as
+D(m, n) =
+�
+�
+�
+D(0, 0)
+. . .
+D(0, Nt − 1)
+...
+...
+...
+D(Nf − 1, 0)
+. . .
+D(Nf − 1, Nt − 1)
+�
+�
+� .
+(12)
+2D-DFT of D(m, n) can be performed to compute the range
+and velocity of the object. It can be implemented in two stages:
+column-by-column 1D-IDFTs of length Nf are proceeded after
+row-by-row 1D-DFTs of length Nt as shown in Fig. 3. Then,
+the 2D range-velocity periodogram can be obtained, giving an
+intuitive indication of the reflecting objects.
+IV. A NOVEL WAVEFORM DESIGN AND SENSING SIGNAL
+PROCESSING ALGORITHM
+A. Challenges
+Despite its performance and practicality, the 2D-DFT-based
+periodogram method suffers from several drawbacks. First,
+the 2D-DFT calculation is computationally expensive. The
+complexity of an N-point DFT is O(N 2). For example,
+to apply 2D-DFT to one matrix D(m, n) produced by the
+sensing block illustrated in Fig. 1(a), 960O(N 2) complex
+multiplications (480 DFTs in row and 480 IDFTs in column)
+are needed to generate one range-velocity estimate. Since
+(a) A diagonal of sensing signals
+(b) Consecutively transmitted diagonals of sensing signals
+Fig. 4: A diagonal structure of sensing signals to meet the
+KPIs in Table I.
+the sensing system is “unintelligent” and cannot predict the
+positions of objects in the detection range, it must scan
+the environment in each direction to localize and track the
+objects using narrow beams, leading to an extremely high
+computational complexity. Second, 2D-DFT is calculated in
+stages where all the results of the row-by-row 1D-DFTs
+must be available before the column-by-column 1D-IDFTs can
+be performed. Therefore, an additional intermediate cache is
+required, thus increasing hardware cost, especially for large
+DFT size. Finally, the sensing signals must be transmitted both
+in the time and frequency domain in a comb structure, which
+induces high sensing overhead.
+B. Method
+To tackle these challenges, a novel OFDM-based sensing
+waveform structure and corresponding signal processing algo-
+rithm are proposed. As shown in Fig. 4(a), a rectangular time-
+frequency block with a bandwidth of 400 MHz contains 3360
+subcarriers in the frequency domain, and the block spans 30 ms
+in the time domain. Uniformly distributed sensing signals are
+allocated along the diagonal of the block. The time domain and
+the frequency domain contain the same number, N, of sensing
+signals. A diagonal of sensing signals produces a transmitted
+modulation symbol sequence dTx(k) of length N, spanning
+240 slots in the time domain and 400 MHz in the frequency
+domain. The normalized modulation symbol vector d(k) can
+be obtained by the element-wise division between the received
+modulation symbol sequence dRx(k) and dTx(k), yielding
+d(k) = dRx(k)
+dTx(k) = yR(k) ⊗ yD(k), k = 0, · · · , N − 1
+(13)
+where
+yR(k) = exp(−j4π∆fRk
+c
+), k = 0, · · · , N − 1
+(14)
+
+D(m,n)
+D(m,n)
+1D-DFT in row
+1D-IDET in column
+m =N-
+Range-
+velocity
+m=1
+profile
+m=0
+n=0n=1
+n=Nt-1
+n=0n=1
+n=Nt-1
+.3359
+......
+.....
+Subcarrier No.
+21
+400 MHz
+14
+7
+0
+Symbol 2
+Symbol 9
+Symbol 2
+Symbol 9
+Slot 0
+Slot 1
+ Slot 239
+Sensing signal symbol
+7 Communication symbol
+30 msBlock 0
+Block 1
+Block 2
+B = 400 MHz
+T
+Tb = 30 msFig. 5: The radar image of an object with range of 40 m and
+velocity of 5 m/s using the proposed algorithm.
+yD(k) = exp(j4πTsymfcvk
+c
+), k = 0, · · · , N − 1
+(15)
+The range R of the object causes the phase shifts of the
+individual elements of vector yR(k). The velocity v of the
+object causes the phase shifts of the individual elements of
+vector yD(k). Since vector yR(k) and vector yD(k) contain the
+same number of elements and they are equally spaced in both
+the frequency domain and time domain, the range and velocity
+of the object can be obtained simultaneously by performing
+DFT of d(k), which yields
+Yrv(l) = DFT(d(k)) =
+N−1
+�
+k=0
+yR(k)yD(k)exp(−j2πkl
+N
+),
+l = 0, · · · , N − 1
+(16)
+Fig. 5 shows the normalized DFT result of a modulation
+symbol vector d(k), which is derived from the sensing signals
+reflected by a moving object with range R = 40 m and veloc-
+ity v = 5 m/s at time t0. The x-axis represents DFT bin indices
+l. The number of bins equals the DFT size N. The integer
+values on the x-axis correspond to the spectral frequencies
+sampled by the DFT. The DFT result shows a dual-peak-like
+profile. The bin indices at two peak locations are l1 = 81 and
+l2 = 134. Two spectral frequencies fH and fL, which carry the
+range and Doppler information of the object, can be calculated
+by
+fH = l1 + l2
+2
+,
+(17)
+fL = l2 − l1
+2
+.
+(18)
+The DFT of d(k) translates the modulation signals in the
+time-frequency domain to two spectral peaks centered at fH
+and spaced at an interval of 2fL in the radar image. One of fH
+and fL contains range information, whereas the other contains
+velocity information. The range and velocity of the object can
+be extracted by
+R =
+cfH
+2∆fNc
+, v =
+cfL
+2TsymfcNc
+,
+(19)
+or
+R =
+cfL
+2∆fNc
+, v =
+cfH
+2TsymfcNc
+.
+(20)
+As a side effect of the proposed algorithm, using (19) and
+(20) yields two rang-velocity estimates from one block of
+sensing signals. These two estimates include a correct range-
+velocity estimate and an incorrect range-velocity estimate. For
+example, two range-velocity estimates can be extracted from
+bin indices l1 = 81 and l2 = 134 shown in Fig. 5, referred to
+as estimate A and estimate B below.
+• Estimate A: R = 40 m and v = 5 m/s
+• Estimate B: R = 10 m and v = 20 m/s
+By using the proposed method, an object with R = 40 m
+and v = 5 m/s produces the same radar image as that of an-
+other object with R = 10 m and v = 20 m/s. The uncertainty
+is caused by the unlabeled bin indices l1 and l2 at peak
+locations. For spectral frequencies fH and fL, it is uncertain
+which one is indicative of the range and which one indicates
+the velocity. These two estimates cannot be discerned using
+information from one sensing block. A multi-temporal data
+fusion method can be performed to filter out the incorrect
+estimate. The main idea of the multi-temporal data fusion
+method is as follows. Based on the two range-velocity esti-
+mates acquired in the recent past, the range-velocity estimate
+in the present can be predicted. The incorrect estimate can be
+identified by comparing the actual estimates obtained in the
+present with the predicted estimates.
+A traffic monitoring scenario is considered to show how the
+incorrect estimate is identified. We assume that the vehicles
+in the scenario comply with the linear motion model, which
+is popularly used in traffic research [9]. The linear motion
+model is a motion model where an object’s velocity or
+acceleration is held constant within a span of time, provided
+that the time is sufficiently short. We assume the JCAS
+system has no prior range-velocity information on the vehicles.
+The vehicle’s maximum acceleration (a) is assumed to be
+5.4 m/s2, corresponding to the acceleration from 0 to 60 mph
+in 5 seconds (acceleration of high-performance cars). The
+consecutive sensing blocks are transmitted at time 0 ms (t0),
+30 ms (t1), 60 ms (t2), 90 ms (t3), 120 ms (t4), and so on. For
+the linear motion model with constant velocity, the range and
+velocity of the vehicle at time t1-t4 can be predicted based on
+estimate A and estimate B obtained at time t0. Similarly, the
+range and velocity of the vehicle at time t1-t4 can be obtained
+for the linear motion model with a constant acceleration of
+5.4 m/s2.
+The amplitude of the spectral peak in the radar image is
+determined by range R as it affects the received signal power
+PR reflected by the object. PR is formulated as
+PR = PTxGTxGRxσλ2
+(4π)3R4f 2c
+,
+(21)
+where PTx is the transmitted power, GTx is the Tx antenna
+gain, GRx is the Rx antenna gain, σ is the RCS of the object,
+λ is the wavelength of the carrier.
+
+0
+Peak 1
+Peak 2
+= 81
+。= 134
+= 107.5
+-10
+-15
+= 26.5
+= 26.5
+-20
+dB
+-25
+0
+in
+results
+-10
+-30
+60
+80
+100
+120-
+140
+-20
+-30
+40
+-50
+0
+50
+100
+150
+200
+Bin indices of /(a) Real radar image for a vehicle with R = 40 m and constant
+v = 5 m/s at time t0, and predicted radar images at time t0-t4
+(b) Real radar image for a vehicle with R = 10 m and constant
+v = 20 m/s at time t0, and predicted radar images at time t0-t4
+(c) Real radar image for a vehicle with R = 40 m, v = 5 m/s
+and constant acceleration 5.4 m/s2 at time t0, and predicted
+radar images at time t0-t4
+(d) Real radar image for a vehicle with R = 10 m, v = 20 m/s
+and constant acceleration 5.4 m/s2 at time t0, and predicted
+radar images at time t0-t4
+Fig. 6: Predicted radar images at time t1-t4 based on estimate A and estimate B obtained at time t0.
+TABLE III: The predicted range and velocity of the vehicle
+at time t1-t4 based on estimate A and estimate B obtained at
+time t0
+Estimate
+t (ms)
+R(t) (m)
+v(t) (m/s)
+Ap(t) (dB)
+t0 = 0
+40
+5
+0
+Estimate A
+t1 = 30
+39.9
+5
+0.06
+with
+t2 = 60
+39.7
+5
+0.13
+constant v
+t3 = 90
+39.6
+5
+0.19
+t4 = 120
+39.4
+5
+0.26
+t0 = 0
+10
+20
+0
+Estimate B
+t1 = 30
+9.4
+20
+1.07
+with
+t2 = 60
+8.8
+20
+2.22
+constant v
+t3 = 90
+8.2
+20
+3.45
+t4 = 120
+7.6
+20
+4.77
+t0 = 0
+40
+5
+0
+Estimate A
+t1 = 30
+39.9
+5.2
+0.06
+with
+t2 = 60
+39.7
+5.3
+0.13
+constant a
+t3 = 90
+39.5
+5.5
+0.2
+t4 = 120
+38.4
+5.7
+0.28
+t0 = 0
+10
+20
+0
+Estimate B
+t1 = 30
+9.4
+20.2
+1.08
+with
+t2 = 60
+8.8
+20.3
+2.24
+constant a
+t3 = 90
+8.2
+20.5
+3.49
+t4 = 120
+7.6
+20.7
+4.86
+Based on estimate A and estimate B obtained at time t0 and
+the linear motion model, the predicted ranges R(t), velocities
+v(t) and normalized peak amplitudes Ap(t) at time t1-t4 are
+tabulated in Table III. Ap(t0) is normalized to 0 dB. The real
+radar images calculated at time t0 and the predicted radar
+images at time t1-t4 in the future are depicted in Fig. 6. The
+blue dashed lines indicate the real radar images yielded by
+the sensing block transmitted at time t0, while the predicted
+radar images at time t1-t4 are visualized with assorted colors.
+It can be observed that although a vehicle with R = 40 m and
+v = 5 m/s produces the same radar image as that of a vehicle
+with R = 10 m and v = 20 m/s at time t0, the following
+radar images at time t1-t4 show different patterns. Fig. 6(a)
+shows minor peak shifts due to the elapsed time. Because
+constant velocity causes unchanged fL, and the quite low
+velocity (5 m/s) causes minor range changes at time t1-t4.
+Hence, the changes in fH are less significant. The bin indices
+at peak locations at time t0-t4 are within 79-81 and 133-134,
+as shown in the zoomed-in sections in Fig. 6(a). Also, the
+peak amplitude increases slightly from 0 dB at t0 to 0.26 dB
+at t4. The minor range changes causes small received signal
+power changes, which, in turn, leads to slight peak amplitude
+fluctuation.
+It can be observed that the radar images in Fig. 6(b)
+are pretty different from the radar images in Fig. 6(a). The
+constant velocity produces an unchanged spectral frequency
+fH in Fig. 6(b). Meanwhile, the high velocity (v = 20 m/s)
+leads to significant changes in the range (major changes in
+spectral frequency fL). The bin indices at peak locations at
+time t0-t4 are within 80-88 and 128-134, which are distinctly
+different from the bin indices in Fig. 6(a). Furthermore, due to
+the significant distance changes, the peak amplitude increases
+rapidly from 0 dB at t0 to 4.77 dB at t4. Therefore, for constant
+velocity, by comparing the bin indices and peak amplitudes
+
+0
+B
+ul 
+0
+-10
+ results
+20
+2
+Normalized DF'
+76
+78
+80
+134
+135
+136
+: R= 40 m, v= 5 m/s
+30
+t,: R= 39.9 m, v= 5.2 m/s
+t.: R= 39.7 m, v= 5.3 m/s
+t.: R= 39.5 m, v= 5.5 m/s
+40
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Indices of /B
+0
+d
+4
+T results 
+2
+-10
+0
+-20
+131132133134
+80
+85
+90
+: R= 10 m, v= 20 m/s
+Normalized
+t,: R= 9.4 m, v= 20.2 m/s
+-30
+t,: R= 8.8 m, v= 20.3 m/s
+t.: R= 8.2 m, v= 20.5 m/s
+-40
+: R= 7.6 m, v= 20.7 m/s
+4
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Indices of /0
+B
+d
+ul 
+0
+-10
+ results
+20
+.2
+79
+80
+81
+133
+134
+DF
+t.: R= 40 m, v= 5 m/s
+0
+30
+t,: R= 39.9 m, v= 5 m/s
+: R= 39.7 m, v= 5 m/s
+t.: R= 39.6 m, v= 5 m/s
+-40
+3
+t.: R= 39.4 m, v= 5 m/s
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Indices of /B
+0
+da
+4
+results
+2
+2
+-10
+0
+0
+.2.
+-2
+-20
+128
+130
+132
+134
+80
+82
+84
+86
+88
+t.: R= 10 m, v= 20 m/s
+Jormalized
+0
+t,: R= 9.4 m, v= 20 m/s
+-30
+t.: R= 8.8 m, v= 20 m/s
+t,: R= 8.2 m, v= 20 m/s
+40
+t,: R= 7.6 m, v= 20 m/s
+4'
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Indices of /from predicted radar images at time t1-t4 with the bin indices
+and peak amplitudes from actual radar images obtained at time
+t1-t4, the correct range-velocity estimate can be identified.
+Although the acceleration of vehicles is utterly unpre-
+dictable in the traffic monitoring scenario, the acceleration
+of vehicles is limited. The maximum acceleration of high-
+performance cars is 5.4 m/s2. For the linear motion model with
+constant acceleration 5.4 m/s2, the predicted radar images at
+time t1-t4 are still significantly different between estimate A
+and estimate B, as shown in Fig. 6(c) and Fig. 6(d). The non-
+zero acceleration will induce changes in both fH and fL. For
+estimate A, the velocity changes slowly due to the limited
+acceleration. The range also changes slowly due to the low
+acceleration and low velocity (5 m/s) at time t0. Therefore,
+the vehicle yields minor changes in both fH and fL at t1-
+t4 as shown in Fig. 6(c). For a vehicle with R = 10 m,
+v = 20 m/s, and a = 5.4 m/s2 at time t0, enormous range
+changes due to high velocity induce major changes in fL at
+time t1-t4. Major changes in fL shift the peaks significantly,
+leading to a sparser peak pattern in Fig. 6(d) than the pattern
+in Fig. 6(c). Therefore, the velocity is crucial to filter out the
+incorrect estimates herein since the range change depends on
+the velocity. The limited acceleration in the traffic scenario is
+a minor factor in changing the peaks in the radar image.
+C. Discussions
+It has been mentioned that the signal processing complexity
+of applying 2D-DFT is computationally high. An OFDM
+sensing system with parameters in Table II needs 2N ×O(N 2)
+complex multiplications (480 DFTs in row and 480 IDFTs
+in column) to generate a range-velocity estimate. In contrast,
+the computational complexity of the proposed algorithm is
+relatively low because it avoids the DFT calculation both on
+all rows and all columns.
+Since the phase shift of modulation signals along the
+frequency axis can extract the delay, and the phase shift of
+modulation signals along the time axis can extract the Doppler,
+the signaling overhead of the proposed diagonal structure is
+reduced by combining the phase shift along both the frequency
+axis and the time axis into a series of sensing signals along the
+diagonal of the resource block. Hence the range and velocity
+can be derived simultaneously. The sensing overhead of the
+conventional comb structure is N 2/N 2
+c . The sensing overhead
+of the proposed diagonal structure is N/N 2
+c , reducing the
+sensing overhead by a factor of N.
+The only minor disadvantage of the proposed algorithm is
+that two unlabeled spectral peaks in the radar image cause
+ambiguity in the range-velocity estimates. The proposed multi-
+temporal data fusion method can be used to resolve ambiguity.
+Furthermore, steady objects do not create ambiguity by using
+the proposed algorithm. For example, Fig. 7 shows the radar
+image of a steady object with a range of 40 m and a
+velocity of 0 m/s. Another object with a range of 0 m and
+a velocity of 20 m/s produces the same radar image shown in
+Fig. 7. However, it directly contacts the antenna of the JCAS
+system due to the range R being zero, which is kinematically
+Fig. 7: The radar image of an object with range of 40 m and
+velocity of 0 m/s.
+infeasible. Therefore, the proposed algorithm can derive the
+range-velocity estimate of a steady object without ambiguity
+by using one diagonal block of sensing signals only.
+V. CONCLUSIONS
+The typical processing algorithm of OFDM radar applies
+2D-DFT for extracting the range and velocity information.
+This paper proposes a diagonal waveform structure and cor-
+responding signal processing algorithm. With this approach,
+two advantages can be achieved. First, the signal processing
+algorithm is simple regarding computational complexity and
+memory requirement. Second, sensing overhead is signifi-
+cantly reduced by the proposed waveform structure. The minor
+disadvantage is that the proposed approach obtains the range
+and velocity in a coupled manner. Therefore, range-velocity
+ambiguity may occur. A multi-temporal data fusion method
+can be performed to resolve ambiguity. The simulations have
+proven the operability of the proposed waveform and signal
+processing algorithm.
+REFERENCES
+[1] H. Wymeersch et al., “Deliverable D3.1 Localisation and sensing use
+cases and gap analysis,” Hexa-X, Dec. 31, 2021.
+[2] F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths and L. Hanzo, “Joint
+Radar and Communication Design: Applications, State-of-the-Art, and
+the Road Ahead,” in IEEE Transactions on Communications, vol. 68,
+no. 6, pp. 3834-3862, June 2020, doi: 10.1109/TCOMM.2020.2973976.
+[3] T. Wild, V. Braun and H. Viswanathan, “Joint Design of Communication
+and Sensing for Beyond 5G and 6G Systems,” in IEEE Access, vol. 9,
+pp. 30845-30857, 2021, doi: 10.1109/ACCESS.2021.3059488.
+[4] C. Sturm and W. Wiesbeck, “Waveform Design and Signal Processing
+Aspects for Fusion of Wireless Communications and Radar Sensing,” in
+Proceedings of the IEEE, vol. 99, no. 7, pp. 1236-1259, July 2011, doi:
+10.1109/JPROC.2011.2131110.
+[5] C. Sturm, Y. Sit, M. Braun and T. Zwick, “Spectrally interleaved multi-
+carrier signals for radar network applications and multi-input multi-
+output radar,” IET Radar, Sonar and Navigation, 2012, doi: 10.1049/iet-
+rsn.2012.0040.
+[6] C. Sturm, E. Pancera, T. Zwick and W. Wiesbeck, “A novel approach
+to OFDM radar processing,” 2009 IEEE Radar Conference, 2009, pp.
+1-4, doi: 10.1109/RADAR.2009.4977002.
+[7] C. Sturm, M. Braun, T. Zwick and W. Wiesbeck, “A multiple target
+doppler estimation algorithm for OFDM based intelligent radar systems,”
+The 7th European Radar Conference, 2010, pp. 73-76.
+[8] A. Behravan et al., “Introducing sensing into future wireless com-
+munication systems,” 2022 2nd IEEE International Symposium on
+Joint
+Communications
+&
+Sensing
+(JC&S),
+2022,
+pp.
+1-5,
+doi:
+10.1109/JCS54387.2022.9743513.
+
+0
+B
+a
+0
+-5
+-10
+-10
+-15
+20
+-20
+-25
+-30
+105
+110
+100
+115
+40
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+Bin indices of /[9] R. Schubert, E. Richter and G. Wanielik, “Comparison and evaluation of
+advanced motion models for vehicle tracking,” 2008 11th International
+Conference on Information Fusion, 2008, pp. 1-6.
+

FtE1T4oBgHgl3EQfqwWe/content/tmp_files/load_file.txt

@@ -0,0 +1,401 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf,len=400
+page_content='A Novel Waveform Design for OFDM-Based  Joint Sensing and Communication System  Yi Geng  Cictmobile, China  gengyi@cictmobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='com  Abstract—The dominating waveform in 5G is orthogonal  frequency division multiplexing (OFDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' OFDM will remain  a promising waveform candidate for joint communication and  sensing (JCAS) in 6G since OFDM can provide excellent  data transmission capability and accurate sensing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  This paper proposes a novel OFDM-based diagonal waveform  structure and corresponding signal processing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' This  approach allocates the sensing signals along the diagonal of the  time-frequency resource block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Therefore, the sensing signals in a  linear structure span both the frequency and time domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The  range and velocity of the object can be estimated simultaneously  by applying 1D-discrete Fourier transform (DFT) to the diagonal  sensing signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Compared to the conventional 2D-DFT OFDM  radar algorithm, the computational complexity of the proposed  algorithm is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' In addition, the sensing overhead can be sub-  stantially reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The performance of the proposed waveform  is evaluated using simulation and analysis of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Index Terms—OFDM, 6G, JCAS, radar, waveform, DFT, IDFT  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' INTRODUCTION  Sixth generation (6G) will not be only about communica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Joint communication and sensing (JCAS), which will en-  able a myriad of new use cases, is a significant area of interest  within 6G research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' To achieve a favorable trade-off between  communication and sensing, the waveforms for 6G need to  be designed for simultaneous communication and sensing  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Orthogonal frequency division multiplexing (OFDM) is a  promising waveform candidate for JCAS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Compared  to other JCAS waveform candidates, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=', frequency modu-  lated continuous wave (FMCW), OFDM waveform naturally  supports MIMO processing and has excellent communication  performance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' In terms of sensing performance, OFDM is  well-suited for range and velocity estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For example,  an OFDM-based periodogram algorithm can obtain range-  velocity estimation by applying 2D-discrete Fourier transform  (DFT) to signals in modulation symbol domain [3] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' On  the other hand, OFDM waveform avoids extra hardware com-  plexity and costs compared to dual-waveform systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=',  time-multiplexing of OFDM and FMCW) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Section II gives an  example of how to design the sensing signal structure ac-  cording to the requirements of a traffic monitoring scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Section III provides an overview of the periodogram algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  In Section IV, we propose a novel diagonal waveform structure  and corresponding signal processing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Section V  concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  TABLE I: Exemplary KPIs for traffic monitoring  Requirement  Symbol  Value  Range resolution  ∆R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m  Velocity resolution  ∆v  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2 m/s  Maximum detection range  Rmax  150 m  Maximum detection velocity  vmax  90 m/s  TABLE II: OFDM system parameters under the condition of  satisfying the KPIs in Table I  System parameter  Symbol  Value  Carrier frequency  fc  28 GHz  Bandwidth  B  400 MHz  Subcarrier spacing  SCS (∆f)  120 KHz  Total subcarriers  Nc  3360  OFDM symbol duration  Tsym  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='92 µs  OFDM slot duration  Ts  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='125 ms  Time-domain duration  Tb  240Ts (30 ms)  Total symbols in Tb  Nsym  3360  Comb size in frequency domain  Cf  7∆f  Comb size in time domain  Ct  7Tsym  Sensing signals in frequency domain  Nf  480  Sensing signals in time domain  Nt  480  Sensing signals along diagonal  N  480  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' WAVEFORM DESIGN FOR JCAS SYSTEM  The waveform design of an OFDM-based JCAS system  depends on the key performance indicators (KPIs) requested  by the sensing applications, such as range resolution ∆R,  velocity resolution ∆v, maximum detection range Rmax, and  maximum detection velocity vmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For example, an OFDM-  based JCAS system at 28 GHz carrier frequency (fc) with  120 kHz subcarrier spacing (SCS), which is deployed for traf-  fic monitoring and communication simultaneously, is designed  to meet the KPIs tabulated in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' To realize such a system,  the OFDM system parameters that will be used in this paper  are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' A waveform structure to meet the KPIs in  Table I is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' To reduce the sensing overhead,  the sensing subcarriers are assigned in a comb structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Nf  sensing signals are uniformly distributed at an interval of comb  size Cf = 7∆f within the band B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Note that the Nf sensing  signals occupy the full bandwidth B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Hence no deterioration  of range resolution occurs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The range resolution of this  waveform ∆R is given by  ∆R =  c  2B ,  (1)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='03347v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='IT]  9 Jan 2023  (a) A comb structure of sensing signals  (b) Consecutively transmitted blocks of sensing signals  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 1: A comb structure of sensing signals to meet the KPIs  in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 2: Tx and Rx scheme of OFDM-based JCAS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  where c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The maximum unambiguous detection range of this inter-  leaved scheme Rmax is reduced by a factor Nc  Nf compared to  the maximum unambiguous range with a classical contiguous  subcarrier allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Rmax is given by  Rmax =  cNf  2∆fNc  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  (2)  Similarly, in the time domain, Nt sensing signals are uni-  formly distributed at an interval of comb size Ct = 7Tsym within  240 OFDM slots (30 ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The sensing signals are transmitted  at symbol 2 and symbol 9 of each slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The velocity resolution  of this waveform ∆v is thus  ∆v =  c  2fcTb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  (3)  The maximum unambiguous velocity vmax can be given by  vmax = c∆fNt  2fcNsym  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  (4)  According to (1)-(4), a sensing system with the parameters  in Table II would be suitable to support the KPIs listed in  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' One sensing block illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 1(a) can be used  to derive one range-velocity estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The sensing blocks can  be transmitted consecutively to track the objects with an update  rate of 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='3 Hz, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' OFDM RANGE-DOPPLER PROCESSING IN THE  MODULATION SYMBOL DOMAIN  To date, various OFDM-based radar algorithms have been  developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' In this section, a periodogram algorithm in the  “modulation symbol” domain [4] is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 2 illus-  trates the transmitting and receiving block diagram of an  OFDM-based JCAS system using the algorithm in the cited  work [4] and using the waveform shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' At Tx,  sensing signals are converted from serial to parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Each  parallel symbol stream modulates a 120 kHz subcarrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Over  30 ms, a 480×480 modulation symbol matrix DTx(m, n) is  processed by Inverse Fast Fourier Transform (IFFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Each  row of DTx(m, n) represents a vector carrying Doppler in-  formation obtained from a sensing subcarrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The indices of  sensing subcarriers m in the frequency domain range from  0 to Nf − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Each column of DTx(m, n) represents a vector  carrying range information obtained from a sensing symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The indices of sensing symbols n range from 0 to Nt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  After IFFT processing, CP insertion, and digital-to-analog  conversion, the matrix DTx(m, n) is transmitted in the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The objects in the detection range affect the propagation of  DTx(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Therefore, the received modulation symbol matrix  DRx(m, n) is the combination of DTx(m, n) and information  of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The user data carried by the sensing signals are  eliminated by element-wise division between DRx(m, n) and  DTx(m, n), yielding  D(m, n) = DRx(m, n)  DTx(m, n) = yR(m) ⊗ yD(n),  (5)  where  yR(m) = exp(−j4π∆fRm  c  ), m = 0, · · · , Nf − 1  (6)  yD(n) = exp(j4πTsymfcvn  c  ), n = 0, · · · , Nt − 1  (7)  where D(m, n) is a Nf × Nt matrix, the operator ⊗ denotes  dyadic product, yR(m) and yD(n) are Nf×1 vector and 1×Nt  vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' R is the range between the JCAS antenna  and the object, v is the radial velocity of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The linear phase shifts of yR(m) and yD(n) carry the range  and Doppler information of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' When a D(m, n)  is extracted from a sensing block, inverse discrete Fourier  transform (IDFT) and DFT are performed in the frequency  domain and time domain, respectively, to derive the range R  as well as the velocity v of the object [6] [7],  Yr(p) = IDFT(yR(m)) = 1  Nf  Nf−1  �  m=0  yR(m)exp(j2πmp  Nf  )  = 1  Nf  Nf−1  �  m=0  exp(−j4π∆fRm  c  )exp(j2πmp  Nf  ),  p = 0, · · · , Nf − 1  (8)  3359  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Subcarrier No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  21  400 MHz  14  0  Symbol 2  Symbol 9  Symbol 2  Symbol 9  Slot 0  Slot 1  Slot 239  Sensing signal symbol  Tb = 30 ms  Communication symbolBlock 0  Block 1  Block 2  B = 400 MHz  t  Tp = 30 ms  toTx  △f  dTx(m,n)  Subcarriers  CP  S/P  IFFT  insertion  to RF Txi  = 1/△f  Sensing symbols  Rx  dRx (m,n)  CP  from RF Rx  FFT  P/S  removal  Sensing symbolsFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 3: Sensing signal processing of a matrix D(m, n) with  application of 2D-DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Yv(q) = DFT(yD(n)) =  Nt−1  �  n=0  yD(n)exp(−j2πnq  Nt  )  =  Nt−1  �  n=0  exp(j4πTsymfcvn  c  )exp(−j2πnq  Nt  ),  q = 0, · · · , Nt − 1  (9)  By performing IDFT and DFT, the phase shifts of yR(m)  and yD(n) are transformed from the time-frequency domain to  spectral peaks in the delay domain and Doppler domain [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Range R can be calculated by the IDFT bin index p in the  delay domain where a peak occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Similarly, velocity v can  be obtained by the DFT bin index q in the Doppler domain  where a peak occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The range and velocity can be calculated  by  R = cppeak  2∆fNf  ,  (10)  v =  cqpeak  2fcTsymNt  ,  (11)  where ppeak is the bin index at peak location in the delay  domain, qpeak is the bin index at peak location in the Doppler  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Matrix D(m, n) in (5) can be rewritten as  D(m, n) =  �  �  �  D(0, 0)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  D(0, Nt − 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  D(Nf − 1, 0)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  D(Nf − 1, Nt − 1)  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  (12)  2D-DFT of D(m, n) can be performed to compute the range  and velocity of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' It can be implemented in two stages:  column-by-column 1D-IDFTs of length Nf are proceeded after  row-by-row 1D-DFTs of length Nt as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Then,  the 2D range-velocity periodogram can be obtained, giving an  intuitive indication of the reflecting objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' A NOVEL WAVEFORM DESIGN AND SENSING SIGNAL  PROCESSING ALGORITHM  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Challenges  Despite its performance and practicality, the 2D-DFT-based  periodogram method suffers from several drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' First,  the 2D-DFT calculation is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The  complexity of an N-point DFT is O(N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For example,  to apply 2D-DFT to one matrix D(m, n) produced by the  sensing block illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 1(a), 960O(N 2) complex  multiplications (480 DFTs in row and 480 IDFTs in column)  are needed to generate one range-velocity estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Since  (a) A diagonal of sensing signals  (b) Consecutively transmitted diagonals of sensing signals  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 4: A diagonal structure of sensing signals to meet the  KPIs in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  the sensing system is “unintelligent” and cannot predict the  positions of objects in the detection range, it must scan  the environment in each direction to localize and track the  objects using narrow beams, leading to an extremely high  computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Second, 2D-DFT is calculated in  stages where all the results of the row-by-row 1D-DFTs  must be available before the column-by-column 1D-IDFTs can  be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Therefore, an additional intermediate cache is  required, thus increasing hardware cost, especially for large  DFT size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Finally, the sensing signals must be transmitted both  in the time and frequency domain in a comb structure, which  induces high sensing overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Method  To tackle these challenges, a novel OFDM-based sensing  waveform structure and corresponding signal processing algo-  rithm are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 4(a), a rectangular time-  frequency block with a bandwidth of 400 MHz contains 3360  subcarriers in the frequency domain, and the block spans 30 ms  in the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Uniformly distributed sensing signals are  allocated along the diagonal of the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The time domain and  the frequency domain contain the same number, N, of sensing  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' A diagonal of sensing signals produces a transmitted  modulation symbol sequence dTx(k) of length N, spanning  240 slots in the time domain and 400 MHz in the frequency  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The normalized modulation symbol vector d(k) can  be obtained by the element-wise division between the received  modulation symbol sequence dRx(k) and dTx(k), yielding  d(k) = dRx(k)  dTx(k) = yR(k) ⊗ yD(k), k = 0, · · · , N − 1  (13)  where  yR(k) = exp(−j4π∆fRk  c  ), k = 0, · · · , N − 1  (14)  D(m,n)  D(m,n)  1D-DFT in row  1D-IDET in column  m =N-  Range-  velocity  m=1  profile  m=0  n=0n=1  n=Nt-1  n=0n=1  n=Nt-1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='3359  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='.  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Subcarrier No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  21  400 MHz  14  7  0  Symbol 2  Symbol 9  Symbol 2  Symbol 9  Slot 0  Slot 1  Slot 239  Sensing signal symbol  7 Communication symbol  30 msBlock 0  Block 1  Block 2  B = 400 MHz  T  Tb = 30 msFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 5: The radar image of an object with range of 40 m and  velocity of 5 m/s using the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  yD(k) = exp(j4πTsymfcvk  c  ), k = 0, · · · , N − 1  (15)  The range R of the object causes the phase shifts of the  individual elements of vector yR(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The velocity v of the  object causes the phase shifts of the individual elements of  vector yD(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Since vector yR(k) and vector yD(k) contain the  same number of elements and they are equally spaced in both  the frequency domain and time domain, the range and velocity  of the object can be obtained simultaneously by performing  DFT of d(k), which yields  Yrv(l) = DFT(d(k)) =  N−1  �  k=0  yR(k)yD(k)exp(−j2πkl  N  ),  l = 0, · · · , N − 1  (16)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 5 shows the normalized DFT result of a modulation  symbol vector d(k), which is derived from the sensing signals  reflected by a moving object with range R = 40 m and veloc-  ity v = 5 m/s at time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The x-axis represents DFT bin indices  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The number of bins equals the DFT size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The integer  values on the x-axis correspond to the spectral frequencies  sampled by the DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The DFT result shows a dual-peak-like  profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The bin indices at two peak locations are l1 = 81 and  l2 = 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Two spectral frequencies fH and fL, which carry the  range and Doppler information of the object, can be calculated  by  fH = l1 + l2  2  ,  (17)  fL = l2 − l1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  (18)  The DFT of d(k) translates the modulation signals in the  time-frequency domain to two spectral peaks centered at fH  and spaced at an interval of 2fL in the radar image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' One of fH  and fL contains range information, whereas the other contains  velocity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The range and velocity of the object can  be extracted by  R =  cfH  2∆fNc  , v =  cfL  2TsymfcNc  ,  (19)  or  R =  cfL  2∆fNc  , v =  cfH  2TsymfcNc  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  (20)  As a side effect of the proposed algorithm, using (19) and  (20) yields two rang-velocity estimates from one block of  sensing signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' These two estimates include a correct range-  velocity estimate and an incorrect range-velocity estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For  example, two range-velocity estimates can be extracted from  bin indices l1 = 81 and l2 = 134 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 5, referred to  as estimate A and estimate B below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Estimate A: R = 40 m and v = 5 m/s  Estimate B: R = 10 m and v = 20 m/s  By using the proposed method, an object with R = 40 m  and v = 5 m/s produces the same radar image as that of an-  other object with R = 10 m and v = 20 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The uncertainty  is caused by the unlabeled bin indices l1 and l2 at peak  locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For spectral frequencies fH and fL, it is uncertain  which one is indicative of the range and which one indicates  the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' These two estimates cannot be discerned using  information from one sensing block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' A multi-temporal data  fusion method can be performed to filter out the incorrect  estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The main idea of the multi-temporal data fusion  method is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Based on the two range-velocity esti-  mates acquired in the recent past, the range-velocity estimate  in the present can be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The incorrect estimate can be  identified by comparing the actual estimates obtained in the  present with the predicted estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  A traffic monitoring scenario is considered to show how the  incorrect estimate is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' We assume that the vehicles  in the scenario comply with the linear motion model, which  is popularly used in traffic research [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The linear motion  model is a motion model where an object’s velocity or  acceleration is held constant within a span of time, provided  that the time is sufficiently short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' We assume the JCAS  system has no prior range-velocity information on the vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The vehicle’s maximum acceleration (a) is assumed to be  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m/s2, corresponding to the acceleration from 0 to 60 mph  in 5 seconds (acceleration of high-performance cars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The  consecutive sensing blocks are transmitted at time 0 ms (t0),  30 ms (t1), 60 ms (t2), 90 ms (t3), 120 ms (t4), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For  the linear motion model with constant velocity, the range and  velocity of the vehicle at time t1-t4 can be predicted based on  estimate A and estimate B obtained at time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Similarly, the  range and velocity of the vehicle at time t1-t4 can be obtained  for the linear motion model with a constant acceleration of  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The amplitude of the spectral peak in the radar image is  determined by range R as it affects the received signal power  PR reflected by the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' PR is formulated as  PR = PTxGTxGRxσλ2  (4π)3R4f 2c  ,  (21)  where PTx is the transmitted power, GTx is the Tx antenna  gain, GRx is the Rx antenna gain, σ is the RCS of the object,  λ is the wavelength of the carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  0  Peak 1  Peak 2  = 81  。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='= 134  = 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5  10  15  = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5  = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5  20  dB  25  0  in  results  10  30  60  80  100  120-  140  20  30  40  50  0  50  100  150  200  Bin indices of /(a) Real radar image for a vehicle with R = 40 m and constant  v = 5 m/s at time t0, and predicted radar images at time t0-t4  (b) Real radar image for a vehicle with R = 10 m and constant  v = 20 m/s at time t0, and predicted radar images at time t0-t4  (c) Real radar image for a vehicle with R = 40 m, v = 5 m/s  and constant acceleration 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m/s2 at time t0, and predicted  radar images at time t0-t4  (d) Real radar image for a vehicle with R = 10 m, v = 20 m/s  and constant acceleration 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m/s2 at time t0, and predicted  radar images at time t0-t4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6: Predicted radar images at time t1-t4 based on estimate A and estimate B obtained at time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  TABLE III: The predicted range and velocity of the vehicle  at time t1-t4 based on estimate A and estimate B obtained at  time t0  Estimate  t (ms)  R(t) (m)  v(t) (m/s)  Ap(t) (dB)  t0 = 0  40  5  0  Estimate A  t1 = 30  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='9  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='06  with  t2 = 60  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='7  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='13  constant v  t3 = 90  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='6  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='19  t4 = 120  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='26  t0 = 0  10  20  0  Estimate B  t1 = 30  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='07  with  t2 = 60  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='8  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='22  constant v  t3 = 90  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='45  t4 = 120  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='6  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='77  t0 = 0  40  5  0  Estimate A  t1 = 30  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='06  with  t2 = 60  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='13  constant a  t3 = 90  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2  t4 = 120  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='28  t0 = 0  10  20  0  Estimate B  t1 = 30  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='08  with  t2 = 60  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='8  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='24  constant a  t3 = 90  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='49  t4 = 120  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='6  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='86  Based on estimate A and estimate B obtained at time t0 and  the linear motion model, the predicted ranges R(t), velocities  v(t) and normalized peak amplitudes Ap(t) at time t1-t4 are  tabulated in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Ap(t0) is normalized to 0 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The real  radar images calculated at time t0 and the predicted radar  images at time t1-t4 in the future are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The  blue dashed lines indicate the real radar images yielded by  the sensing block transmitted at time t0, while the predicted  radar images at time t1-t4 are visualized with assorted colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  It can be observed that although a vehicle with R = 40 m and  v = 5 m/s produces the same radar image as that of a vehicle  with R = 10 m and v = 20 m/s at time t0, the following  radar images at time t1-t4 show different patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(a)  shows minor peak shifts due to the elapsed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Because  constant velocity causes unchanged fL, and the quite low  velocity (5 m/s) causes minor range changes at time t1-t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Hence, the changes in fH are less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The bin indices  at peak locations at time t0-t4 are within 79-81 and 133-134,  as shown in the zoomed-in sections in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Also, the  peak amplitude increases slightly from 0 dB at t0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='26 dB  at t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The minor range changes causes small received signal  power changes, which, in turn, leads to slight peak amplitude  fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  It can be observed that the radar images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(b)  are pretty different from the radar images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The  constant velocity produces an unchanged spectral frequency  fH in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Meanwhile, the high velocity (v = 20 m/s)  leads to significant changes in the range (major changes in  spectral frequency fL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The bin indices at peak locations at  time t0-t4 are within 80-88 and 128-134, which are distinctly  different from the bin indices in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Furthermore, due to  the significant distance changes, the peak amplitude increases  rapidly from 0 dB at t0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='77 dB at t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=" Therefore, for constant  velocity, by comparing the bin indices and peak amplitudes  0  B  ul  0  10  results  20  2  Normalized DF'  76  78  80  134  135  136  : R= 40 m, v= 5 m/s  30  t,: R= 39." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='9 m, v= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2 m/s  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='7 m, v= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='3 m/s  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5 m, v= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5 m/s  40  0  20  40  60  80  100  120  140  160  180  200  Indices of /B  0  d  4  T results  2  10  0  20  131132133134  80  85  90  : R= 10 m, v= 20 m/s  Normalized  t,: R= 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m, v= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2 m/s  30  t,: R= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='8 m, v= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='3 m/s  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2 m, v= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='5 m/s  40  : R= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='6 m, v= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='7 m/s  4  0  20  40  60  80  100  120  140  160  180  200  Indices of /0  B  d  ul  0  10  results  20  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2  79  80  81  133  134  DF  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 40 m, v= 5 m/s  0  30  t,: R= 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='9 m, v= 5 m/s  : R= 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='7 m, v= 5 m/s  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='6 m, v= 5 m/s  40  3  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m, v= 5 m/s  0  20  40  60  80  100  120  140  160  180  200  Indices of /B  0  da  4  results  2  2  10  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  2  20  128  130  132  134  80  82  84  86  88  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 10 m, v= 20 m/s  Jormalized  0  t,: R= 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m, v= 20 m/s  30  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=': R= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='8 m, v= 20 m/s  t,: R= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='2 m, v= 20 m/s  40  t,: R= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content="6 m, v= 20 m/s  4'  0  20  40  60  80  100  120  140  160  180  200  Indices of /from predicted radar images at time t1-t4 with the bin indices  and peak amplitudes from actual radar images obtained at time  t1-t4, the correct range-velocity estimate can be identified." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Although the acceleration of vehicles is utterly unpre-  dictable in the traffic monitoring scenario, the acceleration  of vehicles is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The maximum acceleration of high-  performance cars is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For the linear motion model with  constant acceleration 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m/s2, the predicted radar images at  time t1-t4 are still significantly different between estimate A  and estimate B, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The non-  zero acceleration will induce changes in both fH and fL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For  estimate A, the velocity changes slowly due to the limited  acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The range also changes slowly due to the low  acceleration and low velocity (5 m/s) at time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Therefore,  the vehicle yields minor changes in both fH and fL at t1-  t4 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For a vehicle with R = 10 m,  v = 20 m/s, and a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='4 m/s2 at time t0, enormous range  changes due to high velocity induce major changes in fL at  time t1-t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Major changes in fL shift the peaks significantly,  leading to a sparser peak pattern in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(d) than the pattern  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Therefore, the velocity is crucial to filter out the  incorrect estimates herein since the range change depends on  the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The limited acceleration in the traffic scenario is  a minor factor in changing the peaks in the radar image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Discussions  It has been mentioned that the signal processing complexity  of applying 2D-DFT is computationally high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' An OFDM  sensing system with parameters in Table II needs 2N ×O(N 2)  complex multiplications (480 DFTs in row and 480 IDFTs  in column) to generate a range-velocity estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' In contrast,  the computational complexity of the proposed algorithm is  relatively low because it avoids the DFT calculation both on  all rows and all columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Since the phase shift of modulation signals along the  frequency axis can extract the delay, and the phase shift of  modulation signals along the time axis can extract the Doppler,  the signaling overhead of the proposed diagonal structure is  reduced by combining the phase shift along both the frequency  axis and the time axis into a series of sensing signals along the  diagonal of the resource block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Hence the range and velocity  can be derived simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The sensing overhead of the  conventional comb structure is N 2/N 2  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The sensing overhead  of the proposed diagonal structure is N/N 2  c , reducing the  sensing overhead by a factor of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  The only minor disadvantage of the proposed algorithm is  that two unlabeled spectral peaks in the radar image cause  ambiguity in the range-velocity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The proposed multi-  temporal data fusion method can be used to resolve ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  Furthermore, steady objects do not create ambiguity by using  the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' For example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 7 shows the radar  image of a steady object with a range of 40 m and a  velocity of 0 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Another object with a range of 0 m and  a velocity of 20 m/s produces the same radar image shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' However, it directly contacts the antenna of the JCAS  system due to the range R being zero, which is kinematically  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' 7: The radar image of an object with range of 40 m and  velocity of 0 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Therefore, the proposed algorithm can derive the  range-velocity estimate of a steady object without ambiguity  by using one diagonal block of sensing signals only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' CONCLUSIONS  The typical processing algorithm of OFDM radar applies  2D-DFT for extracting the range and velocity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  This paper proposes a diagonal waveform structure and cor-  responding signal processing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' With this approach,  two advantages can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' First, the signal processing  algorithm is simple regarding computational complexity and  memory requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Second, sensing overhead is signifi-  cantly reduced by the proposed waveform structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The minor  disadvantage is that the proposed approach obtains the range  and velocity in a coupled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Therefore, range-velocity  ambiguity may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' A multi-temporal data fusion method  can be performed to resolve ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' The simulations have  proven the operability of the proposed waveform and signal  processing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content='  REFERENCES  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=' Wymeersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
+page_content=', “Deliverable D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE1T4oBgHgl3EQfqwWe/content/2301.03347v1.pdf'}
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HtAzT4oBgHgl3EQfHvtN/content/tmp_files/2301.01049v1.pdf.txt

@@ -0,0 +1,1293 @@
+Frequency-Domain Detection for Molecular
+Communications
+Meltem Civas∗† Ali Abdali∗ Murat Kuscu∗ Ozgur B. Akan∗†
+∗Center for neXt-generation Communications (CXC)
+Department of Electrical and Electronics Engineering
+Koc¸ University, 34450, Istanbul, Turkey
+{mcivas16, aabdali21, mkuscu, akan}@ku.edu.tr
+†Internet of Everything (IoE) Group
+Electrical Engineering Division, Department of Engineering
+University of Cambridge, CB3 0FA Cambridge, UK
+{mc2365, oba21}@cam.ac.uk
+Abstract—Molecular Communications (MC) is a bio-inspired
+communication paradigm which uses molecules as information
+carriers, thereby requiring unconventional transmitter/receiver
+architectures and modulation/detection techniques. Practical MC
+receivers (MC-Rxs) can be implemented based on field-effect
+transistor biosensor (bioFET) architectures, where surface recep-
+tors reversibly react with ligands, whose concentration encodes
+the information. The time-varying concentration of ligand-bound
+receptors is then translated into electrical signals via field-effect,
+which is used to decode the transmitted information. However,
+ligand-receptor interactions do not provide an ideal molecular
+selectivity, as similar types of ligands, i.e., interferers, co-existing
+in the MC channel can interact with the same type of receptors,
+resulting in cross-talk. Overcoming this molecular cross-talk with
+time-domain samples of the Rx’s electrical output is not always
+attainable, especially when Rx has no knowledge of the interferer
+statistics or it operates near saturation. In this study, we propose
+a frequency-domain detection (FDD) technique for bioFET-based
+MC-Rxs, which exploits the difference in binding reaction rates
+of different types of ligands, reflected to the noise spectrum of the
+ligand-receptor binding fluctuations. We analytically derive the
+bit error probability (BEP) of the FDD technique, and demon-
+strate its effectiveness in decoding transmitted concentration
+signals under stochastic molecular interference, in comparison
+to a widely-used time-domain detection (TDD) technique. The
+proposed FDD method can be applied to any biosensor-based
+MC-Rxs, which employ receptor molecules as the channel-Rx
+interface.
+Index Terms—Molecular communications, receiver, frequency-
+domain detection, biosensor, ligand-receptor interactions
+I. INTRODUCTION
+Using molecules to encode and transfer information, i.e.,
+Molecular Communications (MC), is nature’s way of con-
+necting bio things, such as natural cells, with each other.
+Engineering this unconventional communication paradigm to
+extend our connectivity to synthetic bio-nano things, such as
+nanobiosensors, artificial cells, is the vision that gave rise to
+the Internet of Bio-Nano Things (IoBNT), a novel networking
+framework promising for unprecedented healthcare and envi-
+ronmental applications of bionanotechnology [1], [2].
+Being fundamentally different from the conventional elec-
+tromagnetic communication techniques, MC requires novel
+transceiver architectures along with new modulation, coding,
+and detection techniques that can cope with the highly time-
+varying, nonlinear, and complex channel characteristics in bio-
+chemical environments [3]. The design of MC receivers (MC-
+Rxs) and detection techniques has unquestionably attracted
+the most attention in the literature. However, due to the
+simplicity it provides in modeling, many of the previous
+studies considered passive Rx architectures, that are physically
+unlinked from the MC channel, and thus, of little practical
+relevance [3]. An emerging trend in MC is to model and
+design more practical MC-Rxs that employ ligand receptors on
+their surface as selective biorecognition units, resembling the
+sensing and communication interface of natural cells. One such
+design, which was practically implemented in [4], is based on
+field-effect transistor biosensors (bioFETs), where the ligand-
+receptor (LR) interactions are translated into electrical signals
+via field-effect for the decoding of the transmitted information.
+LR interactions are fundamental to the sensing and commu-
+nication of natural cells. However, the selectivity of biological
+receptors against their target ligands is not ideal, and this so-
+called receptor promiscuity results in cross-talk of other types
+of molecules co-existing in the biochemical environment [5].
+This cross-talk is often dealt with by natural cells through
+intracellular chemical reaction networks and multi-state recep-
+tor mechanisms, such as kinetic proofreading [6]. The same
+molecular interference problem also applies to abiotic MC-Rxs
+that employ ligand receptors, and thus, should be addressed
+in developing reliable detection techniques [7].
+Our previous studies on biosynthetic MC-Rxs have ad-
+dressed the molecular interference problem by developing
+detection techniques based on sampling the bound time inter-
+vals of individual receptors to discriminate between interferer
+and information molecules [6], [7]. However, this approach
+is not plausible for biosensor-based MC-Rxs, which have
+no access to time-trajectory of individual receptor states. On
+the other hand, decoding information from the time-varying
+concentration of bound receptors performs poorly due to the
+indistinguishability of different ligand types in time-domain,
+especially when the Rx does not have any knowledge of the
+statistics of the interferer concentration, and when the Rx
+operates near saturation [7].
+In this paper, we develop a frequency-domain detec-
+tion (FDD) technique for biosensor-based MC-Rxs based on
+arXiv:2301.01049v1  [eess.SP]  3 Jan 2023
+
+LR binding interactions, which can distinguish different types
+of ligands co-existing in the channel and estimate their indi-
+vidual concentrations from the power spectral density (PSD)
+of the fluctuations in receptor occupancy, i.e., binding noise.
+Stochastic and reversible LR interactions can be modeled
+as a two-state continuous-time Markov process at equilibrium
+where the state transition rates are given by the binding and
+unbinding rates of LR pair [5]. Although many different types
+of ligands can interact with the same type of receptors, these
+interactions are typically governed by different binding and
+unbinding rates. This difference in reaction rates is reflected to
+a difference in characteristic frequency fch of the interactions,
+which is the reciprocal of the correlation time τB of the
+Markov process at equilibrium, and also a function of ligand
+concentration and LR reaction rates [7]. The characteristic fre-
+quency of the LR pair manifests itself as a cut-off frequency in
+the Lorentzian-shaped PSD of the binding noise. The proposed
+FDD method exploits this correlation in the frequency domain
+to estimate the concentration of information molecules in a
+Maximum Likelihood (ML) manner, and using the estimated
+concentration, it optimally decodes the transmitted informa-
+tion. We obtained the bit error probability (BEP) for FDD
+in closed form and compared it to the error performance of
+a time-domain detection (TDD) technique, which relies on
+the number of bound receptors, sampled at a single sampling
+point. The results of the performance analysis indicate that the
+proposed FDD method vastly outperforms the TDD method,
+especially at high interference conditions.
+II. SYSTEM MODEL
+We consider a microfluidic MC system utilizing binary
+concentration shift keying (CSK) such that the transmitter (Tx)
+instantly releases Nm|s number of molecules at the beginning
+of each signaling interval [8]. Here m stands for information
+molecules, and s ∈ {0, 1} denotes the transmitted bit. The
+signaling interval is assumed to be large enough to neglect
+inter-symbol interference (ISI). The microfluidic channel is
+abstracted as a 3-dimensional channel with a rectangular cross-
+section, as shown in Fig. 1(a). Tx is located at the channel
+inlet, and the molecules are released instantly and uniformly
+across the cross-section of the channel and propagate through
+unidirectional fluid flow from Tx to Rx, which is located at
+the channel bottom. We consider a two-dimensional graphene
+bioFET-based MC-Rx as illustrated in Fig. 1(b) [4]. There is
+a single type of interferer molecules in the channel, which can
+also bind the receptors on Rx, though with different reaction
+rates. The concentration of the interferer molecules in the Rx’s
+vicinity, ci, at the sampling time is assumed to follow a log-
+normal distribution with mean µci and variance σ2
+ci. We as-
+sume that Rx has the knowledge of the number of information
+molecules transmitted, Nm|s, and the binding/unbinding rates
+of information and interferer molecules.
+The released molecules propagate along the microfluidic
+channel through convection and diffusion. While convection
+results in the uniform and unidirectional drift of the transmitted
+molecules from Tx to Rx, diffusion acts in all directions
+hch
+lch
+x
+�
+z
+Si
+SiO2
+Drain
+electrode
+Source
+electrode
+Receptor
+Information
+molecule
+Interferer
+molecule
+Insulator
+Graphene
+channel
+(a)
+(b)
+MC Receiver
+MC Transmitter
+Flow 
+Direction
+x = xR
+y
+x
+z
+hch
+lch
+Fig. 1: a) 3D view of the microfluidic channel; the locations of
+Tx and Rx are shown. b) Graphene FET-based MC-Rx exposed
+to information and interferer molecules.
+causing the dispersion of the molecules as they propagate. The
+dispersion results in a smooth concentration profile which can
+be approximated by a Gaussian distribution. Assuming that
+the number of ligands binding the receptors is low enough to
+neglect the change of concentration in the channel, the prop-
+agation can be represented as a one-dimensional convection-
+diffusion problem with the following solution [8]:
+cm|s(x, t) =
+Nm|s
+Ach
+√
+4πDt
+exp
+�
+−(x − ut)2
+4Dt
+�
+,
+(1)
+where cm|s(x, t) is the ligand concentration at position x and
+time t, Ach = hch × lch is the cross-sectional area of the
+channel with hch and lch being the channel height and width,
+respectively, u is fluid flow velocity in the x-axis, and D is the
+effective diffusion coefficient. For channels with rectangular
+cross-section, D can be expressed as follows [9]:
+D =
+�
+1 +
+8.5u2h2
+chl2
+ch
+210D2
+0(h2
+ch + 2.4hchlch + l2
+ch)
+�
+D0,
+(2)
+where D0 is the diffusion coefficient of the ligand.
+The peak of ligand concentration profile given by (1)
+reaches the Rx’s center position, xR, at time tD = xR
+u . As
+the MC channel characteristic is similar to a low-pass filter
+due to diffusion, the concentration signal is slowly varying
+around the Rx position, thus allowing equilibrium conditions
+for the LR reactions with steady ligand concentration in a
+short time window around tD [8], [9]. Rx can sample the
+receptor states at time t = tD when the ligand concentration
+is cm|s(xR, tD) =
+Nm|s
+Ach
+√4πDtD [10]. Therefore, the number
+of bound receptors, Nb|s, follows Binomial distribution with
+mean µNb|s = pb|sNr and variance σ2
+Nb|s = pb|s(1 − pb|s)Nr
+[10], where Nr is the number of independent surface receptors.
+Bound state probability of a single receptor, pb|s, in the
+presence of two different types of ligands, i.e., information
+
+and interferer molecules, is given as [6]
+pb|s =
+cm|s/KDm + ci/KDi
+1 + cm|s/KDm + ci/KDi
+,
+(3)
+where KDm = k−
+m/k+
+m and KDi = k−
+i /k+
+i is the dissociation
+constant of information and interferer molecules, respectively.
+The binding of charged ligands to the receptors creates an
+effective charge reflected on the graphene channel as expressed
+by QGr|s = Nb|sqeffNe−, where Ne− is number of free
+electrons per ligand molecules. qeff is the effective charge
+of a single electron of a bound ligand in the presence of
+ionic screening, i.e., Debye screening: qeff = q×exp
+�
+− r
+λD
+�
+,
+where q is the elementary charge, r is the length of a surface
+receptor, and λD is Debye length whose relation is given by
+λD =
+�
+(ϵκBT)/(2NAq2cion), where ϵ is the permittivity of
+the medium, κB is the Boltzmann’s constant and NA is the
+Avogadro’s constant [10]. Then, the mean surface potential
+due to bound molecules can be written as ΨGr|s =
+QGr|s
+CG ,
+where CG=
+�
+1
+CGr+ 1
+CQ
+�−1
+is the total gate capacitance of the
+bioFET. CGr is the electrical double layer capacitance between
+graphene and electrolyte channel, CGr = AGrϵ/λD, with AGr
+being the area of graphene surface exposed to the electrolyte,
+and CQ is quantum capacitance, CQ = cq × AGr, where cq is
+the quantum capacitance of graphene per unit area [4]. The
+deviation in the output current due to bound molecules at
+equilibrium is
+∆Ib|s = g × ΨGr|s,
+(4)
+where g is the bioFET transconductance. For large Nr, the
+number of bound receptors Nb|s at the sampling time can
+be approximated as Gaussian distributed [10], i.e., Nb|s ∼
+N(µNb|s, σ2
+Nb|s). As the transduction process is linear, the
+change in the output current due to bound molecules can also
+be approximated as Gaussian with mean µ∆Ib|s = ζµNb|s and
+variance σ2
+∆Ib|s = ζ2σ2
+Nb|s, where ζ =
+�
+qeff Ne−g
+CG
+�
+.
+Another type of noise that contributes to the overall output
+current fluctuations in low-dimensional semiconductor mate-
+rials is 1/f noise, which depends on the gate voltage and
+is independent of the received signal. We use the commonly
+utilized charge-noise model describing the behavior of 1/f
+noise in graphene FETs [11]: Sf(f) = Sf1Hz/f β where Sf1Hz
+is the noise power at 1 Hz, and the noise exponent β is an
+empirical parameter 0.8 ≤ β ≤ 1.2. As discussed in [10], 1/f
+noise can be approximated as white noise within physically
+relevant observation windows. Based on this, the variance of
+1/f noise can be written as
+σ2
+f =
+� fL
+0
+Sf(fL)df +
+� fH
+fL
+Sf(f)df,
+(5)
+where fL is the lower frequency of the observation window,
+below which the noise power is considered constant, and fH is
+the upper frequency, beyond which the noise power is assumed
+to be negligible. Hence, the variance and mean of total output
+current variance is σ2
+∆Is = ζ2σ2
+Nb|s + σ2
+f and µ∆Is = µ∆Ib|s.
+III. TIME-DOMAIN DETECTION
+Since Rx has no knowledge of the interferer concentration
+statistics, it constructs the optimal ML decision threshold for
+TDD solely based on its knowledge of the received signal
+statistics corresponding to the transmitted concentration of
+information molecules [7]:
+γtd =
+1
+σ2
+∆I1 − σ2
+∆I0
+�
+σ2
+∆I1µ∆I0 − σ2
+∆I0µ∆I1 + σ∆I1σ∆I0
+×
+�
+(µ∆I1 − µ∆I0)2 + 2(σ2
+∆I1 − σ2
+∆I0) ln(σ∆I1/σ∆I0)
+�
+.
+(6)
+As Rx does not account for interference statistics in calculating
+γtd, it uses the bound state probability corresponding to a
+single molecule case, namely, pb|s =
+cm|s/KDm
+1+cm|s/KDm .
+To derive the BEP for TDD, we first obtain the statis-
+tics of the receiver output. By applying the law of total
+expectation, we can express the mean number of bound
+receptors as follows: µNb|s =
+� ∞
+0
+Nrpb|s(ci)f(ci)dci, where
+pb|s(ci) =
+cm|s/KDm+ci/KDi
+1+cm|s/KDm+ci/KDi , and f(·) is the probability
+density function of log-normal distribution. Hence, µ∆Is =
+ζµNb|s. Similarly, by applying the law of total variance, we
+obtain the output current variance as
+σ2
+∆Is = ζ2
+� � ∞
+0
+�
+1 − pb|s(ci)
+�
+pb|s(ci)Nrf(ci)dci
++
+� ∞
+0
+�
+pb|s(ci)Nr
+�2 f(ci)dci
+�
+− µ2
+∆Is + σ2
+f.
+(7)
+Therefore, given the decision threshold γtd, BEP for time
+detection method can be expressed as follows [7]:
+P T DD
+e
+= 1
+4 erfc
+�
+�γtd − µ∆I0
+�
+2σ2
+∆I0
+�
+� + 1
+4 erfc
+�
+�µ∆I1 − γtd
+�
+2σ2
+∆I1
+�
+� .
+(8)
+IV. FREQUENCY-DOMAIN DETECTION
+In this section, we introduce the FDD method utilizing the
+model and observed PSD of the overall noise process (binding
+noise + 1/f noise of the graphene bioFET-based MC-Rx) to
+estimate the received concentration of information molecules
+cm, which will be used in symbol decision. Here, the observed
+PSD is the periodogram of the noise constructed with the time-
+domain samples. In the sequel, we describe the model PSD
+and then introduce the proposed estimation method.
+A. Theoretical Model of Binding Noise PSD
+This section describes the theoretical model of the binding
+noise PSD for a particular pair of information and interference
+concentration, namely λ = [cm, ci]. The binding process of
+receptors can be described by the Langmuir reaction model
+with three states, i.e., unbound (R), bound with information
+molecules (RM) and bound with interferer molecules (RI),
+with state occupation probabilities pR, pRM and pRI, respec-
+tively [12]: R + M
+k−
+m
+⇌
+k+
+m
+RM, and R + I
+k−
+i⇌
+k+
+i
+RI. Hence, the
+
+chemical master equations are expressed as follows:
+�
+�����
+dpRM
+dt
+dpRI
+dt
+dpR
+dt
+�
+�����
+=
+�
+�
+−k−
+m
+0
+k+
+mcm
+0
+−k−
+i
+k+
+i ci
+k−
+m
+k−
+i
+−k+
+mcm − k+
+i ci
+�
+�
+�
+�
+pRM
+pRI
+pR
+�
+� (9)
+The matrix containing reaction rates and the concentrations in
+(9), has rank 2 since one state probability can be written in
+terms of the other two state occupation probabilities as pR +
+pRM + pRI = 1. Therefore, by setting the left-hand side in
+(9) to zero the equilibrium probabilities can be obtained as
+p0
+RM =
+cm/KDm
+1 +
+cm
+KDm +
+ci
+KDi
+, p0
+RI =
+ci/KDi
+1 +
+cm
+KDm +
+ci
+KDi
+(10)
+and p0
+R = 1−(p0
+RM +p0
+RI). In the equilibrium conditions, the
+state occupation probabilities can be expressed in terms of the
+equilibrium state probability and the fluctuations around this
+probability [12], [13] as
+pj(t) = p0
+j + ∆pj(t),
+j ∈ {RM, RI, R}.
+(11)
+Putting (11) into (9) and using Taylor’s expansion, the state
+fluctuations can be expressed as follows [12]:
+d∆p′(t)
+dt
+= Ω∆p′(t).
+(12)
+In (12), ∆p′(t) = [∆pRM(t); ∆pRI(t)] is the reduced form of
+the vector containing the state occupation probabilities, where
+Ω is
+Ω =
+�−k+
+mcm − k−
+m
+−k+
+m
+−k+
+i ci
+−k+
+i ci − k−
+i
+�
+.
+(13)
+The deviation in the output current of the MC-Rx due to
+stochastic binding reactions, i.e., ∆Ib(t), is then obtained as
+∆Ib(t) = qeff g
+CG
+zT R∆p′(t)
+(14)
+where z = [Ne−; Ne−; 0] is the vector containing the number
+of elementary charges corresponding to each state and R is the
+transformation matrix such that ∆p(t) = R∆p′(t). As ∆Ib(t)
+is a stationary process, the theoretical PSD of the binding noise
+fluctuations can be found by setting t = 0 as follows [12]:
+Sb(f) = 2 F{E[∆Ib(t)∆Ib(t + τ)]}
+(15)
+= 2 F{E[∆Ib(0)∆Ib(τ)]}
+= 4Nr
+�qeffg
+CG
+�2
+z⊺RΓ
+�
+Re{(j2πfI2×2 − Ω)−1}
+�⊺ R⊺z
+where F{·} stands for Fourier transform, I2×2 is the identity
+matrix and Γ is the matrix containing the expected state
+probabilities, which is given as follows [12]:
+Γ =
+�p0
+RM
+�
+1 − p0
+RM
+�
+−p0
+RMp0
+RI
+−p0
+RMp0
+RI
+p0
+RI
+�
+1 − p0
+RI
+�
+�
+.
+(16)
+Therefore, the theoretical PSD of the total current noise
+corresponding to a particular (cm, ci) pair can be written as
+S(f) = Sb(f) + Sf(f).
+(17)
+B. Maximum Likelihood Estimation of PSD Parameters
+In the following part, we describe the parameter value
+extraction, namely the estimation of information and inter-
+fering molecule concentrations, λ = [cm, ci], from the noise
+PSD. The detector uses the estimated information molecule
+concentration ˆcm for symbol decision, as will be explained in
+the following section, Sec. IV-C. Our analysis is based on the
+following assumptions:
+• The total noise process, namely the binding fluctuations
+combined with 1/f noise, is stationary, zero-mean with a
+single-sided spectrum.
+• Rx is given the model PSD function expressed by (17), and
+the binding/unbinding rates of information and interferer
+molecules. Rx also has the knowledge of the number of
+information molecules transmitted for bits s = 0 and s = 1
+as mentioned in Sec. II. Therefore, Rx will estimate the
+steady information and interferer concentrations by taking
+time samples from the output current ∆Ib in a sampling
+window, where we consider a single realization of the
+interferer concentration ci following log-normal distribution
+as mentioned in Sec. II. The DC component of ∆Ib is
+discarded to isolate the noise.
+• The information and interferer concentrations are considered
+constant in the sampling window based on the equilibrium
+assumption discussed in Sec. II [10].
+• The observed PSD of time domain samples and the para-
+metric model of the PSD expressed by (17) will be used in
+the ML estimation of λ = [cm, ci]. It is assumed that the
+observed PSD is calculated with the periodogram method.
+For each transmitted symbol, we have N number of noise
+samples x = (x1, x2, ..., xN) taken with the sampling period
+of ∆t. Hence, the total duration of sampling per symbol,
+namely the length of the sampling window, is Td = N∆t.
+Periodogram for the sampled signal can be computed from
+the Discrete Fourier transform (DFT) of the samples x.
+With even N, the periodogram values are then expressed as
+follows: Yk = 2∆t
+N |Xk|2 where k = 1, ..., N/2 − 1, and |Xk|
+DFT components of x.
+For a stochastic time series of length N, the random variable
+Wk = 2
+Yk
+S(fk) follows chi-squared distribution χ2 [14], where
+S(fk) given by Eq. (17) is the true PSD at frequency fk and
+fk =
+k
+N∆t and k = 1, ..., N/2 − 1. The χ2 distribution with
+two degrees of freedom is in fact the exponential distribution
+[15]. Therefore, the periodogram values are exponentially
+distributed about the true PSD with the following probability
+given the model PSD value at a given frequency:
+p(Yk|S(fk)) =
+1
+S(fk)e−
+Yk
+S(fk) ,
+(18)
+following that S(fk) is also expectation value at fk [15].
+Based on (18), the likelihood of observing a pair of particular
+information and interferer concentrations, λ = [cm, ci], is
+L(λ) =
+N/2−1
+�
+k=1
+p(Yk|S(fk, λ)) =
+N/2−1
+�
+k=1
+1
+S(fk, λ)e−
+Yk
+S(fk,λ) ,
+(19)
+
+Algorithm 1 Algorithm for the FDD
+1: Run Newton method with initial guess λ0 = [c0
+m, c0
+i ] to
+find optimal λ∗ = [c∗
+m, c∗
+i ] satisfying (21).
+2: ˆcm ← c∗
+m
+3: Find the decision threshold γfd.
+4: Run threshold operation:
+5: if ˆcm > γfd Estimated bit ˆs ← 1
+6: else Estimated bit ˆs ← 0
+7: end if
+where λ = [cm, ci] is the parameters to be estimated. Here, we
+use Whittle likelihood, which can be a good approximation to
+the exact likelihood asymptotically, and also provide computa-
+tional efficiency, i.e., O(n log n) compared to O(n2) for exact
+likelihood [15], [16]. Accordingly, the quasi-log likelihood can
+be written as follows:
+ln L(λ) = −
+N/2−1
+�
+k=1
+�
+Yk
+S(fk, λ) + ln S(fk, λ)
+�
+.
+(20)
+ML estimator extracts the value of λ, i.e., ˆλ, that maximizes
+(20). Maximizing ln L(λ) is equivalent to minimizing l =
+− ln L(λ) [17], such that
+ˆλ = arg min
+λ {l}.
+(21)
+Eq. (21) can be solved using numerical methods such as
+Newton-Ralphson method attaining the ML within few iter-
+ations [18].
+C. Symbol Detection
+The ML estimator described in Sec. IV-B is asymptotically
+unbiased such that ˆλ tends to have multi-normal distribution
+[19] with E[ˆλ] = λ, and the respective variance of the esti-
+mated parameters, which is the diagonal elements of inverse
+Fisher information matrix (FIM) F(λ),
+σ2
+ˆ
+λi = (F(λ))−1
+(ii),
+F(λ)(ij) = E
+�
+∂2l
+∂λi∂λj
+�
+.
+(22)
+where the expectation is taken with respect to the probabil-
+ity distribution of the observed spectrum p(Y1, Y2, ..., YN).
+Putting l =−lnL(λ) into (22), the FIM can be expanded as:
+F(ij) =
+E
+�
+�
+N/2−1
+�
+k=1
+S(fk) − Yk
+S(fk)2
+∂2S
+∂λi∂λj
++ 2Yk − S(fk)
+S(fk)3
+∂S
+∂λi
+∂S
+∂λj
+�
+� .
+(23)
+Considering that S(f) is a slowly varying function, there
+is no need to calculate individual periodogram values in
+(23). Because periodogram values can be smoothed by
+summing over frequency such that �N/2−1
+n=1
+Ykφ(fk)
+≃
+�N/2−1
+n=1
+S(fk)φ(fk), for any smooth function φ(fk)
+[19],
+[20]. Based on this, Eq. (23) can be simplified as [19]
+F(ij) ≃
+N/2−1
+�
+k=1
+1
+S(fk)2
+∂S
+∂λi
+∂S
+∂λj
+,
+(24)
+where the derivatives are taken at the true value of the
+parameters. This is a good approximation for the large number
+of samples such that periodogram values can be approximated
+as Gaussian by the central limit theorem [21]. Rx decides
+the transmitted bit by applying the ML decision rule on the
+estimated information molecule concentration ˆcm, as described
+by the pseudo-algorithm for FDD in Algorithm 1. The ML
+decision threshold for FDD is
+γfd =
+1
+σ2
+ˆcm|1 − σ2
+ˆcm|0
+�
+σ2
+ˆcm|1cm|0 − σ2
+ˆcm|0cm|1 + σˆcm|1σˆcm|0
+×
+�
+(cm|1 − cm|0)2 + 2(σ2
+ˆcm|1 − σ2
+ˆcm|0) ln
+�
+σˆcm|1/σˆcm|0
+��
+,
+(25)
+where σ2
+ˆcm|s is the variance and cm|s is the expected value of
+estimated information molecule when the transmitted bit is s ∈
+{0, 1}. Since Rx does not know the true value of the interfering
+molecule concentration, it computes the decision threshold as
+if there was no interference. Therefore, the following model
+PSD is used while computing the threshold γfd:
+S(f, cm) = 4Nrζ2
+1
+2πf + 1/τm
+pb(1 − pb) + Sf(f),
+(26)
+where τm = 1/(cmk+
+m + k−
+m) and pb =
+cm
+KDm + cm
+. Hence,
+using (26), and (24) for λ = [cm], the variance of the
+estimated information molecule concentration corresponding
+to the transmitted bit s ∈ {0, 1} can be written as σ2
+ˆcm|s =
+1/ �N/2−1
+k=1
+1
+S(fk)2 ( ∂S
+∂cm )2���cm=cm|s. Note that here the expres-
+sion for σ2
+ˆcm|s does not give the actual asymptotic variances
+since Rx estimates the value of cm based on the model PSD
+described by (17).
+D. Asymptotic Bit Error Probability
+To calculate BEP for FDD, we need the actual values of
+the variance of estimated information molecule concentration
+corresponding to s=0 and s=1, i.e., σ2
+ˆcm|s. Using the model
+PSD S(f, (cm, ci)) given by (17), and (24) with λ = [cm, ci],
+the variance can be expressed as σ2
+ˆcm|s=(Fs(λ))−1
+(11), where
+Fs’s elements are:
+Fs(11) =
+N/2−1
+�
+k=1
+1
+S(fk)2
+� ∂S
+∂cm
+�2���cm=cm|s
+ci=µci
+,
+(27)
+Fs(22) =
+N/2−1
+�
+k=1
+1
+S(fk)2
+� ∂S
+∂ci
+�2���cm=cm|s
+ci=µci
+,
+(28)
+Fs(12),(21) =
+N/2−1
+�
+k=1
+1
+S(fk)2
+� ∂S
+∂cm
+� � ∂S
+∂ci
+����cm=cm|s
+ci=µci
+.
+(29)
+As a result, BEP for FDD can be written as
+P F DD
+e
+= 1
+4 erfc
+�
+�γfd − cm|0
+�
+2σ2
+ˆcm|0
+�
+� + 1
+4 erfc
+�
+�cm|1 − γfd
+�
+2σ2
+ˆcm|1
+�
+� .
+(30)
+Here, it should be noted that (30) is an asymptotic expression
+based on the Gaussian distribution assumption in Sec. IV-C.
+
+10-2
+10-1
+100
+101
+102
+Frequency (Hz)
+10-23
+10-22
+10-21
+PSD (A2/Hz)
+Interferer molecule
+Information molecule
+fch|m
+fch|i
+Fig. 2: The model PSD with highlighted characteristic fre-
+quencies.
+V. PERFORMANCE EVALUATION
+In this section, we analyze the performance of FDD and
+TDD in terms of BEP. The default values of the system
+parameters are given in Table I, with the reaction rates
+adopted from [7]. In the rest of the paper, the saturation
+and the non-saturation corresponds to the Rx’s receptors
+being saturated due to high ligand concentrations and far
+from the saturation, respectively. To simulate the saturation,
+the number of transmitted information molecules is taken as
+Nm|s∈{0,1} = [2, 5] × 104. Otherwise, default values are used.
+We first consider the effect of the mean interference con-
+centration, µci on the BEP performance of TDD and FDD in
+saturation and non-saturation conditions. We define a tuning
+parameter γ such that mean interferer concentration is given
+by µci = γ cm|s=1. As shown in Fig. 3a, FDD outperforms
+TDD in both scenarios. The performance of TDD degrades
+dramatically due to Rx saturation with increasing µci. In non-
+saturation, the performance of FDD improves with increasing
+TABLE I: Default Values of System Parameters
+Temperature (T )
+300K
+Microfluidic channel height (hch), width (lch)
+5 µm, 10 µm
+Average flow velocity (u)
+10 µm/s
+Distance of Rx’s center position to Tx (xR)
+1mm
+Ionic concentration of medium (cion)
+30 mol/m3
+Relative permittivity of medium (ϵ/ϵ0)
+80
+Intrinsic diffusion coefficient (D0)
+2 × 10−11 m2/s
+Binding
+rate
+of
+information
+and
+interferer
+molecules (k+
+m, k+
+i )
+4 × 10−17 m3/s
+Unbinding rate of information molecules (k−
+m)
+2 s−1
+Unbinding rate of interferers (k−
+i )
+8 s−1
+Average # of electrons in a ligand (Ne−)
+3
+Number of independent receptors (Nr)
+120
+Length of a surface receptor (r)
+2 nm
+Transconductance of graphene bioFET (g)
+1.9044 × 10−4 A/V
+Width of graphene in transistor (lgr)
+10µm
+Quant. capacitance of graphene per unit area (cq)
+2 × 10−2 F m−2
+# of transmitted ligands for s = 0, 1 (Nm|s)
+[1, 5] × 103
+# of noise samples (N)
+700
+Sampling period (∆t)
+0.005 s
+Mean interference to information concentration
+ratio (γ = µci/cm|s=1)
+1
+Interference mean/std ratio (µci/σci)
+10
+Power of 1/f noise at 1 Hz (Sf1Hz )
+10−23 A2/Hz
+µci up to a certain point, beyond which further increase in
+µci degrades the performance of FDD because when Rx is not
+saturated, the variance of the estimated information molecule
+concentration σ2
+ˆcm|s is minimized at a certain µci beyond
+which its value increases with increasing µci. In saturation,
+however, σ2
+ˆcm|s monotonically increases with µci.
+Next, we consider the effect of similarity parameter, namely
+the affinity ratio of information and interferer molecules η =
+KDi/KDm, on the BEP performance for saturation and non-
+saturation cases. As displayed in Fig. 3b, FDD outperforms
+TDD in both saturation and non-saturation cases. Regarding
+the non-saturation case, the performances of both detection
+methods improve with increasing similarity up to a certain
+point because the effect of interference on the detection perfor-
+mance weakens; namely, bound state probability for interferer
+molecules decreases. However, when the similarity is further
+increased, the performance of FDD degrades. Intuitively, this
+is because the characteristic frequencies corresponding to bits
+s = 0 and s = 1 [22]
+fch|s = [fch|m, fch|i]
+= 1
+4π
+�
+� 1
+τm|s
++ 1
+τi
+±
+�� 1
+τm|s
+− 1
+τi
+�2
++ 4k+
+mcm|sk+
+i ci
+�
+� ,
+(31)
+where τm|s = 1/(cm|sk+
+m + k−
+m) and τi = 1/(cik+
+i + k−
+i ),
+are approaching each other in the spectrum, making it dif-
+ficult to distinguish the bits. As shown in Fig. 2 for an
+example scenario, two characteristic frequencies, fch|m and
+fch|i, appear in the spectrum for each transmitted bit due to
+the binding of two types of molecules, namely information
+and interferer molecules, with the order depending on the
+concentrations and binding/unbinding rates of the individual
+molecule types. In the non-saturation case, we do not observe
+this phenomenon because the characteristic frequencies do not
+come close to each other to degrade the detection performance
+with increasing similarity. We also consider the effect of the
+number of time samples, N, and the sampling period ∆t on
+the BEP performance. As shown in Fig. 3c, the performance
+of FDD increases with N. This is expected as taking more
+samples decreases the variance of the estimated information
+molecule concentration σ2
+ˆcm|s, hence, decreases the BEP. For
+TDD, the performance does not change with N as Rx takes
+one sample in the sampling window. For varying ∆t, the
+performance of FDD increases with increasing ∆t for the
+non-saturation case. Note that ∆t should be shorter than the
+characteristic time scale of any reactions to be able to capture
+the fluctuations and to satisfy the sampling at the equilibrium
+assumption, which is discussed in Sec. II. Therefore, we
+consider ∆t values satisfying this condition.
+VI. CONCLUSION
+In this paper, we proposed a FDD method for the FET-
+based MC-Rx, which utilizes the output noise PSD to extract
+the transmitted bit. We derived the BEP for the proposed
+method and a one-shot TDD method considering the existence
+
+0
+1
+2
+3
+4
+5
+Interference/Information (.)
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Bit error probability
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Bound state probability, PB
+TDD (Sat)
+FDD (Sat)
+TDD (Non-sat)
+FDD (Non-sat)
+s = 0 (Sat)
+s = 1 (Sat)
+s = 0 (Non-sat)
+s = 1 (Non-sat)
+(a)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Similarity (2)
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Bit error probability
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Bound state probability, PB
+TDD (Sat)
+FDD (Sat)
+TDD (Non-sat)
+FDD (Non-sat)
+s = 0 (Sat)
+s = 1 (Sat)
+s = 0 (Non-sat)
+s = 1 (Non-sat)
+(b)
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+# of Samples (N)
+10-6
+10-4
+10-2
+100
+Bit error probability
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Bound state probability, PB
+TDD (Sat)
+FDD (Sat)
+TDD (Non-sat)
+FDD (Non-sat)
+s = 0 (Sat)
+s = 1 (Sat)
+s = 0 (Non-sat)
+s = 1 (Non-sat)
+(c)
+Fig. 3: BEP for varying (a) mean interference concentration level (b) similarity of affinities for information and interferer
+molecules (c) number of time samples N.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Sampling Period ("t(ms))
+10-8
+10-6
+10-4
+10-2
+100
+Bit error probability
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Bound state probability, PB
+TDD (Sat)
+FDD (Sat)
+TDD (Non-sat)
+FDD (Non-sat)
+s = 0 (Sat)
+s = 1 (Sat)
+s = 0 (Non-sat)
+s = 1 (Non-sat)
+Fig. 4: BEP for varying sampling period.
+of a single type of interferer molecules in a microfluidic
+channel. Our analysis reveals that the proposed detection
+method significantly outperforms the TDD, primarily when
+high interference exists in the channel.
+ACKNOWLEDGMENT
+This work was supported in part by the AXA Research Fund
+(AXA Chair for Internet of Everything at Koc¸ University), the
+Horizon 2020 Marie Skłodowska-Curie Individual Fellowship
+under Grant Agreement 101028935, and by The Scientific and
+Technological Research Council of Turkey (TUBITAK) under
+Grant #120E301, and Huawei Graduate Research Scholarship.
+REFERENCES
+[1] O. B. Akan, et al., “Fundamentals of molecular information and commu-
+nication science,” Proceedings of the IEEE, vol. 105, no. 2, pp. 306–318,
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+[2] I. F. Akyildiz, et al., “Panacea: An internet of bio-nanothings application
+for early detection and mitigation of infectious diseases,” IEEE Access,
+vol. 8, pp. 140 512–140 523, 2020.
+[3] M. Kuscu, et al., “Transmitter and receiver architectures for molecular
+communications: A survey on physical design with modulation, coding,
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+[4] M. Kuscu, et al., “Fabrication and microfluidic analysis of graphene-
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+2013.
+

HtAzT4oBgHgl3EQfHvtN/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf,len=418
+page_content='Frequency-Domain Detection for Molecular  Communications  Meltem Civas∗† Ali Abdali∗ Murat Kuscu∗ Ozgur B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Akan∗†  ∗Center for neXt-generation Communications (CXC)  Department of Electrical and Electronics Engineering  Koc¸ University, 34450, Istanbul, Turkey  {mcivas16, aabdali21, mkuscu, akan}@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='tr  †Internet of Everything (IoE) Group  Electrical Engineering Division, Department of Engineering  University of Cambridge, CB3 0FA Cambridge, UK  {mc2365, oba21}@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='uk  Abstract—Molecular Communications (MC) is a bio-inspired  communication paradigm which uses molecules as information  carriers, thereby requiring unconventional transmitter/receiver  architectures and modulation/detection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Practical MC  receivers (MC-Rxs) can be implemented based on field-effect  transistor biosensor (bioFET) architectures, where surface recep-  tors reversibly react with ligands, whose concentration encodes  the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The time-varying concentration of ligand-bound  receptors is then translated into electrical signals via field-effect,  which is used to decode the transmitted information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' However,  ligand-receptor interactions do not provide an ideal molecular  selectivity, as similar types of ligands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', interferers, co-existing  in the MC channel can interact with the same type of receptors,  resulting in cross-talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Overcoming this molecular cross-talk with  time-domain samples of the Rx’s electrical output is not always  attainable, especially when Rx has no knowledge of the interferer  statistics or it operates near saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' In this study, we propose  a frequency-domain detection (FDD) technique for bioFET-based  MC-Rxs, which exploits the difference in binding reaction rates  of different types of ligands, reflected to the noise spectrum of the  ligand-receptor binding fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We analytically derive the  bit error probability (BEP) of the FDD technique, and demon-  strate its effectiveness in decoding transmitted concentration  signals under stochastic molecular interference, in comparison  to a widely-used time-domain detection (TDD) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The  proposed FDD method can be applied to any biosensor-based  MC-Rxs, which employ receptor molecules as the channel-Rx  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Index Terms—Molecular communications, receiver, frequency-  domain detection, biosensor, ligand-receptor interactions  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' INTRODUCTION  Using molecules to encode and transfer information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=',  Molecular Communications (MC), is nature’s way of con-  necting bio things, such as natural cells, with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Engineering this unconventional communication paradigm to  extend our connectivity to synthetic bio-nano things, such as  nanobiosensors, artificial cells, is the vision that gave rise to  the Internet of Bio-Nano Things (IoBNT), a novel networking  framework promising for unprecedented healthcare and envi-  ronmental applications of bionanotechnology [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Being fundamentally different from the conventional elec-  tromagnetic communication techniques, MC requires novel  transceiver architectures along with new modulation, coding,  and detection techniques that can cope with the highly time-  varying, nonlinear, and complex channel characteristics in bio-  chemical environments [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The design of MC receivers (MC-  Rxs) and detection techniques has unquestionably attracted  the most attention in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' However, due to the  simplicity it provides in modeling, many of the previous  studies considered passive Rx architectures, that are physically  unlinked from the MC channel, and thus, of little practical  relevance [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' An emerging trend in MC is to model and  design more practical MC-Rxs that employ ligand receptors on  their surface as selective biorecognition units, resembling the  sensing and communication interface of natural cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' One such  design, which was practically implemented in [4], is based on  field-effect transistor biosensors (bioFETs), where the ligand-  receptor (LR) interactions are translated into electrical signals  via field-effect for the decoding of the transmitted information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  LR interactions are fundamental to the sensing and commu-  nication of natural cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' However, the selectivity of biological  receptors against their target ligands is not ideal, and this so-  called receptor promiscuity results in cross-talk of other types  of molecules co-existing in the biochemical environment [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  This cross-talk is often dealt with by natural cells through  intracellular chemical reaction networks and multi-state recep-  tor mechanisms, such as kinetic proofreading [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The same  molecular interference problem also applies to abiotic MC-Rxs  that employ ligand receptors, and thus, should be addressed  in developing reliable detection techniques [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Our previous studies on biosynthetic MC-Rxs have ad-  dressed the molecular interference problem by developing  detection techniques based on sampling the bound time inter-  vals of individual receptors to discriminate between interferer  and information molecules [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' However, this approach  is not plausible for biosensor-based MC-Rxs, which have  no access to time-trajectory of individual receptor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' On  the other hand, decoding information from the time-varying  concentration of bound receptors performs poorly due to the  indistinguishability of different ligand types in time-domain,  especially when the Rx does not have any knowledge of the  statistics of the interferer concentration, and when the Rx  operates near saturation [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  In this paper, we develop a frequency-domain detec-  tion (FDD) technique for biosensor-based MC-Rxs based on  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='01049v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='SP]  3 Jan 2023  LR binding interactions, which can distinguish different types  of ligands co-existing in the channel and estimate their indi-  vidual concentrations from the power spectral density (PSD)  of the fluctuations in receptor occupancy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', binding noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Stochastic and reversible LR interactions can be modeled  as a two-state continuous-time Markov process at equilibrium  where the state transition rates are given by the binding and  unbinding rates of LR pair [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Although many different types  of ligands can interact with the same type of receptors, these  interactions are typically governed by different binding and  unbinding rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' This difference in reaction rates is reflected to  a difference in characteristic frequency fch of the interactions,  which is the reciprocal of the correlation time τB of the  Markov process at equilibrium, and also a function of ligand  concentration and LR reaction rates [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The characteristic fre-  quency of the LR pair manifests itself as a cut-off frequency in  the Lorentzian-shaped PSD of the binding noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The proposed  FDD method exploits this correlation in the frequency domain  to estimate the concentration of information molecules in a  Maximum Likelihood (ML) manner, and using the estimated  concentration, it optimally decodes the transmitted informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We obtained the bit error probability (BEP) for FDD  in closed form and compared it to the error performance of  a time-domain detection (TDD) technique, which relies on  the number of bound receptors, sampled at a single sampling  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The results of the performance analysis indicate that the  proposed FDD method vastly outperforms the TDD method,  especially at high interference conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' SYSTEM MODEL  We consider a microfluidic MC system utilizing binary  concentration shift keying (CSK) such that the transmitter (Tx)  instantly releases Nm|s number of molecules at the beginning  of each signaling interval [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Here m stands for information  molecules, and s ∈ {0, 1} denotes the transmitted bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The  signaling interval is assumed to be large enough to neglect  inter-symbol interference (ISI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The microfluidic channel is  abstracted as a 3-dimensional channel with a rectangular cross-  section, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Tx is located at the channel  inlet, and the molecules are released instantly and uniformly  across the cross-section of the channel and propagate through  unidirectional fluid flow from Tx to Rx, which is located at  the channel bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We consider a two-dimensional graphene  bioFET-based MC-Rx as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 1(b) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' There is  a single type of interferer molecules in the channel, which can  also bind the receptors on Rx, though with different reaction  rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The concentration of the interferer molecules in the Rx’s  vicinity, ci, at the sampling time is assumed to follow a log-  normal distribution with mean µci and variance σ2  ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We as-  sume that Rx has the knowledge of the number of information  molecules transmitted, Nm|s, and the binding/unbinding rates  of information and interferer molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  The released molecules propagate along the microfluidic  channel through convection and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' While convection  results in the uniform and unidirectional drift of the transmitted  molecules from Tx to Rx, diffusion acts in all directions  hch  lch  x  �  z  Si  SiO2  Drain  electrode  Source  electrode  Receptor  Information  molecule  Interferer  molecule  Insulator  Graphene  channel  (a)  (b)  MC Receiver  MC Transmitter  Flow  Direction  x = xR  y  x  z  hch  lch  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 1: a) 3D view of the microfluidic channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' the locations of  Tx and Rx are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' b) Graphene FET-based MC-Rx exposed  to information and interferer molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  causing the dispersion of the molecules as they propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The  dispersion results in a smooth concentration profile which can  be approximated by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Assuming that  the number of ligands binding the receptors is low enough to  neglect the change of concentration in the channel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' the prop-  agation can be represented as a one-dimensional convection-  diffusion problem with the following solution [8]:  cm|s(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' t) =  Nm|s  Ach  √  4πDt  exp  �  −(x − ut)2  4Dt  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (1)  where cm|s(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' t) is the ligand concentration at position x and  time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Ach = hch × lch is the cross-sectional area of the  channel with hch and lch being the channel height and width,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' u is fluid flow velocity in the x-axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' and D is the  effective diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' For channels with rectangular  cross-section, D can be expressed as follows [9]:  D =  �  1 +  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='5u2h2  chl2  ch  210D2  0(h2  ch + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='4hchlch + l2  ch)  �  D0,  (2)  where D0 is the diffusion coefficient of the ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  The peak of ligand concentration profile given by (1)  reaches the Rx’s center position, xR, at time tD = xR  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As  the MC channel characteristic is similar to a low-pass filter  due to diffusion, the concentration signal is slowly varying  around the Rx position, thus allowing equilibrium conditions  for the LR reactions with steady ligand concentration in a  short time window around tD [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Rx can sample the  receptor states at time t = tD when the ligand concentration  is cm|s(xR, tD) =  Nm|s  Ach  √4πDtD [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Therefore, the number  of bound receptors, Nb|s, follows Binomial distribution with  mean µNb|s = pb|sNr and variance σ2  Nb|s = pb|s(1 − pb|s)Nr  [10], where Nr is the number of independent surface receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Bound state probability of a single receptor, pb|s, in the  presence of two different types of ligands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', information  and interferer molecules, is given as [6]  pb|s =  cm|s/KDm + ci/KDi  1 + cm|s/KDm + ci/KDi  ,  (3)  where KDm = k−  m/k+  m and KDi = k−  i /k+  i is the dissociation  constant of information and interferer molecules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  The binding of charged ligands to the receptors creates an  effective charge reflected on the graphene channel as expressed  by QGr|s = Nb|sqeffNe−, where Ne− is number of free  electrons per ligand molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' qeff is the effective charge  of a single electron of a bound ligand in the presence of  ionic screening, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', Debye screening: qeff = q×exp  �  − r  λD  �  ,  where q is the elementary charge, r is the length of a surface  receptor, and λD is Debye length whose relation is given by  λD =  �  (ϵκBT)/(2NAq2cion), where ϵ is the permittivity of  the medium, κB is the Boltzmann’s constant and NA is the  Avogadro’s constant [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Then, the mean surface potential  due to bound molecules can be written as ΨGr|s =  QGr|s  CG ,  where CG=  �  1  CGr+ 1  CQ  �−1  is the total gate capacitance of the  bioFET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' CGr is the electrical double layer capacitance between  graphene and electrolyte channel, CGr = AGrϵ/λD, with AGr  being the area of graphene surface exposed to the electrolyte,  and CQ is quantum capacitance, CQ = cq × AGr, where cq is  the quantum capacitance of graphene per unit area [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The  deviation in the output current due to bound molecules at  equilibrium is  ∆Ib|s = g × ΨGr|s,  (4)  where g is the bioFET transconductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' For large Nr, the  number of bound receptors Nb|s at the sampling time can  be approximated as Gaussian distributed [10], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', Nb|s ∼  N(µNb|s, σ2  Nb|s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As the transduction process is linear, the  change in the output current due to bound molecules can also  be approximated as Gaussian with mean µ∆Ib|s = ζµNb|s and  variance σ2  ∆Ib|s = ζ2σ2  Nb|s, where ζ =  �  qeff Ne−g  CG  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Another type of noise that contributes to the overall output  current fluctuations in low-dimensional semiconductor mate-  rials is 1/f noise, which depends on the gate voltage and  is independent of the received signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We use the commonly  utilized charge-noise model describing the behavior of 1/f  noise in graphene FETs [11]: Sf(f) = Sf1Hz/f β where Sf1Hz  is the noise power at 1 Hz, and the noise exponent β is an  empirical parameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='8 ≤ β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As discussed in [10], 1/f  noise can be approximated as white noise within physically  relevant observation windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Based on this, the variance of  1/f noise can be written as  σ2  f =  � fL  0  Sf(fL)df +  � fH  fL  Sf(f)df,  (5)  where fL is the lower frequency of the observation window,  below which the noise power is considered constant, and fH is  the upper frequency, beyond which the noise power is assumed  to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Hence, the variance and mean of total output  current variance is σ2  ∆Is = ζ2σ2  Nb|s + σ2  f and µ∆Is = µ∆Ib|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' TIME-DOMAIN DETECTION  Since Rx has no knowledge of the interferer concentration  statistics, it constructs the optimal ML decision threshold for  TDD solely based on its knowledge of the received signal  statistics corresponding to the transmitted concentration of  information molecules [7]:  γtd =  1  σ2  ∆I1 − σ2  ∆I0  �  σ2  ∆I1µ∆I0 − σ2  ∆I0µ∆I1 + σ∆I1σ∆I0  ×  �  (µ∆I1 − µ∆I0)2 + 2(σ2  ∆I1 − σ2  ∆I0) ln(σ∆I1/σ∆I0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (6)  As Rx does not account for interference statistics in calculating  γtd, it uses the bound state probability corresponding to a  single molecule case, namely, pb|s =  cm|s/KDm  1+cm|s/KDm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  To derive the BEP for TDD, we first obtain the statis-  tics of the receiver output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' By applying the law of total  expectation, we can express the mean number of bound  receptors as follows: µNb|s =  � ∞  0  Nrpb|s(ci)f(ci)dci, where  pb|s(ci) =  cm|s/KDm+ci/KDi  1+cm|s/KDm+ci/KDi , and f(·) is the probability  density function of log-normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Hence, µ∆Is =  ζµNb|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Similarly, by applying the law of total variance, we  obtain the output current variance as  σ2  ∆Is = ζ2  � � ∞  0  �  1 − pb|s(ci)  �  pb|s(ci)Nrf(ci)dci  +  � ∞  0  �  pb|s(ci)Nr  �2 f(ci)dci  �  − µ2  ∆Is + σ2  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (7)  Therefore, given the decision threshold γtd, BEP for time  detection method can be expressed as follows [7]:  P T DD  e  = 1  4 erfc  �  �γtd − µ∆I0  �  2σ2  ∆I0  �  � + 1  4 erfc  �  �µ∆I1 − γtd  �  2σ2  ∆I1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (8)  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' FREQUENCY-DOMAIN DETECTION  In this section, we introduce the FDD method utilizing the  model and observed PSD of the overall noise process (binding  noise + 1/f noise of the graphene bioFET-based MC-Rx) to  estimate the received concentration of information molecules  cm, which will be used in symbol decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Here, the observed  PSD is the periodogram of the noise constructed with the time-  domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' In the sequel, we describe the model PSD  and then introduce the proposed estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Theoretical Model of Binding Noise PSD  This section describes the theoretical model of the binding  noise PSD for a particular pair of information and interference  concentration, namely λ = [cm, ci].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The binding process of  receptors can be described by the Langmuir reaction model  with three states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', unbound (R), bound with information  molecules (RM) and bound with interferer molecules (RI),  with state occupation probabilities pR, pRM and pRI, respec-  tively [12]: R + M  k−  m  ⇌  k+  m  RM, and R + I  k−  i⇌  k+  i  RI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Hence, the  chemical master equations are expressed as follows:  �  �����  dpRM  dt  dpRI  dt  dpR  dt  �  �����  =  �  �  −k−  m  0  k+  mcm  0  −k−  i  k+  i ci  k−  m  k−  i  −k+  mcm − k+  i ci  �  �  �  �  pRM  pRI  pR  �  � (9)  The matrix containing reaction rates and the concentrations in  (9), has rank 2 since one state probability can be written in  terms of the other two state occupation probabilities as pR +  pRM + pRI = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Therefore, by setting the left-hand side in  (9) to zero the equilibrium probabilities can be obtained as  p0  RM =  cm/KDm  1 +  cm  KDm +  ci  KDi  , p0  RI =  ci/KDi  1 +  cm  KDm +  ci  KDi  (10)  and p0  R = 1−(p0  RM +p0  RI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' In the equilibrium conditions, the  state occupation probabilities can be expressed in terms of the  equilibrium state probability and the fluctuations around this  probability [12], [13] as  pj(t) = p0  j + ∆pj(t),  j ∈ {RM, RI, R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (11)  Putting (11) into (9) and using Taylor’s expansion, the state  fluctuations can be expressed as follows [12]:  d∆p′(t)  dt  = Ω∆p′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (12)  In (12), ∆p′(t) = [∆pRM(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' ∆pRI(t)] is the reduced form of  the vector containing the state occupation probabilities, where  Ω is  Ω =  �−k+  mcm − k−  m  −k+  m  −k+  i ci  −k+  i ci − k−  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (13)  The deviation in the output current of the MC-Rx due to  stochastic binding reactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', ∆Ib(t), is then obtained as  ∆Ib(t) = qeff g  CG  zT R∆p′(t)  (14)  where z = [Ne−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Ne−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 0] is the vector containing the number  of elementary charges corresponding to each state and R is the  transformation matrix such that ∆p(t) = R∆p′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As ∆Ib(t)  is a stationary process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' the theoretical PSD of the binding noise  fluctuations can be found by setting t = 0 as follows [12]:  Sb(f) = 2 F{E[∆Ib(t)∆Ib(t + τ)]}  (15)  = 2 F{E[∆Ib(0)∆Ib(τ)]}  = 4Nr  �qeffg  CG  �2  z⊺RΓ  �  Re{(j2πfI2×2 − Ω)−1}  �⊺ R⊺z  where F{·} stands for Fourier transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' I2×2 is the identity  matrix and Γ is the matrix containing the expected state  probabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' which is given as follows [12]:  Γ =  �p0  RM  �  1 − p0  RM  �  −p0  RMp0  RI  −p0  RMp0  RI  p0  RI  �  1 − p0  RI  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (16)  Therefore, the theoretical PSD of the total current noise  corresponding to a particular (cm, ci) pair can be written as  S(f) = Sb(f) + Sf(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (17)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Maximum Likelihood Estimation of PSD Parameters  In the following part, we describe the parameter value  extraction, namely the estimation of information and inter-  fering molecule concentrations, λ = [cm, ci], from the noise  PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The detector uses the estimated information molecule  concentration ˆcm for symbol decision, as will be explained in  the following section, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Our analysis is based on the  following assumptions:  The total noise process, namely the binding fluctuations  combined with 1/f noise, is stationary, zero-mean with a  single-sided spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Rx is given the model PSD function expressed by (17), and  the binding/unbinding rates of information and interferer  molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Rx also has the knowledge of the number of  information molecules transmitted for bits s = 0 and s = 1  as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Therefore, Rx will estimate the  steady information and interferer concentrations by taking  time samples from the output current ∆Ib in a sampling  window, where we consider a single realization of the  interferer concentration ci following log-normal distribution  as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The DC component of ∆Ib is  discarded to isolate the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  The information and interferer concentrations are considered  constant in the sampling window based on the equilibrium  assumption discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' II [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  The observed PSD of time domain samples and the para-  metric model of the PSD expressed by (17) will be used in  the ML estimation of λ = [cm, ci].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' It is assumed that the  observed PSD is calculated with the periodogram method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  For each transmitted symbol, we have N number of noise  samples x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', xN) taken with the sampling period  of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Hence, the total duration of sampling per symbol,  namely the length of the sampling window, is Td = N∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Periodogram for the sampled signal can be computed from  the Discrete Fourier transform (DFT) of the samples x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  With even N, the periodogram values are then expressed as  follows: Yk = 2∆t  N |Xk|2 where k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', N/2 − 1, and |Xk|  DFT components of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  For a stochastic time series of length N, the random variable  Wk = 2  Yk  S(fk) follows chi-squared distribution χ2 [14], where  S(fk) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' (17) is the true PSD at frequency fk and  fk =  k  N∆t and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', N/2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The χ2 distribution with  two degrees of freedom is in fact the exponential distribution  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Therefore, the periodogram values are exponentially  distributed about the true PSD with the following probability  given the model PSD value at a given frequency:  p(Yk|S(fk)) =  1  S(fk)e−  Yk  S(fk) ,  (18)  following that S(fk) is also expectation value at fk [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Based on (18), the likelihood of observing a pair of particular  information and interferer concentrations, λ = [cm, ci], is  L(λ) =  N/2−1  �  k=1  p(Yk|S(fk, λ)) =  N/2−1  �  k=1  1  S(fk, λ)e−  Yk  S(fk,λ) ,  (19)  Algorithm 1 Algorithm for the FDD  1: Run Newton method with initial guess λ0 = [c0  m, c0  i ] to  find optimal λ∗ = [c∗  m, c∗  i ] satisfying (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  2: ˆcm ← c∗  m  3: Find the decision threshold γfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  4: Run threshold operation:  5: if ˆcm > γfd Estimated bit ˆs ← 1  6: else Estimated bit ˆs ← 0  7: end if  where λ = [cm, ci] is the parameters to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Here, we  use Whittle likelihood, which can be a good approximation to  the exact likelihood asymptotically, and also provide computa-  tional efficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', O(n log n) compared to O(n2) for exact  likelihood [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Accordingly, the quasi-log likelihood can  be written as follows:  ln L(λ) = −  N/2−1  �  k=1  �  Yk  S(fk, λ) + ln S(fk, λ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (20)  ML estimator extracts the value of λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', ˆλ, that maximizes  (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Maximizing ln L(λ) is equivalent to minimizing l =  − ln L(λ) [17], such that  ˆλ = arg min  λ {l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (21)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' (21) can be solved using numerical methods such as  Newton-Ralphson method attaining the ML within few iter-  ations [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Symbol Detection  The ML estimator described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' IV-B is asymptotically  unbiased such that ˆλ tends to have multi-normal distribution  [19] with E[ˆλ] = λ, and the respective variance of the esti-  mated parameters, which is the diagonal elements of inverse  Fisher information matrix (FIM) F(λ),  σ2  ˆ  λi = (F(λ))−1  (ii),  F(λ)(ij) = E  �  ∂2l  ∂λi∂λj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (22)  where the expectation is taken with respect to the probabil-  ity distribution of the observed spectrum p(Y1, Y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', YN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Putting l =−lnL(λ) into (22), the FIM can be expanded as:  F(ij) =  E  �  �  N/2−1  �  k=1  S(fk) − Yk  S(fk)2  ∂2S  ∂λi∂λj  + 2Yk − S(fk)  S(fk)3  ∂S  ∂λi  ∂S  ∂λj  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (23)  Considering that S(f) is a slowly varying function, there  is no need to calculate individual periodogram values in  (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Because periodogram values can be smoothed by  summing over frequency such that �N/2−1  n=1  Ykφ(fk)  ≃  �N/2−1  n=1  S(fk)φ(fk), for any smooth function φ(fk)  [19],  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Based on this, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' (23) can be simplified as [19]  F(ij) ≃  N/2−1  �  k=1  1  S(fk)2  ∂S  ∂λi  ∂S  ∂λj  ,  (24)  where the derivatives are taken at the true value of the  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' This is a good approximation for the large number  of samples such that periodogram values can be approximated  as Gaussian by the central limit theorem [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Rx decides  the transmitted bit by applying the ML decision rule on the  estimated information molecule concentration ˆcm, as described  by the pseudo-algorithm for FDD in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The ML  decision threshold for FDD is  γfd =  1  σ2  ˆcm|1 − σ2  ˆcm|0  �  σ2  ˆcm|1cm|0 − σ2  ˆcm|0cm|1 + σˆcm|1σˆcm|0  ×  �  (cm|1 − cm|0)2 + 2(σ2  ˆcm|1 − σ2  ˆcm|0) ln  �  σˆcm|1/σˆcm|0  ��  ,  (25)  where σ2  ˆcm|s is the variance and cm|s is the expected value of  estimated information molecule when the transmitted bit is s ∈  {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Since Rx does not know the true value of the interfering  molecule concentration, it computes the decision threshold as  if there was no interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Therefore, the following model  PSD is used while computing the threshold γfd:  S(f, cm) = 4Nrζ2  1  2πf + 1/τm  pb(1 − pb) + Sf(f),  (26)  where τm = 1/(cmk+  m + k−  m) and pb =  cm  KDm + cm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Hence,  using (26), and (24) for λ = [cm], the variance of the  estimated information molecule concentration corresponding  to the transmitted bit s ∈ {0, 1} can be written as σ2  ˆcm|s =  1/ �N/2−1  k=1  1  S(fk)2 ( ∂S  ∂cm )2���cm=cm|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Note that here the expres-  sion for σ2  ˆcm|s does not give the actual asymptotic variances  since Rx estimates the value of cm based on the model PSD  described by (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Asymptotic Bit Error Probability  To calculate BEP for FDD, we need the actual values of  the variance of estimated information molecule concentration  corresponding to s=0 and s=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=', σ2  ˆcm|s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Using the model  PSD S(f, (cm, ci)) given by (17), and (24) with λ = [cm, ci],  the variance can be expressed as σ2  ˆcm|s=(Fs(λ))−1  (11), where  Fs’s elements are:  Fs(11) =  N/2−1  �  k=1  1  S(fk)2  � ∂S  ∂cm  �2���cm=cm|s  ci=µci  ,  (27)  Fs(22) =  N/2−1  �  k=1  1  S(fk)2  � ∂S  ∂ci  �2���cm=cm|s  ci=µci  ,  (28)  Fs(12),(21) =  N/2−1  �  k=1  1  S(fk)2  � ∂S  ∂cm  � � ∂S  ∂ci  ����cm=cm|s  ci=µci  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (29)  As a result, BEP for FDD can be written as  P F DD  e  = 1  4 erfc  �  �γfd − cm|0  �  2σ2  ˆcm|0  �  � + 1  4 erfc  �  �cm|1 − γfd  �  2σ2  ˆcm|1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  (30)  Here, it should be noted that (30) is an asymptotic expression  based on the Gaussian distribution assumption in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  10-2  10-1  100  101  102  Frequency (Hz)  10-23  10-22  10-21  PSD (A2/Hz)  Interferer molecule  Information molecule  fch|m  fch|i  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 2: The model PSD with highlighted characteristic fre-  quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' PERFORMANCE EVALUATION  In this section, we analyze the performance of FDD and  TDD in terms of BEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The default values of the system  parameters are given in Table I, with the reaction rates  adopted from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' In the rest of the paper, the saturation  and the non-saturation corresponds to the Rx’s receptors  being saturated due to high ligand concentrations and far  from the saturation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' To simulate the saturation,  the number of transmitted information molecules is taken as  Nm|s∈{0,1} = [2, 5] × 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Otherwise, default values are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  We first consider the effect of the mean interference con-  centration, µci on the BEP performance of TDD and FDD in  saturation and non-saturation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We define a tuning  parameter γ such that mean interferer concentration is given  by µci = γ cm|s=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 3a, FDD outperforms  TDD in both scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' The performance of TDD degrades  dramatically due to Rx saturation with increasing µci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' In non-  saturation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' the performance of FDD improves with increasing  TABLE I: Default Values of System Parameters  Temperature (T )  300K  Microfluidic channel height (hch),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' width (lch)  5 µm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 10 µm  Average flow velocity (u)  10 µm/s  Distance of Rx’s center position to Tx (xR)  1mm  Ionic concentration of medium (cion)  30 mol/m3  Relative permittivity of medium (ϵ/ϵ0)  80  Intrinsic diffusion coefficient (D0)  2 × 10−11 m2/s  Binding  rate  of  information  and  interferer  molecules (k+  m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' k+  i )  4 × 10−17 m3/s  Unbinding rate of information molecules (k−  m)  2 s−1  Unbinding rate of interferers (k−  i )  8 s−1  Average # of electrons in a ligand (Ne−)  3  Number of independent receptors (Nr)  120  Length of a surface receptor (r)  2 nm  Transconductance of graphene bioFET (g)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='9044 × 10−4 A/V  Width of graphene in transistor (lgr)  10µm  Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' capacitance of graphene per unit area (cq)  2 × 10−2 F m−2  # of transmitted ligands for s = 0, 1 (Nm|s)  [1, 5] × 103  # of noise samples (N)  700  Sampling period (∆t)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='005 s  Mean interference to information concentration  ratio (γ = µci/cm|s=1)  1  Interference mean/std ratio (µci/σci)  10  Power of 1/f noise at 1 Hz (Sf1Hz )  10−23 A2/Hz  µci up to a certain point, beyond which further increase in  µci degrades the performance of FDD because when Rx is not  saturated, the variance of the estimated information molecule  concentration σ2  ˆcm|s is minimized at a certain µci beyond  which its value increases with increasing µci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' In saturation,  however, σ2  ˆcm|s monotonically increases with µci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  Next, we consider the effect of similarity parameter, namely  the affinity ratio of information and interferer molecules η =  KDi/KDm, on the BEP performance for saturation and non-  saturation cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 3b, FDD outperforms  TDD in both saturation and non-saturation cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Regarding  the non-saturation case, the performances of both detection  methods improve with increasing similarity up to a certain  point because the effect of interference on the detection perfor-  mance weakens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' namely, bound state probability for interferer  molecules decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' However, when the similarity is further  increased, the performance of FDD degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Intuitively, this  is because the characteristic frequencies corresponding to bits  s = 0 and s = 1 [22]  fch|s = [fch|m, fch|i]  = 1  4π  �  � 1  τm|s  + 1  τi  ±  �� 1  τm|s  − 1  τi  �2  + 4k+  mcm|sk+  i ci  �  � ,  (31)  where τm|s = 1/(cm|sk+  m + k−  m) and τi = 1/(cik+  i + k−  i ),  are approaching each other in the spectrum, making it dif-  ficult to distinguish the bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 2 for an  example scenario, two characteristic frequencies, fch|m and  fch|i, appear in the spectrum for each transmitted bit due to  the binding of two types of molecules, namely information  and interferer molecules, with the order depending on the  concentrations and binding/unbinding rates of the individual  molecule types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' In the non-saturation case, we do not observe  this phenomenon because the characteristic frequencies do not  come close to each other to degrade the detection performance  with increasing similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We also consider the effect of the  number of time samples, N, and the sampling period ∆t on  the BEP performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 3c, the performance  of FDD increases with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' This is expected as taking more  samples decreases the variance of the estimated information  molecule concentration σ2  ˆcm|s, hence, decreases the BEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' For  TDD, the performance does not change with N as Rx takes  one sample in the sampling window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' For varying ∆t, the  performance of FDD increases with increasing ∆t for the  non-saturation case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Note that ∆t should be shorter than the  characteristic time scale of any reactions to be able to capture  the fluctuations and to satisfy the sampling at the equilibrium  assumption, which is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Therefore, we  consider ∆t values satisfying this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' CONCLUSION  In this paper, we proposed a FDD method for the FET-  based MC-Rx, which utilizes the output noise PSD to extract  the transmitted bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' We derived the BEP for the proposed  method and a one-shot TDD method considering the existence  0  1  2  3  4  5  Interference/Information (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=')  10-6  10-5  10-4  10-3  10-2  10-1  100  Bit error probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
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+page_content='95  1  Bound state probability, PB  TDD (Sat)  FDD (Sat)  TDD (Non-sat)  FDD (Non-sat)  s = 0 (Sat)  s = 1 (Sat)  s = 0 (Non-sat)  s = 1 (Non-sat)  (a)  0  5  10  15  20  25  30  35  40  Similarity (2)  10-5  10-4  10-3  10-2  10-1  100  Bit error probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='95  1  Bound state probability, PB  TDD (Sat)  FDD (Sat)  TDD (Non-sat)  FDD (Non-sat)  s = 0 (Sat)  s = 1 (Sat)  s = 0 (Non-sat)  s = 1 (Non-sat)  (b)  100  200  300  400  500  600  700  800  900  1000  # of Samples (N)  10-6  10-4  10-2  100  Bit error probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='95  1  Bound state probability, PB  TDD (Sat)  FDD (Sat)  TDD (Non-sat)  FDD (Non-sat)  s = 0 (Sat)  s = 1 (Sat)  s = 0 (Non-sat)  s = 1 (Non-sat)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 3: BEP for varying (a) mean interference concentration level (b) similarity of affinities for information and interferer  molecules (c) number of time samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  1  2  3  4  5  6  7  8  9  Sampling Period ("t(ms))  10-8  10-6  10-4  10-2  100  Bit error probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='95  1  Bound state probability, PB  TDD (Sat)  FDD (Sat)  TDD (Non-sat)  FDD (Non-sat)  s = 0 (Sat)  s = 1 (Sat)  s = 0 (Non-sat)  s = 1 (Non-sat)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' 4: BEP for varying sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  of a single type of interferer molecules in a microfluidic  channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content=' Our analysis reveals that the proposed detection  method significantly outperforms the TDD, primarily when  high interference exists in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work was supported in part by the AXA Research Fund  (AXA Chair for Internet of Everything at Koc¸ University), the  Horizon 2020 Marie Skłodowska-Curie Individual Fellowship  under Grant Agreement 101028935, and by The Scientific and  Technological Research Council of Turkey (TUBITAK) under  Grant #120E301, and Huawei Graduate Research Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAzT4oBgHgl3EQfHvtN/content/2301.01049v1.pdf'}
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I9E0T4oBgHgl3EQfSAAQ/content/tmp_files/2301.02214v1.pdf.txt

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+AUTOMATIC SOUND EVENT DETECTION AND CLASSIFICATION OF
+GREAT APE CALLS USING NEURAL NETWORKS
+Zifan Jiang1,2, Adrian Soldati1, Isaac Schamberg2, Adriano R. Lameira3, Steven Moran1
+1University of Neuchâtel, 2University of Zurich, 3University of Warwick
+[email protected], {adrian.soldati, steven.moran}@unine.ch, [email protected], [email protected]
+ABSTRACT
+We present a novel approach to automatically
+detect and classify great ape calls from continuous
+raw
+audio
+recordings
+collected
+during
+field
+research.
+Our method leverages deep pretrained
+and sequential neural networks, including wav2vec
+2.0 and LSTM, and is validated on three data sets
+from three different great ape lineages (orangutans,
+chimpanzees, and bonobos). The recordings were
+collected by different researchers and include
+different annotation schemes, which our pipeline
+preprocesses and trains in a uniform fashion.
+Our results for call detection and classification
+attain high accuracy. Our method is aimed to be
+generalizable to other animal species, and more
+generally, sound event detection tasks.
+To foster
+future research, we make our pipeline and methods
+publicly available.1
+Keywords: sound event detection, neural networks,
+primatology, phonetics, computational linguistics
+1. INTRODUCTION
+In primatology, as in documentary linguistics, the
+collection, annotation, and analysis of primary
+field data are time-consuming and expensive. The
+recordings of primate calls are also often undertaken
+in not ideal environmental conditions (e.g., in a
+dense and noisy forest where other vocal species are
+present), making the process even more challenging.
+Therefore it is typical that primatologists first
+manually annotate their recordings and then conduct
+acoustic analyses on their species’ specific calls.
+Our goal in this paper is to automatically and
+accurately detect and classify primate calls from
+raw audio data.
+After discussing related work
+(§2), we describe our data sets (§3) and present
+our method that leverages pretrained and sequential
+neural networks (§4).
+We carry out reliable
+and reproducible experiments (§5) to support the
+effectiveness of our approach and compare our
+results between different architectural choices on
+the data sets of three great ape lineages. Overall,
+our models achieve more than 80.0 frame-level
+classification accuracy and weighted F1-score.
+Interestingly, we find that the wav2vec 2.0 [1] model
+– even though pretrained on a large corpus of human
+speech [2] – generalizes surprisingly well as an
+audio feature representation layer for great ape calls
+without additional fine-tuning. This finding perhaps
+bides well with the similar vocal tract morphology
+between extant great apes and humans, and the
+larger prediction that ancestral ape-like calls evolved
+to become the building blocks of human speech.
+2. RELATED WORK
+To the best of our knowledge,
+there is no
+precedent research on the automatic detection and
+classification of great ape calls from raw audio.
+Therefore, we find our work lies in between general-
+purpose sound event detection and human speech
+recognition. We briefly discuss these also in light
+of existing research on animal sound classification.
+Sound
+Event
+Detection
+(SED)
+aims
+at
+automatically
+recognizing
+what
+is
+happening
+in an audio signal and when it is happening [3].
+SED tasks are usually general-purpose (e.g., birds
+singing or footsteps) as opposed to domain-specific
+tasks like human speech or music analysis, and thus
+encounter particular challenges, such as the cocktail
+party effect [4]. Notably, the DCASE challenge that
+takes place in recent years involves relevant SED
+tasks [5]. A variant of SED is the audio tagging task
+[6], in which only what is happening in an audio
+signal is annotated and recognized, but not when.
+Prominent work on both tasks [7, 8] tends to
+consist of common components, including:
+(1)
+an acoustic feature representation layer,
+either
+(a) traditional spectrogram-based approaches or
+(b)
+convolutional
+neural
+networks
+(CNN)
+on
+spectrogram features or on raw waveforms [9]; (2)
+a sequence modeling layer that captures temporal
+interaction between frame-level features, which
+takes the forms of mean/max pooling strategies,
+recurrent neural networks (RNN) or Transformers
+[10]; and (3) an objective function, usually cross-
+arXiv:2301.02214v1  [eess.AS]  5 Jan 2023
+
+data set
+# audio clips
+mean / call / total duration
+# indiv. (♂/ ♀)
+# call types (duration ratio) / units
+chimpanzee
+235
+~ 8s / 1,955s / 1,964s
+11 (11 / 0)
+4 (6:3:3:1) / 686
+orangutan
+65
+~ 74s / 2,793s / 4,817s
+10 (10 / 0)
+7 (700:40:200:100:20:1:1) / 9016
+bonobo
+28
+~ 24s / 62s / 677s
+7 (3 / 4)
+18 (20:40:10:20:...:200:...:5:1) / 356
+Table 1: Data set overview and stats. Shown in the table are the number of audio clips, mean duration of each clip,
+total duration of calls of all clips, total duration of all clips, number of individuals (male and female), number of
+call types (approximate total duration ratio of them, partially omitted for bonobo due to space limit), and number
+of individually annotated call units.
+entropy classification on frame-level (for SED) or
+clip-level (for audio tagging), or occasionally CTC
+[11] for “sequentially labeled data” [12].
+Animal sound classification involves related
+research
+that
+addresses
+specifically
+at
+animal
+sounds, e.g., [13, 14] on the classification of general
+animal species and [15, 16, 17] on species, call types
+and caller identities of primates. While this line of
+research uses similar acoustic feature representation
+techniques as seen in SED tasks, the models tend
+to work on top of individual short audio units of
+animal vocalizations that are manually selected from
+raw audio recordings by human experts, and thus
+fall short on dealing with a continuous audio signal
+that contains many audio events as well as noise.
+Instead, our method detects and classifies great ape
+calls directly from raw recordings ranging from
+seconds to minutes (theoretically recurrent models
+also extend to hours, but it could pose challenges in
+model training time).
+Automatic
+Speech
+Recognition
+(ASR)
+on
+humans – one of the extant five great apes – has
+seen significant progress [18] (while animal calls
+remain mysterious to fully decipher), including the
+recent work wav2vec 2.0 [1].
+Wav2vec 2.0 is
+pretrained on Librispeech [2] in a self-supervised
+way, and it demonstrates the feasibility of ASR with
+limited amounts of labeled data. We are interested
+in whether the speech representation learned by
+wav2vec 2.0 can be applied successfully to great ape
+call detection and classification.
+3. DATA
+Our data consists of recordings of chimpanzee pant-
+hoots, orangutan long calls, and bonobo high-hoots.
+Table 1 provides an overview of these data sets.
+Chimpanzee pant-hoots are vocal sequences
+composed of up to four acoustically distinct phases,
+typically produced in this order: introduction, build-
+up, climax, and let-down [19].
+Most pant-hoots
+contain two or more phases, although single phases
+can be produced in specific contexts [20].
+The
+successive nature and well-balanced phase duration
+facilitate the training of a classifier (§5.1). We use
+annotated recordings of isolated pant-hoots (i.e., no
+temporal overlap with others’ calls) produced during
+feeding, traveling, and resting context.
+Orangutan long calls are composed by a
+full pulse, which is sub-divided into a sub-pulse
+transitory element and a pulse body,
+or into
+sequences of bubble sub-pulse or grumble sub-pulse
+[21, 22].
+The complex temporal interrelationship
+between phases and the overlapping and class-
+unbalanced
+characteristics
+(some
+phases
+are
+extremely short) pose challenges for training a
+multi-class classifier.
+On the other hand, the
+duration of calls and non-calls in the data set are
+well balanced, which opens the gate to a binary
+call detection model (§5.2). Annotated recordings
+include spontaneous long calls and long calls
+produced in response to other males’ vocal presence
+or environmental disturbances.
+Bonobo high-hoots – which make up the
+plurality of our bonobo call data set – are loud,
+tonal vocalizations [23].
+Unlike the chimpanzee
+and orangutan data sets, call types other than the
+homologous loud calls are also present in the data
+set but are rather minor and diverse (e.g., peep-yelp,
+soft barks), which could be challenging to find. The
+total call duration is relatively short.
+4. METHOD
+This section introduces our method,
+which is
+illustrated in Fig. 1. We describe each step in turn.
+4.1. Audio Preprocessing
+First, all audio clips are converted to .wav format and
+resampled to 16 kHz sample rate. We segment them
+into 20ms frames and pad with zeros if necessary.
+4.2. Feature and Label Extraction
+Next, we extract three types of acoustic features with
+the torchaudio Python package. For an audio clip of
+T frames, we get a sequence of frame-level features
+I1:T, in the shape of T × feature_dim:
+
+• Raw waveform: the original amplitude values
+over time. feature_dim = 0.02×16000 = 320.
+• Spectrogram: calculated then from the raw
+waveform with feature_dim = 201.
+• wav2vec 2.0: inferred from the raw waveform
+by the WAV2VEC2_BASE model on a CPU
+device with feature_dim = 768.
+We
+then
+extract
+frame-level
+labels
+from
+our
+annotations. For each annotated unit, we mark all
+frames inside the annotated time span with a positive
+class index indicating the call type (e.g., 1 for intro);
+unmarked frames are 0s by default, indicating non-
+calls. The labels’ shape of a clip L1:T is T ×1.
+4.3. Data Split
+We shuffle the clips (features and labels) of each
+data set and split them into 80%, 10%, and 10%
+for training, validation, and test sets, respectively.
+We make three different splits by three different
+random seeds 0, 42, and 3407 [24] to allow multiple
+experiments and study the effect of randomness.
+4.4. Sequence Modeling
+Since our goal is to learn a function f from I1:T
+to L1:T, we first input I1:T to a sequence modeling
+function f m that outputs a hidden sequence H1:T
+(T × hidden_dim),
+which
+captures
+inter-frame
+interaction. H1:T then goes through a dense linear
+function f d (output D1:T) followed by a Softmax
+function f s to produce the probability distribution
+over target classes, i.e., P1:T (T ×num_class).
+Specifically, f m takes the following forms:
+• RNN: a bidirectional LSTM with hidden size
+1024. hidden_dim = 1024∗2 = 2048.
+• Transformer encoder: a Transformer encoder
+with 8 heads and 6 layers. hidden_dim = 1024.
+• Autoregressive model:
+we optionally add
+autoregressive connections between time steps
+to encourage consistent output labels.
+The
+output of
+f d at time step t,
+i.e.,
+Dt is
+concatenated to the next time step’s input It+1.
+Lastly, a cross-entropy loss is computed between
+the model output P1:T and the gold labels L1:T, then
+backpropagated to train the models. To mitigate the
+class imbalance problem, we set class weights to the
+reciprocal of each class’s occurrence times.
+5. EXPERIMENTS AND RESULTS
+Our experiments were done in PyTorch [25], with
+Python 3.8 on an Nvidia Tesla V100 GPU (32GB
+ram). Most models have ∼40k million parameters
+and finish the training of up to 200 epochs with early
+raw waveform
+spectrogram
+features
+wav2vec 2.0
+features
+autoregressive
+frame-level
+sequence model
+strong labels
+prediction 
+(ar vs. non-ar)
+intro
+build up climax let down
+distribution
+via Softmax 
+(ar vs. non-ar)
++
++
++
+I t+1
+I t+2
+I T
+Figure 1: Our method (bottom-up). The audio
+clip shown in the figure is from the chimpanzee
+data set. (Non-)ar stands for (non-)autoregressive.
+stopping on validation F1-score within one hour (or
+a few hours because autoregressive recurrent models
+run slowly on PyTorch). Table 2 presents our results.
+5.1. Initial Exploration with Chimpanzee Data
+We first test the viability of our approach on
+the chimpanzee data in light of the simplicity
+of the pant-hoot annotation scheme,
+although
+phylogenetically orangutans are closer to humans.
+We start from E1, a simple waveform + LSTM
+baseline, and observe in E2 that wav2vec 2.0
+outperforms the raw waveform and spectrogram by
+a large margin, which demonstrates the power of
+transfer learning from pretraining on human speech.
+In E2.1, we find that the Transformer encoder
+does not outperform LSTM. Hence, we infer that
+
+Dt+1
+Dt+2
+DT
+Dense
+Dense
+Dense
+T1
+RNN
+RNN
+RNN
+D
+t+1
+-18200
+30
+175
+20
+150
+10
+125
+0
+bin
+freg
+100
+-10
+75
+-20
+50 
+-30
+25
+-4010
+700
+600
+0.8
+feature dimension
+500
+0.6
+400
+OOE
+0.4
+200
+ 0.2
+100
+0
+0.0no call
+intro
+build up
+dimax
+let downno call
+intro
+build up
+xeup
+let down
+AULMULMID
+data
+feature
+model
+dev acc.
+dev f1
+test acc.
+test f1
+aucpr
+Explore the best feature and model combination
+E1
+chimp
+waveform
+lstm (baseline)
+51.7±2.1
+35.4±2.5
+51.0±3.6
+34.7±4.3
+-
+E1.1
+chimp
+spectrogram
+lstm
+60.3±1.5
+55.7±2.3
+58.7±4.7
+53.9±5.4
+-
+E2
+chimp
+wav2vec2
+lstm
+81.0±4.0
+79.9±4.6
+79.3±2.3
+77.9±3.6
+-
+E2.1
+chimp
+wav2vec2
+transformer
+71.3±0.6
+68.3±0.2
+75.3±0.6
+72.1±0.5
+-
+Explore the hyper-parameters
+E3.1
+chimp
+wav2vec2
+lstm (E2 + batch_size = 4)
+69.7±1.5
+71.8±2.6
+67.7±4.0
+69.6±4.0
+-
+E3.2
+chimp
+wav2vec2
+lstm (E2 + batch_size = 8)
+63.3±0.6
+62.6±1.0
+62.0±4.4
+61.5±4.0
+-
+E3.3
+chimp
+wav2vec2
+lstm (E2 + dropout = 0.2)
+80.7±3.5
+80.0±4.4
+78.0±1.7
+76.8±2.7
+-
+E3.4
+chimp
+wav2vec2
+lstm (E2 + dropout = 0.1)
+81.0±4.0
+80.2±4.8
+78.7±2.9
+77.3±3.9
+-
+E3.5
+chimp
+wav2vec2
+lstm (E2 - balance_weights)
+81.0±3.6
+79.6±4.4
+79.3±2.3
+78.3±3.6
+-
+Explore autoregressive modeling
+E4
+chimp
+wav2vec2
+lstm (E2 + autoregressive)
+87.7±1.2
+87.1±1.8
+85.7±2.1
+85.6±2.5
+-
+Extend to orangutan long calls and a binary setting
+E5
+orang
+wav2vec2
+lstm (= E4)
+83.0±1.0
+82.7±1.4
+81.7±3.1
+82.0±2.6
+-
+E5.1
+orang
+wav2vec2
+lstm (E5 + binary target)
+92.3±2.5
+92.1±2.5
+92.0±1.0
+91.9±1.1
+0.96
+Extend to bonobo calls and a binary setting
+E6
+bonobo
+wav2vec2
+lstm (= E4)
+87.0±4.6
+85.9±6.3
+83.7±3.8
+82.3±2.2
+-
+E6.1
+bonobo
+wav2vec2
+lstm (E6 + binary target)
+92.0±3.6
+91.9±3.4
+87.7±3.5
+87.8±2.9
+0.87
+Zero-shot transferring from orangutan to bonobo
+E7
+bonobo
+wav2vec2
+lstm (= E5.1)
+63.0±13
+69.2±10
+72.0±4.0
+74.2±3.1
+0.55
+Table 2: Experimental results. We run all experiments three times based on different random seeds and report the
+mean and standard deviation. acc. stands for frame-level accuracy, f1 stands for the frame-level average F1-score
+weighted by the number of true instances per class, and aucpr stands for the area under the precision-recall curve
+for the positive class in the binary case at test time when the random seed is set to 0. For hyper-parameters, we
+start E1 with batch_size = 1, dropout = 0.4 and keep them by default, if not otherwise specified in the table.
+Transformer’s ability to capture arbitrary long-range
+dependency is not beneficial to our task.
+Next, we explore the hyper-parameters in the
+second experimental group and we find in E3.5 that
+balancing class weights (§4.4) has a small impact.
+Lastly, we show in E4 that the autoregressive
+connections are beneficial for consistent output, as
+illustrated by the tiny gaps in non-ar output in Fig. 1.
+5.2. Extending to Orangutan and Bonobo Data
+We successfully extend the model in E4 to as trained
+on the orangutan (E5) and bonobo (E6) data sets. We
+note that some minority classes perform less well
+due to data scarcity, in contrast to the well-balanced
+situation in the chimpanzee data set (see Table 1).
+We further reduce the task to a binary (call vs.
+non-call) classification that resembles voice activity
+detection of human speech.
+This is a useful tool
+to automatically extract calls from raw recordings
+for further bioacoustic analysis (e.g., studying the
+repertoire of a given species).
+Finally, to understand the generalizability of our
+models, we try zero-shot transferring the model
+trained in E5.1 on orangutans directly to unseen
+bonobo data. The results show that it is promising
+to build a potential general-purpose sound event
+detection model for all great ape calls.
+6. DISCUSSION
+We have addressed a gap in the bioacoustics
+research of non-human great apes by developing
+an approach for automatically identifying sound
+events and classifying great ape calls using a neural
+network architecture. Our method successfully and
+accurately identifies and classifies calls in three
+species of non-human great apes, and provides a tool
+for primatologists to bootstrap call identification and
+analysis from raw unannotated audio recordings.
+Our method also shows the general applicability
+of the wav2vec 2.0 model trained on human speech
+for identifying vocalizations and call types in other
+species. Future work may apply our approach to
+more animals, as part of the goal to decode the
+communication systems of great apes and other non-
+human animals more broadly.2
+1 https://github.com/J22Melody/sed_great_ape
+2 For example: https://www.earthspecies.org
+
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+

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+page_content='AUTOMATIC SOUND EVENT DETECTION AND CLASSIFICATION OF  GREAT APE CALLS USING NEURAL NETWORKS  Zifan Jiang1,2, Adrian Soldati1, Isaac Schamberg2, Adriano R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Lameira3, Steven Moran1  1University of Neuchâtel, 2University of Zurich, 3University of Warwick  jiang@cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='ch, {adrian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='soldati, steven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='moran}@unine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='ch, isaac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='schamberg@uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='ch, adriano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='lameira@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='uk  ABSTRACT  We present a novel approach to automatically  detect and classify great ape calls from continuous  raw  audio  recordings  collected  during  field  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Our method leverages deep pretrained  and sequential neural networks, including wav2vec  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 and LSTM, and is validated on three data sets  from three different great ape lineages (orangutans,  chimpanzees, and bonobos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' The recordings were  collected by different researchers and include  different annotation schemes, which our pipeline  preprocesses and trains in a uniform fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Our results for call detection and classification  attain high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Our method is aimed to be  generalizable to other animal species, and more  generally, sound event detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  To foster  future research, we make our pipeline and methods  publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  Keywords: sound event detection, neural networks,  primatology, phonetics, computational linguistics  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' INTRODUCTION  In primatology, as in documentary linguistics, the  collection, annotation, and analysis of primary  field data are time-consuming and expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' The  recordings of primate calls are also often undertaken  in not ideal environmental conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', in a  dense and noisy forest where other vocal species are  present), making the process even more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Therefore it is typical that primatologists first  manually annotate their recordings and then conduct  acoustic analyses on their species’ specific calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Our goal in this paper is to automatically and  accurately detect and classify primate calls from  raw audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  After discussing related work  (§2), we describe our data sets (§3) and present  our method that leverages pretrained and sequential  neural networks (§4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  We carry out reliable  and reproducible experiments (§5) to support the  effectiveness of our approach and compare our  results between different architectural choices on  the data sets of three great ape lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Overall,  our models achieve more than 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 frame-level  classification accuracy and weighted F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Interestingly, we find that the wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 [1] model  – even though pretrained on a large corpus of human  speech [2] – generalizes surprisingly well as an  audio feature representation layer for great ape calls  without additional fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' This finding perhaps  bides well with the similar vocal tract morphology  between extant great apes and humans, and the  larger prediction that ancestral ape-like calls evolved  to become the building blocks of human speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' RELATED WORK  To the best of our knowledge,  there is no  precedent research on the automatic detection and  classification of great ape calls from raw audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Therefore, we find our work lies in between general-  purpose sound event detection and human speech  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' We briefly discuss these also in light  of existing research on animal sound classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Sound  Event  Detection  (SED)  aims  at  automatically  recognizing  what  is  happening  in an audio signal and when it is happening [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  SED tasks are usually general-purpose (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', birds  singing or footsteps) as opposed to domain-specific  tasks like human speech or music analysis, and thus  encounter particular challenges, such as the cocktail  party effect [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Notably, the DCASE challenge that  takes place in recent years involves relevant SED  tasks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' A variant of SED is the audio tagging task  [6], in which only what is happening in an audio  signal is annotated and recognized, but not when.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Prominent work on both tasks [7, 8] tends to  consist of common components, including:  (1)  an acoustic feature representation layer,  either  (a) traditional spectrogram-based approaches or  (b)  convolutional  neural  networks  (CNN)  on  spectrogram features or on raw waveforms [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' (2)  a sequence modeling layer that captures temporal  interaction between frame-level features, which  takes the forms of mean/max pooling strategies,  recurrent neural networks (RNN) or Transformers  [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' and (3) an objective function, usually cross-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='02214v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='AS]  5 Jan 2023  data set  # audio clips  mean / call / total duration  # indiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' (♂/ ♀)  # call types (duration ratio) / units  chimpanzee  235  ~ 8s / 1,955s / 1,964s  11 (11 / 0)  4 (6:3:3:1) / 686  orangutan  65  ~ 74s / 2,793s / 4,817s  10 (10 / 0)  7 (700:40:200:100:20:1:1) / 9016  bonobo  28  ~ 24s / 62s / 677s  7 (3 / 4)  18 (20:40:10:20:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=':200:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=':5:1) / 356  Table 1: Data set overview and stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Shown in the table are the number of audio clips, mean duration of each clip,  total duration of calls of all clips, total duration of all clips, number of individuals (male and female), number of  call types (approximate total duration ratio of them, partially omitted for bonobo due to space limit), and number  of individually annotated call units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  entropy classification on frame-level (for SED) or  clip-level (for audio tagging), or occasionally CTC  [11] for “sequentially labeled data” [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Animal sound classification involves related  research  that  addresses  specifically  at  animal  sounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', [13, 14] on the classification of general  animal species and [15, 16, 17] on species, call types  and caller identities of primates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' While this line of  research uses similar acoustic feature representation  techniques as seen in SED tasks, the models tend  to work on top of individual short audio units of  animal vocalizations that are manually selected from  raw audio recordings by human experts, and thus  fall short on dealing with a continuous audio signal  that contains many audio events as well as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Instead, our method detects and classifies great ape  calls directly from raw recordings ranging from  seconds to minutes (theoretically recurrent models  also extend to hours, but it could pose challenges in  model training time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Automatic  Speech  Recognition  (ASR)  on  humans – one of the extant five great apes – has  seen significant progress [18] (while animal calls  remain mysterious to fully decipher), including the  recent work wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 is  pretrained on Librispeech [2] in a self-supervised  way, and it demonstrates the feasibility of ASR with  limited amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' We are interested  in whether the speech representation learned by  wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 can be applied successfully to great ape  call detection and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' DATA  Our data consists of recordings of chimpanzee pant-  hoots, orangutan long calls, and bonobo high-hoots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Table 1 provides an overview of these data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Chimpanzee pant-hoots are vocal sequences  composed of up to four acoustically distinct phases,  typically produced in this order: introduction, build-  up, climax, and let-down [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Most pant-hoots  contain two or more phases, although single phases  can be produced in specific contexts [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  The  successive nature and well-balanced phase duration  facilitate the training of a classifier (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' We use  annotated recordings of isolated pant-hoots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', no  temporal overlap with others’ calls) produced during  feeding, traveling, and resting context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Orangutan long calls are composed by a  full pulse, which is sub-divided into a sub-pulse  transitory element and a pulse body,  or into  sequences of bubble sub-pulse or grumble sub-pulse  [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  The complex temporal interrelationship  between phases and the overlapping and class-  unbalanced  characteristics  (some  phases  are  extremely short) pose challenges for training a  multi-class classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  On the other hand, the  duration of calls and non-calls in the data set are  well balanced, which opens the gate to a binary  call detection model (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Annotated recordings  include spontaneous long calls and long calls  produced in response to other males’ vocal presence  or environmental disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Bonobo high-hoots – which make up the  plurality of our bonobo call data set – are loud,  tonal vocalizations [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Unlike the chimpanzee  and orangutan data sets, call types other than the  homologous loud calls are also present in the data  set but are rather minor and diverse (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', peep-yelp,  soft barks), which could be challenging to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' The  total call duration is relatively short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' METHOD  This section introduces our method,  which is  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' We describe each step in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Audio Preprocessing  First, all audio clips are converted to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='wav format and  resampled to 16 kHz sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' We segment them  into 20ms frames and pad with zeros if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Feature and Label Extraction  Next, we extract three types of acoustic features with  the torchaudio Python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' For an audio clip of  T frames, we get a sequence of frame-level features  I1:T, in the shape of T × feature_dim:  Raw waveform: the original amplitude values  over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' feature_dim = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='02×16000 = 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Spectrogram: calculated then from the raw  waveform with feature_dim = 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0: inferred from the raw waveform  by the WAV2VEC2_BASE model on a CPU  device with feature_dim = 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  We  then  extract  frame-level  labels  from  our  annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' For each annotated unit, we mark all  frames inside the annotated time span with a positive  class index indicating the call type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', 1 for intro);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  unmarked frames are 0s by default, indicating non-  calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' The labels’ shape of a clip L1:T is T ×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Data Split  We shuffle the clips (features and labels) of each  data set and split them into 80%, 10%, and 10%  for training, validation, and test sets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  We make three different splits by three different  random seeds 0, 42, and 3407 [24] to allow multiple  experiments and study the effect of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Sequence Modeling  Since our goal is to learn a function f from I1:T  to L1:T, we first input I1:T to a sequence modeling  function f m that outputs a hidden sequence H1:T  (T × hidden_dim),  which  captures  inter-frame  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' H1:T then goes through a dense linear  function f d (output D1:T) followed by a Softmax  function f s to produce the probability distribution  over target classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', P1:T (T ×num_class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Specifically, f m takes the following forms:  RNN: a bidirectional LSTM with hidden size  1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' hidden_dim = 1024∗2 = 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Transformer encoder: a Transformer encoder  with 8 heads and 6 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' hidden_dim = 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Autoregressive model:  we optionally add  autoregressive connections between time steps  to encourage consistent output labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  The  output of  f d at time step t,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=',  Dt is  concatenated to the next time step’s input It+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Lastly, a cross-entropy loss is computed between  the model output P1:T and the gold labels L1:T, then  backpropagated to train the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' To mitigate the  class imbalance problem, we set class weights to the  reciprocal of each class’s occurrence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' EXPERIMENTS AND RESULTS  Our experiments were done in PyTorch [25], with  Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8 on an Nvidia Tesla V100 GPU (32GB  ram).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Most models have ∼40k million parameters  and finish the training of up to 200 epochs with early  raw waveform  spectrogram  features  wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  features  autoregressive  frame-level  sequence model  strong labels  prediction  (ar vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' non-ar)  intro  build up climax let down  distribution  via Softmax  (ar vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' non-ar)  +  +  +  I t+1  I t+2  I T  Figure 1: Our method (bottom-up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' The audio  clip shown in the figure is from the chimpanzee  data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' (Non-)ar stands for (non-)autoregressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  stopping on validation F1-score within one hour (or  a few hours because autoregressive recurrent models  run slowly on PyTorch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Table 2 presents our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Initial Exploration with Chimpanzee Data  We first test the viability of our approach on  the chimpanzee data in light of the simplicity  of the pant-hoot annotation scheme,  although  phylogenetically orangutans are closer to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  We start from E1, a simple waveform + LSTM  baseline, and observe in E2 that wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  outperforms the raw waveform and spectrogram by  a large margin, which demonstrates the power of  transfer learning from pretraining on human speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  In E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1, we find that the Transformer encoder  does not outperform LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Hence, we infer that  Dt+1  Dt+2  DT  Dense  Dense  Dense  T1  RNN  RNN  RNN  D  t+1  18200  30  175  20  150  10  125  0  bin  freg  100  10  75  20  50  30  25  4010  700  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8  feature dimension  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  400  OOE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0no call  intro  build up  dimax  let downno call  intro  build up  xeup  let down  AULMULMID  data  feature  model  dev acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  dev f1  test acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  test f1  aucpr  Explore the best feature and model combination  E1  chimp  waveform  lstm (baseline)  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3 E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  chimp  spectrogram  lstm  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4 E2  chimp  wav2vec2  lstm  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6 E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  chimp  wav2vec2  transformer  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5 Explore the hyper-parameters  E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  chimp  wav2vec2  lstm (E2 + batch_size = 4)  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2  chimp  wav2vec2  lstm (E2 + batch_size = 8)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0 E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3  chimp  wav2vec2  lstm (E2 + dropout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7 E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4  chimp  wav2vec2  lstm (E2 + dropout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9 E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  chimp  wav2vec2  lstm (E2 - balance_weights)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6 Explore autoregressive modeling  E4  chimp  wav2vec2  lstm (E2 + autoregressive)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5 Extend to orangutan long calls and a binary setting  E5  orang  wav2vec2  lstm (= E4)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6 E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  orang  wav2vec2  lstm (E5 + binary target)  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='96  Extend to bonobo calls and a binary setting  E6  bonobo  wav2vec2  lstm (= E4)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2 E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  bonobo  wav2vec2  lstm (E6 + binary target)  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='6  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='7±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='87  Zero-shot transferring from orangutan to bonobo  E7  bonobo  wav2vec2  lstm (= E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±13  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2±10  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='55  Table 2: Experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' We run all experiments three times based on different random seeds and report the  mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' stands for frame-level accuracy, f1 stands for the frame-level average F1-score  weighted by the number of true instances per class, and aucpr stands for the area under the precision-recall curve  for the positive class in the binary case at test time when the random seed is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' For hyper-parameters, we  start E1 with batch_size = 1, dropout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4 and keep them by default, if not otherwise specified in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Transformer’s ability to capture arbitrary long-range  dependency is not beneficial to our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Next, we explore the hyper-parameters in the  second experimental group and we find in E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='5 that  balancing class weights (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='4) has a small impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Lastly, we show in E4 that the autoregressive  connections are beneficial for consistent output, as  illustrated by the tiny gaps in non-ar output in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Extending to Orangutan and Bonobo Data  We successfully extend the model in E4 to as trained  on the orangutan (E5) and bonobo (E6) data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' We  note that some minority classes perform less well  due to data scarcity, in contrast to the well-balanced  situation in the chimpanzee data set (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  We further reduce the task to a binary (call vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  non-call) classification that resembles voice activity  detection of human speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  This is a useful tool  to automatically extract calls from raw recordings  for further bioacoustic analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=', studying the  repertoire of a given species).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Finally, to understand the generalizability of our  models, we try zero-shot transferring the model  trained in E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='1 on orangutans directly to unseen  bonobo data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' The results show that it is promising  to build a potential general-purpose sound event  detection model for all great ape calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' DISCUSSION  We have addressed a gap in the bioacoustics  research of non-human great apes by developing  an approach for automatically identifying sound  events and classifying great ape calls using a neural  network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content=' Our method successfully and  accurately identifies and classifies calls in three  species of non-human great apes, and provides a tool  for primatologists to bootstrap call identification and  analysis from raw unannotated audio recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
+page_content='  Our method also shows the general applicability  of the wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E0T4oBgHgl3EQfSAAQ/content/2301.02214v1.pdf'}
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INE0T4oBgHgl3EQfRwA9/content/tmp_files/2301.02211v1.pdf.txt

@@ -0,0 +1,1107 @@
+Teaching Computer Vision for Ecology
+Elijah Cole1
+Suzanne Stathatos1
+Bj¨orn L¨utjens2
+Tarun Sharma1
+Justin Kay1,3
+Jason Parham4
+Benjamin Kellenberger5
+Sara Beery1,2
+1Caltech
+2MIT
+3Ai.Fish
+4Wild Me
+5Yale University
+https://cv4ecology.caltech.edu/
+Abstract
+Computer vision can accelerate ecology research by
+automating the analysis of raw imagery from sen-
+sors like camera traps, drones, and satellites. How-
+ever, computer vision is an emerging discipline that is
+rarely taught to ecologists. This work discusses our
+experience teaching a diverse group of ecologists to
+prototype and evaluate computer vision systems in
+the context of an intensive hands-on summer work-
+shop.
+We explain the workshop structure, discuss
+common challenges, and propose best practices. This
+document is intended for computer scientists who
+teach computer vision across disciplines, but it may
+also be useful to ecologists or other domain experts
+who are learning to use computer vision themselves.
+1
+Introduction
+Extracting important information from images and
+videos normally requires painstaking manual effort
+from human annotators. Computer vision algorithms
+can automate this process. This is especially impor-
+tant when manually reviewing the data is not feasible,
+either because the amount of data is too large (e.g.
+the >100TB of satellite imagery collected daily) or
+the number of annotators is too small (e.g. when ex-
+pertise is required to identify a species in an image).
+Both of these challenges are common in ecology.
+Ecology presents a particularly compelling use case
+for computer vision.
+Due to the effects of cli-
+mate change, we need to monitor animal popula-
+tions, vegetation properties, and other indicators
+of ecosystem health at a large scale [41].
+Ecolo-
+gists are collecting vast amounts of raw data with
+camera traps, drones, and satellites, but there are
+not enough experts to annotate the data.
+Com-
+puter vision algorithms can accelerate the pace of re-
+search in ecology by efficiently transforming this raw
+data into useful knowledge.
+Encouraging progress
+is already being made in areas like animal detec-
+Git
+Overfitting
+Generalization
+Evaluation
+Deep 
+Learning
+Python
+OOP
+AWS
+Unix
+PyTorch
+Data Splits
+Ablations
+Model 
+Development
+Loss 
+Functions
+Software Engineering
+Machine Learning
+Namespaces
+Data 
+Structures
+Data Types
+Mutability
+Figure 1: A simplified dependency graph depicting
+some of the skills required to develop a computer vi-
+sion system. These software engineering and machine
+learning topics are rarely included in ecology train-
+ing. See Appendix B for a catalog of key topics and
+their significance.
+tion [18,25,29], fine-grained species recognition [42],
+individual re-identification [38], species distribution
+modeling [19,21], and land cover mapping [32]. These
+efforts can be viewed in the broader context of compu-
+tational sustainability [23] and efforts to use machine
+learning to combat the effects of climate change [33].
+To build on this progress, we must equip ecologists
+with the skills they need to understand and apply
+computer vision methods in their research.
+While
+ecologists often have training in statistics and pro-
+gramming, they are rarely exposed to the inter-
+connected web of software engineering and machine
+learning topics necessary for computer vision.
+We
+illustrate a few of these topics in Figure 1.
+In this work, we discuss the process of teaching com-
+puter vision to ecologists in the context of the Resnick
+Sustainability Institute Summer Workshop on Com-
+puter Vision Methods for Ecology (CV4E Workshop),
+an intensive 3-week workshop held at Caltech in
+2022 [4]. We review related work in Section 2 before
+1
+arXiv:2301.02211v1  [cs.CY]  5 Jan 2023
+
+describing the workshop in Section 3, discussing key
+take-aways in Section 4, and outlining educational
+techniques we found useful in Section 5.
+2
+Related Work
+There is an emerging literature devoted to teach-
+ing machine learning [20, 34, 37], deep learning [30],
+and computer vision [24, 26, 31, 36]. [35] more nar-
+rowly focuses on common errors in machine learning
+course projects. However, most of these works con-
+cern efforts to teach students from computer science
+or related disciplines. There is prior work discussing
+the specific challenge of teaching machine learning
+to cross-disciplinary audiences such as non-CS un-
+dergraduates [39], business students [43], artists [22],
+materials scientists [40], and biologists [28]. Our work
+is complementary, focusing on the process of teaching
+computer vision to ecologists (mostly Ph.D. students
+and postdocs – see Figure 2) who have background
+knowledge in statistics and programming but little
+prior experience in machine learning. In addition, we
+consider an immersive workshop in which researchers
+build prototypes using their own research data, not a
+traditional classroom environment.
+3
+The CV4E Workshop
+The inaugural CV4E Workshop was held at Caltech
+from August 1 - 19, 2022. The program was designed
+to train ecologists to use computer vision in their own
+research. Here we outline the stages of the workshop.
+Application. The application had five components:
+(i) a one-page project proposal, (ii) a one-page per-
+sonal statement, (iii) a programming example, (iv)
+one letter of reference, and (v) a CV. The most im-
+portant element was the project proposal, in which
+participants described the problem they wanted to
+solve with computer vision, the potential impact of a
+working solution, and the available data and labels.
+Selection process. The CV4E staff recruited appli-
+cation reviewers from the machine learning and ecol-
+ogy communities. Each application received two re-
+views. Final decisions were made by the CV4E staff.
+The primary criteria were: (i) goal clarity, (ii) project
+feasibility, (iii) potential impact, and (iv) candidate
+preparation. Details about the 2022 cohort can be
+found in Figure 2 and Appendix A. To maximize
+accessibility, all participants were funded for travel,
+room, and board for the duration of the program.
+Pre-workshop preparation. All participants were
+Gov't / NGO
+11.1%
+Postdoc
+27.8%
+M.S. Student
+5.6%
+Ph.D. Student
+55.6%
+Participant Type
+Super-resolution
+5.6%
+Clustering
+5.6%
+Segmentation
+16.7%
+Detection
+22.2%
+Re-ID
+11.1%
+Regression
+5.6%
+Classification
+33.3%
+Project Type
+Figure 2:
+Summary of the 2022 CV4E Workshop
+participant backgrounds (top) and project categories
+(bottom). Full details can be found in Appendix A.
+added to a Slack workspace which served as the pri-
+mary communication channel for the workshop. Each
+participant was assigned to a working group overseen
+by a CV4E instructor. During the ∼6 months be-
+tween participant selection and the beginning of the
+workshop, participants met with their instructors to
+finalize project plans and address any data or label is-
+sues. Participants were also expected to learn Python
+during this period. Instructors assisted by providing
+Python resources and holding biweekly office hours.
+In-person workshop. Figure 3 gives a representa-
+tive weekly schedule for the CV4E Workshop. Par-
+ticipants received classroom instruction from Lectures
+and Invited Speakers. Each participant joined a Read-
+ing Group on a topic of their choice (see Appendix D),
+which met twice weekly for a guided discussion of re-
+search papers. During the Work Time, participants
+worked on their projects independently, with CV4E
+staff and working groups peers available for questions.
+Each working group discussed their progress and ob-
+stacles during the Group Updates.
+Outcomes. All 18 of our participants had trained
+models for their projects by the end of the work-
+2
+
+shop. Some of these models were already achieving
+high performance, while others needed more investi-
+gation. In addition, the participants and staff formed
+a community that has endured beyond the workshop
+through the Slack workspace and ongoing projects.
+MONDAY
+TUESDAY
+WEDNESDAY
+THURSDAY
+FRIDAY
+9:00 AM
+Lecture
+Invited Speaker
+Invited Speaker
+Lecture
+Invited Speaker
+10:00 AM
+Work Time
+Lecture
+Lecture
+ Work Time
+Lecture
+11:00 AM
+Lunch
+Lunch
+Lunch
+Lunch
+Lunch
+12:00 PM
+1:00 PM
+Invited Speaker
+Reading Groups
+Invited Speaker
+Reading Groups
+Work Time
+2:00 PM
+Work Time
+Work Time
+Work Time
+Work Time
+3:00 PM
+4:00 PM
+Group Updates
+Group Updates
+5:00 PM
+Break
+Break
+Break
+Break
+Break
+6:00 PM
+Dinner
+Dinner
+Dinner
+Dinner
+Dinner
+7:00 PM
+Figure 3: The weekly schedule for the 2022 CV4E
+Workshop was roughly evenly split between in-
+structional time (Lectures, Invited Speakers, Reading
+Groups, Group Updates) and instructor-supervised
+working time (Work Time). In practice, portions of
+the Group Updates, Lunch, Break, and Dinner slots
+were often used by participants as extra work time.
+4
+Lessons Learned
+Enforce structured Python preparation. The
+primary obstacle for most participants was insuffi-
+cient Python preparation. While participants were
+not required to know Python before applying, they
+were asked to learn Python before arriving.
+To
+facilitate this process, the staff provided resources
+for learning Python and hosted office hours in the
+months leading up to the CV4E Workshop.
+How-
+ever, many participants (even capable R program-
+mers) still struggled with Python issues throughout
+the workshop.
+In hindsight, we overestimated the
+extent to which R experience is helpful for quickly
+learning Python. In the future we will enforce more
+structured Python preparation before the workshop.
+Start simple. It is challenging to build a working
+computer vision system from scratch in 3 weeks. To
+maximize the probability of success, it is important to
+start simple. When appropriate, we encouraged par-
+ticipants to use standard well-understood pipelines
+e.g. fine-tuning an ImageNet-pretrained ResNet-50.
+Work in long blocks.
+Participants made much
+more progress during long blocks of work time (3+
+hours) than during shorter work blocks (1-2 hours).
+Collect similar projects in working groups.
+Participants were often eager to help each other, es-
+pecially when they were deploying similar techniques.
+Working groups should be constructed to maximize
+opportunities for such collaborations.
+Mix experience levels in working groups. Some
+participants had significant experience with machine
+learning or programming, enabling them to make
+swift progress on their projects with minimal assis-
+tance from instructors. Experienced participants rou-
+tinely volunteered to assist less experienced partici-
+pants, which seemed mutually beneficial. In the fu-
+ture, we plan to ensure that each working group has
+a mix of experienced and inexperienced participants.
+Make unambiguous infrastructure recommen-
+dations.
+There are many reasonable ways to set
+up the infrastructure necessary for computer vision
+work. For instance, consider the problem of develop-
+ing code which is meant to be executed on a VM. One
+approach is to edit the code locally in a text editor
+and move it to the VM using rsync, handling revi-
+sion control locally. Another approach is to use a tool
+like VSCode [17] which allows code on the VM to be
+edited directly via SSH. In this case, revision control
+would be handled on the VM. A third approach is to
+edit code locally, push the code to GitHub, and pull
+the code to the VM. Revision control is “built in”
+for this workflow. The instructors had different pref-
+erences, and no workflow was clearly superior. Par-
+ticipants did not benefit from being asked to make
+their own choice about which workflow to use.
+In
+the future we will provide unambiguous and unified
+recommendations for development infrastructure.
+Avoid deep learning library wrappers. There
+are many “wrappers” for deep learning libraries
+which are meant to make deep learning tools easier to
+use. Some are general-purpose (e.g. PyTorch Light-
+ning [13]) while others are domain-specific (e.g. Open-
+Soundscape [11], DeepForest [5], TorchGeo [16]).
+While these wrappers are undoubtedly useful, they
+are not ideal for our participants for two reasons.
+First, they conceal too much complexity which hin-
+ders the process of learning about e.g. training loops
+and data flow. Second, they were more difficult to
+customize and debug, even with instructor assistance.
+In the future, we will encourage all participants to
+work directly with deep learning libraries.
+Avoid Jupyter Notebooks.
+Jupyter Notebooks
+provide capabilities familiar to experienced R users,
+such as the ability to run sections of code interac-
+tively. However, participants who relied on Jupyter
+Notebooks while learning Python often struggled to
+transition to more traditional command line work-
+flows when developing their computer vision systems.
+We now believe that learning to work with Python
+3
+
+through the command line provides a better founda-
+tion for understanding machine learning workflows.
+Make sure GPUs are available. Cloud computing
+services like AWS and Azure often provide free credits
+for education and research. However, GPUs may not
+be available depending on customer demand.
+It is
+important to confirm with cloud providers that GPUs
+will be made available. Alternatively, consider using
+local computing resources or university clusters.
+5
+Educational Techniques
+In this section we describe a few educational tech-
+niques we found helpful for the CV4E Workshop.
+Guided troubleshooting. Troubleshooting and de-
+bugging are vital skills in machine learning, and it
+was important to provide participants with opportu-
+nities to hone these abilities.
+However, due to the
+tight schedule of the CV4E Workshop, we did not
+want participants to be stuck on any one problem
+for too long. To balance these objectives, instructors
+tried to walk participants through the troubleshoot-
+ing process by asking leading questions about the
+problems they were encountering. For unusual prob-
+lems of limited educational value (i.e. complex config-
+uration or installation issues), instructors intervened
+to resolve the issue as quickly as possible.
+Pair pseudocoding. Most of our participants were
+not comfortable writing Python code at the beginning
+of the CV4E Workshop, so we wanted to provide fre-
+quent opportunities for hands-on coding. Whenever
+possible, instructors avoided writing code for the par-
+ticipants. To prevent participants from getting stuck
+on code design issues, we used pair pseudocoding:
+1. The instructor asks the participant to explain
+what they would like to accomplish, discussing
+until the goal is clear to both parties.
+2. The instructor writes pseudocode that solves the
+problem and walks through it with the partici-
+pant to help them understand the logic of the
+solution. The pseudocode can be more specific
+or vague depending on the participant’s needs.
+3. The participant writes Python code to solve the
+problem, while the instructor remains available
+to answer questions as they arise.
+Goal statements. During the initial stages of the
+project, some of our participants felt like progress was
+not being made because the code didn’t “work” yet.
+To make their progress more salient, some instructors
+asked participants to make a goal statement at the be-
+ginning of each work session, and to check progress
+towards that goal at the end of each work session.
+This strategy helped participants to maintain moti-
+vation until more tangible results were obtained.
+Contextualized lectures.
+Maintaining interest
+during lectures was not a significant problem for the
+CV4E Workshop due to the enthusiasm of the par-
+ticipants. However, it is easy for lectures on machine
+learning topics to become too abstract. We tried to
+ensure that the lectures remained grounded in ap-
+plications and examples. Since each participant had
+their own applied problem in mind, we often paused
+lectures to ask participants to reflect on how the lec-
+ture topic applied to their individual projects. Partic-
+ipants shared their answers with the class, providing
+concrete examples that illustrated the lecture topic.
+6
+Conclusion
+We have described our experience at the inaugural
+Resnick Sustainability Institute Summer Workshop
+on Computer Vision Methods for Ecology. We con-
+sider the format to be a success, as all of our partici-
+pants trained models for their projects by the end of
+the workshop. However, we have also discussed some
+challenges we encountered and identified opportuni-
+ties to improve the CV4E Workshop. We hope these
+observations will be useful for others who teach com-
+puter vision across disciplines.
+7
+Acknowledgements
+We would like to thank the Resnick Sustainability
+Institute, Caroline Murphy, Xenia Amashukeli, and
+Pietro Perona for making the CV4E Workshop pos-
+sible. Computing credits were provided by Amazon
+AWS and Microsoft Azure. We also thank the inau-
+gural cohort of the CV4E Workshop: Ant´on ´Alvarez,
+Carly Batist, Peggy Bevan, Catherine Breen, Anna
+Boser, Tiziana Gelmi Candusso, Melanie Clapham,
+Rowan Converse, Roni Goldshmid, Natalie Imirzian,
+Brian Lee, Francesca Ponce, Alixandra Prybyla,
+Rachel Renne, Felix Rustemeyer, Taiki Sakai, Ethan
+Shafron, and Casey Youngflesh.
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+6
+
+A
+2022 Cohort
+A.1
+Participant Backgrounds
+The inaugural 2022 CV4E Workshop had 18 partic-
+ipants. Broken down by current occupation, our co-
+hort consisted of:
+• 1 Master’s student;
+• 10 Ph.D. students;
+• 5 post-doctoral researchers; and
+• 2 researchers from government agencies or non-
+governmental organizations.
+Broken down geographically, our cohort consisted of:
+• 11 participants from 7 different states in the
+U.S.;
+• 5 participants from European countries; and
+• 2 participants from Canada.
+Participants
+came
+from
+diverse
+academic
+back-
+grounds, including conservation biology, biological
+anthropology,
+geography,
+mechanical engineering,
+civil engineering, neuroscience, and ecology.
+A.2
+Participant Projects
+Projects fell into seven main categories.
+1. Individual
+Re-Identification:
+Associating
+images of the same animal taken from various
+cameras, locations, and times. The two relevant
+projects were: (1) re-identifying bears, (2) re-
+identifying Iberian Lynx.
+2. Regression: Assigning a continuous number to
+an image or video. The one relevant project was:
+(1) analyzing wind speed from overhead drone
+video of trees.
+3. Classification: Categorizing or labeling images
+or parts of images from a fixed collection of cat-
+egories. The six relevant projects were: (1) de-
+termining presence or absence of lemur vocal-
+izations, (2) beaked whale species classification
+from echolocation clicks, (3) bumblebee species
+and caste classification from flight sounds, (4)
+assigning ants to size categories, (5) identify-
+ing weather conditions from camera trap images,
+and (6) species identification in urban camera
+traps. Note that projects (1), (2), and (3) used
+computer vision techniques to classify images
+(spectrograms) that represent audio signals.
+4. Object Detection: Locating instances of ob-
+jects in images or videos.
+The four relevant
+projects were detecting:
+(1) piospheres, (2)
+woodland draws, (3) flies, and (4) waterfowl.
+Projects (3) and (4) use detection as an inter-
+mediate step towards counting.
+5. Segmentation:
+Classifying pixels based on
+their semantic characteristics.
+The three rele-
+vant projects were segmenting: (1) walrus groups
+in the Arctic, (2) permafrost, and (3) trees. All
+projects were based on remote sensing imagery.
+6. Clustering: Grouping objects together accord-
+ing to some notion of similarity.
+The one rel-
+evant project was: (1) determining the species
+richness of an area using the number of clusters
+in a collection of camera trap imagery.
+7. Super-resolution: Increasing the resolution of
+an image.
+The one relevant project was: (1)
+increasing the resolution of land surface temper-
+ature data using satellite imagery.
+B
+Key Topics
+In this section we catalog topics that many of our par-
+ticipants learned during the workshop, either through
+formal instruction or on their own.
+We emphasize
+tools and concepts that were initially unfamiliar to
+most participants. For each topic, we describe the
+content and explain why it was important for our par-
+ticipants. See also the list of lectures in Appendix C.
+B.1
+Tools
+B.1.1
+Annotation Tools
+Content: Using annotation tools to label image [3,8]
+or audio [2] data.
+Motivation: Labeled data is essential for training
+and evaluating computer vision algorithms.
+Since
+CV4E Workshop participants were using their own
+data, many of them needed to learn to use some sort
+of annotation tool. Furthermore, many of these tools
+can export labels in the standard formats expected
+by open-source computer vision libraries.
+B.1.2
+Unix Commands
+Content: Common Unix commands like ls, pwd, rm,
+mkdir, rmdir, mv, cat, head, tail etc. Occasionally,
+less common commands like chmod or grep.
+Motivation: Facility with Unix commands is cru-
+cial for installing packages, working with virtual ma-
+7
+
+chines, and using revision control.
+Understanding
+Unix commands also helps to build intuition for core
+concepts like absolute vs. relative paths.
+B.1.3
+Terminal-Based Text Editing
+Content: Tools like nano for editing text that is
+stored on a server from the command line.
+Motivation: When configuring SSH authentication
+it is often necessary to edit text files on the VM (e.g.
+∼/.ssh/config).
+B.1.4
+Terminal Multiplexing
+Content: Tools like tmux or screen for managing
+terminal sessions.
+Motivation: Long-running code (e.g. model training
+in PyTorch) should be executed in a terminal session
+that is decoupled from the SSH connection to avoid
+being terminated when a laptop is closed or internet
+connection is lost.
+B.1.5
+SSH
+Content: The ssh command and SSH keys. Occa-
+sionally, SSH tunneling.
+Motivation: The ssh command is used to create a
+terminal session connected to a VM. Related topics
+like SSH keys are also important for e.g. authenticat-
+ing terminal-based file transfers and enabling GitHub
+access. SSH tunneling can be necessary for setting up
+tools like TensorBoard [14].
+B.1.6
+Terminal-Based File Transfers
+Content: Tools like scp or rsync for transferring
+files.
+Motivation: Command line tools are the most re-
+liable way to move large amounts of data from one
+place to another. This is useful for local transfers (e.g.
+from one hard drive to another) and remote transfers
+(e.g. from a local hard drive to a storage volume at-
+tached to a virtual machine).
+B.1.7
+Revision Control
+Content: Using GitHub for tracking changes made
+to code.
+Motivation:
+Code for computer vision projects
+tends to quickly grow in complexity, and it is easy to
+forget what has changed since the last working ver-
+sion. Tools like GitHub allow earlier versions of the
+code to be revisited easily if a bug was introduced
+by some change. In addition, GitHub can be used to
+move code from a local machine (git push) to a vir-
+tual machine (git pull) along with allowing users to
+download (git clone) open-sourced computer vision
+repositories.
+B.1.8
+Cloud Computing
+Content:
+Interacting with the web interfaces of
+cloud computing providers. Creating a virtual ma-
+chine with appropriate resources e.g. GPUs, storage.
+Estimating and managing cost.
+Motivation: One of the most common ways to ac-
+cess GPU resources for computer vision work is to use
+a VM from a cloud computing provider like Amazon
+Web Services (AWS) [1] or Microsoft Azure [9]. It
+is important to understand the benefits (scalability,
+reliability) and drawbacks (cost) of cloud computing.
+B.1.9
+Virtual Environments
+Content: Creating and managing virtual environ-
+ments.
+Motivation: Computer vision projects typically rely
+on large pre-existing codebases, which may require
+particular versions of certain packages to be installed.
+While the user could change their base installations,
+a better solution is to create a virtual environment
+(through e.g. conda) in which the dependencies of the
+codebase can be installed. Virtual environments are
+also useful if a “clean reinstall” becomes necessary,
+because they are easy to create and delete.
+B.1.10
+Python
+Content: Basic syntax, conditionals, loops, string
+parsing, file I/O, functions, classes and data struc-
+tures.
+Motivation: Facility with Python is crucial for effi-
+ciently working with Python-based deep learning li-
+braries, which the computer vision community uses
+almost exclusively.
+B.1.11
+Python Libraries
+Content:
+Common libraries like numpy, pandas,
+ipdb, sklearn, and matplotlib.
+Motivation: Python has many stable, high-quality
+libraries for numerical computing and data analysis.
+Libraries like ipdb allow for in-line debugging.
+8
+
+B.1.12
+Deep Learning Libraries
+Content: Preferably PyTorch [12] and alternatively
+TensorFlow [15] for building deep learning systems.
+Motivation: Modern deep learning libraries are in-
+dispensable for developing and training computer vi-
+sion systems.
+B.1.13
+Image Processing Libraries
+Content:
+Libraries and command-line tools like
+OpenCV [10], ImageMagick [7], and FFmpeg [6].
+Motivation: These tools are often used for efficient
+data augmentation and visualization.
+B.2
+Computer Science Concepts
+There are a few core concepts from computer science
+that came up frequently throughout the program.
+B.2.1
+Object Oriented Programming
+Content: Classes and objects. Inheritance, encap-
+sulation, polymorphism.
+Motivation: Many important libraries assume an
+understanding of object oriented programming con-
+cepts. For instance, one common point of confusion
+for our participants was the difference between the
+PyTorch dataset class and a dataset object from that
+class.
+Understanding object oriented programming
+also makes it easier to understand data structures.
+B.2.2
+Data Structures
+Content:
+Common data structures (e.g. list, tu-
+ple, dictionary, NumPy array, PyTorch tensor) and
+their methods, casting from one data type to another,
+checking data structures.
+Motivation:
+Unexpected behavior differences be-
+tween e.g. Python lists, NumPy arrays, and PyTorch
+tensors can cause significant frustration if data struc-
+tures are not well understood.
+B.2.3
+Data Types
+Content: Common data types e.g. int, float, double,
+string, and bool.
+Motivation:
+Understanding data types increases
+context understanding and can significantly impact
+data storage size.
+B.2.4
+Namespaces
+Content:
+The built-in, global, and local names-
+paces.
+Motivation: Namespaces are the answer to many
+common questions e.g. why variables defined inside a
+function are not accessible outside the function.
+B.2.5
+Mutability
+Content: Mutable and immutable objects. In-place
+operations.
+Motivation: Mutable objects can be changed in-
+place while mutable objects cannot. This is the basis
+for understanding whether changes made to an object
+inside a function will affect the object outside of the
+function.
+B.3
+Machine Learning Concepts
+Participants learned different practical and concep-
+tual aspects of computer vision and machine learning
+depending on their project. However, all participants
+had to engage with a few core concepts.
+B.3.1
+Generalization
+Content: The concept of generalization, different
+types of generalization, identifying a type of general-
+ization that reflects the goals of a project.
+Motivation: In ecology there are many different no-
+tions of generalization, and it is important to choose
+one that reflects the goals of a project. For instance,
+in camera trap image classification it might be impor-
+tant to generalize to new locations or to future data
+from the same locations. These different notions of
+generalization need to be measured in different ways.
+B.3.2
+Data Splits
+Content: The role of training, validation, and test-
+ing data. Designing appropriate splits to measure the
+chosen type of generalization.
+Motivation:
+Training,
+validation,
+and
+testing
+splits should be designed to capture an appropri-
+ate problem-specific notion of generalization. These
+splits must then be handled appropriately (e.g. no hy-
+perparameter tuning on the test split) to ensure that
+performance measurements reflect generalization.
+B.3.3
+Overfitting
+Content: Defining and recognizing overfitting. Mit-
+igating overfitting using regularization techniques.
+9
+
+Motivation:
+All participants were working with
+deep learning, for which overfitting is always a sig-
+nificant concern.
+B.3.4
+Evaluation Metrics
+Content: Common evaluation metrics for different
+tasks, their strengths and limitations, choosing met-
+rics that reflect high-level goals.
+Motivation: Appropriate metrics are vital for de-
+termining which approaches work best and deciding
+if a computer vision system is “good enough” to be
+used for a real application.
+B.3.5
+Deep Learning
+Content: Neural networks, loss functions, minibatch
+gradient descent.
+Motivation: All modern computer vision methods
+rely on deep learning.
+Since our participants were
+building and troubleshooting computer vision sys-
+tems, they needed to understand deep learning ba-
+sics as well. Loss functions were a particular focus,
+since changing the loss is one of the primary ways of
+adapting an existing method to a new problem.
+B.3.6
+Representations
+Content: Image embeddings, distances in embed-
+ding space, pretraining, transfer learning.
+Motivation: ImageNet pretraining is ubiquitous in
+modern computer vision, but many of our partici-
+pants work in specialized domains for which Ima-
+geNet pretraining may not be appropriate. Domain-
+specific pretraining requires an understanding of rep-
+resentation learning. The concept of image embed-
+dings is also useful for understanding many common
+computer vision algorithms (e.g. metric learning) and
+visualization techniques (e.g. t-SNE).
+B.4
+Other Skills
+B.4.1
+Critically Reading Machine Learning
+Papers
+Content:
+Understanding machine learning termi-
+nology and paper structure, critically interpreting
+claims, evaluating complexity vs. performance trade-
+offs.
+Motivation: Exploring the literature in a new field
+is always daunting.
+This is particularly challeng-
+ing in machine learning where papers may be over-
+enthusiastically written, necessitating extra vigilance
+from the reader to clearly understand the drawbacks
+and benefits of a method [27].
+B.4.2
+Selecting
+“Good”
+Open
+Source
+Li-
+braries
+Content: Recognizing markers of quality in open
+source code.
+Motivation: There is plenty of open-source com-
+puter vision code, but not all of it is reliable or well-
+maintained. Participants must learn to check indica-
+tors of code quality e.g. how many users a library has
+or how often the developers fix bugs.
+B.4.3
+Digging in to Libraries
+Content: Reading documentation, finding the code
+that handles a certain task, understanding how com-
+ponents of a codebase interact.
+Motivation: Computer vision projects depend on
+numerous complex but (generally) well-documented
+libraries. It is important to be able to understand
+the documentation. Sometimes it also becomes nec-
+essary to locate and inspect the piece of code being
+documented (e.g. a function from some library) to
+understand how it works in detail.
+B.4.4
+General Troubleshooting
+Content: Errors vs. warnings, searching for more
+information about error messages.
+Motivation: Errors and warnings are common when
+e.g. installing packages or testing new code. One of
+the most important skills in any programming activ-
+ity is the ability to use a search engine to understand
+an error message. This involves locating the appro-
+priate part of an error message to use as a search
+term, reading through the results, and choosing an
+appropriate next step.
+B.4.5
+Debugging Python Code
+Content: Types of errors, finding the source of an
+error, print statement debugging.
+Motivation: For a given line of code, any number of
+errors could arise. Understanding the different types
+of Python errors is helpful for pinpointing the root
+cause. Print statement debugging is also extremely
+useful for troubleshooting code running on a remote
+machine.
+10
+
+C
+List of Lectures
+1. Intro and Logistics (Sara Beery)
+2. Dataset Prototyping and Visualization (Jason
+Parham)
+3. Working on the Cloud (Suzanne Stathatos)
+4. Data Splitting and Avoiding Data Poisoning
+(Sara Beery)
+5. Training your Model:
+Deciding on Configura-
+tions, Launching, Monitoring, Checkpointing,
+and Keeping Runs Organized (Benjamin Kellen-
+berger)
+6. Working
+with
+Open-Source
+CV
+Codebases:
+Choosing a Baseline Model and Custom Data
+Loading (Sara Beery)
+7. Evaluation Metrics (Elijah Cole)
+8. Offline Evaluation and Analysis (Sara Beery)
+9. What’s next? Rules of Thumb to Improve Re-
+sults (Benjamin Kellenberger)
+10. Data Augmentation (Bj¨orn L¨utjens)
+11. Expanding and Improving Training Datasets
+with Models: Weak Supervision, Self Supervi-
+sion, Targeted Relabeling, and Anomaly Detec-
+tion (Tarun Sharma)
+12. Fair Comparisons and Ablation Studies: Under-
+standing What is Important (Elijah Cole)
+13. Efficient Models: Speed vs.
+Accuracy (Justin
+Kay)
+14. Serving, Hosting, and Deploying Models and
+Quality Control (Jason Parham)
+D
+List of Reading Groups
+1. Time Series, Spectral Transforms, and Remote
+Sensing
+2. Data Imbalance & Long Tail Distributions
+3. Weak Supervision, Unsupervised Learning, Fine-
+tuning & Transfer Learning
+4. Bias & Domain Shift and Generalization
+11
+

KtFIT4oBgHgl3EQfaisW/content/tmp_files/2301.11257v1.pdf.txt

@@ -0,0 +1,1740 @@
+ 
+ 
+A Benchmark Study by using various Machine Learning  
+Models for Predicting Covid-19 trends 
+ 
+D.Kamelesun, R.Saranya, P.Kathiravan 
+Department of Computer Science, Central University of Tamil Nadu, India 
+[email protected] 
+ 
+Abstract 
+Machine learning and deep learning play vital roles in predicting diseases in the medical 
+field. Machine learning algorithms are widely classified as supervised, unsupervised, and 
+reinforcement learning. This paper contains a detailed description of our experimental research 
+work in that we used a supervised machine-learning algorithm to build our model for outbreaks 
+of the novel Coronavirus that has spreaded over the whole world and caused many deaths, 
+which is one of the most disastrous Pandemics in the history of the world. The people suffered 
+physically and economically to survive in this lockdown. This work aims to understand better 
+how machine learning, ensemble, and deep learning models work and implements in the real 
+dataset. In our work, we are going to analyze the current trend or pattern of the coronavirus and 
+then predict the further future of the covid-19 confirmed cases or new cases by training the past 
+Covid-19 dataset by using the machine learning algorithm such as Linear Regression, 
+Polynomial Regression, K-nearest neighbor, Decision Tree, Support Vector Machine and 
+Random forest algorithm are used to train the model. 
+The decision tree and the Random Forest algorithm perform better than SVR in this 
+work. The performance of SVR and lasso regression are low in all prediction areas Because 
+the SVR is challenging to separate the data using the hyperplane for this type of problem. So 
+SVR mostly gives a lower performance in this problem. Ensemble (Voting, Bagging, and 
+Stacking) and deep learning models(ANN) also predict well. After the prediction, we evaluated 
+the model using MAE, MSE, RMSE, and MAPE. This work aims to find the trend/pattern of 
+the covid-19. 
+1. Introduction 
+ 
+The First known Coronavirus was discovered in Wuhan, China. World Health 
+Organisation(WHO) was informed about the case of an unknown virus caused in Wuhan in 
+December 2019. The Symptoms of Coronavirus are cough and fever. The cough spreads this 
+virus (Jignesh Chowdary et al., 2020; Liu, 2020; Su et al., 2022; Tiwari et al., 2020). This virus 
+spread over most countries in the world. In our country, people are also terribly affected by this 
+virus. In the last 20 years, As per the national center for Biotechnology Information record, 
+they have recorded various coronaviruses such as SARS-Cov in 2002-2004, HINI Influenza in 
+2009, and MERS-Cov in Saudi Arabic in 2012, declared a Pandemic by WHO. 
+         In this work, we took the covid-19 dataset from the Kaggle site. We built a Machine 
+Learning model to predict the current trend/pattern of  covid-19  and number of confirmed 
+cases using machine-learning algorithms such as linear regression (Mojjada et al., 2021), 
+Polynomial Regression (Shaikh et al., 2021; Yadav et al., 2020), SVR (Rivas-Perea et al., 
+2013), Lasso Regression (Jolliffe et al., 2003; Mojjada et al., 2021; Rustam et al., 2020), K-
+
+ 
+ 
+NN, Decision tree (DT), Random Forest, ensemble techniques such as voting, bagging, and 
+stacking, and deep learning also used to build the model by current past dataset covid-19. After 
+training, we evaluated the model using some evaluation metrics to find the accuracy of the 
+models using MAE, MSE, RMSE, and MAPE (Kanimozhi et al., 2020; Liu, 2020; Verma & 
+Pal, 2020). Following objectives were addressed in this paper. 
+Objective 1: To analyze the current trend of covid-19 confirmed cases by the past covid-19 
+data. 
+Objective 2: To predict the future of confirmed cases of covid-19 by training the model using 
+the past covid-19 dataset. 
+ 
+ The related works proposed methodologies, the model evaluations using various matrices, and 
+the conclusion are in the rest of this paper. 
+ 
+ 
+2. Related work 
+S.no 
+Authors 
+Methodology 
+Finding 
+1 
+(Gambhir et 
+al., 2020) 
+SVR, 
+Polynomial 
+regression 
+ 
+Analyzes Current / patterns of covid-19 
+ 
+2 
+(Mandayam 
+et al., 2020) 
+Linear Regression 
+and SVR 
+ 
+To predict the future number of positive cases 
+ 
+3 
+(Rustam et 
+al., 2020) 
+Linear Regression 
+LASSO Regression 
+SVR 
+ 
+No. of newly infected cases, the no. of deaths, 
+and the no. of recoveries in the next 10 days. 
+ 
+4 
+(Nikhil et 
+al., 2021) 
+Polynomial 
+Regression 
+Predicting the upcoming cases for next 25 days 
+ 
+5 
+(Liu, 2020) 
+Linear Regression 
+Logistic Regression 
+and RNN 
+predict pandemic data of the U.S 
+6 
+(Shaikh et 
+al., 2021) 
+Linear Regression 
+and Polynomial 
+Regression 
+ 
+Analysis, prediction and Time series forecasting 
+7 
+(Painuli et 
+al., 2021) 
+Random Forest and 
+extra tree classifiers 
+ one for predicting the chance of being infected 
+and other for forecasting the number of positive 
+cases 
+8 
+ 
+(Mojjada et 
+al., 2021) 
+Linear Regression, 
+Lasso Regression 
+SVM, 
+Exponential 
+Smoothing, 
+The number of newly infected COVID 19 
+people, mortality rates and the recovered 
+COVID-19 estimates in the next 10 days. 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+3 Proposed methodology 
+ 
+In our Research, This research contains two main phases. The first phase is the training, 
+and another one is testing. In the first phase, my dataset had many null or missing values, so 
+we used pre-processing to clean the data. And then, we used Feature Scaling to modify the data 
+from its original form because the machine can't be trained well with its original form of data 
+(MinMax, AbslouteMax, Normalization, Standardization, Robust Scaling) (Boente & Salibian-
+Barrera, 2015; Lau & Baldwin, 2016; Lin et al., 2016; Storcheus, Dmitry; Rostamizadeh, 
+Afshin; Kumar, 2015), so by the Feature scaling, we can modify the data for our convince to 
+build the model. After the feature scaling, we segregated the data to train and test data. For 
+training data, we prepared the model and the test data to evaluate the trained model, so we used 
+the split function to split the data for training data is 80%, and test data is 20% of the datasets. 
+For our experimental research work, we took the date and country attributes as the independent 
+variable from the dataset. Both date and country attribute values are in String type. So we have 
+to convert the value of the date attribute into numeric data by splitting the date into the date, 
+month, and year and putting them into the different attributes, and then convert the country 
+attributes data into numeric with the help of Label Encoding, which gives the numeric value 
+for each country, Example India-1, USA-2, UK-3 like this it will create the numeric values for 
+all the countries in the dataset. We analyzed the current trend or pattern of the coronavirus and 
+then predict the further future of the covid-19 confirmed cases or new cases by Training the 
+past Covid-19 dataset using the machine learning algorithm, ensembling models, and deep 
+learning such as Linear Regression, Polynomial Regression, K-nearest neighbor, Lasso 
+Regression, Decision Tree, Support Vector Machine, Decision tree, Random Forest algorithm, 
+Artificial Neural Networks(ANN) and Improve the model evaluation by the Ensemble methods 
+(Voting, Bagging, Stacking). 
+ 
+ 
+In the second phase, we first validated the model using the metrics of (MAE, MSE, 
+RMSE, and MAPE) to observe the model's accuracy. The decision tree and the Random Forest 
+algorithm perform better than these algorithms. The performance of SVR is low in all 
+prediction areas because the SVR is challenging to separate the data using the hyperplane for 
+this type of problem. So SVR mostly gives a low performance in this problem. What is the use 
+of this paper? If the covid -19 comes, we have to know the current trend/pattern of the covid-
+19, so we need that current past dataset to train the model. After preparing the model to analyze 
+the current trend of the covid-19, predict the future number of confirmed coronavirus cases in 
+a day; how it works is shown in the below architecture Diagram (Fig. 1). 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+                   
+                         
+ 
+Covid-19 Dataset               
+ Data Pre-Processing                                  Feature Scaling 
+                                  
+                                         
+ 
+                                                          Data Splitting 
+                
+     
+            
+                  Machine Learning and Deep Learning                 Ensemble Model            
+                                                              Trained Models                                                                                                
+   
+Data
+Preprocessig
+Label 
+Encoding
+remove the 
+misssing 
+values
+Feature 
+Scaling
+MinMax
+AbslouteMax
+Standazation
+Normalization
+Robust
+Linear 
+Regression
+Polynomial
+Regression
+K-NN
+SVR
+Random 
+Forest
+Decsion Tree
+LASSO
+ANN
+Ensembling
+Voting
+Stacking
+Bagging
+
+ 
+ 
+                       
+ 
+                                                                Figure 1. Architecture Diagram 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+3.1 Dataset Collection 
+ 
+1. Dataset Description 
+Dataset-1 
+In this research, The dataset used was taken from the Kaggle Website (Gambhir et al., 
+2020), and contained the COVID-19 details. In this dataset, we have 201 countries' COVID -
+19 data. Each country has data from 15 February 2020 to 02nd 22 December 2021 in our 
+dataset. And the attributes are date, country, how many people were affected by COVID-19 
+daily (daily-new-cases), the total number of affected people on that date(cumulative-total-
+cases), how many people are under treatment on that date(active-cases), and the total number 
+of death cases (cumulative-total- deaths). How many people die daily? (daily-new-deaths), 
+These are the attributes we have. The entire length of the dataset is 1,45,220. 
+ 
+TOP 10 Countries          Cumulative total                  TOP 10 Countries           Cumulative total 
+confirmed Cases                                                             Death Cases 
+ 
+Global                            264440620                          Global                         5249736 
+ 
+USA                                        49716825                                USA                                  806398 
+ 
+INDIA                                     34615757                                BRAZIL                             615225 
+BRAZIL                                   22118782                                INDIA                               470115 
+Evatuation
+Metrics
+MAE
+MSE
+RMSE
+MAPE
+
+ 
+ 
+UK                                           10329063                               MEXICO                           294428 
+RUSSIA                                   9703107 
+                         RUSSIA                            277640 
+TURKEY                                  8839891                                  PERU                               201282 
+FRANCE                                  7773530                                  UK                                   145281 
+IRAN 
+                                  6125596                                  INDONESIA                    143850 
+GERMANY                              6026796                                  ITALY                              134003 
+ARGENTINA                           5335310                                  IRAN                               129988 
+ 
+Dataset-2 
+In this research, The dataset used was taken from the Kaggle Website (Mandayam et 
+al., 2020), and contained the COVID-19 details. In this dataset, we have 193 countries' COVID 
+-19 data. Each country has data from 16 November 2020 to 12 September 2021 in our dataset. 
+And the attributes are date, country, CountryAlphaCode, Confirmed cases maximum of 
+41million, Death cases maximum of 660k, Recoveries maximum of 31 million, ECR, 
+GRTStringencyIndex, 
+DaySinceFirstCases, 
+DaySince100th 
+Cases, 
+ConfirmedPopPct, 
+DeathPopPct, RecoveriesPopPct. These are the attributes we have. The total length of the 
+dataset is 2952600. 
+TOP 10         Cumulative total           TOP 10             Cumulative total       TOP 10        Cumulative total                
+Countries      confirmed Cases            Countries         Death Cases              Countries      Recovery Cases    
+Global         3.03868E+13 
+        Global            6.18299E+11            Global         4.08442E+12   
+ 
+ 
+US                   2.65162E+13                   US                      5.10454E+11             Brazil              2.67528E+12 
+ 
+Brazil              3.68155E+12                   Brazil                 1.01685E+11             US                   1.34381E+12  
+ 
+Canada           68033025900                 China                2712603648              China              45989925888  
+ 
+China              55937676288                 Canada             1666076244              India               4859387857 
+ 
+United            34461082300                 United              1067745075              Russia            1128064202 
+Kingdom                                                  Kingdom  
+ 
+India                6529623206                   India                  87802534                 Turkey            919259007 
+ 
+Russia              1562989024                  Mexico              68604089                  Italy                759237934 
+ 
+France             1511222838                  Peru                   55929784                  Colombia       735029503 
+ 
+Turkey             1228359396                  Italy                    39566917                  Argentina      711610324 
+ 
+Spain                1100245487                 France                34621623                  Germany       69524824 
+ 
+We had missing or null values in my dataset, so we eliminated the missing values from the 
+dataset to build the good machine learning model.      
+
+ 
+ 
+                         
+ 
+                            
+I used the 
+dropna() function to delete the missing value from the dataset. Figure 3 shows how many 
+daily cases appear in each country. Below you will see how many null values are in the 
+dataset by the graph.   
+    
+Dataset 
+      Before Eliminating Null Values              After Eliminating Null Values 
+Dataset -1                 
+                    
+                                               
+ 
+Dataset-2                  
+                         
+         
+Table 2. Before and After eliminating null values 
+2 Summary of the dataset 
+The covid-19 datasets are mixed of integer and real-valued attribute characteristics 
+and this is used to evaluate the machine leaning models. 
+ 
+Figure 2. Null values in dataset-1 
+Figure3.  Null values of dataset-2 
+ 
+ 
+
+<AxesSubplot:>
+10
+0.8
+0.6
+0.4
+0.2
+0.0
+Country
+CountryAlpha3Code
+Date
+confirmed
+deaths
+recoveries
+confirmed_inc
+deaths_inc
+recoveries_inc
+ECR
+GRTStringencylndex
+DaysSince1Cases
+DaysSince100Cases
+confirmed_PopPct
+deaths_PopPct
+recoveries_PopPctdate
+0
+country
+0
+cumulative total cases
+0
+daily new cases
+7491
+active_cases
+4599
+cumulative total deaths
+7227
+daily new deaths
+22791
+dtype: int64date
+0
+country
+0
+cumulative total cases
+0
+daily new_cases
+0
+active_cases
+0
+cumulative total deaths
+0
+daily new deaths
+0
+dtype: int64Country
+0
+CountryAlpha3Code
+0
+Date
+0
+confirmed
+0
+deaths
+0
+recoveries
+0
+confirmed_inc
+0
+deaths_inc
+0
+recoveries_inc
+0
+ECR
+0
+GRTStringencyIndex
+130634
+DaysSince1Cases
+0
+DaysSince100Cases
+0
+confirmed_PopPct
+1200
+deaths_PopPct
+1200
+recoveries_PopPct
+1200
+dtype: int64Country
+?
+CountryA1pha3Code
+Date
+confirmed
+deaths
+recoveries
+0
+confirmed_inc
+deaths_inc
+recoveries_inc
+0
+ECR
+0
+GRTStringencyIndex
+0
+DaysSince1Cases
+0
+DaysSince100Cases
+confirmed_PopPct
+deaths_PopPct
+recoveries_PopPct
+dtype:
+int64<AxesSubplot:>
+1.0
+80
+0.6
+0.4
+0.2
+0.0
+date
+country
+cumulative_total_cases 
+daily_new_cases 
+active_cases 
+cumulative_total_deaths 
+ 
+Attributes                                             Dataset-1                                Dataset-2      
+       
+Data set characteristics                          Covid-19                                 Covid-19 
+Attribute characteristics                         Integer and alphabetic            Integer and alphabetic  
+                                                               values                                      values 
+No .of .Attributes                                   07                                            16 
+No .of .instance                                      145220                                    2952600  
+Missing values or errors present            yes                                           yes 
+Table3. Attributes and Details of the dataset 
+ 
+3 Data Visualisation  
+Dataset-1 
+ 
+ 
+ 
+The covid-19 global dataset created this Map, showing the Total confirmed cases of 
+each country worldwide, separated by the colors depending upon the confirmed cases and the 
+Total confirmed cases on each date. This plot was created by this (import plotly. express as px) 
+(Plotly Python Graphing Library, n.d.; Preacher et al., 2006). This scale, from blue to yellow, 
+refers to the range of covid-19 confirmed cases. If it shows dark blue, it has zero cases. If it 
+shows yellow, it will range between 15 million to 20 million cases. 
+ 
+ 
+Figure 4. Shows the Confirmed in all over the world dataset-1 
+ 
+ 
+
++二网
+Global Spread of Coronavirus
+cumulative_total_cases
+20M
+15M
+Australia
+10M
+date=01-01-2021
+cumulative total cases=28.427k
+5M
+date=01-01-2021
+01-01-2021 03-06-2020 05-11-2021 08-06-2020 10-11-2021
+13-06-2020
+15-11-2021 18-06-2020 20-11-2021
+23-05-2021 25-10-2020 28-03-2020 30-09-2020 
+ 
+This Map also worked like the previous Confirmed cases Map. This Map shows the 
+Total death cases of each country worldwide, each country separated by different colors 
+depending upon the death cases and the Total death cases on each date.  
+ 
+Figure 5. Shows the Death Cases in all over World dataset-1 
+Dataset-2 
+ 
+The covid-19 global dataset created this Map, showing the Total confirmed cases of 
+each country worldwide, separated by the colors depending upon the confirmed cases. This 
+plot was created by this (import plotly. express as px). This scale, from blue to yellow, refers 
+to the range of covid-19 confirmed cases. If it shows light sandals, it has zero cases. If it shows 
+yellow, it will range between 1.5 billion to 2 billion cases. 
+ 
+ 
+Figure 6. Shows the Confirmed cases in all over world dataset-2 
+ 
+This Map also worked like the previous Confirmed cases Map. This Map shows the 
+Total death cases of each country worldwide, separated by the different colours depending 
+
++-网
+Global Spread of Coronavirus Death Cases
+cumulative_total_deaths
+600k
+date=04-11-2021
+country=Japan
+400k
+200k
+date=04-11-2021
+01-01-2021 03-06-2021 06-01-2021 08-07-2021 11-02-2021
+13-08-2021 16-03-2021 18-09-2021 21-04-2021 23-10-2020 26-03-2021 28-08-2021 31-05-2021+二网
+Confirmed Cases
+2B
+2.12967BUS
+1.5B
+1B
+0.5B 
+ 
+upon the range of death cases in each country.
+ 
+Figure 7. Shows the Death cases in all over world dataset-2 
+ 
+  
+ 
+Global Spread of covid-19 top 5 countries in the graph 
+Dataset-1 
+We took five countries from this dataset to plot this graph with confirmed cases data 
+on the y-axis and the date on the x-axis. From this graph, we can see the confirmed cases on 
+the respective date for these countries. 
+ 
+Figure 8. Five countries confirmed cases data show in graph for dataset-1 
+ 
+ 
+Dataset-2 
+
+Deaths Cases
+30M
+25M
+20M
+16.43183
+Brazi
+15M
+10M
+5MCovid-19bycountry
+USA
+400k
+UK
+(0.9-05-2021.366.499k)
+India
+350k
+Italy
+Russia
+300k
+Spain
+250k
+200k
+150k
+100k
+50k
+NN
+Date 
+ 
+  
+From this graph, you can see the data from the five countries of confirmed cases with 
+the respective date(X-axis). Confirmed cases in Y-axis and date in X-axis. If we drag the 
+mouse on the graph line, it will show the confirmed cases on respective dates for a country. 
+ 
+Figure 9. Five countries confirmed cases data show in graph for dataset-1 
+3.2.Data Pre-processing  
+1 Feature Scaling 
+a)  Absolute Maximum Scaling 
+Find the absolute maximum value in the column and divide all the values in the queue 
+by that maximum value. Our data will lie between -1 and 1 (Mojjada et al., 2021; Rustam et 
+al., 2020). 
+Absolute Maximum Scaling= Each value in the column / Maximum value of the column (1) 
+In our experimental research work, we used this Feature scaling for the dependent and 
+independent variables to enhance the model by changing the values of the dataset into the 
+range of -1 to 1 because these data will understand by engines easily. If I do the model 
+training with my original data, the model may not be trained well. 
+b)  MinMax Scaling 
+Subtract each column value by the minimum value of the column and then divide this 
+by the subtracted value of the maximum and minimum value of the column. By this method, 
+our data will lie between 0 and 1. if we have negative values in the dataset. This Scaling will 
+compress all the inliers in the narrow range [0, 0.005] . 
+ 
+MinMax Scaling= (X - Min(X)) / (Max(X) - Min(X)) 
+ 
+ 
+(2) 
+We used this feature scaling to enhance the model of my experimental research work by 
+changing the values of the dependent and independent variables into the range -1 to 1 for 
+machine-readable (Sklearn.Preprocessing.Minmax_scale — Scikit-Learn 1.1.3 
+Documentation, n.d.) . 
+
+Covid-19 by country
+US
+40M
+ Brazil
+ India
+ Italy
+ Russia
+WSe
+Spain
+30M
+(Mar 22,2021,29.87613M)US
+25M
+Confirmed
+20M
+15M
+Wot
+5M
+Mar 2020
+May 2020
+Jul 2020
+Sep 2020
+Nov 2020
+Jan 2021
+Mar 2021
+May 2021
+Jul 2021
+Sep 2021
+Date 
+ 
+c)  Normalization 
+In this case, we used the average value of the column instead of the minimum value. 
+Normalization Scaling = (X - Mean(X)) / (Max(X) - Min(X))  
+ 
+ 
+(3) 
+ In my Experimental research work, I have the columns with different values, not in sequential 
+order. I used this Scaling to enhance the model by changing the values of the dataset into the 
+range of 0 to 1 (Lin et al., 2016; Storcheus, Dmitry; Rostamizadeh, Afshin; Kumar, 2015). 
+d)  Standardization 
+Each column value subtracts the average value of the column and then is divided by the 
+Standard deviation value and is not restricted to a specific range. In this experimental 
+research work. We  used this to enhance the model by changing the values of the dataset into 
+the range of [0, 1]. (Sklearn.Preprocessing.StandardScaler — Scikit-Learn 1.1.3 
+Documentation, n.d.). 
+Standardization Scaling = (X – Mean(X)) / σ  
+ 
+ 
+ 
+(4) 
+e)  Robust Scaling 
+Subtract the data points with the median value and divide them by the Inter Quartile 
+Range value (IQR). IQR means the difference between the dataset's first and third quartile 
+(Barros & Hirakata, 2003; Boente & Salibian-Barrera, 2015). 
+Robust Scaling = X – Median(X) / IQR. 
+ 
+ 
+ 
+ 
+(5) 
+ 
+2 Date Splitting 
+ 
+In our dataset, The date in the String value can't be readable by the machine, so we 
+have changed the date into an understandable format by splitting the date into three columns 
+date, day, and year.  
+ 
+ 
+ 
+3 Label Encoding 
+
+date
+country
+cumulative_total_ cases
+daily_new_cases
+active_cases
+cumulative_ total_ deaths
+daily_new_deaths
+Dates
+Month
+Year
+Country
+10
+25-02-2020
+Afghanistan
+1
+0.0 
+1.0
+0.0
+0.0
+25 
+2
+2020
+0
+11
+26-02-2020
+Afghanistan
+1
+0.0
+1.0
+0.0 
+0.0
+26 
+2020
+0
+12
+27-02-2020
+Afghanistan
+1
+0.0
+1.0
+0.0
+0.0
+ 27
+2020
+0
+13
+28-02-2020
+Afghanistan
+1
+0.0 
+1.0
+00
+0.0
+28 
+2020
+0
+14
+29-02-2020
+Afghanistan
+1
+0.0
+1.0
+0.0 
+0.0
+29
+2020 
+0
+.
+..
+..
+...
+...
+".
+:
+145216
+28-11-2021
+Zimbabwe
+133991
+40.0
+631.0
+4705.0
+00
+28
+11
+2021
+200 
+145217
+29-11-2021
+Zimbabwe
+134226
+235.0
+817.0
+4706.0
+1.0
+29 
+11
+2021
+200
+145218
+30-11-2021
+Zimbabwe
+134625
+399.0
+1171.0
+4707.0
+1.0
+11
+2021
+200
+145219
+01-12-2021
+Zimbabwe
+135337
+712.0
+1846.0
+4707.0
+00
+12 
+2021
+200
+145220
+02-12-2021
+Zimbabwe
+136379
+1042.0
+2843.0
+4707.0
+00
+12
+2021
+200
+119189 rows 
+ 
+Converting the alphabetic values into a numeric form converts them into a machine-
+readable 
+now 
+machine-learning 
+algorithm 
+that 
+can 
+easily 
+understand 
+it(Sklearn.Preprocessing.LabelEncoder — Scikit-Learn 1.1.3 Documentation, n.d.) . Because 
+it transformed into categorical values, for example, India, USA, U.K… by this method, it 
+will change like 0, 1, 2, … and so on. In our Experimental research work, we used the country 
+column. Because the machine learning algorithm can't read the alphabetic values, we used 
+this method to change them into categorical values. 
+ 
+ 
+ 
+ 
+4  Model Building 
+4.1 Machine learning 
+ 
+a) Linear Regression 
+This algorithm is used to find the changes of the dependent variable according to the 
+independent variable and shows the difference between dependent and independent variables, 
+known as Linear regression (Mojjada et al., 2021; Rustam et al., 2020; Sardinha & Catalán, 
+2018; Yadav et al., 2020). It plots the straight line and covers most of the data points in the 
+dependent variable. For continuous/real/numeric variables like sales, salary, age, and product 
+price, among others, linear regression makes predictions. 
+Two types of linear regression  
+Simple linear regression 
+- predict the value by a single independent variable. 
+Multiple linear regression 
+- predict the value by more than one independent variable. 
+ Y = mX + b  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(6) 
+Y is the dependent variable  
+X is the independent variable  
+
+date
+country
+cumulative_total_cases
+daily_new_cases
+active_cases
+cumulative_total_ deaths
+daily_new_deaths
+Datess
+Month
+Year
+Country
+10
+25-02-2020
+Afghanistan
+1
+0'0
+1.0
+0'0
+0'0
+25 
+2
+2020
+0
+11
+26-02-2020
+Afghanistan
+1
+0'0
+1.0
+0'0
+0.0 
+26 
+2
+2020
+0
+12
+27-02-2020
+Afghanistan
+1
+0.0
+1.0
+0.0
+0.0
+ 27 
+2 
+2020
+0
+13
+28-02-2020
+Afghanistan
+1
+0.0
+1.0
+0.0
+0.0
+28 
+2 
+2020
+0
+14
+29-02-2020
+Afghanistan
+1
+0.0
+1.0
+0.0
+0.0
+29 
+2 
+2020
+0
+...
+"
+..
+...
+.
+...
+...
+..
+...
+...
+145216
+28-11-2021
+Zimbabwe
+133991
+40.0
+631.0
+4705.0
+0.0
+28 
+11
+2021
+200
+145217
+29-11-2021
+Zimbabwe
+134226
+235.0
+817.0
+4706.0
+1.0
+29 
+11
+2021
+200 
+145218
+30-11-2021
+Zimbabwe
+134625
+399.0
+1171.0
+4707.0
+1.0
+30 
+11
+2021
+200 
+145219
+01-12-2021
+Zimbabwe
+135337
+712.0
+1846.0
+4707.0
+0.0
+12 
+1
+2021
+200
+145220
+02-12-2021
+Zimbabwe
+136379
+1042.0
+2843.0
+4707.0
+0.0
+12
+2 
+2021
+200
+119189
+ rows × 11 columns 
+ 
+m is the slope of the line 
+b is the intercept  
+In this experimental research work, we used  Multiple Linear Regression (Rath et al., 
+2020; Sardinha & Catalán, 2018). We took the date and country as the independent variable 
+and the confirmed cases as the dependent variable. By this algorithm, we can predict the 
+confirmed cases of covid-19 by giving the value of the input of the independent variable to 
+the trained model and predicting the output, respectively.  
+ 
+b)  Polynomial Regression 
+It is also known as the Multiple Linear Regression Special Case in Machine Learning. 
+Because transforming the equation for multiple linear regression into polynomial regression by 
+adding specific polynomial terms. It is a linear model that has been modified in several ways 
+to improve accuracy. The training dataset for polynomial regression is non-linear. To fit the 
+complex and non-linear functions and datasets instead of using linear regression. This 
+statement leads to the polynomial regression: transformed original features into polynomial 
+features of the essential degree (2,3,..,n) and then used in a linear model (Nikhil et al., 2021; 
+Shaikh et al., 2021; Yadav et al., 2020). 
+y= b0+b1x+ b2x2+ b3x3+....+ bnxn  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(7) 
+This formula also works like the linear regression formula but changes the nth degree 
+to plot the line closure to the data points. 
+This algorithm also works like linear regression, but we can change the predicted output 
+by changing the degree value of polynomial regression. By this change, we can give the 
+optimized predicted output for the given input.  
+ 
+c)  K-NN  
+ 
+KNN - K-Nearest Neighbour 
+The K-NN algorithm first finds the similarity between the new case and the existing 
+cases, and then it places the new case in a category that is quite similar to the existing 
+categories. After storing all of the previous data, a new data point is categorized using the K-
+NN Algorithm based on the similarity or distance of the data points. This model will quickly 
+recognize and place new cases in the most related category. 
+The Euclidean distance between the data points will be calculated. The distance 
+between two points is known as the Euclidean distance. The data points are classified based 
+on this Euclidean distance. 
+ 
+ 
+  d=√((x2-x1)²+(y2-y1)²)  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(8) 
+ By this example, you can understand the formula, 
+d=√((age2-age1)²+(gender2-gender1)²) 
+ 
+ 
+ 
+ 
+ 
+ 
+(9) 
+
+ 
+ 
+This algorithm is typically used for classification problems, but we used it in a 
+regression-type problem this time, and it predicted the confirmed cases but did not perform 
+well. 
+ 
+d)  Decision tree 
+A decision tree is one of the most popular supervised learning methods to solve 
+classification and regression problems (Deng et al., 2019; Safavian & Landgrebe, 1991). 
+Structured classifier, 
+where internal nodes stand for a dataset's features, branches for the decision-making 
+process, and each leaf node for the classification result (Alghamdi & Alfalqi, 2015; Verma & 
+Pal, 2020). It is used to split the datasets into tree-like structures for decision-making.  
+The Decision Node and Leaf Node are the two decision 
+tree 
+nodes. 
+While Leaf 
+nodes are the results of decisions and do not have any more branches,  
+Decision nodes are 
+used to create decisions and have numerous branches. The CART algorithm divides a node 
+into sub-nodes based on the Gini Index value. It starts with the training set as a root node, and 
+after successfully splitting it in two, it breaks the subsets using the same logic and then splits 
+the sub-subsets again, recursively, until it discovers that further splitting will not result in any 
+pure sub-nodes. 
+ 
+We used this algorithm in my experimental research work to build the model and 
+predict the confirmed cases of the covid-19. The evaluation metrics proved that this algorithm 
+worked well for our experimental dataset. 
+ 
+e) Random Forest 
+Random Forest is a classifier. It also works like the decision tree algorithm, but it uses 
+many decision trees on different subsets of the input dataset. Each tree has the predicted output 
+by voting averages of the results to increase the predicted accuracy of the model (Ankit & 
+Saleena, 2018; Hossain et al., 2021; Painuli et al., 2021; Pokharel & Deardon, 2014). This 
+algorithm can improve the accuracy and performance of the model from the decision tree 
+model. More trees in the random forest result in increased accuracy and high performance and 
+prevent (or) avoid the problem of overfitting. In my experimental research work, I used this 
+algorithm to build the model and predict the confirmed cases of the covid-19. This algorithm 
+also worked well than the decision tree because and proved by the evaluation metrics. 
+ 
+f) SVM 
+Support Vector Machine (SVM)  is a popular Supervised Learning algorithm used for 
+classification and regression problems. However, it is primarily used in Machine Learning for 
+Classification problems. The SVM algorithm aims to find the best line or decision boundary 
+for categorizing n-dimensional space to place new data points in the correct category easily. A 
+hyperplane is the best decision boundary (Education, 2021; Oumina et al., 2020; Rustam et al., 
+2020; Suhasini et al., n.d.).  
+Support Vector Regression (SVR) is a supervised learning algorithm for predicting 
+discrete values (Rivas-Perea et al., 2013). Support Vector Regression operates on the same 
+principles as SVMs. SVR's basic concept is to find the best fit line. The best fit line in SVR is 
+the hyperplane with the most significant number of points. This algorithm was used to create 
+
+ 
+ 
+the model and predict the confirmed cases of the covid-19. However, the SVR performance is 
+also low because it is challenging to construct the hyperplane for all data points in the dataset. 
+ 
+ 
+4.2 Ensembling Models 
+ 
+a) Voting 
+Voting classifiers and regressors are both ensemble methods; the predictions of these 
+models are simply an aggregation of ensemble predictions (Ankit & Saleena, 2018; Hammar 
+et al., 2019). An ensemble is a collection of predictors. As a result, these models are made up 
+of multiple predictors. The model aggregates each of these predictors' predictions into a final 
+one (Agnihotri et al., 2019). In this experimental research work, we used voting for the 
+regression model. I took the three machine learning models, the K-NN model, the Decision 
+tree model, and the Random Forest Model; we calculated the result by averaging those outputs 
+from these model outputs. 
+ 
+ 
+b) Stacking 
+ 
+Stacking is a popular ensemble modeling technique for predicting multiple nodes to 
+generate a new model and enhance model performance (Kim et al., 2021; Verma & Pal, 2020). 
+We used the stacking technique in my experimental research to improve the model. We 
+combined the three machine learning models, the K-NN model, the Decision tree model, and 
+Random Forest Model by combining these models to build the new model and improve the 
+model performance. 
+ 
+c) Bagging 
+ 
+Bagging, also known as Bootstrap aggregation, is an ensemble learning technique that 
+helps machine learning algorithms to improve their performance and accuracy. It is used to 
+deal with bias-variance trade-offs and reduces a prediction model's variance. Bagging prevents 
+data overfitting and is used in both regression and classification models, particularly 
+for decision tree algorithms (Dong & Qian, 2022). 
+ 
+In this experimental research work, we are going to use bagging for the regression 
+model to enhance the previously used model performance. We took the Decision tree algorithm 
+to improve the model performance of this algorithm. 
+ 
+4.3 Deep Learning  
+ 
+a) ANN 
+ 
+Artificial Neural Networks (ANN) are multi-layer fully-connected neural nets. They 
+are made up of an input layer, several hidden layers, and an output layer. Every node in one 
+layer is linked to every node. A structure like the human brain Artificial neural networks, like 
+the human brain, have neurons that are interconnected to one another in various layers of the 
+networks. These neurons are called nodes (Kathiravan & Saranya, 2021; Nan & Gao, 2018; 
+Oumina et al., 2020; Ozturk et al., 2020). 
+ 
+
+ 
+ 
+In this Experimental research work, we used this Deep learning Algorithm to build the 
+model and predict the confirmed cases of covid-19. The Performance of this Algorithm is also 
+very well for our experimental dataset. 
+ 
+ 
+5.Evaluation  
+                                    
+USED 
+ALGORITHMS 
+SCALING 
+Dataset—1                                             
+Dataset--2 
+ 
+ 
+MAE 
+MSE 
+RMSE 
+MAPE 
+MAE 
+MSE 
+RMSE MAPE 
+Linear 
+Regression 
+Absolute 
+Maximum 
+0.00773 
+0.00059 
+0.02430 
+0.77380 
+0.00327 0.00019 0.01381 0.32732 
+Min Max 
+0.00786 
+0.00076 
+0.02759 
+0.78608 
+0.00318 0.00018 0.01357 0.31863 
+Normalization 
+0.27527 
+0.13680 
+0.36987 
+27.5272 
+0.44087 0.22091 0.47001 44.0871 
+Standardization 
+0.29441 
+0.95309 
+0.97626 
+29.4419 
+0.24278 1.02367 1.01176 24.2787 
+Robust 
+3.97267 
+173.280 
+13.1636 
+397.267 
+7.86555 1051.98 32.4343 786.555 
+Polynomial  
+Regression 
+Absolute 
+Maximum 
+0.00772 
+0.00058 
+0.02426 
+0.77240 
+0.00322 0.00018 0.01374 0.32228 
+Min Max 
+0.00782 
+0.00075 
+0.02755 
+0.78286 
+0.00313 0.00018 0.01351 0.31378 
+Normalization 
+0.27342 
+0.13595 
+0.36872 
+27.3423 
+0.43739 0.21898 0.46796 43.7397 
+Standardization 
+0.29424 
+0.94983 
+0.97459 
+29.4241 
+0.23882 1.01550 1.00772 23.8825 
+Robust 
+3.96498 
+172.826 
+13.1463 
+396.498 
+7.73825 1041.97 32.2795 773.825 
+K-NN 
+Absolute 
+Maximum 
+0.00779 
+0.00076 
+0.02758 
+0.77987 
+0.00339 0.00020 0.01441 0.33969 
+Min Max 
+0.00735 
+0.00065 
+0.02563 
+0.73524 
+0.00308 0.00018 0.01342 0.30884 
+Normalization 
+0.20626 
+0.10418 
+0.32277 
+20.6264 
+0.29956 0.14464 0.38032 29.9567 
+Standardization 
+0.28259 
+0.95238 
+0.97590 
+28.2595 
+0.23513 1.00703 1.00351 23.5139 
+Robust 
+3.83587 
+184.926 
+13.5987 
+383.587 
+7.65189 1051.12 32.4210 765.189 
+SVR 
+Absolute 
+Maximum 
+0.09660 
+0.00961 
+0.09803 
+9.66083 
+0.09847 0.00979 0.09894 9.84745 
+Min Max 
+0.09678 
+0.00979 
+0.09895 
+9.6785 
+0.09860 0.00979 0.09897 9.86086 
+Normalization 
+0.23199 
+0.14200 
+0.37683 
+23.1999 
+0.37507 0.28580 0.53461 37.5072 
+Standardization 
+0.23072 
+1.03625 
+1.01796 
+23.072 
+0.20443 0.85370 0.92396 20.4438 
+
+ 
+ 
+Robust 
+2.72609 
+199.169 
+14.1127 
+272.60 
+4.75392 948.336 30.7950 475.392 
+Decision Tree 
+Absolute 
+Maximum 
+0.00122 
+6.42435 
+0.00801 
+0.12245 
+0.00050 2.37978 0.00487 0.05063 
+Min Max 
+0.00127 
+7.28606 
+0.00853 
+0.12702 
+0.00049 2.23365 0.00472 0.04979 
+Normalization 
+0.12954 
+0.12982 
+0.36030 
+12.954 
+0.12124 0.12215 0.34950 12.1241 
+Standardization 
+0.04880 
+0.11049 
+0.33241 
+4.88080 
+0.03811 0.17476 0.41805 3.81192 
+Robust 
+0.65679 
+16.8198 
+4.10119 
+65.6796 
+1.26035 192.319 13.8679 126.035 
+Random Forest 
+Absolute 
+Maximum 
+0.00113 
+3.71692 
+0.00609 
+0.11386 
+0.00046 1.89923 0.00435 0.04655 
+Min Max 
+0.00116 
+4.25899 
+0.00652 
+0.11669 
+0.00046 2.23436 0.00472 0.04675 
+Normalization 
+0.13557 
+0.08211 
+0.28656 
+13.5574 
+0.12545 0.07804 0.27936 12.5452 
+Standardization 
+0.04479 
+0.07184 
+0.26804 
+4.47991 
+0.03484 0.12288 0.35054 3.48449 
+Robust 
+0.62770 
+16.1979 
+4.02466 
+62.7707 
+1.13672 139.042 11.7916 113.672 
+Voting  
+Ensemble 
+Absolute 
+Maximum 
+0.00288 
+9.84888 
+0.00992 
+0.28810 
+0.00127 5.05432 0.00710 0.12745 
+Min Max 
+0.00278 
+9.67740 
+0.00983 
+0.27831 
+0.00116 3.73179 0.00610 0.11633 
+Normalization 
+0.15719 
+0.08449 
+0.29067 
+15.7198 
+0.18187 0.08527 0.29202 18.1877 
+Standardization 
+0.10422 
+0.15918 
+0.39897 
+10.4228 
+0.08832 0.17096 0.41348 8.83253 
+Robust 
+1.45676 
+35.3470 
+5.94533 
+145.676 
+2.82742 176.511 13.2857 282.742 
+Bagging  
+Ensemble 
+Absolute 
+Maximum 
+0.00115 
+3.58553 
+0.00598 
+0.11504 
+0.00047 2.39693 0.00489 0.04786 
+Min Max 
+0.00130 
+7.81970 
+0.00884 
+0.13041 
+0.00047 2.17509 0.00466 0.04714 
+Normalization 
+0.13660 
+0.07791 
+0.27912 
+13.6604 
+0.12664 0.07389 0.27183 12.6646 
+Standardization 
+0.04498 
+0.07324 
+0.27064 
+4.49816 
+0.03515 0.11348 0.33686 3.51532 
+Robust 
+0.63517 
+16.8967 
+4.11056 
+63.5177 
+1.07096 84.6469 9.20037 107.096 
+Stacking 
+Ensemble 
+Absolute 
+Maximum 
+0.00072 
+1.78910 
+0.00422 
+0.07217 
+0.00035 6.49338 0.00254 0.03549 
+Min Max 
+0.00067 
+1.53106 
+0.00391 
+0.06747 
+0.00023 3.76875 0.00194 0.02359 
+Normalization 
+0.09456 
+0.02263 
+0.15046 
+9.4561 
+0.07794 0.01525 0.12349 7.79464 
+Standardization 
+0.01444 
+0.00667 
+0.08172 
+1.44478 
+0.01977 0.01457 0.12073 1.97717 
+
+ 
+ 
+ 
+ 
+6. Conclusion 
+ 
+ 
+In this experimental research work, we have done predictive analysis by proposing a 
+new model to predict the recent covid-19 cases. Before training, we made some changes in the 
+dataset for our convenience by pre-processing techniques such as Dropna() function to delete 
+the null values in the dataset to avoid wrong prediction performance and separate the date 
+column into the day, month and year because the string can not load in the machine learning. 
+And then we used scaling to change the data in the range between 0 to 1 for easy understanding 
+of the Machine Learning model because the original data is tough to train and predict, so we 
+made these changes in the original data from 0 to 1, such as Absolute maximum, Min-Max, 
+Normalization, Standardization,  and Robust. Then, in the  Data visualization phase, we used 
+maps and graphs for an easy understanding of the data those we used in this experimental 
+research. After that, we built the model using different machine learning algorithms such as 
+Linear Regression (L.R.), Polynomial Regression (PR), K-nearest neighbor (KNN), Decision 
+Tree (D.T.), Random Forest (R.F.), Lasso Regression (LR), Support Vector Regression (SVR), 
+and we used the ensembling model also to improve the model performance. In ensembling 
+techniques, we used voting, Stacking, and Bagging. 
+In deep learning Algorithms, the evaluation metrics such as MAE, MSE, RSME, and 
+MAPE are used in our prediction model. Out of this decision tree and Random Forest algorithm 
+performed better than SVR and all other algorithms. This model helps in predicting trends and 
+patterns of covid-19. 
+ 
+Reference 
+Agnihotri, D., Verma, K., Tripathi, P., & Singh, B. K. (2019). Soft voting technique to 
+improve the performance of global filter based feature selection in text corpus. Applied 
+Intelligence, 49(4), 1597–1619. https://doi.org/10.1007/S10489-018-1349-1 
+Alghamdi, R., & Alfalqi, K. (2015). A Survey of Topic Modeling in Text Mining. 
+International Journal of Advanced Computer Science and Applications, 6(1), 147–153. 
+https://doi.org/10.14569/IJACSA.2015.060121 
+Ankit, & Saleena, N. (2018). An Ensemble Classification System for Twitter Sentiment 
+Robust 
+0.23091 
+1.68392 
+1.29766 
+23.0917 
+0.60178 19.0907 4.36929 60.1785 
+ANN 
+Absolute 
+Maximum 
+0.00524 
+0.00067 
+0.02597 
+0.52460 
+0.00222 0.00022 0.01500 0.22281 
+Min Max 
+0.00521 
+0.00067 
+0.02597 
+0.52182 
+0.00203 0.00017 0.01328 0.20345 
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+16.4920 
+0.34391 0.34419 0.58668 34.3914 
+Standardization 
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+1.09775 
+1.04773 
+19.644 
+0.15029 0.87252 0.93408 15.0297 
+Robust 
+2.76430 
+214.265 
+14.6378 
+276.43 
+4.74052 948.755 30.8018 474.052 
+
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+ 
+

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@@ -0,0 +1,1084 @@
+Grokking modular arithmetic
+Andrey Gromov
+Meta AI
+Meta Platforms, Inc.
+Menlo Park, California 94025
+&
+Department of Physics,
+Condensed Matter Theory Center,
+University of Maryland
+College Park, Maryland 20740
+[email protected]
+Abstract
+We present a simple neural network that can learn modular arithmetic tasks and
+exhibits a sudden jump in generalization known as “grokking”. Concretely, we
+present (i) fully-connected two-layer networks that exhibit grokking on various
+modular arithmetic tasks under vanilla gradient descent with the MSE loss function
+in the absence of any regularization; (ii) evidence that grokking modular arithmetic
+corresponds to learning specific feature maps whose structure is determined by the
+task; (iii) analytic expressions for the weights – and thus for the feature maps –
+that solve a large class of modular arithmetic tasks; and (iv) evidence that these
+feature maps are also found by vanilla gradient descent as well as AdamW, thereby
+establishing complete interpretability of the representations learnt by the network.
+1
+Introduction and overview of literature
+Grokking is an effect discovered empirically in [11]. Its phenomenology is characterized by a
+steep and delayed rise in generalization from 0% to a fixed value, as depicted in Fig. 1b. Beyond
+that observation, however, there are no clear characteristics of grokking that are reproduced across
+different works. Here, we start with a lightening review of various claims made in the literature.
+In the original work [11], the authors studied how a shallow transformer learns data distributions
+that are generated by simple deterministic rules (termed ‘algorithmic datasets’). Examples of such
+datasets include modular arithmetic, finite groups, bit operations and more. Specifically, in [11] the
+data took a form of a string “a ◦ b = c”, where c was masked and had to be predicted by a two-layer
+decoder-only transformer. In that study, the following empirical facts were observed:
+• Generalization occurs long after training accuracy reached 100%. The jump in generalization
+is quite rapid and occurs after a large number of epochs (cf. Fig. 1).
+• There is a minimal amount of data (dependent on the task) that needs to be included into the
+training set in order for generalization to occur (cf. Fig. 4b).
+• Various forms of regularization improve how quickly grokking happens. Weight decay
+included in AdamW optimizer showed to be particularly effective (cf. Fig. 4b).
+In subsequent work [7], the authors simplified the architecture to a single linear learnable encoder
+followed by a multilayer perceptron (MLP) decoder and showed that, even if the task is recast as
+a classification problem, grokking persists. They also interpret grokking as a competition between
+encoder and decoder, and developed a toy model of grokking as dynamics of the embeddings only.
+Preprint. Under review.
+arXiv:2301.02679v1  [cs.LG]  6 Jan 2023
+
+(a)
+(b)
+(d)
+(c)
+Figure 1: Dynamics under GD for the minimal model (4) with MSE loss and α = 0.49. (a) Train
+and test loss. Train loss generally decays monotonically, while test loss reaches its maximum right
+before the onset of grokking. (b) Norms of weight matrices during training. We do not observer a
+large increase in weight norms as in [14], but we do see that weight norms start growing at the onset
+of grokking. (c) Train and test accuracy showing the delayed and sudden onset of generalization. (d)
+Norms of gradient vectors. The dynamics accelerates until the test loss maximum is reached and then
+slowly decelerates.
+This model indeed leads to some quantitative predictions such as the critical amount of data needed
+for grokking to happen relatively fast.
+In more recent work [14], it was argued that if the Adam optimizer is used, then in order for grokking
+to happen without regularization, the training dynamics must undergo a slingshot – a sudden explosion
+in the training loss – which was followed by the rise of generalization. It was further shown that those
+slingshots and grokking can be turned on and off by tuning the ϵ parameter of the Adam optimizer.
+In a blogpost [9], it was argued that the algorithm for modular addition learnt by a single-layer
+transformer can be reverse-engineered and is human-interpretable. It was further argued that (i)
+regularization is required for grokking and (ii) there should be no grokking in the infinite-data regime.
+Furthermore, many other algorithmic datasets were considered.
+On a theoretical front, the authors of [1] studied online learning of the (k, n) sparse parity problem
+where the network function is asked to compute parity of k bits in a length-n string of random bits. In
+particular, they observed grokking both in under- and over-parametrized regimes. For large minibatch
+sizes, generalization was attributed to amplification of the information already present in the initial
+gradient (called Fourier gap) rather than to the diffusive search by stochastic gradient descent, and
+derived the scaling of grokking time with n, k to be nO(k).
+Finally, [8] studied grokking for non-algorithmic datasets and its dependence on the initialization,
+while [16] developed a solvable model of grokking in the teacher-student setup.
+To summarize, the available results, although undoubtedly inspiring, leave grokking on algorithmic
+datasets as a somewhat mysterious effect. Furthermore, the empirical results suggest that grokking
+provides a fascinating platform for quantitatively studying many fundamental questions of deep
+learning in a controlled setting. These include: (i) the precise role of regularization in deep nonlinear
+neural networks; (ii) feature learning; (iii) the role of training data distributions in optimization
+dynamics and generalization performance of the network; (iv) data-, parameter- and compute-
+efficiency of training; (v) interpretability of learnt features; and (vi) expressivity of architectures and
+complexity of tasks.
+2
+
+Train accuracy
+Test accuracyTrain loss
+Test lossWeight norm layer 1
+Weight norm layer 2grad norm layer 1
+grad norm layer 2(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+(i)
+Figure 2: Preactivations. First row: Preactivation h(2)
+6 (n, m). Second row: Fourier image of the
+Preactivation h(2)
+6 (n, m). Third row: Preactivation h(1)
+6 (n, m) or h(1)
+30 (n, m). First column: At
+initialization. Second column: Found by vanilla GD. The Fourier image shows a single series of
+peaks corresponding to m + n = 6 mod 97. Third column: Evaluated using the analytic solution
+(6)-(7). The Fourier image shows the same peak as found by GD, but also weak peaks corresponding
+to 2m = 6 mod 97, 2n = 6 mod 97 and m − n = 6 mod 97 that were suppresed by the choice of
+phases via (12).
+This motivates the present study, proposing and analyzing a minimal yet realistic model and opti-
+mization process that lead to grokking on modular arithmetic tasks.
+2
+Set up and overview of results
+In this Section we describe a very simple, solvable, setting where grokking takes place and learnt
+features can be understood analytically. We consider a two-layer MLP network without biases, given
+by
+h(1)
+k (x) =
+�
+1
+D
+D
+�
+j=1
+W (1)
+kj xj ,
+z(1)
+i
+(x) = φ(h(1)
+i (x)) ,
+(1)
+h(2)
+q (x) = 1
+N
+N
+�
+k=1
+W (2)
+qk z(1)
+k (x) ,
+(2)
+where N is the width of the hidden layer, D is the input dimension, and φ is an activation function. At
+initialization the weights are sampled from the standard normal distribution W (1), W (2) ∼ N(0, 1).
+In Eqs. (1)–(2), we have chosen to follow the mean-field parametrization [13]: this parametrization
+ensures that the analytic solution presented in the next Section remains finite in the large-N limit1.
+Given this architecture, we then set up modular arithmetic tasks as classification problems. To this
+end, we fix an integer p (that does not have to be prime) and consider functions over Zp. Each input
+integer is encoded as a one-hot vector. The output integer is also encoded as a one-hot vector. For the
+task of learning bivariate functions over Zp the input dimension is 2p, the output dimension is p, the
+total number of points in the dataset is p2, while the model (1)–(2) has 3Np parameters. Finally, we
+1In the limit of infinite width, the meanfield parametrization allows for feature learning.
+3
+
+hg'(n,m)hg'(n,m)hg(n,m)hg'(n,m)hg(n,m)hg'(n,m)h?'(n,m)hg'(n,m)hg(n,m)split the dataset D into train Dtrain and test Dtest subsets, and furnish this setup with the MSE loss
+function.2
+Under this minimal setting grokking occurs consistently for many modular functions, provided
+enough epochs of training have taken place and the fraction of data used for training,
+α ≡ |Dtrain|
+|D|
+,
+(3)
+is sufficiently large (if α is too small, generalization is not possible even after long training time).
+By adjusting width N, at fixed α, we can tune between underparametrized and overparametrized
+regimes. The ‘simplest’ optimizer that leads to grokking is the full-batch gradient descent. No explicit
+regularization is necessary for grokking to occur. We have tried other optimizers and regularization
+methods such as AdamW, GD with weight decay and momentum, SGD with Batchnorm, and GD
+with Dropout. Generally, regularization and the use of adaptive optimizers produce two effects: (i)
+grokking happens after a smaller number of epochs and (ii) grokking happens at smaller α. See
+Fig. 4.
+In passing, we note that, in the case of quadratic activation the full network function takes an even
+simpler form
+f(x) =
+1
+DN W (2) �
+W (1)x
+�2
+.
+(4)
+This function is cubic in parameters and quadratic in its inputs. Eq. (4) is the simplest possible
+nonlinear generalization of the ‘u-v’ model studied in [6]. The exact results are derived for this
+particular choice (and can be generalized to other monomials if wished) while empirical results are
+only mildly sensitive to the choice of activation function.
+Whether grokking happens or not depends on the modular function at hand assuming the architecture
+and optimizer are fixed. We show that for any function of the form f(n, m) = f1(n) + f2(m) mod p
+as well as ˜f(n, m) = F(f1(n) + f2(m)) mod p one can present an analytic solution for the weights
+that yield 100% accuracy and these weights are approximately found by various optimizers with
+and without regularization. Functions of the form g(n, m) = g1(n) · g2(m) mod p can also be
+grokked, however we have failed to find the analytic expression for the weights. Functions of the
+form f(n, m) + g(n, m) mod p are more difficult to grok: they require more epochs and larger α.
+In summary, our setup is simple enough to be analytically tractable but complex enough to exhibit
+representation learning and, consequently, grokking.
+3
+Interpretability: analytic expression for the weights
+3.1
+Modular addition
+In this Section we will exhibit the analytic expression for the weights that solve the modular addition
+task. Namely, the network supplied with these weights implements the following modular function
+f(n, m) = n + m mod p .
+(5)
+This solution is approximate and can be made increasingly more accurate (meaning the test loss
+can be made arbitrarily close to 0) by increasing the width N. To simplify the presentation, we
+will discuss modular addition at length and then generalize the solution to a broad class of modular
+functions. In the next Section we will provide evidence that the GD and AdamW find the same
+solution.
+Claim I. If the network function has the form (4) then the weights W (1)
+kn and W (2)
+qk solving the
+modular addition problem are given by
+W (1)
+kn =
+�
+�cos
+�
+2π k
+pn1 + ϕ(1)
+k
+�
+cos
+�
+2π k
+pn2 + ϕ(2)
+k
+�
+�
+�
+T
+,
+n = (n1, n2)
+(6)
+W (2)
+qk = cos
+�
+−2π k
+pq − ϕ(3)
+k
+�
+,
+(7)
+2CSE loss can be used, if desired.
+4
+
+where we represent W (1)
+kn as a row of two N × p matrices and n1, n2 = 0, 1, . . . , p − 1. The full size
+of W (1)
+kn is N × 2p. The phases ϕ(1)
+k , ϕ(2)
+k
+and ϕ(3)
+k
+are random, sampled from a uniform distribution
+and satisfy the constraint (12).
+Reasoning.
+Here we explain why and how the solution (6)-(7) works. There are two important
+ingredients in (6)-(7). The first ingredient is the periodicity of weights with respect to the indices
+n1, n2, q. The set of frequencies is determined by the base of Zp. The full set of independent
+frequencies is obtained by varying k from 0 to p−1
+2
+if p is odd and to p
+2 if p is even. The second
+ingredient is the set of phases ϕ(1)
+k , ϕ(2)
+k , ϕ(3)
+k . Indeed, Eqs. (6)-(7) solve modular addition only after
+these phases are chosen appropriately. We will discuss the choice shortly.
+To show that (6)-(7) solve modular addition we will perform the inference step analytically. Consider
+a general input (n, m) represented as a pair of one-hot vectors stacked into a single vector of size
+2p × 1.
+The preactivations in the first layer are given by (we drop the normalization factors)
+h(1)
+k (n, m) = cos
+�
+2π k
+pn + ϕ(1)
+k
+�
++ cos
+�
+2π k
+pm + ϕ(2)
+k
+�
+.
+(8)
+The activations in the first layer are given by
+z(1)
+k (n, m) =
+�
+cos
+�
+2π k
+pn + ϕ(1)
+k
+�
++ cos
+�
+2π k
+pm + ϕ(2)
+k
+��2
+,
+(9)
+which, after some trigonometry, becomes
+z(1)
+k (n, m)
+=
+1 + 1
+2
+�
+cos
+�
+2π k
+p2n + 2ϕ(1)
+k
+�
++ cos
+�
+2π k
+p2m + 2ϕ(2)
+k
+��
++
+cos
+�
+2π k
+p(n + m) + ϕ(1)
+k
++ ϕ(2)
+k
+�
++ cos
+�
+2π k
+p(n − m) + ϕ(1)
+k
+− ϕ(2)
+k
+�
+.(10)
+Finally, the preactivations in the second layer take form
+h(2)
+q (n, m)
+=
+1
+4
+N
+�
+k=1
+cos
+�
+2π k
+p(2n − q) + 2ϕ(1)
+k
+− ϕ(3)
+k
+�
++ cos
+�
+2π k
+p(2n + q) + 2ϕ(1)
+k
++ ϕ(3)
+k
+�
++
+1
+4
+N
+�
+k=1
+cos
+�
+2π k
+p(2m − q) + 2ϕ(1)
+k
+− ϕ(3)
+k
+�
++ cos
+�
+2π k
+p(2m + q) + 2ϕ(1)
+k
++ ϕ(3)
+k
+�
++
+1
+2
+N
+�
+k=1
+cos
+�
+2π k
+p(n + m − q) + ϕ(1)
+k
++ ϕ(2)
+k
+− ϕ(3)
+k
+�
++
+1
+2
+N
+�
+k=1
+cos
+�
+2π k
+p(n + m + q) + ϕ(1)
+k
++ ϕ(2)
+k
++ ϕ(3)
+k
+�
++
+1
+2
+N
+�
+k=1
+cos
+�
+2π k
+p(n − m − q) + ϕ(1)
+k
+− ϕ(2)
+k
+− ϕ(3)
+k
+�
++
+1
+2
+N
+�
+k=1
+cos
+�
+2π k
+p(n − m + q) + ϕ(1)
+k
+− ϕ(2)
+k
++ ϕ(3)
+k
+�
++
+N
+�
+k=1
+cos
+�
+2π k
+pq + ϕ(3)
+k
+�
+.
+(11)
+Expression (11) does not yet perform modular addition. Observe that each term in (11) is a sum
+of waves with different phases, but systematically ordered frequencies. We are going to choose
+the phases ϕ(1)
+k , ϕ(2)
+k , ϕ(3)
+k
+to ensure constructive interference in the third line of (11). The simplest
+choice is to take
+ϕ(1)
+k
++ ϕ(2)
+k
+= ϕ(3)
+k
+.
+(12)
+5
+
+Then the term in the third line of (11) takes form
+1
+2
+N
+�
+k=1
+cos
+�
+2π k
+p(n + m − q)
+�
+= N
+2 δ(n + m − q) ,
+(13)
+where δ(n+m−q) is the modular version of the δ-function. It is equal to 1 when n+m−q = 0 mod p
+and is equal to 0 otherwise. This concludes the constructive part of the interference.
+Next, we need to ensure that all other waves (i.e. all terms, but the third term in (11)) interfere
+destructively. Fortunately, this can be accomplished by observing that the constraint (12) leaves some
+phases in every single term in (11) apart from the third one. We will spare the reader the explicit
+expression. Every remaining term takes form
+1
+2
+N
+�
+k=1
+cos
+�
+2π k
+ps + ϕk
+�
+,
+(14)
+where s is an integer and ϕk is a linear combination of ϕ(1)
+k
+and ϕ(2)
+k . We now assume that ϕ(1)
+k
+and
+ϕ(2)
+k
+are uniformly distributed random numbers. Then so are ϕk. For any appreciable N (see Fig. 4b)
+we have
+N
+�
+k=1
+cos
+�
+2π k
+ps + ϕk
+�
+≪ N ,
+(15)
+which implies that every term in (11) can be neglected compared to the third term. Thus, for
+reasonable values of N (and restoring normalisation) the network function h(2)
+q (n, m) takes form
+h(2)
+q (n, m) ≈ 1
+2
+N
+�
+k=1
+cos
+�
+2π k
+p(n + m − q)
+�
+=
+1
+2Dδ(n + m − q) .
+(16)
+In the limit of large N the approximation becomes increasingly more accurate. Note that h(2)
+q (n, m)
+is finite in the infinite width limit.
+The test accuracy of the solution (6)-(7) increases with width. For larger N the interference is
+stronger leading to the better approximation of the δ-function and, ultimately, to better accuracy. In
+this example we clearly see that larger width does not imply a larger number of relevant features.
+Instead, it introduces redundancy: each frequency appears several times with different random phases
+ultimately leading to a better wave interference.
+We emphasize that, the weights (6)-(7) are not iid. At fixed k the weights W (1)
+kn , W (2)
+qk are strongly
+correlated with each other. This provides a non-trivial yet analytically tractable example of a
+correlated, non-linear, network far away from the Gaussian limit.
+The weights (6)-(7) also work for other activation functions, including ReLU, however the 100%
+accuracy is achieved at higher width compared to quadratic activation function (more details in
+Appendix B).
+3.2
+General modular functions and complexity
+The solution (6)-(7) can be easily generalized to represent a general modular function of the form
+f(n, m) = f1(n) + f2(m) mod p ,
+(17)
+where f1, f2 are arbitrary modular functions of a single variable. The generalization becomes obvious
+once we observe that the proof presented in Section 3 holds verbatim upon replacing n → f1(n) and
+m → f2(m) leading to a δ-function supported on f1(n) + f2(m) − q = 0 mod p. These solutions
+are also found by the optimizer just like in the case of modular addition. More precisely, we claim
+Claim II. If the network function has the form (4) then the weights W (1)
+kn and W (2)
+qk solving the
+modular task f(n, m) = f1(n) + f2(m) mod p are given by
+W (1)
+kn =
+�
+�cos
+�
+2π k
+pf1(n1) + ϕ(1)
+k
+�
+cos
+�
+2π k
+pf2(n2) + ϕ(2)
+k
+�
+�
+� ,
+n = (n1, n2)
+(18)
+6
+
+ ReLU
+Quadratic
+GD
+AdamW
+Figure 3: Solutions found by the optimizer. In all cases distribution of ϕ(1)
+k
++ ϕ(2)
+k
+− ϕ(3)
+k
+is strongly
+peaked around 0. The solutions found by AdamW are closer to the analytic ones because the phases
+are peaked stronger around 0. Note that for solutions found by the optimizer the phases are not iid
+which leads to the better accuracy.
+and Eq. (7). The weights depend on the modular arithmetic task at hand. Furthermore, for this class
+of tasks the weights in the readout layer are unchanged. A simple example is f(n, m) = n2 + m2.
+The activations for this task are presented in the Appendix C.
+Corollary. Given the Claim II, a more general modular task ˜f(n, m) = F(f1(n) + f2(m)) mod p,
+can be solved, assuming that F is invertible. This is accomplished by modifying the readout layer
+weights as follows
+W (2)
+qk = cos
+�
+−2π k
+pF −1(q) − ϕ(3)
+k
+�
+.
+(19)
+This solution approximates δ(f1(n) + f2(m) − F −1(q)), which is equivalent to the δ-function
+supported on the claimed modular task δ(F(f1(n) + f2(m)) − q) assuming F −1 is single-valued.
+Note that application of F −1 must follow modular arithmetic rules. If F −1 is not single-value then
+the accuracy will be approximately 100%/b, where b is the number of branches. A simple example is
+f(n, m) = (n + m)2. The activations for this task are presented in the Appendix. Analytic solution
+has accuracy ≈ 50% since F −1(x) = x
+1
+2 mod p, which has two branches.
+The architecture (4) can also learn modular multiplication, however we do not posses an analytic
+solution for that case.
+Broadly speaking, a bivariate modular function is a p × p table where each entry can take values
+between 0 and p−1. There are pp2 such tables. Clearly, grokking is not possible on the overwhelming
+majority of such functions, because this set includes placing random integers in each entry of the
+table. Some modular functions, namely the ones that involve addition and multiplication, and are not
+of the form ˜f are substantially harder to learn. They require more data, more time and do not always
+yield 100% test accuracy after grokking. One particularly interesting example was found by [11],
+f(n, m) = n3 + nm2 + m, which does not generalize even for α > 0.9, both for transformer and
+MLP architectures. Some examples are discussed in Appendix. It is not clear how to predict which
+functions will generalize and which will not given an architecture.
+7
+
+4
+Properties of solutions found by gradient descent
+4.1
+General properties
+In this Section we show that optimization of the network (1)-(2) yields a solution that is very close to
+the one we proposed in the previous Section.
+(a)
+(b)
+Figure 4: Scaling with width and data. (a) Grokking time vs. the amount of training data for various
+optimizers. The abrupt change in grokking time is observed at different α. Momentum appears to
+play a major role both in reducing grokking time and α. (b): Test accuracy as a function of width
+for the solution found by GD, AdamW and for the analytic solution (6)–(7). The optimizer can tune
+phases better than random uniform distribution in order to ensure better cancellations. The shape of
+the curves also depends on the amount of data used for training and number of epochs. Here we took
+α = 0.5 and trained longer for GD.
+As can be seen in Fig. 1 during the optimization the network first overfits the train data. The periodic
+structure in weights and activations does not form at that point. Train loss slowly gets smaller until
+it either (i) saturates leading to a memorizing solution without grokking the problem, or (ii) after a
+period of slow decrease, it slightly accelerates. It is during that time grokking and feature formation
+take place. The test loss is non-monotonic and reaches a local maximum right before grokking
+happens. In the memorizing phase test loss never leaves this local maximum. This general behaviour
+appears to be insensitive to either optimizer used, loss function or modular function (i.e. dataset) in
+question.
+We then show empirically that independently of the optimizer and the loss function the features
+found by optimization in the grokking phase are indeed periodic functions with frequencies 2πk
+p
+where k = 0, . . . , p − 1. If the width is larger than p−1
+2
+then multiple copies of these functions are
+found with different phases. The phases are approximately random and satisfy the constraint (12)
+approximately as we show in Fig.3. Given the simplicity of the setup, the basic explanation for
+grokking must be quite banal. At some point in training, the only way to decrease training loss is to
+start learning the “right” features.
+4.2
+Scaling
+Scaling with width and dataset size are presented on Fig. 4. The accuracy of solution (6)-(7) favorably
+scales with width. This stems from the simple fact that destructive interference condition (15)
+becomes increasingly more accurate with larger N. The test accuracy of trained network also
+8
+
+GD
+GD, momentum 0.9
+GD, wd
+GD,momentum 1.0
+GD, momentum 1.0 + wd
+Adam
+AdamWDGD
+Adamw
+Analyticincreases with the width, reaching perfect accuracy before the analytic solution does, which is not
+surprising because optimizer can tune the individual phases to ensure better performance.
+The grokking time scales with the amount of data. Both for GD and AdamW there is a critical amount
+of data αc such that grokking is possible. The precise value of αc is hard to determine because of
+the long time scales needed for grokking close to αc. This is clearly seen on Fig.4. AdamW appears
+to be more data-efficient than GD, however it is difficult to rule out the possibility that for α ≈ 0.2
+GD requires extremely long time scales to show grokking. The value of αc also depends on how
+the training set is sampled. One can imagine a random sampling or a guided algorithmic choice of
+training examples. The latter will lead to smaller αc.
+4.3
+Dynamics
+In this Section we introduce an empirical measure that quantifies the feature learning for the modular
+addition task. To define such measure we turn to the exact solution (6)- (7). We will utilize the fact
+that periodic weights are peaked in Fourier space, while random weights are not.
+To define the measure of feature learning, we first transform the weights W (1)
+nk to a Fourier space
+with respect to index n. Denote the transformed weights ˜W (1)
+νk . If the weights are periodic, then
+Fourier-transformed weights are localized in ν, i.e. for most values of ν we have ˜W (1)
+νk ≈ 0 except
+for a few values determined by the frequency 2π
+p k. At initialization, when the weights are random
+the Fourier-transformed weights are delocalized, i.e. will take roughly equal values for any ν.
+We introduce a measure of localization known as the inverse participation ratio (IPR). It is routinely
+used in localization physics [3] as well as network theory [10]. We define IPR in terms of the
+normalized Fourier-transformed weights
+IPRr(k) =
+D
+�
+ν=1
+| ˜w(1)
+νk |2r ,
+where
+˜w(1)
+νk =
+˜W (1)
+νk
+��D
+ν=1( ˜W (1)
+νk )2
+,
+(20)
+and r is a parameter traditionally taken to be 2. It follows from the definition that IPR1(k) = 1 for
+any k. Unfortunately, IPRr(k) is defined per neuron. We would like a single measure for all of the
+weights in a given layer. Thus, we introduce the average IPR
+IPRr = 1
+N
+N
+�
+k=1
+IPRr(k) .
+(21)
+Larger values of IPRr indicate that the weights are more periodic, while the smaller values indicate
+that the weights are more random.
+We plot IPR2 as a function of time in Fig. 5. It is clear that there is an upward trend from the very
+beginning of training. Onset of grokking is correlated with the sharp increase of rate of IPR growth.
+5
+Conclusions and discussions
+5.1
+Conclusions
+We have presented a simple architecture that exhibits grokking on a variety of modular arithmetic
+problems. The architecture (4) is simple enough to determine the weights and features that solve
+modular addition problems analytically, leading to complete interpretability of what was learnt by the
+model: the network is learning a δ-function represented by a complete set of trigonometric functions
+with frequencies determined by the base of modular addition; the phases are chosen to ensure that
+waves concentrated on m + n = q mod p interfere constructively.
+As suggested in some literature before, we reiterate that grokking is likely to be intimately connected
+to feature learning. In particular, random feature models such as infinitely-wide neural networks (in
+the NTK regime) do not exhibit grokking, at least on the tasks that involve modular functions. In
+addition, Ref. [7] argued that grokking is due to the competition between encoder and decoder. While
+it is certainly true in their model, in the present case there is no learnable encoder but grokking is still
+present. In our minimal setup, the simplest explanation for grokking is that once training loss reached
+a certain value, the only way to further minimize it is to start learning the right features.
+9
+
+Figure 5: Inverse participation ratio. IPR plotted against the dynamics (under AdamW) of train and
+test accuracy. Empirically, we see 4 regimes: (i) early training when IPR grows linearly and slowly;
+(ii) transition from slow liner growth to fast linear growth. This transition period coincides with
+grokking; (iii) fast linear growth, that starts after 100% test accuracy was reached; (iv) saturation,
+once weights became periodic. The dashed line indicates IPR2 for the exact solution (6)-(7). The gap
+between the two indicates that even in the final solution there is quite a bit of noise leading do some
+mild delocalization in Fourier space. More training and more data helps to reduce the gap.
+5.2
+Discussions
+We close with a discussion of open problems and directions.
+Different modular functions clearly fit into different complexity classes: (i) functions that can be
+learnt easily; (ii) functions that can be learnt with a lot of data and training time; and (iii) functions
+that cannot be learnt at all (at least within the class of architectures we and [11] have considered). It
+would be interesting to (1) define the notion of complexity rigorously as a computabe quantity and
+(2) construct architectures/optimizers that can learn more complex modular functions (or argue that it
+cannot be done).
+A neural network can learn a smooth approximation to complicated modular operations, such as mod-
+ular square root and modular logarithm. It would be interesting to determine if these approximations
+provide any practical gain over known algorithms that perform these operations as well as to enable
+the networks to operate over large numbers.
+The critical amount of data needed for generalization, αc, is likely to be computable as well, and is a
+measure of complexity of a modular function. We would like to have an expression for the absolute
+minimal value of αc (i.e. minimized over all possible ML methods). This value is also an implicit
+function of modulus p, and the modular functions with larger modulus are likely simpler since we find
+empirically that αc is a decreasing function of p. The value of αc further depends on how training set
+is sampled from the entire dataset; the appropriate choice of the sampling method may thus improve
+the data efficiency.
+While grokking happens in a very simple setting described here, adaptive methods and regularization
+improve both speed and data efficiency. It might be possible to characterize these improvements
+quantitatively.
+Modular functions of many variables can be grokked as well and, in some cases, the corresponding
+analytic solution can be constructed. It is possible that the analytic solution can inform a type of
+architecture one should be using, e.g., in applications of deep learning to cryptography.
+10
+
+IPR
+IPR exact
+Tain accuracy
+Test accuracyPresented solutions only work for a single-hidden-layer neural network. To quantify the role of depth,
+we would like to have examples of algorithmic tasks that require a deeper architecture. For instance,
+it is possible that deep convolutional architectures, given an appropriate algorithmic dataset with
+hierarchical structure, would admit a solution in terms of wavelets rather than Fourier components [2].
+In real-world datasets and tasks that require feature learning, it is possible that grokking is happening
+but the jumps in generalization after learning a new feature may be so small that we perceive a
+continuous learning curve. To elucidate this point further, it is important to construct a realistic model
+of datasets and tasks with controllable amount of hierarchical structure. More broadly, it would be
+very interesting to characterize grokking in terms that are not specific to a particular problem or a
+particular model and to establish whether it occurs in more traditional ML settings.
+Given the simplicity of our model (4), loss function (MSE) and optimization algorithm (vanilla GD),
+it is plausible that some aspects of the training dynamics – not just the solution at the end of training
+– can be treated analytically. As the training and test losses show peculiar dynamics, it would be
+interesting to understand the structure of the loss landscape to explain the dynamics, in particular
+what happens at the onset of generalization and why it is so abrupt. Perhaps methods described in
+[12] – where the feature kernel and the neural tangent kernel can be computed analytically throughout
+the training – will take a particularly simple form in this setting.
+There are certainly many other directions that the reader may be interested in exploring.
+Acknowledgments and Disclosure of Funding
+Discussions with N. Ardalani, Y. Bahri, M. Barkeshli, L. Bottou, T. Can, F. Charton, D. Doshi,
+S. Ganguli, P. Glorioso, B. Hanin, T. He, I. Molybog, M. Paul, D. Roberts, A. Saxe, D. Schwab and
+S. Yaida are acknowledged. I am particularly grateful to S. Yaida, D. Roberts, and B. Hanin for
+encouraging, detailed and insightful feedback on the manuscript. A.G.’s work at the University of
+Maryland was supported in part by NSF CAREER Award DMR-2045181, Sloan Foundation and the
+Laboratory for Physical Sciences through the Condensed Matter Theory Center.
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+12
+
+A
+Complex network
+A simpler network that solves modular addition problem can be phrased using complex weights. This
+structure would also be more friendly to physicists. The complex solution takes form
+W (1)
+kn =
+�
+e2πi k
+p n1+iϕ(1)
+k
+e2πi k
+p n2+iϕ(2)
+k
+�
+,
+n = (n1, n2)
+(22)
+W (2)
+qk = e−2πi k
+p q−iϕ(3)
+k ,
+(23)
+We can take quadratic activation function that simply squares the preactivations. The first preactivation
+and activation are given by
+h(1)(n, m) = e2πi k
+p n+iϕ(1)
+k
++ e2πi k
+p m+iϕ(2)
+k ,
+(24)
+z(1)(n, m) = e2πi k
+p 2n+iϕ(1)
+k
++ e2πi k
+p 2m+iϕ(2)
+k
++ 2e2πi k
+p (n+m)+i(ϕ(1)
+k +ϕ(2)
+k ) .
+(25)
+The final activation is given by
+h(2)(n, m)
+=
+N
+�
+k=1
+�
+e2πi k
+p (2n−q)+i(ϕ(1)
+k −ϕ(3)
+k ) + e2πi k
+p (2m−q)+i(ϕ(2)
+k −ϕ(3)
+k )
+(26)
++
+2e2πi k
+p (n+m−q)+i(ϕ(1)
+k +ϕ(2)
+k −ϕ(3)
+k )�
+.
+(27)
+Similarly setting
+ϕ(1)
+k
++ ϕ(2)
+k
+− ϕ(3)
+k
+= 0
+(28)
+yields the constructive interference for the output supported on (n + m − q) = 0 mod p.
+B
+Other activations
+Remarkably, the weights (6)-(7) also solve the modular addition problem for networks (1)-(2) with
+other activation functions. That is, the function
+f(x) =
+1
+D
+√
+N
+W (2)φ
+�
+W (1)x
+�
+(29)
+approximates the δ-function concentrated on the modular addition problem. This also holds for the
+generalizations discussed in the main text. We do not have an analytic proof of this fact, so we
+provide the evidence in Fig. 6.
+C
+Some other modular functions
+We show a few examples of the modular functions for which the exact solutions discussed in the
+main text apply.
+• f(n, m) = n2 + m2 mod p. Full solution is available and gives 100% accuracy. The first
+layer weights are given by
+W (1)
+kn =
+�
+�cos
+�
+2π k
+pn2
+1 + ϕ(1)
+k
+�
+cos
+�
+2π k
+pn2
+2 + ϕ(2)
+k
+�
+�
+� ,
+n = (n1, n2) ,
+(30)
+while the second layer weights remain unmodified.
+• f(n, m) = (n + m)2 mod p. The weights in the first layer are unmodified, while the
+weights in the second layer are given by
+W (2)
+qk = cos
+�
+−2π k
+pq
+1
+2 − ϕ(3)
+k
+�
+.
+(31)
+Note that q
+1
+2 must be understood in the modular sense, that is r = q
+1
+2 is a solution to
+r2 = q mod p.
+13
+
+Figure 6: Accuracy for various activation functions. Test accuracy vs. width for different activation
+functions for f(n, m) = n+m mod p. The weights are given by (6)-(7). GELU activation eventually
+reaches 100% accuracy, but at very large width.
+• f(n, m) = nm. We do not have an analytic solution. The activations are presented in Fig. 9
+• f(n, m) = n2 + m2 + nm mod p. We do not have an analytic solution. This generalization
+on this function never reaches 100% unless most of the data is utilized, α > 0.95. See the
+learning curve in Fig. 10. Note that although generalization accuracy is very high: ≈ 97%,
+there is a large gap between train and test loss. This is to be contrasted with Fig. 2, where
+the gap disappears over time.
+• f(n, m) = n3 + nm2 + m. We do not have an analytic solution. The generalization never
+rises above 1%. See the learning curve in Fig. 10.
+We show the corresponding activations on Fig. 7 - Fig. 9
+14
+
+Quadratic
+Quartic
+Absolute value
+ReLU
+GELUFigure 7: Top: Preactivations h(1)
+k
+and h(2)
+q
+found by the AdamW for f(n, m) = (n + m)2 mod p.
+Note that h(1)
+k
+is the same as for f(n, m) = (n + m) mod p as expected. Bottom: Analytic solution
+for the same function. Note that since square root is not invertible – because it has two branches
+– the accuracy of analytic solution is ≈ 51%. It can be clearly seen in the form of h(2)
+q : there are
+4 activation lines in the top plots and only 2 in the bottom. Each pair corresponds to a branch of
+square root. The noisy preactivations h(2)
+q
+correspond to the values of q that cannot be represented as
+(n + m)2 mod p.
+Figure 8: Top: Preactivations h(1)
+k
+and h(2)
+q
+found by the AdamW for f(n, m) = n2 + m2 mod p.
+Bottom: Analytic solution for the same function. Both solutions have 100% accuracy.
+15
+
+888
+8888888888
+880036:
+..
+.Figure 9: Preactivations h(1)
+k
+and h(2)
+q
+found by the AdamW for f(n, m) = nm mod p.
+Figure 10: The learning curves for f(n, m) = n2 + m2 + nm mod p and f(n, m) = n3 + nm2 +
+m mod p at α = 0.73 and α = 0.9 correspondingly. Note the gap between train and test loss in the
+former case. Although test accuracy is almost 100%, it is clear that the network did not grok all the
+right features.
+16
+
+Tain loss
+Test lossTain accuracy
+Test accuracyTain loss
+Test lossTrain accuracy
+Test accuracy..

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@@ -0,0 +1,2891 @@
+arXiv:2301.01295v1  [math.CV]  3 Jan 2023
+NEVANLINNA THEORY ON COMPLETE K¨AHLER
+MANIFOLDS WITH NON-NEGATIVE RICCI CURVATURE
+XIANJING DONG
+Abstract. This paper considers the equiv-distribution of meromorphic
+mappings from a complete K¨ahler manifold with non-negative Ricci cur-
+vature into a complex projective manifold. When the domain manifold
+is of maximal volume growth, one establishes a second main theorem in
+Nevanlinna theory with a refined error term. As an important result, we
+prove a sharp defect relation. Furthermore, we apply our main theorems
+to the problems on the propagation of algebraic dependence, then finally
+we obtain some unicity results for dominant meromorphic mappings.
+Contents
+1.
+Introduction
+3
+1.1.
+Motivation
+3
+1.2.
+Main results
+4
+2.
+Preliminaries
+5
+2.1.
+Curvatures in differential geometry
+5
+2.2.
+Comparison theorems on Riemannian manifolds
+7
+2.3.
+Existence of positive global Green functions
+9
+2.4.
+Positivity, ampleness and bigness for Q-line bundles
+10
+2.5.
+Poincar´e-Lelong formula and Jensen-Dynkin formula
+11
+3.
+Nevanlinna theory on complete K¨ahler manifolds
+13
+3.1.
+Nevanlinna’s functions
+13
+3.2.
+Calculus lemma and logarithmic derivative lemma
+15
+3.3.
+Second main theorem and defect relation
+25
+2010 Mathematics Subject Classification. 32H30; 32H04.
+Key words and phrases. Nevanlinna theory; second main theorem; defect relation; al-
+gebraic dependence; unicity theorem.
+
+2
+X.-J. DONG
+3.4.
+The case for singular divisors
+29
+4.
+Application to algebraic dependence problems
+31
+4.1.
+Consequences of Theorems 3.1 and 3.10
+31
+4.2.
+Propagation of algebraic dependence
+32
+4.3.
+Unicity theorems
+38
+4.4.
+Targets are compact Riemann surfaces and Pn(C)
+39
+References
+42
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+3
+1. Introduction
+1.1. Motivation.
+In 1972, Carlson-Griffiths [9] devised an equiv-distribution theory of holo-
+morphic mappings from Cm into a complex projective manifold. This theory
+is undoubtedly a great progress in the study of Nevanlinna theory [37], which
+was further generalized by Griffiths-King [22] to the affine algebraic varieties.
+Many famous scholars made efforts towards an extension of Carlson-Griffiths
+theory. For example, W. Stoll [45, 46] extended it to the parabolic manifolds,
+Lang-Cherry [33] promoted it onto a covering space of Cm. More extensions
+and developments refer to J. Noguchi [34, 35], M. Ru [38], B. Shiffman [39],
+F. Sakai [40, 41], Wong-Stoll [49], Wong-Wong [50] and Yang-Ru [51], etc.
+The study of Nevanlinna theory on K¨ahler manifolds via Brownian motion
+(initialized by T. K. Carne [12]) can be traced back to the work of A. Atsuji
+[1] in 1995. Later, Atsuji wrote a series of papers (cf. [2, 3, 4, 5]) to study the
+value distribution of meromorphic functions on a complete K¨ahler manifold.
+His excellent contribution to Nevanlinna theory seems to be the second main
+theorem of meromorphic functions on a complete K¨ahler manifold with non-
+positive sectional curvature. Along the line of Atsuji, X. J. Dong [17] further
+extended the notion for Nevanlinna’s functions and generalized the Carlson-
+Griffiths theory from Cm to the complete K¨ahler manifolds with non-positive
+sectional curvature. More details about the Nevanlinna theory via Brownian
+motion refer to Dong-He-Ru [16], Dong-Yang [18] and X. J. Dong [19], etc.
+Although the Nevanlinna theory on K¨ahler manifolds has been studied for
+a long time, we know little about this theory if domain is not non-positively
+curved, particularly, if domain is of non-negative Ricci curvature. In fact, it
+is a long-standing problem. Refer to [42], we see that such class of manifolds
+are in abundance. Motivated by those, we shall focus on the Carlson-Griffith
+theory on complete K¨ahler manifolds with non-negative Ricci curvature, and
+we would like to derive a sharp defect relation in Nevanlinna theory.
+Another motivation of this paper is the propagation problem of algebraic
+dependence of meromorphic mappings, which was first studied by L. Smiley
+[43] in 1979. This problem has aroused widespread concern among scholars,
+referred to Y. Aihara [6, 7, 8], Dulock-Ru [14], S. Drouilhet [15], H. Fujimote
+[20], S. Ji [24, 25] and W. Stoll [47], etc. So far, the best result for propaga-
+tion theorems may be due to Y. Aihara [8] who obtained a beautiful criteria
+for the propagation of algebraic dependence of dominant meromorphic map-
+pings from an analytic finite covering space of Cm into a complex projective
+manifold. Aihara’s criteria improved the criteria of W. Stoll [47]. In fact, he
+used the estimate for the ramification term obtained by J. Noguchi [34]. In
+Aihara’s criteria, however, there still exist some unnatural restrictions, such
+as “fibers separated by mappings”, etc.
+
+4
+X.-J. DONG
+In this paper, we shall apply our second main theorem to the propagation
+problems. The purpose is to prove some propagation theorems for algebraic
+dependence of dominant meromorphic mappings on a complete K¨ahler man-
+ifold. We not only generalize Aihara’s criteria to complete K¨ahler manifolds,
+but also remove those restrictions such as “ramification estimate” and “fibers
+separated by mappings” appeared in Aihara’s criteria.
+1.2. Main results.
+In Atsuji [5] and Dong [17], the Nevanlinna’s functions (i.e., characteristic
+function, proximity function and counting function) are defined on a geodesic
+ball of a complete K¨ahler manifold. Thus, in order to establish a second main
+theorem, they must estimate the local Green functions for the geodesic balls.
+A reasonable estimate was obtained under a non-positive sectional curvature
+condition. But, the estimate is not so accurate enough. That’s why a growth
+condition was assumed in their defect relations. Not only that, their methods
+fail to the complete K¨ahler manifolds with non-negative Ricci curvature. It’s
+because that we don’t seem to obtain a suitable estimate (in terms of integral
+forms) on local Green functions for the geodesic balls. It is a great difficulty
+faced with in the study of Nevanlinna theory on a complete K¨ahler manifold
+with non-negative Ricci curvature. To overcome this difficulty, our strategy
+is to use the global Green function instead of local Green functions. Roughly
+speaking, we apply the global Green function to define the relatively compact
+r-domains which exhaust the manifold, and we then estimate the local Green
+functions for those domains and the harmonic measures for their boundaries.
+Define the Nevanlinna’s functions on r-domains and use the approach in this
+paper, we establish the Carlson-Griffiths theory on such class of manifolds.
+Let M be a non-compact complete K¨ahler manifold with maximal volume
+growth and non-negative Ricci curvature, and let X be a complex projective
+manifold of complex dimension not greater than that of M. A meromorphic
+mapping f : M → X is said to be differentiably non-degenerate, if the rank
+of differential df equals dimC X at some point in M \I(f), where I(f) is the
+indeterminacy set of f. In this case, we also call f a dominant meromorphic
+mapping. Next, let us state the main results in this paper. Some notations
+will be introduced later.
+We obtain the following second main theorem:
+Theorem I. Let D ∈ |L| be a divisor of simple normal crossing type, where
+L is a positive line bundle over X. Let f : M → X be a differentiably non-
+degenerate meromorphic mapping. Then for any δ > 0, there exists a subset
+Eδ ⊆ (0, ∞) with finite Lebesgue measure such that
+Tf(r, L) + Tf(r, KX) + T(r, R) ≤ N f(r, D) + O
+�
+log+ Tf(r, L) + δ log r
+�
+holds for all r > 0 outside Eδ.
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+5
+Later, we will note that Tf(r, L) ≥ O(log r) as r → ∞ for any nonconstant
+meromorphic mapping, which means that the error term δ log r in the above
+Theorem I is refined. This yields a defect relation:
+¯δf(D) ≤
+�c1(K∗
+X)
+c1(L)
+�
+− lim inf
+r→∞
+T(r, R)
+Tf(r, L) ≤
+�c1(K∗
+X)
+c1(L)
+�
+.
+We give an application of Theorem I to the propagation problems of alge-
+braic dependence. Let S = S1 ∪ · · · ∪ Sq, where S1, · · · , Sq are hypersurfaces
+of M such that dimC Si ∩Sj ≤ dimC M −2 for i ̸= j. Let D = D1 +· · · +Dq,
+where D1, · · · , Dq ∈ |L| such that D has simple normal crossings, here L is
+an ample line bundle over X. Now, given l dominant meromorphic mappings
+f1, · · · , fl : M → X. Assume that there are integers k1, · · · , kq (may be +∞)
+such that
+Sj = Suppkjf ∗
+i Dj;
+1 ≤ i ≤ l,
+1 ≤ j ≤ q.
+Set k0 = max{k1, · · · , kq}. Given an indecomposable hypersurface Σ of M1×
+· · · × Ml. Moreover, define a Q-line bundle L0 ∈ Pic(X) ⊗ Q by
+L0 =
+�
+q
+�
+j=1
+kj
+kj + 1
+�
+L ⊗
+�
+− ˜γlk0
+k0 + 1F0
+�
+,
+where F0 is some big line bundle over X and ˜γ is a positive rational number
+depending only on Σ and F0.
+Employing Theorem I, we obtain some propagation theorems of algebraic
+dependence in this paper. For example, we show that
+Theorem II. Let f1, · · · , fl : M → X be dominant meromorphic mappings
+given as above. Assume that f1, · · · , fl are Σ-related on S. If L0 ⊗ KX is
+big, then f1, · · · , fl are Σ-related on M.
+The propagation theorems of algebraic dependence can apply to the unic-
+ity problems. For example, Theorem II derives a unicity theorem:
+Theorem III. Let f1, f2 : M → P1(C) be nonconstant holomorphic map-
+pings. Let a1, · · · , aq be distinct points in P1(C). We have
+(a) Assume that Suppf ∗
+1 aj = Suppf ∗
+2 aj for all j. If q ≥ 5, then f1 ≡ f2;
+(b) Assume that Supp1f ∗
+1 aj = Supp1f ∗
+2 aj for all j. If q ≥ 7, then f1 ≡ f2.
+2. Preliminaries
+2.1. Curvatures in differential geometry.
+In order to simplify formulas, the Einstein’s convenience is used.
+
+6
+X.-J. DONG
+2.1.1. Curvatures of Riemannian manifolds.
+Let (M, g) be a Riemannian manifold with Lapalce-Beltrami operator ∆.
+Let ∇ be the Levi-Civita connection of g on M. Recall that the Riemannian
+curvature tensor is defined by
+R = Rijkldxi ⊗ dxj ⊗ dxk ⊗ dxl,
+where ∂j = ∂/∂xj and Rijkl = glpRp
+ijk with
+Γk
+ij = 1
+2gkl (∂jgil + ∂igjl − ∂lgij) ,
+Rl
+ijk = ∂kΓl
+ji − ∂jΓl
+ki + Γl
+kpΓp
+ji − Γl
+jpΓp
+ki.
+For any point x ∈ M and every tangent 2-plane σ to M at x, the sectional
+curvature of σ is defined by
+K(X, Y ) = R(X, Y, Y, X)
+∥X ∧ Y ∥2
+,
+where X, Y ∈ TxM is a basis of σ. Define the Ricci curvature tensor by
+Ric(X, Y ) = R(X, ej, ej, Y ),
+where {e1, · · · , en} is an orthonormal basis of TxM. We can write Ric as
+Ric = Rijdxi ⊗ dxj,
+where
+Rij = Rp
+ipj.
+For 0 ̸= X ∈ TxM, the Ricci curvature at x in the direction X is defined by
+Ric(X, X)/∥X∥2. Moreover, the scalar curvature of M is defined by
+s = gijRij,
+which is the trace of Ricci curvature tensor.
+2.1.2. Curvatures of K¨ahler manifolds.
+Now, we turn to Hermitian manifolds. Let (M, h) be a Hermitian manifold
+with Hermitian connection ˜∇. Note that M can be regarded as a Riemannian
+manifold with Riemannian metric g = ℜh, where g is extended linearly over
+C. We have the Levi-Civita connection ∇ on M as a Riemannian manifold.
+Also, we extend ∇ linearly over C to TM = TM ⊗C. In general, ˜∇ ̸= ∇ since
+the torsion tensor of ˜∇ may not vanish for the general Hermitian manifolds.
+Therefore, the Laplacian ˜∆ of ˜∇ does not coincide with the Laplace-Beltrami
+operator ∆ of ∇. However, the case when ˜∇ = ∇ happens provided that M
+is a K¨ahler manifold. Consequently,
+∆ = ˜∆ = 2hi¯j
+∂2
+∂zi∂¯zj
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+7
+acting on a C 2-class function when M is K¨ahlerian, where (hi¯j) is the inverse
+of metric matrix (hi¯j) of h.
+Suppose that M is a K¨ahler manifold with K¨ahler metric h. The K¨ahlerness
+of h means that ⟨JX, JY ⟩ = ⟨X, Y ⟩ and ∇X(JY ) = J∇XY for X, Y ∈ TM,
+where J is the canonical complex structure of M. Extending J and R linearly
+over C to TM = T 1,0
+M ⊕ T 0,1
+M . Let X, Y ∈ T 1,0
+M,x. The holomorphic bisectional
+curvature H(X, Y ) of M at x in the holomorphic directions X, Y is defined
+by
+H(X, Y ) = R(X, X, Y, Y )
+∥X ∧ Y ∥2
+.
+When X = Y, we call H(X) = H(X, X) the holomorphic sectional curvature
+of M at x in the holomorphic direction X. The Ricci curvature tensor RicC
+of M is defined by
+RicC(X, Y ) = R(X, Y , ej, ej),
+where {e1, · · · , em} is an orthonormal basis of T 1,0
+M,x. Applying the K¨ahlerness
+of M, RicC can be computed in such way: RicC = Ri¯jdzi ⊗ d¯zj, where
+Ri¯j = −
+∂2
+∂zi∂¯zj log det(hs¯t).
+Naturally, it defines the following Ricci form
+R = −ddc log det(hs¯t) =
+√−1
+2π Ri¯jdzi ∧ d¯zj,
+where
+d = ∂ + ∂,
+dc =
+√−1
+4π (∂ − ∂) so that ddc =
+√−1
+2π ∂∂.
+A well-known theorem by S. S. Chern asserts that R defines a cohomology
+class in the de Rham cohomology group H2
+DR(M, R). We can define the Ricci
+curvature at x in the holomorphic direction X by RicC(X, X)/∥X∥2. Indeed,
+we have the scalar curvature sC of M defined by
+(1)
+sC = hi¯jRi¯j = −1
+2∆ log det(hs¯t).
+A comparison gives that (cf. e.g., [52])
+(2)
+s = 2sC.
+2.2. Comparison theorems on Riemannian manifolds.
+Let M be a Riemannian manifold. Fix a point o ∈ M, define
+ρ(x) = dist(o, x),
+∀x ∈ M.
+
+8
+X.-J. DONG
+It is well-known that ρ is Lipschitz-continuous and thus differentiable almost
+everywhere. Let Cut(o) be the cut locus of o, then Cut(o) has measure zero
+and ρ is smooth on M \ Cut(o).
+A space form is a simply-connected complete Riemannian manifold with
+constant sectional curvature. Let ˜
+M be a space form and ˜ρ(˜x) be the distance
+function of ˜x ∈ ˜
+M from a fixed point ˜o ∈ ˜
+M. Let ˜∆ be the Laplace-Beltrami
+operator on
+˜
+M. The Laplacian comparison theorem (cf., e.g., [44]) states
+that
+Theorem 2.1 (Laplacian comparison theorem). Let M be an n-dimensional
+complete Riemannian manifold with Ric ≥ −(n − 1)K, where K is a non-
+negative constant.
+Let
+˜
+M be an n-dimensional space form with sectional
+curvature −K. Assume that x ∈ M and ˜x ∈ ˜
+M such that ρ(x) = ˜ρ(˜x). If x
+is not a cut point of ρ, then
+∆ρ(x) ≤ ˜∆ρ(˜x).
+By finding the Jacobian field along a minimal geodesic, one can compute
+˜∆˜ρ. It follows from Theorem 2.1 that
+Corollary 2.2. Let M be an n-dimensional complete Riemannian manifold
+with Ric ≥ −(n − 1)K, where K is a non-negative constant. Then
+∆ρ ≤ (n − 1)
+�√
+K + 1
+ρ
+�
+on M \ Cut(o). In particular, if Ric(M) ≥ 0, then
+∆ρ ≤ n − 1
+ρ
+in the sense of distributions on M.
+We also state the following Bishop’s volume comparison theorem (cf. e.g.,
+[44]):
+Theorem 2.3 (Volume comparison theorem). Let M be an n-dimensional
+complete Riemannian manifold with Ric ≥ (n−1)K, where K is a constant.
+Let ˜
+M be an n-dimensional space form with sectional curvature K. Then for
+r > 0, Vx(r)/V (K, r) is non-increasing in r and
+Vx(r) ≤ V (K, r),
+where Vx(r) is the volume of geodesic ball centered at x with radius r in M,
+and V (K, r) is the volume of geodesic ball with radius r in ˜
+M.
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+9
+2.3. Existence of positive global Green functions.
+Let M be an n-dimensional non-compact complete Riemannian manifold,
+whose Laplace-Beltrami operator is ∆. Set ρx(y) = dist(x, y). In this paper,
+a positive global Green function (if it exists) for M means a Green function
+G(x, y) of ∆/2 on M ×M, which is a smooth function on M ×M \diag(M ×
+M) satisfying that (cf. e.g., [44])
+1◦ G(x, y) ≥ 0;
+2◦ G(x, y) = G(y, x);
+3◦ ∆yG(x, y) = 0 for y ̸= x;
+4◦ Fix any x, when y → x
+G(x, y) ∼ C2 log ρ−1
+x (y),
+n = 2;
+G(x, y) ∼ Cnρ2−n
+x
+(y),
+n > 2,
+where Cn (n ≥ 2) is a positive constant such that
+−1
+2∆yG(x, y) = δx(y)
+in the sense of distributions. Note that G(x, y) with n = 2 may change sign,
+thus the positivity of G(x, y) in this case should be understood to be outside
+a compact subset of M.
+It is known that there always exists a global Green function for any non-
+compact complete Riemannian manifold M, this fact was confirmed for the
+first time by M. Malgrange [32], while a constructive proof, and best suited
+for applications, which was presented by Li-Tam [28]. Moreover, if M is non-
+parabolic (it admits a positive global Green function), Li-Tam’s construction
+produces a unique minimal positive global Green function for M. In case that
+M has a nonconstant positive harmonic function, Schoen-Yau [44] confirmed
+that M admits a positive global Green function. In particular, they proved
+that
+Theorem 2.4 (Schoen-Yau). Let M be a non-compact complete Riemann-
+ian manifold with lower bounded Ricci curvature. Then M admits a positive
+global Green function, and thus there exists a unique minimal positive global
+Green function for M.
+There are necessary conditions for the existence of a positive global Green
+function for M. Cheng-Yau [13] gave the first result in this direction, which
+involves only the volume growth of M. The first major result for the suffi-
+ciency was due to Li-Yau [31] and N. Varopoulos [48]. Utilizing the estimates
+of the heat kernel, Li-Yau [31] proved the following theorem:
+
+10
+X.-J. DONG
+Theorem 2.5 (Li-Yau). Let M be a non-compact complete Riemannian
+manifold with non-negative Ricci curvature. If
+� ∞
+0
+t
+Vx(t)dt < ∞,
+then there exists a positive global Green function for M. Furthermore, the
+unique minimal positive global Green function G(x, y) for M satisfies that
+C−1
+� ∞
+ρx(y)
+t
+Vx(t)dt ≤ G(x, y) ≤ C
+� ∞
+ρx(y)
+t
+Vx(t)dt
+for x ̸= y, where Vx(t) is the volume of geodesic ball centered at x with radius
+t, and C is a positive constant depending only on the dimension of M.
+2.4. Positivity, ampleness and bigness for Q-line bundles.
+Let M be a compact complex manifold. We recall that a holomorphic line
+bundle L over M is said to be positive, if there exists a Hermitian metric h
+on L such that the Chern form c1(L, h) > 0; L is said to be very ample, if
+L provides a holomorphic embedding of X into a complex projective space,
+namely, there exist holomorphic sections e0, · · · , eN in H0(M, L) (the space
+of all holomorphic sections of L over M) such that the mapping
+[e0 : · · · : eN] : M ֒→ PN(C)
+is a holomorphic embedding. It is said that L is ample if L⊗ν (written as νL
+for short) is very ample whenever ν is a sufficiently large positive integer. A
+known fact asserts that L is ample if and only if L is positive. A holomorphic
+line bundle F over M is said to be big, if
+dim H0(M, νL) ≥ Cνdim M
+for all sufficiently large integers ν > 0 and some constant C > 0.
+We introduce two well-known theorems by K. Kodaira (cf. e.g., [21, 27]).
+Theorem 2.6 (Kodaira). Let F be a holomorphic line bundle over M. Then
+F is big if and only if there exists a singular metric e−ψ on F such that
+ddc[ψ] ≥ δωL
+in the sense of currents for an arbitrary ample line bundle L over M, where
+ωL is a Hermitian metric on L and δ is a sufficiently small positive number
+depending on ωL.
+As a matter of convenience, instead of ddc[ψ] ≥ δωL in Theorem 2.6, one
+writes F ≥ δL in short. Indeed, we sometimes write F ⊗L = F +L in short
+for two Q-line bundles F, L.
+Corollary 2.7. Let L be an ample line bundle over M. If F is a big line
+bundle over M, then F − δL is big for any sufficiently small number δ > 0.
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+11
+Theorem 2.8 (Kodaira). Let F be a big line bundle over M, and let L be a
+holomorphic line bundle over M. Then there exists a positive integer µ such
+that µF ⊗ L is big, and
+H0(X, µF ⊗ L) ̸= 0
+for any sufficiently large positive integer µ.
+In general, a Q-line bunde is defined as an element belonging to Pic(M)⊗
+Q, here Pic(M) is the Picard group over M. Let F ∈ Pic(M)⊗Q be a Q-line
+bundle. Then F is said to be positive (resp. ample and big), if νF ∈ Pic(M)
+is positive (resp. ample and big) for some integer ν > 0.
+By Corollary 2.9 and Theorem 2.10, it is trivial to see that
+Corollary 2.9. Let L be an ample Q-line bundle over M. If F is a big
+Q-line bundle over M, then F − δL is big for any sufficiently small positive
+number δ.
+Corollary 2.10. Let F be a big Q-line bundle, and L be a Q-line bundle
+over M. Then there exist positive integers µ, ν such that µF, νL ∈ Pic(M)
+and µF ⊗ νL is big, and
+H0(X, µF ⊗ νL) ̸= 0
+for some sufficiently large positive integers µ, ν.
+2.5. Poincar´e-Lelong formula and Jensen-Dynkin formula.
+Let M be a m-dimensional K¨ahler manifold with Laplace-Beltrami oper-
+ator ∆ associated with the K¨ahler metric h. Let us introduce two important
+formulas as follows.
+2.5.1. Poincar´e-Lelong formula.
+Let (L, hL) be a Hermitian holomorphic line over M. We define the Chern
+form of L associated to hL by
+c1(L, hL) = −ddc log hL.
+If s is the canonical section of L over M with zero divisor D, then s defines
+a positive (1, 1)-current of integration [D] over D by
+[D](ϕ) =
+�
+D
+ϕ
+for any (m − 1, m − 1)-form ϕ with compact support on M.
+Let u be a plurisubharmonic function u on M. If u is of C 2-class, then
+ddcu =
+∂2u
+∂zi∂¯zj
+√−1
+2π dzi ∧ d¯zj ≥ 0,
+
+12
+X.-J. DONG
+by which we mean that the matrix of all the second order partial derivatives
+of u is semi-positive definite (cf. e.g., [36], Section 2.1), i.e.,
+∂2u
+∂zi∂¯zj ξi ¯ξj ≥ 0
+for every complex vector (ξ1, · · · , ξm). If u is not of C 2-class, one can use the
+notation ∂2[u]/∂zi∂¯zj in the sense of (Schwartz) distributions, which defines
+a positive Radon measure, so that
+ddc[u] = ∂2[u]
+∂zi∂¯zj
+√−1
+2π dzi ∧ d¯zj ≥ 0
+which is a positive (1, 1)-current:
+ddc[u](ϕ) =
+�
+M
+ddc[u] ∧ ϕ
+for any (m−1, m−1)-form ϕ with compact support on M. Using the K¨ahler
+property of M, we see that
+∆u = 2hi¯j ∂2[u]
+∂zi∂¯zj ≥ 0
+in the sense of distributions. Thus, u is subharmonic on M as a Riemannian
+manifold. In particular, if f is a nonzero meromorphic function on M, then
+log |f|2 is a plurisubharmonic function and it defines a positive (1, 1)-current
+ddc[log |f|2].
+We state the Poincar´e-Lelong formula (cf. e.g., [9]) as follows:
+Theorem 2.11 (Poincar´e-Lelong formula). Let f be a nonzero meromorphic
+function on M. Then
+ddc �
+log |f|2�
+= [(f = 0)] − [(f = ∞)].
+Corollary 2.12. Let (L, hL) be a Hermitian holomorphic line bundle over
+M. Let s be the canonical section of L over M with zero divisor D. Then
+ddc �
+log ∥s∥2
+hL
+�
+= [D] − c1(L, hL).
+Remark that the K¨ahler condition for M is not necessary in both theorems
+above. In fact, we only need to require that M is a complex manifold.
+2.5.2. Jensen-Dynkin formula.
+Let Ω ⊊ M be a relatively compact domain with piecewise smooth bound-
+ary ∂Ω. Fix a point o ∈ Ω. Let gΩ(o, x) denote the positive Green function
+of ∆/2 for Ω with a pole at o, satisfying Dirichlet boundary condition. Note
+that gΩ(o, x) defines a unique harmonic measure π∂Ω for ∂Ω with respect to
+o, say
+dπ∂Ω(x) = −1
+2
+∂gΩ(o, x)
+∂⃗ν
+dσ∂Ω(x),
+∀x ∈ ∂Ω,
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+13
+where ∂/∂⃗ν is the inward normal derivative on ∂Ω, dσ∂Ω is the Riemannian
+area element of ∂Ω.
+A δ-subharmonic function is defined as the difference of two subharmonic
+functions. We introduce the following Jensen-Dynkin formula that is viewed
+as a generalization of Green-Jensen formula. The formula plays an important
+role in the study of Nevanlinna theory on complex manifolds.
+The Jensen-Dynkin formula (cf. e.g., [17, 18]) reads that
+Theorem 2.13 (Jensen-Dynkin formula). Let u be a δ-subharmonic func-
+tion on M such that u(o) ̸= ∞. Then
+�
+∂Ω
+u(x)dπ∂Ω(x) − f(o) = 1
+2
+�
+Ω
+gΩ(o, x)∆u(x)dv(x),
+where dv is the Riemannian volume element of M.
+We remark that Theorem 2.13 remains true when M is just a Riemannian
+manifold. Recall that the K¨ahler metric form of M is defined by
+α =
+√−1
+π
+hi¯jdzi ∧ d¯zj.
+By Wirtinger’s formula, dv = πmαm/m!. Let u be a plurisubharmonic func-
+tion on M. By the K¨ahler property of h, we see that u is subharmonic and
+∆u = 4mddc[u] ∧ αm−1
+αm
+in the sense of distributions or currents. The following consequence follows
+from the above arguments and Theorem 2.13.
+Corollary 2.14. Let u be a plurisubharmonic function on M such that
+u(o) ̸= ∞. Then
+�
+∂Ω
+udπ∂Ω − f(o) =
+2πm
+(m − 1)!
+�
+Ω
+gΩ(o, x)ddc[u] ∧ αm−1.
+3. Nevanlinna theory on complete K¨ahler manifolds
+3.1. Nevanlinna’s functions.
+Let M be a non-compact complete K¨ahler manifold of complex dimension
+m, with Laplace-Beltrami operator ∆ associated to K¨ahler metric h. Assume
+that M has non-negative Ricci curvature as a Riemannian manifold. Indeed,
+assume that M is of maximal volume growth, i.e.,
+(3)
+lim inf
+r→∞ r−2mV (r) > 0,
+where V (r) is the volume of geodesic ball B(r) centered at a reference point
+o with radius r.
+
+14
+X.-J. DONG
+If m ≥ 2, using Theorem 2.4 or Theorem 2.5, then we see that there exists
+the unique minimal positive global Green function G(x, y) of ∆/2 for M. For
+r > 0, define the r-domain ∆(r) by
+∆(r) =
+�
+x ∈ M : G−1(o, x) < r
+�
+.
+Note that {∆(rn)}∞
+1 exhausts M compactly for a strictly increasing sequence
+{rn}∞
+1 with rn → ∞ as n → ∞. Set
+gr(o, x) = G(o, x) − r−1.
+Evidently, gr(o, x) defines the positive Green function of ∆/2 for ∆(r), with a
+pole at o satisfying Dirichlet boundary condition. Denote by πr the harmonic
+measure for ∂∆(r) with respect to o, which is defined by gr(o, x) as follows:
+(4)
+dπr(x) = −1
+2
+∂gr(o, x)
+∂⃗ν
+dσr(x),
+∀x ∈ ∂∆(r),
+where ∂/∂⃗ν is the inward normal derivative on ∂∆(r), dσr is the Riemannian
+area element of ∂∆(r).
+If m = 1, the curvature assumption implies that M is a parabolic Riemann
+surface and each global Green function for M must change sign. Again, since
+M is of maximal volume growth, we see that M is conformally equivalent to
+C. Equip a conformal metric ds2 = λds2
+0 on M, where λ is a positive smooth
+function and ds2
+0 is the standard Euclidean metric on C. It is noted that the
+Laplacian ∆ in real dimension 2 is conformally invariant, thus it produces a
+global Green function G(x, y) of ∆/2 for M:
+(5)
+G(x, y) = 1
+π log d(x, y),
+where d(x, y) is the Riemannian distance between x, y. In this case, we define
+the r-domain ∆(r) by
+∆(r) = {x ∈ M : d(o, x) < r} .
+It is clear that
+(6)
+gr(o, x) = 1
+π log
+r
+d(o, x)
+gives the Green function of ∆/2 for ∆(r), with a pole at o satisfying Dirichlet
+boundary condition. By (4), the harmonic measure πr for ∂∆(r) with respect
+to o is
+dπr(x) =
+1
+2πrdσr(x),
+∀x ∈ ∂∆(r).
+Let X be a complex projective manifold where we put an ample Hermitian
+line bunlde (L, hL) so that its Chern form c1(L, hL) > 0. Let f : M → X be a
+meromorphic mapping, by which we mean that f is defined by a holomorphic
+mapping f0 : M \ I → X such that the closure G(f0) of the graph G(f0) of
+f0 is an analytic subset of M ×X, and the natural projection π : G(f0) → M
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+15
+is a proper mapping, where I is an analytic subset (called the indeterminacy
+set of f) of M satisfying dimC I ≤ m − 2. Assume that o ̸∈ I. Set
+ef = 2mf ∗c1(L, h) ∧ αm−1
+αm
+= −1
+2∆ log(h ◦ f),
+where
+α =
+√−1
+π
+m
+�
+i,j=1
+hi¯jdzi ∧ d¯zj
+is the K¨ahler form of M. For any analytic hypersuface A of M, we put
+N(r, A) =
+πm
+(m − 1)!
+�
+A∩∆(r)
+gr(o, x)αm−1.
+The Nevanlinna’s functions (characteristic function, proximity function and
+counting function) are defined respectively by
+Tf(r, L) = 1
+2
+�
+∆(r)
+gr(o, x)efdv,
+mf(r, D) =
+�
+∂∆(r)
+log
+1
+∥sD ◦ f∥dπr,
+Nf(r, D) = N(r, f ∗D),
+where sD is the canonical section of L with zero divisor D. Moreover, define
+the simple counting function by
+N f(r, D) = N(r, Suppf ∗D).
+Using Jensen-Dynkin formula and Poincar´e-Lelong formula (cf. Theorems
+2.11 and 2.13), we have the first main theorem (cf. [17] also):
+Theorem 3.1. Let f : M → X be a meromorphic mapping such that f(o) ̸∈
+SuppD. Then
+mf(r, D) + Nf(r, D) = Tf(r, L) + O(1).
+3.2. Calculus lemma and logarithmic derivative lemma.
+3.2.1. Estimate on harmonic measures.
+Let M be a m-dimensional complete Hermitian manifold with non-negative
+Ricci curvature, here m ≥ 2. Fix a point o ∈ M. For a convenience, we denote
+by ρ(x) the Riemannian distance function of x from o, and by V (r), A(r) the
+volume and area of B(r), ∂B(r), respectively, where B(r) is the geodesic ball
+centered at o with radius r. Assume that M is of maximal volume growth.
+Set
+θ(r) = (2m)−1r1−2mA(r).
+
+16
+X.-J. DONG
+The Bishop’s volume comparison theorem (cf. Theorem 2.3) says that there
+exists a number θ > 0, independent of o, such that (cf. [29] also)
+(7)
+θ(r) ց θ,
+r−2mV (r) ց θ
+as r → ∞. By Laplacian comparison theorem (cf. Corollary 2.2), one obtains
+∆ρ(x) ≤ 2m − 1
+ρ
+,
+∆ρ1−2m(x) ≥ 0
+in the sense of distributions. Recall that the notation G(o, x) is the minimal
+positive global Green function of ∆/2 for M with a pole at o. According to
+the estimations of Li-Yau (cf. Theorem 2.5) and (7), there exists a constant
+A(m, θ) > 0 depending only on m, θ such that
+A−1(m, θ)ρ2−2m(x) ≤ G(o, x) ≤ A(m, θ)ρ2−2m(x).
+In further, Colding-Minicozzi II [11] (cf. Li-Tam-Wang [29] also) obtained
+the asymptotic behavior of G(o, x). In fact, they showed that
+Lemma 3.2. Let M be a m-dimensional (m ≥ 2) Hermitian manifold with
+non-negative Ricci curvature. If M has maximal volume growth, then G(o, x)
+satisfies the asymptotic behavior
+(8)
+lim
+x→∞
+2m(m − 1)θG(o, x)
+ρ2−2m(x)
+= 1.
+If m = 1, it was shown by Li-Tam [30] that there also has the asymptotic
+behavior
+lim
+x→∞
+θG(o, x)
+log ρ(x) = 1.
+Next, we give a gradient estimate on G(o, x).
+Theorem 3.3. We have
+(a) If m = 1, then
+∥∇G(o, x)∥ = π−1r−1,
+x ∈ ∂∆(r);
+(b) If m ≥ 2, then there exists a constant Cm > 0 such that
+max
+x∈∂∆(r) ∥∇G(o, x)∥ ≤ Cmr− 2m−1
+2m−2 .
+Proof. If m = 1, then ∆(r) is just the geodesic disc centered at o with radius
+r. It means that ρ(x) = r for x ∈ ∂∆(r). Hence, it follows from (5) that
+∥∇G(o, x)∥ = ∂G(o, x)
+∂⃗ν
+= π−1r−1,
+x ∈ ∂∆(r),
+where ∂/∂⃗ν is the inward normal derivative on ∂∆(r). Thus, (a) holds.
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+17
+Next, we show that (b) holds. Since G(o, x) → 0, ρ2−2m(x) → 0 as x → ∞,
+utilizing L’Hˆospital’s rule, then it yields from (8) that for x ∈ ∂∆(r) (without
+loss of generality, we may assume that x is not a cut point of o):
+lim
+r→∞
+2m(m − 1)θ∂G(o, x)/∂⃗ν
+∂ρ2−2m(x)/∂⃗ν
+= lim
+r→∞
+−2m(m − 1)θ∥∇G(o, x)∥
+(2 − 2m)ρ1−2m(x)∂ρ(x)/∂⃗ν = 1,
+where ∂/∂⃗ν stands for the inward normal derivative on ∂∆(r). On the other
+hand, (8) gives that
+lim
+r→∞
+2m(m − 1)θ
+rρ2−2m(x) = 1,
+x ∈ ∂∆(r).
+Or equivalently,
+lim
+r→∞
+(2m(m − 1)θ)
+2m−1
+2m−2 r− 2m−1
+2m−2
+ρ1−2m(x)
+= 1,
+x ∈ ∂∆(r).
+Hence, we see that ∆(r) turns increasingly to a geodesic ball centered at o,
+as r → ∞. This implies that
+lim
+r→∞
+∂ρ(x)
+∂⃗ν
+= 1,
+x ∈ ∂∆(r).
+Combining the above, we get
+lim
+r→∞
+2m(m − 1)θ∥∇G(o, x)∥
+(2m − 2)ρ1−2m(x)
+= lim
+r→∞
+2m(m − 1)θ∥∇G(o, x)∥
+(2m − 2)(2m(m − 1)θ)
+2m−1
+2m−2 r− 2m−1
+2m−2
+= 1
+for x ∈ ∆(r). Therefore, there exists a constant Cm > 0 such that
+max
+x∈∂∆(r) ∥∇G(o, x)∥ ≤ Cmr− 2m−1
+2m−2 .
+This proves the theorem.
+□
+Corollary 3.4. We have
+(a) If m = 1, then
+dπr =
+1
+2πrdσr;
+(b) If m ≥ 2, then
+dπr ≤ Cm2−1r− 2m−1
+2m−2 dσr.
+In the above, dσr is the Riemannian area element of ∂∆(r).
+
+18
+X.-J. DONG
+Proof. The corollary follows immediately from the following relation
+dπr(x) = 1
+2∥∇G(o, x)∥dσr(x).
+The proof is completed.
+□
+3.2.2. Calculus lemma.
+To establish the desired calculus lemma which plays a key role in deriving
+the second main theorem, we still need a lower bound of gr(o, x) in terms of
+integral forms. Set
+ρr =
+max
+x∈∂∆(r) ρ(x).
+Lemma 3.5. We have
+(a) If m = 1, then
+gr(o, x) = 1
+π
+� r
+ρ(x)
+t−1dt;
+(b) If m ≥ 2, then for any ǫ > 0, there exists ρ0 > 0 such that for all
+sufficiently large r > 0 and x ∈ ∆(r) with ρ(x) > ρ0, we have
+gr(o, x) ≥
+� 1
+mθ − ǫ
+� � ρr
+ρ(x)
+t1−2mdt.
+Proof. When m = 1, it follows immediately from (6) that (a) holds. In what
+follows, we show that (b) holds. In view of (8), we obtain
+G(o, x) =
+ρ2−2m(x)
+2m(m − 1)θ + o
+� ρ2−2m(x)
+2m(m − 1)θ
+�
+as x → ∞. Note that ∂∆(r) is a compact set. Using the definition of ∂∆(r),
+the above equality implies that
+1
+r =
+ρ2−2m
+r
+2m(m − 1)θ + o
+�
+ρ2−2m
+r
+2m(m − 1)θ
+�
+as r → ∞. Thus, for any ǫ0 > 0, there exists ρ0 > 0 such that for sufficiently
+large r > 0 and x ∈ ∆(r) with ρ(x) > ρ0
+gr(o, x) = G(o, x) − r−1
+≥
+�
+1
+2m(m − 1)θ − ǫ0
+� �
+ρ2−2m(x) − ρ2−2m
+r
+�
+≥
+� 1
+mθ − (2m − 2)ǫ0
+� � ρr
+ρ(x)
+t1−2mdt
+=
+� 1
+mθ − ǫ
+� � ρr
+ρ(x)
+t1−2mdt,
+where ǫ = (2m − 2)ǫ0. This completes the proof.
+□
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+19
+Now, we prove the following so-called calculus lemma:
+Theorem 3.6. Let k ≥ 0 be a locally integrable function on M. Assume that
+k is locally bounded at o. Then for any δ > 0, there exist a positive constant
+C and a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure such that
+(a) If m = 1, then
+�
+∂∆(r)
+k(x)dπr(x) ≤ Crδ
+� �
+∆(r)
+gr(o, x)k(x)dv(x)
+�(1+δ)2
+holds for all r > 0 outside Eδ;
+(b) If m ≥ 2, then
+�
+∂∆(r)
+k(x)dπr(x) ≤ Cr
+2m−1
+2m−2 δ
+� �
+∆(r)
+gr(o, x)k(x)dv(x)
+�(1+δ)2
+holds for all r > 0 outside Eδ.
+Proof. If m = 1, then ∆(r) is the geodesic disc centered at o with radius r.
+It yields from (6) that
+�
+∆(r)
+gr(o, x)k(x)dv(x) =
+� r
+0
+dt
+�
+∂∆(t)
+gr(o, x)k(x)dσt(x)
+= 1
+π
+� r
+0
+� � r
+t
+s−1ds
+�
+∂∆(t)
+k(x)dσt(x)
+�
+dt.
+Set
+Γ(r) =
+� r
+0
+� � r
+t
+s−1ds
+�
+∂∆(t)
+k(x)dσt(x)
+�
+dt.
+Then
+dΓ(r)
+dr
+= r−1
+� r
+0
+� �
+∂∆(t)
+k(x)dσt(x)
+�
+dt,
+which yields from Corollary 3.4 that
+(9)
+d
+dr
+�
+rΓ′(r)
+�
+=
+�
+∂∆(r)
+k(x)dσr(x) = 2πr
+�
+∂∆(r)
+k(x)dπr(x).
+On the other hand, applying Borel lemma to (rΓ′)′ twice, then for any δ > 0,
+there exists a subset Eδ ⊆ (0, ∞) with finite Lebesque measure such that
+(10)
+d
+dr
+�
+rΓ′(r)
+�
+≤ r1+δΓ(1+δ)2(r)
+holds for all r > 0 outside Eδ. Combining (9) with (10), we get
+�
+��∆(r)
+k(x)dπr(x) ≤
+1
+2πrr1+δΓ(1+δ)2(r).
+By the definition of Γ, we have (a) holds.
+
+20
+X.-J. DONG
+Next, we show that (b) holds. By Lemma 3.5 and (8), for any ǫ > 0, there
+is a sufficiently large number r0 > 0 such that
+gr(o, x) ≥
+� 1
+mθ − ǫ
+� � ρr
+ρt
+s1−2mds
+holds for all x ∈ ∂∆(t) with r0 < t ≤ r. Thus,
+�
+∆(r)
+gr(o, x)k(x)dv(x)
+(11)
+=
+� r
+r0
+dt
+�
+∂∆(t)
+gr(o, x)k(x)dσt(x) + O(1)
+≥ C(m, ǫ)
+� r
+r0
+� � ρr
+ρt
+s1−2mds
+�
+∂∆(t)
+k(x)dσt(x)
+�
+dt,
+where
+C(m, ǫ) =
+1
+mθ − ǫ.
+Set
+Λ(r) =
+� r
+r0
+� � ρr
+ρt
+s1−2mds
+�
+∂∆(t)
+k(x)dσt(x)
+�
+dt.
+A direct computation leads to
+dΛ(r)
+dr
+= ρ1−2m
+r
+dρr
+dr
+� r
+r0
+� �
+∂∆(t)
+k(x)dσt(x)
+�
+dt.
+In further, we have
+(12)
+d
+dr
+�ρ2m−1
+r
+Λ′(r)
+ρ′r
+�
+=
+�
+∂∆(r)
+k(x)dσr(x).
+Since
+dσr ≥ 2C−1
+m r
+2m−1
+2m−2 dπr
+due to Corollary 3.4, then it follows from (12) that
+(13)
+�
+∂∆(r)
+k(x)dπr(x) ≤ Cm2−1r− 2m−1
+2m−2 d
+dr
+�ρ2m−1
+r
+Λ′(r)
+ρ′r
+�
+.
+Using Borel lemma twice, for any δ > 0, we have
+(14)
+d
+dr
+�ρ2m−1
+r
+Λ′(r)
+ρ′r
+�
+≤ ρ(2m−1)(1+δ)
+r
+(ρ′r)1+δ
+Λ(1+δ)2(r)
+holds for all r > 0 outside a subset Fδ ⊆ (0, ∞) with finite Lebesque measure.
+Combining (13) with (14), we get
+�
+∂∆(r)
+k(x)dπr(x) ≤ Cm2−1r− 2m−1
+2m−2 ρ(2m−1)(1+δ)
+r
+(ρ′r)1+δ
+Λ(1+δ)2(r).
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+21
+Since (8) implies that
+ρr ∼ (2m(m − 1)θ)−
+1
+2m−2 r
+1
+2m−2 ,
+ρ′
+r ∼ 1
+as r → ∞, then for r0 > 0 sufficiently large, we have (0 < δ < 1)
+�
+∂∆(r)
+k(x)dπr(x)
+≤
+Cm2−1r− 2m−1
+2m−2
+�
+2 (2m(m − 1)θ)−
+1
+2m−2 r
+1
+2m−2
+�(2m−1)(1+δ)
+(2−1)2
+Λ(1+δ)2(r)
+≤ Cm24m−1 (2m(m − 1)θ)− 2m−1
+2m−2 r
+2m−1
+2m−2 δΛ(1+δ)2(r)
+holds for all r > r0. Hence, by this with (11)
+�
+∂∆(r)
+k(x)dπr(x)
+≤ Cm24m−1 (2m(m − 1)θ)− 2m−1
+2m−2 r
+2m−1
+2m−2 δ
+C(1+δ)2(m, ǫ)
+��
+∆(r)
+gr(o, x)k(x)dv(x)
+�(1+δ)2
+≤ Cm24m−1 (2m(m − 1)θ)− 2m−1
+2m−2 r
+2m−1
+2m−2 δ
+(2−1m−1θ−1)4
+��
+∆(r)
+gr(o, x)k(x)dv(x)
+�(1+δ)2
+= Cr
+2m−1
+2m−2 δ
+��
+∆(r)
+gr(o, x)k(x)dv(x)
+�(1+δ)2
+holds for all r > 0 outside Eδ = Fδ ∪ (0, r0], where
+C = Cm24m+3m4θ4 (2m(m − 1)θ)− 2m−1
+2m−2 .
+This shows that (b) holds. The proof is completed.
+□
+3.2.3. Logarithmic derivative lemma.
+To establish the second main theorem, we still need to prove a logarithmic
+derivative lemma. Let ∇ be the gradient operator on M associated with the
+K¨ahler metric h. Let ψ be a meromorphic function on M. The norm of the
+gradient of ψ is defined by
+∥∇ψ∥2 = 2
+m
+�
+i,j=1
+hij ∂ψ
+∂zi
+∂ψ
+∂zj ,
+where (hij) is the inverse of (hij). Define
+T(r, ψ) = m(r, ψ) + N(r, ψ),
+
+22
+X.-J. DONG
+where
+m(r, ψ) =
+�
+∂∆(r)
+log+ |ψ|dπr,
+N(r, ψ) =
+πm
+(m − 1)!
+�
+ψ∗∞∩∆(r)
+gr(o, x)αm−1.
+It is trivial to show that
+T
+�
+r,
+1
+ψ − ζ
+�
+= T(r, ψ) + O(1).
+On P1(C), take a singular metric
+Ψ =
+1
+|ζ|2(1 + log2 |ζ|)
+√−1
+4π2 dζ ∧ d¯ζ,
+which gives that
+�
+P1(C)
+Ψ = 1.
+Lemma 3.7. We have
+1
+4π
+�
+∆(r)
+gr(o, x)
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dv ≤ T(r, ψ) + O(1).
+Proof. Since
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|) = 4mπψ∗Ψ ∧ αm−1
+αm
+,
+then it yields from Fubini’s theorem that
+1
+4π
+�
+∆(r)
+gr(o, x)
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dv
+= m
+�
+∆(r)
+gr(o, x)ψ∗Ψ ∧ αm−1
+αm
+dv
+=
+πm
+(m − 1)!
+�
+P1(C)
+Ψ(ζ)
+�
+ψ−1(ζ)∩∆(r)
+gr(o, x)αm−1
+=
+�
+P1(C)
+N
+�
+r, 1/(ψ − ζ)
+�
+Ψ(ζ)
+≤
+�
+P1(C)
+�
+T(r, ψ) + O(1)
+�
+Ψ
+= T(r, ψ) + O(1).
+□
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+23
+Lemma 3.8. Assume that ψ ̸≡ 0. Then for any δ > 0, there exists a subset
+Eδ ⊆ (0, ∞) with finite Lebesgue measure such that
+(a) If m = 1, then
+�
+∂∆(r)
+log+
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr ≤ (1 + δ)2 log+ T(r, ψ) + δ log r
+holds for all r > 0 outside Eδ;
+(b) If m ≥ 2, then
+�
+∂∆(r)
+log+
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr ≤ (1 + δ)2 log+ T(r, ψ) + 2m − 1
+2m − 2δ log r
+holds for all r > 0 outside Eδ.
+Proof. Using the concavity of log, we have
+�
+∂∆(r)
+log+
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr
+≤ log
+�
+∂∆(r)
+�
+1 +
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)
+�
+dπr
+≤ log+
+�
+∂∆(r)
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr + O(1).
+By Theorem 3.6, for m = 1
+log+
+�
+∂∆(r)
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr
+≤ (1 + δ)2 log+
+�
+∆(r)
+gr(o, x)
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dv + δ log r + O(1)
+≤ (1 + δ)2 log+ T(r, ψ) + δ log r + O(1)
+for all r > 0 outside Eδ. Similarly, for m ≥ 2
+log+
+�
+∂∆(r)
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr
+≤ (1 + δ)2 log+
+�
+∆(r)
+gr(o, x)
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dv + 2m − 1
+2m − 2δ log r + O(1)
+≤ (1 + δ)2 log+ T(r, ψ) + 2m − 1
+2m − 2δ log r + O(1)
+for all r > 0 outside Eδ. Adjust Eδ so that O(1) is absorbed.
+□
+Define
+m
+�
+r, ∥∇ψ∥
+|ψ|
+�
+=
+�
+∂∆(r)
+log+ ∥∇ψ∥
+|ψ| dπr.
+
+24
+X.-J. DONG
+Theorem 3.9. Let ψ be a nonconstant meromorphic function on M. Then
+for any δ > 0, there exist a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure
+such that
+(a) If m = 1, then
+m
+�
+r, ∥∇ψ∥
+|ψ|
+�
+≤ 2 + (1 + δ)2
+2
+log+ T(r, ψ) + δ
+2 log r
+holds for all r > 0 outside Eδ;
+(b) If m ≥ 2, then
+m
+�
+r, ∥∇ψ∥
+|ψ|
+�
+≤ 2 + (1 + δ)2
+2
+log+ T(r, ψ) + 2m − 1
+4m − 4δ log r
+holds for all r > 0 outside Eδ,
+Proof. Note that
+m
+�
+r, ∥∇ψ∥
+|ψ|
+�
+≤ 1
+2
+�
+∂∆(r)
+log+
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr
++1
+2
+�
+∂∆(r)
+log
+�
+1 + log2 |ψ|
+�
+dπr
+= 1
+2
+�
+∂∆(r)
+log+
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr
++1
+2
+�
+∂∆(r)
+log
+�
+1 +
+�
+log+ |ψ| + log+ 1
+|ψ|
+�2�
+dπr
+≤ 1
+2
+�
+∂∆(r)
+log+
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr
++ log
+�
+∂∆(r)
+�
+log+ |ψ| + log+ 1
+|ψ|
+�
+dπr + O(1)
+≤ 1
+2
+�
+∂∆(r)
+log+
+∥∇ψ∥2
+|ψ|2(1 + log2 |ψ|)dπr + log+ T(r, ψ) + O(1).
+By Lemma 3.8, for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite
+Lebesgue measure such that, for m = 1
+m
+�
+r, ∥∇ψ∥
+|ψ|
+�
+≤ 2 + (1 + δ)2
+2
+log+ T(r, ψ) + δ
+2 log r + O(1)
+holds for all r > 0 outside Eδ; and for m ≥ 2
+m
+�
+r, ∥∇ψ∥
+|ψ|
+�
+≤ 2 + (1 + δ)2
+2
+log+ T(r, ψ) + 2m − 1
+4m − 4δ log r + O(1)
+holds for all r > 0 outside Eδ. Adjust Eδ so that O(1) is absorbed.
+□
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+25
+3.3. Second main theorem and defect relation.
+Let D = D1+· · ·+Dq be a reduced divisor on X. D is said to be of simple
+normal crossing type if every Dj is smooth and; at every point x ∈ X, there
+exist a local holomorphic coordinate neighborhood U(z1, · · · , zm) and a non-
+negative integer k with 0 ≤ k ≤ m such that
+U ∩ D =
+�
+z1 · · · zk = 0
+�
+for U ∩ D ̸= ∅.
+We establish the following second main theorem:
+Theorem 3.10. Let M be a non-compact complete K¨ahler manifold with
+maximal volume growth and non-negative Ricci curvature. Let X be a com-
+plex projective manifold of complex dimension not greater than that of M.
+Let D ∈ |L| be a divisor of simple normal crossing type, where L is a positive
+line bundle over X. Let f : M → X be a differentiably non-degenerate mero-
+morphic mapping. Then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞)
+with finite Lebesgue measure such that
+Tf(r, L) + Tf(r, KX) + T(r, R) ≤ N f(r, D) + O
+�
+log+ Tf(r, L) + δ log r
+�
+holds for all r > 0 outside Eδ.
+Proof. Equip every holomorphic line bundle O(Dj) with a Hermitian metric
+hj. It can induce a Hermitian metric hL = h1⊗· · ·⊗hq on L with c1(L, hL) >
+0. We see that Ω = cn
+1(L, hL) is a volume form on X. Pick sj ∈ H0(X, O(Dj)
+such that (sj) = Dj and ∥sj∥ < 1. On X, define a singular volume form
+Φ =
+Ω
+�q
+j=1 ∥sj∥2 .
+Set
+f ∗Φ ∧ αm−n = ξαm.
+Note that
+αm = m! det(hi¯j)
+m
+�
+j=1
+√−1
+π
+dzj ∧ d¯zj.
+It is clear that
+ddc[log ξ] ≥ f ∗c1(L, hL) − f ∗Ric(Ω) + R − [Suppf ∗D]
+in the sense of currents, where
+R = −ddc log det(hi¯j)
+is the Ricci form of M. It yields that
+1
+4
+�
+∆(r)
+gr(o, x)∆ log ξdv
+(15)
+≥ Tf(r, L) + Tf(r, KX) + T(r, R) − N f(r, D).
+
+26
+X.-J. DONG
+Next, we need to estimate the upper bound of the first term in (15). Since
+D is of simple normal crossing type, then there exist a finite open covering
+{Uλ} of X and finitely many rational functions wλ1, · · · , wλn on X such that
+wλ1, · · · , wλn are holomorphic on Uλ satisfying that
+dwλ1 ∧ · · · ∧ dwλn(x) ̸= 0,
+∀x ∈ Uλ;
+D ∩ Uλ =
+�
+wλ1 · · · wλhλ = 0
+�
+,
+∃hλ ≤ n.
+Moreover, we can require that O(Dj)|Uλ ∼= Uλ × C for λ, j. On Uλ, write
+Φ =
+eλ
+|wλ1|2 · · · |wλhλ|2
+n�
+k=1
+√−1
+π
+dwλk ∧ d ¯wλk,
+where eλ is a positive smooth function on Uλ. Set
+Φλ =
+φλeλ
+|wλ1|2 · · · |wλhλ|2
+n�
+k=1
+√−1
+π
+dwλk ∧ d ¯wλk,
+where {φλ} is a partition of unity subordinate to {Uλ}. Set fλk = wλk ◦ f.
+On f −1(Uλ), we have
+f ∗Φλ =
+φλ ◦ f · eλ ◦ f
+|fλ1|2 · · · |fλhλ|2
+n�
+k=1
+√−1
+π
+dfλk ∧ d ¯fλk
+= φλ ◦ f · eλ ◦ f
+�
+1≤i1̸=···̸=in≤m
+���∂fλ1
+∂zi1
+���
+2
+|fλ1|2 · · ·
+���
+∂fλhλ
+∂z
+ihλ
+���
+2
+|fλhλ|2
+����
+∂fλ(hλ+1)
+∂zihλ+1
+����
+2
+· · ·
+����
+∂fλn
+∂zin
+����
+2
+·
+�√−1
+π
+�n
+dzi1 ∧ d¯zi1 ∧ · · · ∧ dzin ∧ d¯zin.
+Fix any x0 ∈ M, we take local holomorphic coordinates z1, · · · , zm near
+x0 and local holomorphic coordinates ζ1, · · · , ζn near f(x0) such that
+α|x0 =
+√−1
+π
+m
+�
+j=1
+dzj ∧ d¯zj,
+c1(L, hL)|f(x0) =
+√−1
+π
+n
+�
+j=1
+dζj ∧ d¯ζj.
+Put f ∗Φλ ∧ αm−n = ξλαm. Clearly, we have ξ = � ξλ and
+ξλ = φλ ◦ f · eλ ◦ f
+�
+1≤i1̸=···̸=in≤m
+��� ∂fλ1
+∂zi1
+���
+2
+|fλ1|2 · · ·
+���
+∂fλhλ
+∂z
+ihλ
+���
+2
+|fλhλ|2
+����
+∂fλ(hλ+1)
+∂zihλ+1
+����
+2
+· · ·
+����
+∂fλn
+∂zin
+����
+2
+≤ φλ ◦ f · eλ ◦ f
+�
+1≤i1̸=···̸=in≤m
+��∇fλ1
+��2
+|fλ1|2
+· · ·
+��∇fλhλ
+��2
+|fλhλ|2
+·
+��∇fλ(hλ+1)
+��2 · · ·
+��∇fλn
+��2
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+27
+at x0. Define a non-negative function ̺ on M by
+(16)
+f ∗c1(L, hL) ∧ αm−1 = ̺αm.
+Set fj = ζj ◦ f for 1 ≤ j ≤ n. Then, at x0
+f ∗c1(L, hL) ∧ αm−1 = (m − 1)!
+2
+m
+�
+j=1
+��∇fj
+��2αm.
+That is,
+̺ = (m − 1)!
+n
+�
+i=1
+m
+�
+j=1
+��� ∂fi
+∂zj
+���
+2
+= (m − 1)!
+2
+n
+�
+j=1
+��∇fj
+��2
+at x0. Combining the above, we are led to
+ξλ ≤ φλ ◦ f · eλ ◦ f · (2̺)n−hλ
+(m − 1)!n−hλ
+�
+1≤i1̸=···̸=in≤m
+��∇fλ1
+��2
+|fλ1|2
+· · ·
+��∇fλhλ
+��2
+|fλhλ|2
+on f −1(Uλ). Note that φλ ◦f ·eλ ◦f is bounded on M, whence it yields from
+log+ ξ ≤ �
+λ log+ ξλ + O(1) that
+(17)
+log+ ξ ≤ O
+�
+log+ ̺ +
+�
+k,λ
+log+ ∥∇fλk∥
+|fλk|
+�
++ O(1)
+on M. By Jensen-Dynkin formula (cf. Theorem 2.13)
+(18)
+1
+4
+�
+∆(r)
+gr(o, x)∆ log ξdv = 1
+2
+�
+∂∆(r)
+log ξdπr + O(1).
+Combining (17) with (18) and using Theorem 3.9,
+1
+4
+�
+∆(r)
+gr(o, x)∆ log ξdv
+≤ O
+� �
+k,λ
+m
+�
+r, ∥∇fλk∥
+|fλk|
+�
++ log+
+�
+∂∆(r)
+̺dπr
+�
++ O(1)
+≤ O
+� �
+k,λ
+log+ T(r, fλk) + log+
+�
+∂∆(r)
+̺dπr
+�
++ O(1)
+≤ O
+�
+log+ Tf(r, L) + log+
+�
+∂∆(r)
+̺dπr
+�
++ O(1).
+Using Theorem 3.6 and (16), for any δ > 0, there exists a subset Eδ ⊆ (0, ∞)
+with finite Lebesgue measure such that
+log+
+�
+∂∆(r)
+̺dπr ≤ O
+�
+log+ Tf(r, L) + δ log r
+�
+
+28
+X.-J. DONG
+holds for all r > 0 outside Eδ. Thus, we conclude that
+(19)
+1
+4
+�
+∆(r)
+gr(o, x)∆ log ξdv ≤ O
+�
+log+ Tf(r, L) + δ log r
+�
++ O(1)
+for all r > 0 outside Eδ. Combining (15) with (19), we prove the theorem.
+□
+In view of (1) and (2), we see that T(r, R) has the alternative expressions
+T(r, R) = 1
+2
+�
+∆(r)
+gr(o, x)sCdv = 1
+4
+�
+∆(r)
+gr(o, x)sdv.
+The curvature condition implies that T(r, R) ≥ 0, thus we deduce that
+Corollary 3.11. Assume the same conditions as in Theorem 3.10. Then for
+any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue measure
+such that
+Tf(r, L) + Tf(r, KX) ≤ N f(r, D) + O
+�
+log+ Tf(r, L) + δ log r
+�
+holds for all r > 0 outside Eδ.
+We continue to consider a defect relation. Define the defect and the simple
+defect ¯δf(D) of f with respect to D, respectively by
+δf(D) = 1 − lim sup
+r→∞
+Nf(r, D)
+Tf(r, L) ,
+¯δf(D) = 1 − lim sup
+r→∞
+N f(r, D)
+Tf(r, L) .
+By the first main theorem, we have 0 ≤ δf(D) ≤ ¯δf(D) ≤ 1.
+For any two holomorphic line bundles L1, L2 over X, define
+�c1(L2)
+c1(L1)
+�
+= inf
+�
+t ∈ R : ω2 < tω1; ∃ω1 ∈ c1(L1), ∃ω2 ∈ c1(L2)
+�
+.
+Invoking the volume growth condition (3) and the Green function estimate
+(8) (cf. Lemma 3.5 also) for M, one can deduce that Tf(r, L) ≥ O(log r) for
+a nonconstant meromorphic mapping f. Hence, we obtain a defect relation:
+Theorem 3.12. Assume the same conditions as in Theorem 3.10. Then
+¯δf(D) ≤
+�c1(K∗
+X)
+c1(L)
+�
+− lim inf
+r→∞
+T(r, R)
+Tf(r, L) ≤
+�c1(K∗
+X)
+c1(L)
+�
+.
+Note that the above defect relation in Theorem 3.12 is sharp.
+Corollary 3.13. There exist no differentiably non-degenerate meromorphic
+mappings f : M → X satisfying the growth condition
+lim inf
+r→∞
+T(r, R)
+Tf(r, L) >
+�c1(K∗
+X)
+c1(L)
+�
+.
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+29
+3.4. The case for singular divisors.
+We consider a general hypersurface D of X. Let S be the set of the points
+of D at which D has a non-normal-crossing singularity. Hironaka’s resolution
+of singularities (cf. [23]) says that there exists a proper modification
+τ : ˜X → X
+such that ˜X \ ˜S is biholomorphic to X \S through τ, and ˜D has only normal
+crossing singularities, where ˜S = τ −1(S) and ˜D = τ −1(D). Write ˆD = ˜D \ ˜S
+and denote by ˜Sj the irreducible components of ˜S. Put
+(20)
+τ ∗D = ˆD +
+�
+pj ˜Sj = ˜D +
+�
+(pj − 1) ˜Sj,
+Rτ =
+�
+qj ˜Sj,
+where Rτ is ramification divisor of τ, and pj, qj are positive integers. Set
+(21)
+S∗ =
+�
+ςj ˜Sj,
+ςj = max
+�
+pj − qj − 1, 0
+�
+.
+Endow O(S∗) with a Hermitian metric ∥ · ∥ and take a holomorphic section
+σ of O(S∗) such that the zero divisor (σ) = S∗ and ∥σ∥ < 1.
+Let f : M → X be a meromorphic mapping such that f(M) ̸⊆ D. Define
+the proximity function of f with respect to the singularities of D by
+mf(r, Sing(D)) =
+�
+∂∆(r)
+log
+1
+∥σ ◦ τ −1 ◦ f∥dπr.
+Consider the lift ˜f : M → ˜X defined via τ ◦ ˜f = f. Then, ˜f is a holomorphic
+mapping on M \ ˜I, where ˜I = I ∪ f −1(S) with the indeterminacy set I of f.
+One can similarly define the Nevanlinna’s functions of ˜f through the lift of
+f via τ. It is not difficult to verify that
+(22)
+mf(r, Sing(D)) = m ˜f(r, S∗) =
+�
+ςjm ˜f(r, ˜Sj).
+The following second main theorem is an extension of Theorem 3.10.
+Theorem 3.14. Let M be a non-compact complete K¨ahler manifold with
+maximal volume growth and non-negative Ricci curvature. Let X be a com-
+plex projective manifold of complex dimension not greater that of M. Let
+D ∈ |L| be a hypersurface of X, where L is a positive line bundle over X.
+Let f : M → X be a differentiably non-degenerate meromorphic mapping.
+Then for any δ > 0, there exists a subset Eδ ⊆ (0, ∞) with finite Lebesgue
+measure such that
+Tf(r, L) + Tf(r, KX) + T(r, R)
+≤ N f(r, D) + mf(r, Sing(D)) + O
+�
+log+ Tf(r, L) + δ log r
+�
+holds for all r > 0 outside Eδ.
+
+30
+X.-J. DONG
+Proof. We first show the case where D is the union of smooth hypersurfaces,
+namely, no irreducible component of ˜D crosses itself. Let E be the union of
+generic hyperplane sections of X so that the set A = ˜D ∪E has only normal
+crossing singularities. By (20) and K ˜
+X = τ ∗KX + O(Rτ)
+K ˜
+X + O( ˜D) = τ ∗KX + τ ∗O(D) + (1 − pj + qj)O( ˜Sj)
+(23)
+= τ ∗KX + τ ∗L + (1 − pj + qj)O( ˜Sj).
+Applying Theorem 3.10 to ˜f, we obtain
+T ˜f(r, O(A)) + T ˜f(r, K ˜
+X) + T(r, R)
+≤ N ˜f(r, A) + O
+�
+log+ T ˜f(r, τ ∗O(A)) + δ log r
+�
+.
+The First Main Theorem gives that
+T ˜f(r, O(A)) = m ˜f(r, A) + N ˜f(r, A) + O(1)
+= m ˜f(r, ˜D) + m ˜f(r, E) + N ˜f(r, A) + O(1)
+≥ m ˜f(r, ˜D) + N ˜f(r, A) + O(1)
+= T ˜f(r, O( ˜D)) − N ˜f(r, ˜D) + N ˜f(r, A) + O(1),
+which yields
+T ˜f(r, O(A)) − N ˜f(r, A) ≥ T ˜f(r, O( ˜D)) − N ˜f(r, ˜D) + O(1).
+Since T ˜f(r, τ ∗L) = Tf(r, L) and N ˜f(r, ˜D) = N f(r, D), then
+T ˜f(r, O( ˜D)) + T ˜f(r, K ˜
+X) + T(r, R)
+(24)
+≤ N ˜f(r, ˜D) + O
+�
+log Tf(r, L) + δ log r
+�
+.
+It follows from (23) that
+T ˜f(r, O( ˜D)) + T ˜f(r, K ˜
+X)
+(25)
+= T ˜f(r, τ ∗L) + T ˜f(r, τ ∗KX) +
+�
+(1 − pj + qj)T ˜f(r, O( ˜Sj))
+= Tf(r, L) + Tf(r, KX) +
+�
+(1 − pj + qj)T ˜f(r, O( ˜Sj)).
+Note that N ˜f(r, ˜S) = 0, thus it yields from (21) and (22) that
+�
+(1 − pj + qj)T ˜f(r, O( ˜Sj)) =
+�
+(1 − pj + qj)m ˜f(r, ˜Sj) + O(1)
+(26)
+≤
+�
+ςjm ˜f(r, ˜Sj) + O(1)
+= mf(r, Sing(D)) + O(1).
+Combining (24) and (25) with (26), we have the theorem proved in this case.
+For the general case, according to those arguments stated above, it suffices
+to prove the case that a hypersurface D is of normal crossing type. Using the
+arguments from [[39], pp. 175], there exists a proper modification τ : ˜X → X
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+31
+such that ˜D = τ −1(D) is the union of a collection of hypersurfaces of simple
+normal crossing type. It means that mf(r, Sing(D)) = 0. Invoking Theorem
+3.10, we can prove the theorem in general.
+□
+Corollary 3.15. Assume the same conditions as in Theorem 3.14. Then
+¯δf(D) ≤
+�c1(K∗
+X)
+c1(L)
+�
++ lim sup
+r→∞
+mf(r, Sing(D)) − T(r, R)
+Tf(r, L)
+.
+4. Application to algebraic dependence problems
+4.1. Consequences of Theorems 3.1 and 3.10.
+The purpose of this section is to apply the Nevanlinna theory established
+in Section 3 to the study of algebraic dependence problems on the dominant
+meromorphic mappings. We shall point out that some arguments essentially
+follow Y. Aihara [8].
+Let X be a complex projective manifold. Let L be an ample line bundle
+over X. For any holomorphic line bundle F over X, define
+�F
+L
+�
+= inf
+�
+γ ∈ Q : γL ⊗ F −1 is big
+�
+.
+According to Corollary 2.9, we see that [F/L] < 0 if and only if F −1 is big.
+Let f : M → X be a meromorphic mapping, where M is a K¨ahler manifold.
+For F ∈ Pic(X) ⊗ Q, we define
+Tf(r, F) = 1
+ν Tf(r, νF),
+where ν is a positive integer such that νF ∈ Pic(X). Evidently, this is well
+defined. Next, we would like to consider another defect relation. The second
+main theorem (i.e., Theorem 3.10) yields that
+Theorem 4.1. Let M be a non-compact complete K¨ahler manifold with
+maximal volume growth and non-negative Ricci curvature. Let X be a com-
+plex projective manifold of complex dimension not greater than that of M.
+Let D ∈ |L| be a divisor of simple normal crossing type, where L is an ample
+line bundle over X. Let f : M → X be a dominant meromorphic mapping.
+Then
+¯δf(D) ≤
+�
+K−1
+X
+L
+�
+.
+Proof. It follows from the definition of [K−1
+X /L] that ([K−1
+X /L] + ǫ)L ⊗ KX
+is big for any rational number ǫ > 0. Using Corollary 2.9, we obtain
+��
+K−1
+X /L
+�
++ ǫ
+�
+L ⊗ KX ≥ δL
+
+32
+X.-J. DONG
+for a sufficiently small rational number δ > 0. This implies that
+Tf(r, K−1
+X ) ≤
+��
+K−1
+X /L
+�
+− δ + ǫ
+�
+Tf(r, L) + O(1).
+Invoking Theorem 3.10, we conclude that
+¯δf(D) ≤
+�
+K−1
+X
+L
+�
+.
+□
+The first main theorem (i.e., Theorem 3.1) also yields that
+Theorem 4.2. Let M be a K¨ahler manifold and X be a complex projective
+manifold. Let f : M → X be a dominant meromorphic mapping. Assume
+that µF ⊗L−1 is big for some positive integer µ, where F is a big line bundle
+over X. Then
+Tf(r, L) ≤ µTf(r, F) + O(1).
+Proof. By Theorem 2.8, the bigness of µF ⊗ L−1 implies that there exists a
+nonzero holomorphic section s ∈ H0(X, ν(µF ⊗L−1)) for a sufficiently large
+positive integer ν. Note that Theorem 3.1 remains true for a general K¨ahler
+manifold M, whence
+Nf(r, (s)) ≤ Tf(r, ν(µF ⊗ L−1)) + O(1)
+= µνTf(r, F) − νTf(r, L) + O(1).
+This leads to the desired inequality.
+□
+4.2. Propagation of algebraic dependence.
+Let M be a non-compact complete K¨ahler manifold with maximal volume
+growth and non-negative Ricci curvature, and let X be a complex projective
+manifold with complex dimension not greater than that of M. Fix an integer
+l ≥ 2. A proper algebraic subset Σ of Xl is said to be decomposible, if there
+exist positive integers l1, · · · , ls with l = l1 + · · · + ls for some integer s ≤ l
+and algebraic subsets Σj ⊆ Xlj for 1 ≤ j ≤ s, such that Σ = Σ1×· · ·×Σs. If
+Σ is not decomposable, we say that Σ is indecomposable. For l meromorphic
+mappings f1, · · · , fl : M → X, there is a meromorphic mapping f1×· · ·×fl :
+M → Xl, defined by
+(f1 × · · · × fl)(x) = (f1(x), · · · , fl(x)),
+∀x ∈ M \
+l�
+j=1
+I(fj),
+where I(fj) denotes the indeterminacy set of fj for 1 ≤ j ≤ l. As a matter
+of convenience, set
+˜f = f1 × · · · × fl.
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+33
+Definition 4.3. Let S be an analytic subset of M. The nonconstant mero-
+morphic mappings f1, · · · , fl : M → X are said to be algebraically dependent
+on S, if there exists a proper indecomposable algebraic subset Σ of Xl such
+that ˜f(S) ⊆ Σ. In this case, we say that f1, · · · , fl are Σ-related on S.
+Let L be an ample line bundle over X, and let D1, · · · , Dq ∈ |L| such that
+D1 + · · · + Dq has only simple normal crossings. Set
+G =
+�
+f : M → X is a dominant meromorphic mapping
+�
+.
+Let S1, · · · , Sq be hypersurfaces of M such that dimC Si ∩ Sj ≤ m − 2 for all
+i ̸= j. Fix q positive integers k1, · · · , kq which may be +∞. Denote by
+(27)
+F = F
+�
+f ∈ G ; {kj}; (M, {Sj}); (X, {Dj})
+�
+the set of all f ∈ G such that
+Sj = Suppkjf ∗Dj,
+1 ≤ j ≤ q.
+Let ˜L be a big line bundle over Xl. In general, we have
+˜L ̸∈ π∗
+1Pic(X) ⊕ · · · ⊕ π∗
+l Pic(X),
+where πk : Xl → X is the natural projection on the k-th factor for 1 ≤ k ≤ l.
+Let F1, · · · , Fl be big line bundles over X. Then, it defines a line bundle over
+Xl by
+˜F = π∗
+1F1 ⊗ · · · ⊗ π∗
+l Fl.
+If ˜L ̸= ˜F, we assume that there is a rational number ˜γ > 0 such that
+˜γ ˜F ⊗ ˜L−1 is big.
+If ˜L = ˜F, we shall take ˜γ = 1. In further, assume that there is a line bundle
+F0 ∈ {F1, · · · , Fl} such that F0 ⊗ F −1
+j
+is either big or trivial for 1 ≤ j ≤ l.
+Let H be the set of all indecomposable hypersurfaces Σ in Xl satisfying
+Σ = Supp ˜D for some ˜D ∈ |˜L|. Set
+S = S1 ∪ · · · ∪ Sq,
+k0 = max{k1, · · · , kq}.
+Lemma 4.4. Let f1, · · · , fl be meromorphic mappings in F. Assume that
+˜f(S) ⊆ Σ and ˜f(M) ̸⊆ Σ for some Σ ∈ H . Then
+N(r, S) ≤ ˜γ
+l
+�
+j=1
+Tfj(r, Fj) + O(1) ≤ ˜γ
+l
+�
+j=1
+Tfj(r, F0) + O(1).
+Proof. Take ˜D ∈ |˜L| such that Σ = Supp ˜D. As mentioned earlier, ˜γ ˜F ⊗ ˜L−1
+is big for ˜γ ̸= 1 and trivial for ˜γ = 1. Then, by conditions with Theorem 3.1
+
+34
+X.-J. DONG
+and Corollary 4.2, we conclude that
+N(r, S) ≤ T ˜f(r, ˜L) + O(1)
+≤ ˜γT ˜f(r, ˜F ) + O(1)
+≤ ˜γ
+l
+�
+j=1
+Tfj(r, Fj) + O(1)
+≤ ˜γ
+l
+�
+j=1
+Tfj(r, F0) + O(1).
+The proof is completed.
+□
+Lemma 4.5. Let f be a meromorphic mapping in F. Then
+qTf(r, L) + Tf(r, KX)
+≤
+k0
+k0 + 1N(r, S) +
+q
+�
+j=1
+1
+kj + 1Nf(r, Dj) + o
+�
+Tf(r, L)
+�
+.
+Proof. By Sj = Suppkjf ∗Dj, we have
+N f(r, Dj) ≤ N(r, Sj)
+≤
+kj
+kj + 1N(r, Sj) +
+1
+kj + 1Nf(r, Dj)
+≤
+k0
+k0 + 1N(r, Sj) +
+1
+kj + 1Nf(r, Dj).
+Combining this with the second main theorem (cf. Theorem 3.10), then one
+can easily prove the lemma.
+□
+Define a Q-line bundle L0 ∈ Pic(X) ⊗ Q by
+(28)
+L0 =
+�
+q
+�
+j=1
+kj
+kj + 1
+�
+L ⊗
+�
+− ˜γlk0
+k0 + 1F0
+�
+.
+Again, for an arbitrary Q-line bundle H ∈ Pic(X) ⊗ Q, define
+T(r, H) =
+l
+�
+j=1
+Tfj(r, H).
+In what follows, we are to show a theorem for the propagation of algebraic
+dependence.
+Theorem 4.6. Let f1, · · · , fl be meromorphic mappings in F. Assume that
+f1, · · · , fl are Σ-related on S for some Σ ∈ H . If L0 ⊗ KX is big, then
+f1, · · · , fl are Σ-related on M.
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+35
+Proof. It suffices to prove ˜f(M) ⊆ Σ. Otherwise, we assume that ˜f(M) ̸⊆ Σ.
+According to Theorem 3.1 and Lemma 4.5, for i = 1, · · · , l
+qTfi(r, L) + Tfi(r, KX)
+≤
+k0
+k0 + 1N(r, S) +
+q
+�
+j=1
+1
+kj + 1Tfi(r, L) + o
+�
+Tfi(r, L)
+�
+,
+which yields from Lemma 4.4 that
+q
+�
+j=1
+kj
+kj + 1Tfi(r, L) + Tfi(r, KX) ≤
+k0
+k0 + 1N(r, S) + o
+�
+Tfi(r, L)
+�
+≤
+˜γk0
+k0 + 1
+l
+�
+j=1
+Tfj(r, F0) + o
+�
+Tfi(r, L)
+�
+=
+˜γk0
+k0 + 1T(r, F0) + o
+�
+Tfi(r, L)
+�
+.
+Thus, we get
+q
+�
+j=1
+kj
+kj + 1T(r, L) + T(r, KX) ≤
+˜γlk0
+k0 + 1T(r, F0) + o
+�
+T(r, L)
+�
+.
+It follows that
+(29)
+T(r, L0) + T(r, KX) ≤ o
+�
+T(r, L)
+�
+.
+On the other hand, the bigness of L0⊗KX implies that there exists a positive
+integer µ such that µ(L0 ⊗ KX) ⊗ L−1 is a big line bundle. By Theorem 4.2
+T(r, L) ≤ µ
+�
+T(r, L0) + T(r, KX)
+�
++ O(1),
+which contradicts with (29). We conclude that ˜f(M) ⊆ Σ. This completes
+the proof.
+□
+Set
+γ0 =
+�
+L−1
+0
+⊗ K−1
+X
+L
+�
+.
+Then, L0 ⊗ KX is big if and only if γ0 < 0. Whence, we obtain
+Corollary 4.7. Let f1, · · · , fl be meromorphic mappings in F. Assume that
+f1, · · · , fl are Σ-related on S for some Σ ∈ H . If γ0 < 0, then f1, · · · , fl are
+Σ-related on M.
+Remark 4.8. In the case where γ0 ≥ 0, we cannot conclude the propagation
+of algebraic dependence in general. Particularly, for γ0 > 0, the propagation
+of algebraic dependence does not occur even if one assumes the existence of
+Picard’s deficient divisors. For instance, consider the situation that M = C,
+X = P1(C) and l = 2. Set L = F1 = F2 = O(1), where O(1) stands for the
+
+36
+X.-J. DONG
+hyperplane line bundle over P1(C), which means that F0 = O(1). Then, one
+can take ˜L = ˜F = π∗
+1O(1) ⊗ π∗
+2O(1) which implies that ˜γ = 1. Let
+f1(z) = ez, f2(z) = e−z; D1 = 0, D2 = ∞, D3 = −1, D4 = 1.
+Then, D1, D2 are Picard’s deficient values of f1. Put Sj = Supp1f ∗
+1 Dj, which
+means that kj = 1 for 1 ≤ j ≤ 4. Let S = S1 ∪ · · · ∪ S4. It is trivial to check
+that f2 ∈ F. A direct computation leads to
+L0 =
+�
+4
+�
+j=1
+1
+1 + 1
+�
+O(1) ⊗
+�
+−
+2
+1 + 1O(1)
+�
+= O(1).
+It yields that
+γ0 =
+�L−1
+0
+⊗ K−1
+P1(C)
+L
+�
+=
+�−O(1) ⊗ (2O(1))
+O(1)
+�
+= 1 > 0.
+Taking Σ as the diagonal of P1(C) × P1(C), then we have Σ ∈ H . It is clear
+that (f1 ×f2)(S) ⊆ Σ and (f1 ×f2)(C) ̸⊆ Σ. Therefore, one cannot conclude
+the propagation of algebraic dependence if γ0 > 0. However, for γ0 = 0, the
+propagation of algebraic dependence may occur under a defect condition. In
+fact, we have the following theorem:
+Theorem 4.9. Let f1, · · · , fl be meromorphic mappings in F. Assume that
+f1, · · · , fl are Σ-related on S for some Σ ∈ H . If γ0 = 0 and δfi(Dj) > 0 for
+some i ∈ {1, · · · , l} and some j ∈ {1, · · · , q}, then f1, · · · , fl are Σ-related
+on M.
+To prove Theorem 4.9, we need a lemma:
+Lemma 4.10. Let f1, · · · , fl be meromorphic mappings in F. Assume that
+f1, · · · , fl are Σ-related on S for some Σ ∈ H . If γ0 = 0 and ˜f(M) ̸⊆ Σ,
+then there exist positive constants C1, C2 such that for j = 1, · · · , l
+C1 ≤ Tfj(r, L)
+Tf1(r, L) ≤ C2
+holds for all sufficiently large r ̸∈ I, where I = ∪l
+j=1I(fj).
+Proof. Let ν be a rational number with 0 < ν < 1. We claim that
+(30)
+˜γνT(r, F0) ≤ N(r, S) + O(1)
+for all sufficiently large r ̸∈ I. Otherwise, there exists a monotone increasing
+sequence {rn}∞
+n=1 outside I with rn → ∞ as n → ∞, such that
+˜γνT(rn, F0) > N(rn, S) + O(1).
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+37
+Using Lemmas 4.4 and 4.5,
+q
+�
+j=1
+kj
+kj + 1T(rn, L) + T(rn, KX) ≤ ˜γν0lk0
+k0 + 1 T(rn, F0) + o
+�
+T(rn, L)
+�
+.
+This yields that
+(31)
+T(rn, L0 ⊗ KX) + λT(rn, F0) ≤ o
+�
+T(rn, L)
+�
+,
+where
+λ = ˜γlk0(1 − ν)
+k0 + 1
+> 0.
+Since γ0 = 0, the bigness of F0 implies the bigness of L0 ⊗KX ⊗λF0. Hence,
+L0 ⊗ KX ⊗ λF0 ≥ δL for a sufficiently small rational number δ > 0. Thus,
+δT(rn, L) ≤ T(rn, L0) + T(rn, KX) + λT(rn, F0) + O(1).
+By this with (31)
+δT(rn, L) ≤ o(T(rn, L)),
+which is a contradiction and it thus confirms (30). By Theorem 3.1
+N(r, S) =
+q
+�
+j=1
+N(r, Sj) ≤
+q
+�
+j=1
+Nf1(r, Dj) ≤ qTf1(r, L) + O(1).
+Combining this with (30), we get
+˜γνT(r, F0) ≤ qTf1(r, L) + O(1).
+Since F0 is big, for a sufficiently small rational number ǫ > 0
+ǫ˜γνTfj(r, L) ≤ ǫ˜γνT(r, L) ≤ q˜γνT(r, F0) ≤ Tf1(r, L) + O(1).
+This proves the lemma.
+□
+Proof of Theorem 4.9
+It suffices to show that ˜f(M) ⊆ Σ. Otherwise, we assume that ˜f(M) ̸⊆ Σ.
+From the definition of the defect, for any ǫ > 0, there is a subset Eǫ ⊆ (0, ∞)
+with finite Lebesgue measure such that
+Nfs(r, Dj) < (1 − δfs(Dj) + ǫ)Tfs(r, L)
+for all r > 0 outside Eǫ. By (28) and Lemma 4.4 and Lemma 4.5
+T(r, L0) + T(r, KX) ≤ −
+l
+�
+s=1
+q
+�
+j=1
+1
+kj + 1(δfs(Dj) − ǫ)Tfs(r, L) + o
+�
+T(r, L)
+�
+≤ qǫT(r, L) −
+1
+kj + 1δfi(Dj)Tfi(r, L) + o
+�
+T(r, L)
+�
+,
+which yields that
+1
+kj + 1δfi(Dj)Tfi(r, L) + T(r, L0) + T(r, KX) − qǫT(r, L) ≤ o
+�
+T(r, L)
+�
+.
+
+38
+X.-J. DONG
+Since [L−1
+0 ⊗K−1
+X /L] = 0, we have L0 +KX ≥ −γL for an arbitrary rational
+number γ > 0. It implies that
+−γT(r, L) ≤ T(r, L0) + T(r, KX) + O(1).
+Whence,
+1
+kj + 1δfi(Dj)Tfi(r, L) − (γ + qǫ)T(r, L) ≤ o
+�
+T(r, L)
+�
+.
+Note that δfi(Dj) > 0 and ǫ, γ can be be small arbitrarily. However, Lemma
+4.10 implies that the above inequality does not hold if ǫ, γ are small enough.
+This is a contradiction. Thus, we have ˜f(M) ⊆ Σ. The proof is completed.
+4.3. Unicity theorems.
+We apply the propagation theorems of algebraic dependence to the unicity
+problems for meromorphic mappings from a complete K¨ahler manifold into
+a complex projective manifold.
+Since X is projective, there is a holomorphic embedding Φ : X ֒→ PN(C).
+Let O(1) be the hyperplane line bundle over PN(C). Now, take l = 2 and
+F1 = F2 = Φ∗O(1) which gives that F0 = Φ∗O(1) and
+˜F = π∗
+1 (Φ∗O(1)) ⊗ π∗
+2 (Φ∗O(1)) .
+Again, take ˜L = ˜F, then ˜γ = 1. In view of (28), we have
+(32)
+L0 =
+�
+q
+�
+j=1
+kj
+kj + 1
+�
+L ⊗
+�
+− 2k0
+k0 + 1Φ∗O(1)
+�
+.
+Suppose that D1, · · · , Dq ∈ |L| satisfy that D1 + · · · + Dq has only simple
+normal crossings. For f0 ∈ F, one says that the set {Dj}q
+j=1 is generic with
+respect to f0 and φ, if
+ˆ
+Ms = f0(M − I(f0)) ∩ SuppDs ∩ {x ∈ X : rank(dφ(x)) = dimC X} ̸= ∅
+for at least one s ∈ {1, · · · , q}. Next, assume that the set {Dj}q
+j=1 is generic
+with respect to f0 and φ. Denote by F0 the set of all meromorphic mappings
+f ∈ F such that f = f0 on the hypersuface S.
+Theorem 4.11. If L0 ⊗ KX is big, then F0 contains exactly one element.
+Proof. Let f ∈ F0 be an arbitrary meromorphic mapping, it suffices to show
+that f ≡ f0. Recall that Φ : X ֒→ PN(C) is a holomorphic embedding. Since
+f = f0 on S, we have Φ◦f = Φ◦f0 on S. Now, we assert that Φ◦f ≡ Φ◦f0.
+Otherwise, we may assume that Φ◦f ̸≡ Φ◦f0. Let ∆ denote the diagonal of
+PN(C)×PN(C). Put ˜Φ = Φ×Φ and ˜f = f ×f0. Then, it gives a meromorphic
+mapping
+φ = ˜Φ ◦ ˜f : M → PN(C) × PN(C).
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+39
+Again, set ˜
+O(1) = π∗
+1O(1)⊗π∗
+2O(1), which is a holomorphic line bundle over
+PN(C) × PN(C), where O(1) is the hyperplane line bundle over PN(C). It is
+easy to see that ˜L = ˜Φ∗ ˜
+O(1). Since Φ◦f ̸≡ Φ◦f0, there exists a holomorphic
+global section ˜σ of ˜
+O(1) which satisfies that φ∗˜σ ̸= 0 and ∆ ⊆ Supp(˜σ). Take
+Σ = Supp˜Φ∗(˜σ), it yields that ˜f(S) ⊆ Σ and ˜f(M) ̸⊆ Σ. On the other hand,
+with the aid of Theorem 4.6, the bigness of L0 ⊗KX implies that ˜f(M) ⊆ Σ,
+which is a contradiction. Thus, we obtain Φ◦f ≡ Φ◦f0. By the assumption,
+ˆ
+Ms ̸= ∅ for some s ∈ {1, · · · , q}. Take a point P ∈ ˆ
+Ms, then there is an open
+neighborhood UP of P such that Φ|UP : UP → Φ(UP) is biholomorphic. Let
+V = UP . Since Φ ◦ f = Φ ◦ f0 and f = f0 on S, we deduce that f = f0 on V.
+By the uniqueness theorem on analytic functions, we see that f ≡ f0.
+□
+Using the similar arguments as in the proof of Theorem 4.11 and Theorem
+4.9, we can also prove the following theorem:
+Theorem 4.12. Assume that [L−1
+0
+⊗ K−1
+X /L] = 0. If δf0(Ds) > 0 for some
+s ∈ {1, · · · , q}, then F0 contains exactly one element.
+4.4. Targets are compact Riemann surfaces and Pn(C).
+In this section, we consider some consequences of Theorems 4.11 and 4.12
+(and Theorems 4.6, 4.9 too) in the case that X is a compact Riemann surface
+or a complex projective space.
+4.4.1. X is a compact Riemann surface.
+Let R be a compact Riemann surface with genus g. Note that R is viewed
+as P1(C) for g = 0, and a torus for g = 1. Let f : M → R be a nonconstant
+holomorphic mapping. Let a1, · · · , aq be distinct points in R. Theorem 3.12
+gives a defect relation:
+(33)
+q
+�
+j=1
+¯δf(aj) ≤ 2 − 2g.
+A1 The cases g = 0 and g ≥ 2
+In the case of g = 0, we have
+Theorem 4.13. Let f1, f2 : M → P1(C) be nonconstant holomorphic map-
+pings. Let a1, · · · , aq be distinct points in P1(C). We have
+(a) Assume that Suppf ∗
+1 aj = Suppf ∗
+2 aj for all j. If q ≥ 5, then f1 ≡ f2;
+(b) Assume that Supp1f ∗
+1 aj = Supp1f ∗
+2 aj for all j. If q ≥ 7, then f1 ≡ f2.
+Proof. In Theorem 4.11, we put X = P1(C) and L = O(1) as well as kj = ∞
+for all j. Note that KP1(C) = −2O(1). It yields from (32) that
+L0 ⊗ KP1(C) = qO(1) ⊗ (−2O(1)) ⊗ (−2O(1)) = (q − 4)O(1).
+
+40
+X.-J. DONG
+Hence, L0 ⊗KP1(C) is big provided that q ≥ 5. Using Theorem 4.11, we have
+(a) holds. For (b), we just need to put kj = 1 for all j. In this case,
+L0 ⊗ KP1(C) = q
+2O(1) ⊗ (−O(1)) ⊗ (−2O(1)) = q − 6
+2
+O(1).
+Clearly, L0⊗KP1(C) is big provided that q ≥ 7. This completes the proof.
+□
+In the case that g ≥ 2, it yields from (33) that each holomorphic mapping
+f : M → R is a constant mapping. Therefore, this case is actually trivial.
+A2 The case g = 1
+By (33), each nonconstant holomorphic mapping f : M → R is surjective.
+Here, we only consider the case where the torus R is a smooth elliptic curve,
+denoted by E. When M is a finite analytic ramified covering of Cm, Aihara
+proved a unicity theorem (cf. [8], Theorem 5.1). In particular, when M =
+Cm, he showed that (cf. [8], Theorem 5.2)
+Theorem 4.14 (Aihara). Let f1, f2 : Cm → E be nonconstant holomorphic
+mappings. Let D1 = {a1, · · · , aq} be a set of points in E. Set D2 = φ(D1)
+with #(D2) = p, here φ ∈ End(E). Assume that Supp1f ∗
+1D1 = Supp1f ∗
+2D2.
+If pq > (p + q)(deg ψ + 1), then f2 = φ(f1).
+Using the theorems on the propagation of algebraic dependence obtained
+in Section 4.2, we see that those arguments in the proof of Theorem 4.14 can
+be carried to our situation where the domain is M. Hence, there is no need
+to state the details. We give a generalization of Theorem 4.14 as follows:
+Theorem 4.15. Let f1, f2 : M → E be nonconstant holomorphic mappings.
+Let D1 = {a1, · · · , aq} be a set of points in E. Set D2 = φ(D1) with #(D2) =
+p, here φ ∈ End(E). Assume that Supp1f ∗
+1 D1 = Supp1f ∗
+2 D2. If pq > (p +
+q)(deg ψ + 1), then f2 = φ(f1).
+In particular, when φ = Id, we have deg φ = 1. Thus, it yields that
+Corollary 4.16. Let f1, f2 : M → E be nonconstant holomorphic mappings.
+Let a1, · · · , aq be distinct points in E. Assume that Supp1f ∗
+1 aj = Supp1f ∗
+2 aj
+for all j. If q ≥ 5, then f1 ≡ f2.
+4.4.2. X is a complex projective space.
+We consider the case X = Pn(C). For a Q-line bundle F ∈ Pic(Pn(C))⊗Q,
+note that F is big if and only if F is ample. Let F1, F2 be ample line bundles
+over Pn(C). Since Pic(Pn(C)) ∼= Z, then there are positive integers d1, d2 such
+that F1 = d1O(1) and F2 = d2O(1). Again, define a holomorphic line bundle
+˜F over Pn(C) × Pn(C) by ˜F = π∗
+1F1 ⊗ π∗
+2F2. A well-known fact says that
+Pic (Pn(C) × Pn(C)) = π∗
+1Pic(Pn(C)) ⊕ π∗
+2Pic(Pn(C)).
+
+NEVANLINNA THEORY AND ALGEBRAIC DEPENDENCE
+41
+Thus, we may assume that ˜L = ˜F. Let σ be a holomorphic section of ˜L over
+Pn(C)×Pn(C). Note that σ can be identified with a homogeneous polynomial
+P(ξ, ζ) of degree d1 in ξ = [ξ0; · · · ; ξn] and degree d2 in ζ = [ζ0; · · · ; ζn]. Set
+d0 = max{d1, d2}. Let L be an ample line bundle over Pn(C) with L = dO(1).
+Let S = S1 ∪ · · · ∪ Sq, where S1, · · · , Sq are q hypersufaces of M such that
+dimC Si ∩Sj ≤ m−2 for i ̸= j. Let D1, · · · , Dq ∈ |L| such that D1 +· · ·+Dq
+has simple normal crossings.
+Based on the above notations, we give the following theorem:
+Theorem 4.17. Let f1, f2 : M → Pn(C) be dominant meromorphic map-
+pings. Assume that Suppkf ∗
+1 Dj = Suppkf ∗
+2Dj = Sj with 1 ≤ k ≤ +∞ for
+all j. Suppose, in addition, that P(f1, f2) = 0 on S. If
+q > 2d0 + (1 + n)(1 + k−1)
+d
+,
+then P(f1, f2) ≡ 0.
+Proof. Recall that F0 is defined as an element in {F1, F2} such that F0⊗F −1
+j
+is either big or trivial for j = 1, 2. Hence, we have F0 = d0O(1). Since ˜L = ˜F,
+then ˜γ = 1. We also note that l = 2, k0 = kj = k. Thus, it follows from (28)
+that
+L0 =
+�
+q
+�
+j=1
+k
+k + 1
+�
+dO(1) ⊗
+�
+− 2kd0
+k + 1O(1)
+�
+= k(dq − 2d0)
+k + 1
+O(1).
+By KPn(C) = −(n + 1)O(1), we get
+L0 ⊗ KPn(C) = k(dq − 2d0) − (k + 1)(n + 1)
+(k + 1)
+O(1).
+Hence, L0 ⊗ KPn(C) is big if k(dq − 2d0) − (k + 1)(n + 1) > 0, i.e.,
+q > 2d0 + (1 + n)(1 + k−1)
+d
+.
+Invoking Theorem 4.6, we have the theorem proved.
+□
+Take L = F1 = F2 = O(1), then d = d1 = d2 = 1. In this case, D1, · · · , Dq
+reduce to q hyperplanes H1, · · · , Hq. Using Theorem 4.17, it yields that
+Corollary 4.18. Let f1, f2 : M → Pn(C) be dominant meromorphic map-
+pings. Assume that Suppkf ∗
+1 Hj = Suppkf ∗
+2Hj = Sj with 1 ≤ k ≤ +∞ for
+all j. Suppose, in addition, that P(f1, f2) = 0 on S. If
+q > 2 + (1 + n)(1 + k−1)
+then P(f1, f2) ≡ 0.
+Keep the same notations as in Theorem 4.17, we get that
+
+42
+X.-J. DONG
+Theorem 4.19. Let f1, f2 : M → Pn(C) be dominant meromorphic map-
+pings. Assume that Suppkf ∗
+1 Dj = Suppkf ∗
+2Dj = Sj with 1 ≤ k ≤ +∞ for
+all j. Suppose, in addition, that P(f1, f2) = 0 on S. If δf1(Dj) > 0 for some
+j ∈ {1, · · · , q} and
+q = 2d0 + (1 + n)(1 + k−1)
+d
+∈ Z,
+then P(f1, f2) ≡ 0.
+Proof. The given relation between q, d, d0 implies that L0 ⊗KPn(C) is trivial.
+Invoking Theorem 4.9, the theorem can be proved.
+□
+Similarly, take L = F1 = F2 = O(1). Theorem 4.19 yields that
+Corollary 4.20. Let f1, f2 : M ��� Pn(C) be dominant meromorphic map-
+pings. We have
+(a) Assume that Suppf ∗
+1 Dj = Suppf ∗
+2 Dj = Sj for all j, and assume that
+P(f1, f2) = 0 on S. If δf1(Dj) > 0 for some j ∈ {1, · · · , q} with q = n + 3,
+then P(f1, f2) ≡ 0;
+(b) Assume that Supp1f ∗
+1Dj = Supp1f ∗
+2 Dj = Sj for all j, and assume that
+P(f1, f2) = 0 on S. If δf1(Dj) > 0 for some j ∈ {1, · · · , q} with q = 2n + 4,
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+Email address: [email protected]
+

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+arXiv:2301.01321v1  [cs.CR]  3 Jan 2023
+Cheesecloth: Zero-Knowledge Proofs of Real-World Vulnerabilities
+Santiago Cuéllar∗
+Galois, Inc.
+Bill Harris
+Galois, Inc.
+James Parker
+Galois, Inc.
+Stuart Pernsteiner
+Galois, Inc.
+Eran Tromer
+Columbia University
+Abstract
+Currently, when a security analyst discovers a vulnerabil-
+ity in critical software system, they must navigate a fraught
+dilemma: immediately disclosing the vulnerability to the
+public could harm the system’s users; whereas disclosing the
+vulnerability only to the software’s vendor lets the vendor
+disregard or deprioritize the security risk, to the detriment of
+unwittingly-affected users.
+A compelling recent line of work aims to resolve this by
+using Zero Knowledge (ZK) protocols that let analysts prove
+that they know a vulnerability in a program, without reveal-
+ing the details of the vulnerability or the inputs that exploit
+it. In principle, this could be achieved by generic ZK tech-
+niques. In practice, ZK vulnerability proofs to date have been
+restricted in scope and expressibility, due to challenges re-
+lated to generating proof statements that model real-world
+software at scale and to directly formulating violated proper-
+ties.
+This paper presents
+CHEESECLOTH, a
+novel proof-
+statement compiler, which proves practical vulnerabilities in
+ZK by soundly-but-aggressively preprocessing programs on
+public inputs, selectively revealing information about exe-
+cuted control segments, and formalizing information leak-
+age using a novel storage-labeling scheme. CHEESECLOTH’s
+practicality is demonstrated by generating ZK proofs of
+well-known vulnerabilities in (previous versions of) critical
+software, including the Heartbleed information leakage in
+OpenSSL and a memory vulnerability in the FFmpeg graph-
+ics framework.
+1
+Introduction
+Ideally, programs that process sensitive information would
+always execute safely and securely. With this ideal remain-
+ing difficult to achieve for the foreseeable future, it is criti-
+cal that that when programs are found to be vulnerable, the
+program’s affected users are alerted quickly and safely. This
+∗Authors listed alphabetically.
+requirement presents a challenge: convincingly disclosing a
+vulnerability appears to require sharing the vulnerability’s
+details (such as an exploit that triggers it), thereby placing
+users at greater risk in the short term.
+A promising approach to disclosing vulnerabilities con-
+vincingly yet safely is to leverage Zero-Knowledge (ZK)
+proofs: protocols in which one party—designated as the
+prover—convinces another party—designated as the veri-
+fier—of the validity of a claim without revealing any addi-
+tional information about the claim’s evidence.
+Such a use of ZK proofs has arguably been a conceptual
+possibility ever since the initial fundamental results establish-
+ing that they exist for all problems in NP [26]. It has become
+more realistic with improvements to underlying ZK proto-
+cols and with the emergence of schemes for encoding knowl-
+edge of executions of programs written in convenient lan-
+guages (starting with [10,24] and discussed further below).
+In order to prove vulnerabilities in ZK about practical soft-
+ware, several open problems remaing to be addressed. First,
+proof frameworks must scale to compile proofs of vulner-
+abilities that require considerably more steps of execution
+and space. TinyRAM [10–12] is sufficiently flexible to val-
+idate the executions of applications, but it is expensive, in
+part due to the fact that it simulates every instruction in the
+modeled CPU’s ISA in each step. TinyRAM’s performance
+is surpassed by those of Pantry [17] and Buffet [52], but both
+frameworks require loops to be unrolled to a public bound:
+publicly revealing these bounds leaks information about the
+underlying vulnerability.
+A second open problem is to efficiently compile state-
+ments from an understandable form. One immediate ap-
+proach is to execute a program under a dynamic safety moni-
+tor for well-understood safety properties, such as those those
+implemented in Valgrind [41]; however, directly encoding
+the additional monitoring would induce prohibitively large
+overhead. Approaches for verifying low-level exploits in
+ZK [27] rely on being able to efficiently compile directly-
+understandable properties into statements of control-location
+reachability.
+1
+
+To address these problems, we present CHEESECLOTH, an
+optimizing ZK proof-statement generator that efficently en-
+codes vulnerabilities in practical software. The contributions
+behind CHEESECLOTH’s design include:
+1. Optimizations of ZK statements that verify the executions
+of programs, taking advantage of program structure but
+without revealing additional information about the exe-
+cution. Specifically, Public-PC segments construct execu-
+tion traces from segments with public program counters,
+thus enabling aggressive constant folding, without leak-
+ing information about the overall execution trace. Simi-
+larly, instructions which are publicly-determined to be ex-
+ecuted infrequently are sparsely supported (i.e. can’t be
+executed at every step), making the statement smaller.
+2. Novel, efficient ZK encodings of memory errors preva-
+lent in practical software, specifically out-of-bound ac-
+cess, use-after-free, free-after-free, and uninitialized ac-
+cess. Previous related work focused primarily on proving
+knowledge of a valid execution without proving existence
+of a vulnerability [10,11] or encoded proofs of vulnerabil-
+ity using a less efficient memory model [29].
+3. A novel, efficient encoding of statements that a program
+always leaks data (when given an exploit as a secret in-
+put). Our scheme enables proofs of program properties
+that are related to, but critically distinct from, existing pro-
+gram monitors and type systems that prove that a program
+may leak data [20,39], optionally in ZK [21].
+We implemented these optimizations and encodings in
+CHEESECLOTH, a full compilation toolchain for encoding
+vulnerabilities of real-world programs into efficient ZK
+proofs. The toolchain extends previous approaches based
+on TinyRAM, and includes a full definition of a novel
+TinyRAM extension (named MicroRAM) and a compiler to
+MicroRAM from the LLVM intermediate language,enabling
+proofs of vulnerabilities in programs provided in C, C++, or
+Rust.
+We evaluated our implementation by proving in ZK the
+existence of three vulnerabilities in practical systems soft-
+ware. Specifically, we proved that previous versions of the
+GRIT and FFmpeg [2] graphics processing libraries con-
+tained buffer-overflow vulnerabilites, and that the OpenSSL
+cryptograhy toolkit [3] was vulnerable to the notorious Heart-
+bleed vulnerability [5]. CHEESECLOTH takes the software
+C/C++ source code and a flag denoting a vulnerabilities
+class; it combines these with an emulation of the runtime
+environment (operating system and libraries), and applies
+the aforementioned techniques, to derive a statement directly
+provable in ZK. The ZK proof can then be given, as a wit-
+ness, the concrete exploit used to demonstrate the original at-
+tack. CHEESECLOTH contains implementations of powerful
+program analyses that, when combined with manual program
+partitioning in some cases, dramatically increase the scale of
+programs that it can process, compared to a more naive com-
+piler.
+The remainder of this paper is organized as follows: Sec. 2
+reviews the background that this work builds upon. Sec. 3
+presents the implementation details of our CHEESECLOTH
+compilation pipeline; Sec. 4 covers the critical and aggres-
+sive optimizations we make to verify the ZK execution of a
+program; Sec. 5 describes our ZK encodings to efficiently de-
+tect memory and information leakage vulnerabilities; Sec. 6
+describes our practical experience using CHEESECLOTH to
+prove vulnerabilities; Sec. 7 compares our approach to re-
+lated work and Sec. 8 concludes.
+2
+Background
+In this section, we review prior work on which our contribu-
+tion builds upon, specifically Zero-Knowledge (ZK) proofs of
+program executions (Sec. 2.1), information leakage by pro-
+grams (Sec. 2.2), and partial program evaluation (Sec. 2.3).
+2.1
+Zero-Knowledge Proofs
+Zero-knowledge proofs enable a prover party to prove to
+a verifier party that the prover knows the correctness of a
+computational statement (e.g., that a given Boolean circuit
+is satisfiable), without revealing information about their evi-
+dence for the claim (e.g., the witness that satisfied the circuit).
+There exist ZK protocols for proving knowledge of solutions
+to all problems in NP [26], and in recent years, numerous ef-
+ficient protocols have been developed and implemented for
+ZK proofs of general statements (e.g., [8, 10, 12, 14–16, 22,
+24,28,29,31,32,38,43,51]).
+Some of these works specifically address statements about
+correct execution of programs running on a general-purpose
+architecture that include Random Access Memory (RAM),
+where the program is expressed in low-level machine code
+or a high-level language [9,10,12,14,16,17,22,27,29,31,32,
+38,51,52].
+Our compiler uses a hybrid of step-by-step CPU emula-
+tion, similar to TinyRAM [10–12], a MIPS-like CPU that
+can simulate programs in C and similar low-level languages
+that access RAM. The TinyRAM encoder, given a public
+TinyRAM program and bound on the number of steps of ex-
+ecution to simulate and a private program input, generates
+an R1CS that is satisfied by encodings of the input. The con-
+straint system consists of (1) a family of constraint systems
+that validate computations purely over registers in each step
+and (2) a novel memory-checking sub-circuit that verifies the
+correctness of RAM operations using a permutation network.
+This CPU-unrolling technique is excellent for supporting lan-
+guage features such as data-dependent loops, control-flow
+and self-modifying code. The technique can also naturally
+leverage existing tools such as compilers front-ends and li-
+braries.
+2
+
+We combine TinyRAM-style emulation with direct com-
+pilation of program blocks into circuit gates [17, 24, 52]
+(Sec. 4). The compiler’s output is a circuit whose satisfia-
+bility is equivalent to the existence of a vulnerability in the
+source program, and whose structure does not reveal the vul-
+nerability or how it may be triggered.
+In our evaluation, the underlying ZK protocol is the
+Mac’n’Cheese [9] protocol for proving circuit satisfiability,
+as implemented by the Swanky [23] library. This is an inter-
+active protocol, where the prover and verifier engage in mul-
+tiple rounds of communication to evaluate the circuit, at the
+end of which the verifier learns that the circuit accepted the
+secret witness provided by the prover (and nothing else).
+2.2
+Information Flow
+One core contribution of our work is a practical scheme for
+proving in zero knowledge that a program leaks data, which
+we have applied to prove that previous versions of OpenSSL
+leak private data, as triggered by the Heartbleed vulnerabil-
+ity (described in Sec. 5.2 and Sec. 6.3). The scheme’s design
+requires a formal treatment of information flow: specifically,
+a treatment sufficiently formal that we could generate logical
+circuits that would be satisfied only by witnesses to leakage.
+In the interest of space and clarity, we will omit a definition
+of information flow and leakage for a full programming lan-
+guage, but we will describe ours in sfufficient datail to com-
+municate the key challenges and approaches.
+A labeling L is a subset of a program’s input variables I
+designated as the private inputs, and a subset of its output
+variables O denoted as public outputs. Program P satisfies
+noninterference with respect to L if each pair of inputs that
+are only distinct at private inputs result in values that are the
+same at all public outputs; P leaks with respect to L if, with
+respect to L, it does not satisfy noninterference. It follows
+from the above definition that a leak is witnessed by a pair
+of executions that differ only at L-labeled inputs and produce
+distinct L-labeled outputs.
+Noninterference has a precise but accessible formal defi-
+nition that can capture the flow requirements of some criti-
+cal software [20], but its shortcomings in practice are well
+known [39, 40, 45]: the complete information flow spec-
+ifications of practical programs often are not noninterfer-
+ence properties, intuitively because programs that take sen-
+sitive inputs typically do need to reveal some partial infor-
+mation about them; and even when desired flow properties
+are noninterference properties, proving that a program sat-
+isfies the property in general can involve careful reasoning
+about unbounded data and control. A rich body of prior
+work [13, 18, 19, 25] has considered generalizations of non-
+interference involving equivalences over observable events,
+along with rich programming languages and type systems
+and attempt to prove their satisfaction. However, noninterfer-
+ence properties still constitute aspects of a program’s com-
+plete information flow requirements that unfortunately are
+both critical and are violated in practice (Heartbleed being
+a prominent example). This pattern justisfies the current
+work’s primary focus on proving noninterference violations.
+2.2.1
+Labeled Programs and Executions
+In their most general form, information flow and leakage are
+defined over pairs of executions. Practical program moni-
+tors [20,49] and type systems [39] prove facts about all exe-
+cution pairs, by labeling the program’s data and control struc-
+tures with metadata which is tracked through the execution.
+These approaches can be carried out by a programmer or au-
+tomated analysis that directly annotates the program or exe-
+cution. However, the requisite guarantees are different in our
+proof-of-vulnerability context compared to their usual appli-
+cations, as seen next.
+At a high level, the guarantees provided by dynamic infor-
+mation flow monitors are as follows. A labeling of a program
+execution over n steps is an assignment from each program
+variable and step 0 ≤ i < n to a sensitivity label. A label-
+ing over-approximates information flow if, from any two ex-
+ecutions starting from states that only differ at high inputs,
+the program produces results that differ only at high-labeled
+timestamped storage cells (static analyses and type systems
+lift this property to be defined over all pairs of executions
+that differ only at sensitive inputs).
+Such over-approximation allows for “false positives” in
+identifying information flows. For example, in a context
+where only input x is sensitive and the return value is pub-
+lic, the following function always_true does not leak any
+information about its input secret because it returns true
+for each input value:
+bool always_true(bool secret) {
+if (secret) return secret;
+else return !secret; }
+However, many natural taint analyses would label the re-
+turned values as sensitive because it is computed from
+secret.
+Over-approximation of potential leaks is often still valu-
+able for aiding programmers to ensure that their program
+does not leak: falsely determining that a secure program may
+leak may constitute a nuisance, and may need be mitigated
+to ensure practicality, but can to some degree to tolerated.
+However, in our setting of proving a leak in ZK, it is un-
+acceptable for the verifier to learn only that a program may
+leak. The whole point is to prove that it does leak (given the
+purported exploit). We will thus create a labeling which is an
+under-approximation, i.e., when the labels say so, a leakage
+is present. It will then remain to empirically show that label-
+ing indeed detects leakage for the vulnerabilities of interest.
+3
+
+2.3
+Partial Evaluation
+In many practical contexts, a program may receive different
+subsets of its input at different times after it has been written
+and compiled: e.g., after being installed, a configuration file
+may be included that remains the same over all executions
+on distinct inputs subsequently received from a network. A
+natural objective is, given a program and a subset of its in-
+puts that can be fixed, to generate a specialized program that
+processes the remaining inputs with improved performance.
+Stated more precisely, for program P(X,Y) with input vari-
+ables X and Y, a partial evaluation of P on an assignment
+A : X → Words from X to data values Words is a program PA
+such that P(A,B) is the same as PA(B) for each assignment
+B : Y → Words.
+Partially evaluating programs in a practical language
+brings several complexities [35]; the underlying technique
+amounts to: (1) evaluate the program under a symbolic state,
+in which registers and memory addresses may be mapped
+either to memory addresses or terms defined over symbolic
+variables that denoted unknown values; (2) using computed
+symbolic states that describe all possible states at each con-
+trol point, simplify the control structure at each point. Vari-
+ations of this technique may be viewed as aggressive exten-
+sions and generalizations of the constant propagation analy-
+sis and constant folding transformation implemented in con-
+ventional optmizing compilers [7].
+3
+CHEESECLOTH Implementation
+CHEESECLOTH produces ZK proofs of real world vulnera-
+bilities. It takes as input a public LLVM program (typically
+compiled from C, C++, or Rust) and, when run as the prover,
+a secret exploit that triggers a vulnerability in the program.
+CHEESECLOTH outputs a ZK circuit that verifies the execu-
+tion trace of the program and checks whether or not a vulner-
+ability occurred during that execution. The pipeline enables a
+prover to demonstrate to a verifier that there is a vulnerability
+in a program while keeping the vulnerability and triggering
+exploit secret.
+CHEESECLOTH produces ZK circuits in multiple stan-
+dard representations including Rank-1 Constraints Systems
+(R1CS) [10] and SIEVE IR [6]. Because the circuits are seri-
+alized in standardized formats, CHEESECLOTH is agnostic to
+the ZK protocol applied. When run as the prover, CHEESE-
+CLOTH outputs the accompanying witness for the circuit.
+CHEESECLOTH can be extended to check different proper-
+ties about a program’s execution. Users can selectively en-
+able which extensions to run by providing different input
+flags to the compilation pipeline. These extensions are how
+the memory and information leakage vulnerability detection
+checks described in Sec. 5 are implemented. This section
+covers the baseline design of the CHEESECLOTH compila-
+tion pipeline which includes (1) the MicroRAM assembly
+language, (2) the MicroRAM Compiler, and (3) the Witness
+Checker Generator. Sec. 4 describes optimizations for this
+design that enable it to scale to real world vulnerabilities.
+3.1
+MicroRAM
+The MicroRAM assembly language is critical to CHEESE-
+CLOTH. It is the core IR language that CHEESECLOTH op-
+erates on and is the language that the MicroRAM Compiler
+compiles LLVM programs to. The Witness Checker Genera-
+tor produces ZK circuits that verify program executions ac-
+cording to MicroRAM’s architecture.
+MicroRAM is heavily inspired by TinyRAM [10, 11],
+which is a practical and efficient assembly language with
+a simple transition function that is ideal for ZK execution
+verification. We describe MicroRAM and its architecture be-
+low, and we precisely describe how its design diverges from
+TinyRAM in Sec. 3.1.1.
+MicroRAM is a random-access machine designed to effi-
+ciently detect vulnerabilities in program executions. It is a
+reduced instruction set computer (RISC) with a Harvard ar-
+chitecture and byte-addressable random-access memory.
+MicroRAM instructions are relatively simple and include
+4 boolean operations, 8 arithmetic operations for signed and
+unsigned integers, 2 shift operations, 5 compare operations,2
+move operations, 3 jump instructions, 2 operations for read-
+ing and writing to memory, and 1 answer operation that re-
+turns and halts the execution. Floating-point and vector arith-
+metic are not directly supported in the MicroRAM machine
+and must be implemented in software. Instructions take two
+registers and one operand (either a register or an immediate)
+as arguments. As an example, instruction xor ri rj 255
+writes to register ri the exclusive-orof register rj and the im-
+mediate 255. CHEESECLOTH extensions like those described
+in Sec. 5 can introduce additional instructions as needed.
+The state of the MicroRAM machine consists of the pro-
+gram counter (pc), k 64-bit registers, a memory of 264 64-bit
+words, a flag indicating whether or not the execution so far is
+valid (inv_flag), and a flag tracking whether a vulnerability
+has occurred (vuln_flag). CHEESECLOTH extensions can
+extend the state of the MicroRAM machine as well.
+To demonstrate the existence of a vulnerability in a pro-
+gram, a prover must present a secret input that results in a
+valid execution trace that triggers a vulnerability. Formally,
+given a MicroRAM program, P, and an initial memory, m0,
+P(m0) demonstrates a vulnerability in T steps if inv_flag
+is false and vuln_flag is true in the final MicroRAM
+state of the program’s execution trace. inv_flag is set to
+false if any of the checks validating the program’s execu-
+tion fails. The extensions implementing the vulnerability de-
+tection checks set vuln_flag to true if they observe a vul-
+nerability during the program’s execution.
+4
+
+3.1.1
+Beyond TinyRAM
+As mentioned above, our MicroRAM machine is inspired by
+TinyRAM. Here we report on how MicroRAM’s design de-
+parts from the TinyRAM model.
+• MicroRAM’s memory model is byte-addressable while
+TinyRAM is word-addressable. Byte-addressable memory
+is necessary to support functionality like string manipula-
+tions and packed structs, without adding subroutines to ac-
+cess bytes within full words.
+• TinyRAM receives input via input tapes. In MicroRAM,
+input is passed directly in memory, which saves many cy-
+cles that TinyRAM spends copying input to memory. A Mi-
+croRAM program can request non-deterministic advice in
+several ways, however the prover does not have to commit
+to the advice ahead of time on a tape; instead they provide
+the advice upon request. This approach is better suited to
+support backends that exploit parallelism or streaming, and
+it results in smaller circuits.
+• TinyRAM uses a 1-bit condition flag for branching while
+MicroRAM does not. This is advantageous since Micro-
+RAM targets a variety of backends including non-boolean
+arithmetic circuits where the flag is more expensive than a
+regular register1. In addition, the semantics without a flag
+are much simpler so the compiler, interpreter, and circuit
+generator are simpler as well. We found that even when
+targeting boolean circuits, the benefits of having a condi-
+tion flag are outweighed by the extra complexity.
+• We have not yet explored using a von Neumann architec-
+ture [12] for MicroRAM because, despite the asymptotic
+benefits, the instruction fetching circuit is not yet a limit-
+ing factor in our ZK statements.
+3.2
+MicroRAM Compiler
+The MicroRAM Compiler is implemented as a LLVM back-
+end that takes LLVM IR programs as input and produces Mi-
+croRAM assembly as output. We currently support C, C++,
+and Rust programs by compiling them to LLVM IR with the
+Clang and rustc compiler frontends. Support for other lan-
+guages such as C#, Haskell, or Scala can be added in the
+future by connecting their appropriate LLVM frontends and
+writing the appropriate standard libraries.
+Our compiler backend supports a large subset of the
+LLVM IR language. The compiler supports all boolean and
+arithmetic operations for integers of different sizes, bitwise
+operations, all non-concurrent memory operations including
+pointer arithmetic with getelementptr, conversion opera-
+tions, function calls, variable arguments, comparisons, and
+phi nodes. Complex operations like floating-point opera-
+tions are implemented in software via a LLVM compiler
+1If full words fit in a field element, then the flag is the same size as a
+register, but requires special circuitry and has more restrictions.
+pass.
+Exceptions, and all exception handling instructions, are
+not supported; but we we can still tolerate programs with ex-
+ceptions as long as the prover is disclose that the execution of
+interest, which triggers a vulnerability,does not throw any ex-
+ceptions. This is since the MicroRAM Compiler translates all
+exception handling instructions to traps that mark the trace as
+invalid by setting the inv_flag flag. By inserting traps, the
+MicroRAM Compiler can process programs with any num-
+ber of unsupported features, as long as the prover is will-
+ing to reveal that those features are not involved in the vul-
+nerable execution. With this simple trick, users can compile
+real-world programs without having to manually remove un-
+supported features. When enabling traps, provers must take
+care not to reveal too much information about the underlying
+vulnerability. Sec. 3.2.3 presents a more detailed discussion
+about the security implications of how proof statements can
+reveal information about their witnesses.
+3.2.1
+Standard library
+MicroRAM supports a significant portion of the C standard
+library and POSIX system calls, using Picolibc [4]: a library
+that offers the C standard library APIs and was originally de-
+signed for small embedded systems with limited RAM. Picol-
+ibc supports multiple widely deployed target architectures,
+including ARM, RISC-V, and x86-64.
+We implemented MicroRAM as a target architecture for
+Picolibc. This enables the MicroRAM compiler to support
+most of the C standard library and POSIX system calls. It
+is also convenient as it allows prover to publicly customize
+the behavior of system calls. For example, in our case study
+of OpenSSL, the victim server receives the malicious re-
+quest from the attacker over the network. We customized the
+behavior of read when compiled natively to intercept and
+record all data received over the network. When compiled
+for MicroRAM, read returns the previously recorded exploit
+request, which is loaded from secret memory. We also cus-
+tomize the implementation of malloc and free to efficiently
+detect memory vulnerabilities (Sec. 5.1.1).
+3.2.2
+Generating advice
+As we will see in later sections, CHEESECLOTH requires non-
+deterministic advice to efficiently generate a ZK circuit that
+verifies the consistency of memory in an execution (Sec. 3.3)
+and the presence of a vulnerability (Sec. 5.1). To aid the
+prover in producing that advice, the MicroRAM compiler
+runs two interpreter passes. The first pass executes the pro-
+gram without any advice and records the necessary advice.
+The second execution runs the nondeterministic semantics
+and records the trace, which is passed to the Witness Checker
+Generator to produce the witness for the prover.
+5
+
+3.2.3
+Security
+MicroRAM produces zero-knowledge proofs which ensure
+that no additional information is revealed about the wit-
+ness. However, the proof statement itself can reveal informa-
+tion about the secret input. For example, in MicroRAM and
+TinyRAM the circuit reveals a time bound T on the execution
+length. In Pantry/Buffet, the circuit discloses an upper bound
+Ti on every loop (and recursive function) in the execution. In
+vRAM [53], every instruction run during the execution is re-
+vealed to the verifier. We argue that a formalization of this
+information leakage is necessary. Interesting and important
+future work will be to define a formal framework to analyse
+how secure these encodings are.
+3.2.4
+Preprocessing public inputs
+One opportunity for aggressive optimization is to publicly
+evaluate logic that is determined by the program’s public in-
+puts. Many practical programs collect inputs from multiple
+sources, some of which are not secret (i.e., irrelevant to the
+vulnerability). If the prover and verifier agree when defining
+a proof statement that only some inputs are sensitive secrets
+(e.g., data packets received from a network connection) while
+others are not (e.g., straightforward configuration options),
+then the resulting proof statement could be immediately op-
+timized by generating the proof statement and partially eval-
+uating the resulting circuit on its input wires corresponding
+to non-sensitive inputs.
+CHEESECLOTH supports such cases with a compiler pass
+that determines the largest program prefix in which no op-
+eration depends on secret inputs. The MicroRAM compiler
+then separates the public prefix from the remaining program
+suffix and compiles them separately. When the interpreter is
+executed by both the prover and verifier, it executes the pre-
+fix and defines a public snapshot of the resulting state, includ-
+ing both registers and memory. When executed by the prover,
+the interpreter then executes the remaining suffix using both
+the snapshot and their private input generate to generate the
+statement’s witness. In practice, this simple optimization has
+significant impact, reducing the number of execution steps in
+OpenSSL’s ZK proof statement from 25M to 1.3M (Sec. 6.3).
+The compiler optimization implements a relatively re-
+stricted form of partial evaluation and constant folding
+(Sec. 2.3). Our initial experience indicates that further ex-
+tensions could improve CHEESECLOTH’s performance dras-
+tically: a key technical challenge is that while programs may
+perform much processing of public data over the course of
+the entire execution, the processing is often interleaved with
+computation over sensitive inputs. Evaluating each of the in-
+terleaved phases of public computation is sound in principle,
+but can only be automated by ensuring that regions of storage
+used by public and secret phases are disjoint. Such automa-
+tion could potentially be achieved by applying points-to and
+shape analyses [46–48], including separation logic [44].
+3.3
+Witness Checker Generator
+The Witness Checker Generator takes as input a MicroRAM
+program and generates a ZK circuit, serialized in standard-
+ized formats including R1CS and SIEVE IR. It also accepts
+nondeterministic advice as input and outputs the secret wit-
+ness to the circuit when run as the prover.
+The Witness Checker Generator builds arithmetic circuits
+for the prime field 2128−159. As an optimization, it automat-
+ically constant folds gates that are independent of secret in-
+puts. To scale to large circuits and avoid running out of mem-
+ory, it streams the circuit serialization to a file. This stream-
+ing is independent of secret witnesses, so the same circuit is
+generated for the prover and verifier.
+The nondeterministic advice the Witness Checker Genera-
+tor accepts provides a description of a program’s execution
+together with the advice necessary to run it. Concretely, the
+advice for an execution of T steps contains the initial pro-
+gram memory, the T MicroRAM states making up the execu-
+tion trace, and a mapping from step number to additional ad-
+vice given at each step. This additional advice includes mem-
+ory ports for what is read or written to memory and stutters
+that indicate the execution should pause for the current step.
+The Witness Checker Generator produces a ZK circuit that
+verifies that the witness describes a valid execution trace for
+the program and that a vulnerability occurs. The circuit is
+split into four key pieces: (1) the transition function circuit,
+(2) the memory consistency circuit, (3) a state transition net-
+work, and (4) public-pc segments. We describe the first two
+here, which follow a similar structure to the circuit construc-
+tion for TinyRAM [10]. The other two are described later in
+Sec. 4.1.
+Transition function circuit. The transition function circuit
+checks a single step of execution. These checks are chained
+together to validate the entire execution trace. Fig. 1 shows
+pseudocode for the transition function circuit. It takes as
+input the circuit’s wire representation of the current Micro-
+RAM state, the next state, and any additional advice needed
+for the current step. The circuit then fetches the instruction
+to execute based on the program counter and pulls out the
+instruction’s argument values by indexing into machine reg-
+isters. It calculates the expected result of the step by multi-
+plexing over the instruction. Finally, the circuit ensures that
+the calculated expected state matches the next state provided
+as advice.
+Memory consistency circuit. The memory consistency cir-
+cuit is similar to TinyRAM’s except addresses are byte-
+addressable instead of word-addressable. Each step may
+have a corresponding memory port advice that states the ad-
+dress and what was read or written to memory. The transi-
+tion function circuit verifies that the execution trace matches
+the memory port advice. All of the memory ports are sorted
+by address and step number. The memory consistency cir-
+6
+
+1
+fn transition_func(circuit, current_st, next_st) {
+2
+let expected_st = current_st.clone();
+3
+let instr = fetch_instr(circuit, current_st.pc);
+4
+let arg1 = index(circuit, current_st.regs, instr.op1);
+5
+let arg2 = index(circuit, current_st.regs, instr.op2);
+6
+7
+let result = circuit.mux(instr.opcode == XOR,
+8
+xor(circuit, arg1, arg2), ...);
+9
+expected_st.pc = circuit.mux(
+10
+is_jump(circuit, instr.opcode),
+11
+result,
+12
+circuit.add(current_st.pc, 1));
+13
+write_index(circuit, expected_st.regs, instr.dest,
+14
+result);
+15
+16
+circuit.assert(expected_st == next_st); }
+Figure 1: Pseudocode for the transition function circuit that
+validates a single MicroRAM step.
+cuit linearly scans the memory ports to ensure that all reads
+and writes to a given address are consistent with the previ-
+ous memory operation. For example, a read should return
+the same value that was previously written to an address. Fi-
+nally, the memory consistency circuit checks that the sorted
+memory ports are a permutation of the memory ports used by
+the transition function circuit. Sec. 5.1 describes how these
+checks are enhanced to efficiently detect memory vulnerabil-
+ities.
+4
+Optimizations
+This section describes two of CHEESECLOTH’s key optimiza-
+tions: constructing executions with public program coun-
+ters (Sec. 4.1) and tuning steps based on instruction sparsity
+(Sec. 4.2). Sec. 6.4 contains an empirical evaluation of the
+optimizations’ effectiveness.
+4.1
+Public-PC segments
+The MicroRAM machine is design to minimize the size of
+the circuit that checks the transition function circuit. How-
+ever, even with MicroRAM’s small instruction set, the tran-
+sition function circuit is still large. This is due to the fact
+that every transition function must support every operation
+in every step of execution. What if we could remove all the
+unused functionality? This is the approach of vRAM [53],
+where the circuit is tuned to check the instruction that is exe-
+cuted at each step. The resulting circuit is much smaller, but
+unfortunately the trace of executed instructions is revealed.
+The values in memory and registers would still be kept secret,
+however a verifier could easily discover where the vulnerable
+code is in the program. In this section, we present public-pc
+segments which generate much smaller circuits without re-
+vealing the trace.
+A public-pc segment is a sequence of transition circuits
+with a hardcoded and public program counter. Using con-
+stant folding, all the instruction fetches of the public-pc seg-
+ments are known and the unused functionality of every step
+can be removed. For example, with a public program counter,
+the fetch_instr and subsequent mux operations over the in-
+struction in Fig. 1 can be constant folded away. We gener-
+ate public-pc segments for all straightline code segments in
+a program, and we implemented a compiler pass that uses a
+naive control-flow analysis to estimate how many times each
+segment will be used. The analysis takes a global bound spec-
+ifying how many times to unroll loops and estimates how
+many times a function will be called by counting the number
+of call sites for that function in the program.
+To preserve the security of the trace, the cycle counter of
+segments is kept private. In addition, we introduce a state
+routing network so that the end state of a segment could be
+the initial state for any other segment in the circuit. Just like
+the memory routing network, the routing information for the
+state routing network is given by the prover and kept secret.
+As a further optimization, we avoid using the state rout-
+ing network when possible. For example, when a public-pc
+segment branches to two statically known locations, we di-
+rectly connect the end state of that segment to the segments
+representating those two locations.
+It is possible that the bound for unrolling loops is not large
+enough to support certain executions, so the pipeline would
+not generate enough public-pc segments for a section of code.
+As backup, the pipeline also produces private-pc segments
+which are just like public-pc segments except the program
+counter is not revealed. Private-pc segments look similar to a
+much smaller TinyRAM circuit with the difference that their
+start and end states come from the state routing network. The
+circuits for these segments are significantly larger, but can ex-
+ecute any part of the program at any point during execution.
+4.2
+Sparsity
+With the naive CPU unrolling described in Sec. 3.3, ev-
+ery transition function must contain a memory port, which
+causes the memory consistency network to grow at a rate of
+O(T logT), where T is the number of steps executed. Unfor-
+tunately, most of those gates are wasted by execution steps
+that do not access memory. CHEESECLOTH mitigates this ex-
+cess by removing some of the unused memory ports, thereby
+reducing the size of the memory consistency circuit.
+The key observation for this optimization is that memory
+operations are rarely contiguous. Even when a program per-
+forms a memory-intensive operation, other instructions are
+often interleaved between memory instructions. For exam-
+ple, when adding the values in a buffer, it takes some steps
+to increment the pointer and add the values between memory
+reads. This enables us to share one memory port among s
+contiguous steps, shrinking the memory consistency network
+7
+
+by a factor of s.
+We define the memory sparsity, s, as the number of steps
+that share a single memory port. CHEESECLOTH chooses s
+based on a static analysis of the code. The analysis deter-
+mines the minimum distance between two memory opera-
+tions in any possible execution. Across statically-unknown
+jumps (e.g. calling a function from a pointer dereference),
+the analysis naively considers all the instructions the control
+flow can possibly jump to. This memory sparsity number s is
+then used by the MicroRAM Compiler and Witness Checker
+Generator to generate the optimized circuit.
+Given a memory sparsity s, the Witness Checker Generator
+will group s consecutive steps and create a single memory
+port for all of these steps. A multiplexer connects the single
+memory port to the entire group and sends the result, using
+nondeterministic advice, to the right step (if any).
+If s is larger than the actual sparsity displayed by a trace,
+then (if unlucky) multiple memory accesses can fall into
+the same group of steps, which has a single memory port.
+CHEESECLOTH handles this situation by inserting stutter
+instructions that delay memory operations until they are
+pushed into the next group with separate memory ports. In-
+serting stutter instructions can be expensive, but reducing the
+size of the memory consistency circuit is more beneficial
+(Sec. 6.4). In future work, we will explore swapping program
+instructions to reduce stutter instructions and determine the
+optimal s parameter for most programs.
+5
+Encoding Vulnerabilities
+This section describes how CHEESECLOTH encodes two
+prevalent and critical classes of software vulnerabilities:
+memory unsafety (Sec. 5.1) and data leakage (Sec. 5.2).
+5.1
+Memory unsafety
+We now describe how CHEESECLOTH efficiently models
+memory and represents memory vulnerabilties. In CHEESE-
+CLOTH, memory is an array of 264 bytes with half reserved
+for the heap and the rest for global variables and the stack.
+Our approach is to keep track of valid memory (e.g. allo-
+cated arrays) and report a vulnerability (i.e., set bug_flag)
+when the program accesses non-valid memory. At the start
+of the execution, the only valid memory is where the global
+variables are stored and, during execution,malloc makes allo-
+cated regions valid and free makes them invalid again. With
+this technique we can catch the following memory errors:
+• Uninitialized access. All uninitialized memory is invalid,
+so any use triggers a bug.
+• Use-after-free. When a region is freed it becomes invalid,
+so any use triggers a bug.
+• Free-after-free. The implementation of free starts by read-
+ing a word from the region to be freed, if the region is not
+1
+void* malloc(size_t size) {
+2
+// Get pointer from advice
+3
+char* addr = __cc_malloc(size);
+4
+/* Compute and validate the size of
+5
+* the allocation provided by the
+6
+* prover. */
+7
+size_t region_size =
+8
+1ull << ((addr >> 58) & 63);
+9
+/* The allocated region must have
+10
+* space for `size` bytes, plus
+11
+* an additional word for metadata.
+12
+*/
+13
+__cc_valid_if(
+14
+region_size >= size + sizeof(uintptr_t),
+15
+"allocated region size is too small");
+16
+/* `region_size` is always a power of
+17
+* two and is at least the word size,
+18
+* so the address must be a multiple
+19
+* of the word size. */
+20
+__cc_valid_if(addr % region_size == 0,
+21
+"allocated address is misaligned for"
+22
+"its region size");
+23
+/* Write 1 (allocated) to the metadata
+24
+* field, and poison it to prevent
+25
+* tampering, invalidating the trace
+26
+* if the metadata word is already
+27
+* poisoned (this happens if the
+28
+* prover tries to return the same
+29
+* region for two separate
+30
+* allocations). */
+31
+uintptr_t* metadata = (uintptr_t*)
+32
+(addr + region_size - sizeof(uintptr_t));
+33
+__cc_write_and_poison(metadata, 1);
+34
+35
+// further computation...
+36
+return (void*)addr; }
+Figure 2: Implementation of non-deterministic malloc.
+valid it triggers a bug.
+• Out-of-bound access. If the program accesses an address
+out of bounds, that new location might (see below) not be
+valid and this triggers a bug.
+It is clear that a normal execution with such bound check-
+ing might miss out-of-bound access bugs, when the ac-
+cess happens to fall on another valid region, and free-after-
+free/use-after-free bugs, if an intermediate malloc makes the
+region valid before the bug is triggered. However, we only
+need to show that the bug exists in one execution, so we im-
+plement a malloc guided by nondeterministicadvice; this lets
+the prover choose the allocation layout to ensure the bug is
+triggered.
+While the techniques described here are specific to heap
+memory bugs, the same ideas can be applied to the stack.
+5.1.1
+Encoding dynamic memory allocation
+An implementation of malloc with nondeterminism poses
+its own challenges. If left unchecked, the prover could man-
+ufacture an execution that triggers a false bug. For example
+the prover could malloc overlapping regions such that if one
+8
+
+is freed and the other one is accessed a false bug is triggered.
+Thus, our implementation of malloc and free (Fig. 2) focuses
+on verifying that the nondeterminisitc choices are legal. If
+foul play is detected, the execution is flagged as invalid with
+inv_flag and will not be accepted by the verifier.
+To ensure that malloc never returns overlapping regions,
+we predetermine aligned non-overlapping regions of differ-
+ent sizes for malloc to choose from. Concretely, we divide
+memory into 26 pools of size 258, then subdivide pool i into
+regions of size 2i. malloc rounds up the requested size to the
+next power of two, then returns the start of an unallocated
+region of that size. For example, malloc(15) must return a
+region in the 4th pool and be 16-byte aligned. In fact, we can
+easily verify that malloc has allocated a correct region just
+by looking at the pointer returned: the first 6 bits determine
+the pool and the rest the alignment.
+Finally, malloc must not return the same pointer twice
+without it being freed in between. To do so, we add to each
+region one word reserved for metadata that is marked and
+made invalid when the region is allocated. If the region was
+allocated again, the invalid metadata would be made invalid
+again which makes the trace invalid by setting inv_flag.
+Furthermore, an implementation of malloc/free that tracks
+the validity of all memory locations would be quite inef-
+ficient. Luckily, the prover knows exactly where the bug
+will happen and thus the malloc/free implementation only
+needs to track the status of that location. At the beginning
+of the execution, the prover commits to a secret location
+stored in the global variable __cc_memory_error_address
+and then malloc/free only track the validity of that location.
+In particular, if an allocated/freed region does not contain
+__cc_memory_error_address then malloc/free do not check
+for errors, and run in constant time.
+5.2
+Data leakage
+A straightforward approach to proving leakage would be to
+directly encode the definition of noninterference in the ZK
+circuit. This could be accomplished by verifying two pro-
+gram executions where only sensitive inputs are distinct but
+public outputs are distinct. However, suchan approach would
+result in a statement of twice the size required for validating
+a single execution. Instead, we might hope to prove a leak
+using a single execution in which storage is annotated with
+labels (Sec. 2.2). However, such systems tranditionally have
+only been designed to prove that a program may leak infor-
+mation, which is unacceptable for definitively proving a leak
+without providing a violating execution directly (Sec. 2.2.1).
+Specifying leakage To identify sensitive sources and sinks,
+the instructions source and sink are added to the Mi-
+croRAM instruction set, and are directly wrapped by user-
+level functions taintSource and taintSink, respectively.
+source annotates that a given byte of data carries sensitive
+data; sink annotates that a given byte is output to a channel.
+Instantiating the general definitions of information flow and
+leakage (Sec. 2.2) for this extended ISA, a MicroRAM pro-
+gram leaks if it has two executions whose inputs only differ
+at addresses given to source, but result in different values
+at an address given in calls to sink. Leakage is established
+by the prover and verifier collaborating to extend the subject
+program to call the taintSource and taintSink to anno-
+tate sensitive sources and sinks.
+Proving leakage To soundly and precisely prove leakage, we
+propose a novel labeling system that tracks what program
+storage may and must hold secret. There are four labels, de-
+noted and partially ordered as
+⊥ ⊑ ℓ0,ℓ1 ⊑ ⊤
+with a least-upper bound (i.e., join) operation denoted ⊔. La-
+bels ℓ0 and ℓ1 annotates data that must belong to one of two
+principals; ⊤ denotes that the data’s sensitivity is unknown;
+⊥ denotes data that must not be influenced by a principal.
+With this labeling scheme, leakage of ℓ-labeled data written
+to a ℓc-labeled sink must occur when ℓ ̸= ⊤ ∧ℓ ̸⊑ ℓc.
+MicroRAM state is extended so that every register and
+byte of memory is associated with a label, similar to previ-
+ous leakage monitors [20, 42, 49, 50]. Two additional labels
+model effects of instructions other than register arithmetic.
+The control context label γ is maintained to be ⊥ if the pro-
+gram execution has not branched on secret data, and ⊤ other-
+wise; similarly, the storage context label σ is maintained to
+be ⊥ if the program has not stored to a secret address, and ⊤
+otherwise.
+Each assignment x:=e sets the label of x to L(e) ⊔ γ ⊔ σ
+(where L(e) is the label of e, defined below); thus, if the pro-
+gram has branched on secret data or written to a secret ad-
+dress, the label of x is set to ⊤. If e is an arithmetic/logical
+operation f(y), then L(e) is ⊥ when L(y) is ⊥ and ⊤ other-
+wise; L(e) = L(y) if f is a bijection: our current implemen-
+tation conservatively only labels single-register expressions
+(i.e., copy sources) as L(y). If e is a load *p, then L(e) is
+L(*p). Conditional branches update γ and memory stores up-
+date σ according to the labels’ descriptions; we omit formal
+descriptions here, due to space constraints.
+Plenty of natural programs leak but cannot be proved to
+do so by this labeling system, potentially because a leakage
+happens after branching or storing to a ⊤-labeled value, or
+because a secret value is propagated over an operation not
+recognized as a bijection. Such cases restrict the situations in
+which the labeling scheme can be applied to prove leakage,
+but do not threaten its validity when it claims that a given
+program leaks. They might be addressed in future work that
+refines instruction interpretations using valid logical axioms
+(e.g., the fact that for each value x, x + 0 = x). Such threats
+and mitigations are dual to threats to precision, and possi-
+ble refinements when proving that a program execution may
+leak.
+9
+
+6
+Evaluation
+We evaluate CHEESECLOTH with three case studies that
+demonstrate ZK proofs of real world software vulnerabilities.
+The vulnerabilities scale by code size and execution trace
+length to showcase the capabilities of CHEESECLOTH. We
+also benchmark the optimizations (Sec. 4) to evaluate their
+effectiveness.
+Tab. 1 presents the results of using CHEESECLOTH to pro-
+duce ZK proofs for our case studies which include GRIT,
+FFMPEG, and OpenSSL. For each case study, we report
+the size of the program in terms of the number of Micro-
+RAM instructions, the number of execution steps required
+to demonstrate the vulnerability, and the number of multipli-
+cation gates in the resulting ZK circuit. We prove satisfia-
+bility of the ZK circuit using the Mac’n’Cheese [9] interac-
+tive ZK protocol, as implemented by the Swanky [23] library.
+We record the protocol running time and communication cost
+between the prover and verifier. All measurements were per-
+formed on a 128 core Intel Xeon E7-8867 CPU with 2 TB of
+RAM running Debian 11, although our implementation typi-
+cally uses considerably less memory (Tab. 1).
+6.1
+Memory unsafety in GRIT
+The GBA Raster Image Transmogrifier (GRIT) [34] converts
+bitmap image files to a graphics format that is readable by
+the Game Boy Advance. A bitmap image includes headers, a
+palette array indicating the colors in the image, and the pixels
+for the image. For 24bpp images, GRIT’s parser assumes the
+palette size is zero and allocates a buffer without space for
+the palette. When populating the buffer, it checks the image
+header for the numberof palette entries without checking that
+this matches the assumed palette size that was used during
+allocation. As a result, a malformed 24bpp image can write
+an arbitrary amount of data (up to the length of the file) past
+the allocated buffer.
+To demonstrate this memory error, we construct a 24bpp
+exploit image with 0x3000 bytes of pixel data and 12 bytes of
+palette data. On Linux, the 12 byte overflow overwrites heap
+metadata and triggers an assertion failure in the memory allo-
+cator. When run through CHEESECLOTH, we generate a ZK
+proof that a memory error is triggered within six thousand
+steps of GRIT’s execution without revealing the triggering
+image or where the error occurred in the code.
+6.2
+Memory unsafety in FFmpeg
+FFmpeg is a tool for recording, converting, and streaming au-
+dio and video [2], and is used in popular software projects
+such as Chrome, Firefox, iTunes, VLC, and YouTube. FFm-
+peg is written in C and has been plagued by vulnerabilities
+that compromise memory safety, enabling attackers to exe-
+cute code and share local files over the network. Versions of
+FFmpeg prior to v1.1.2 contained a vulnerability [1] caused
+by the memory error in the function gif_copy_img_rect,
+which copies the frame of a GIF file between buffers. Previ-
+ous versions of gif_copy_img_rect insecurely calculated
+a pointer to the end of a memory buffer by directly using
+the input image’s height. This calculation allowed an attacker
+to provide a carefully crafted GIF which causes FFmpeg to
+write to memory outside of an array’s bounds.
+To prove memory unsafety of FFmpeg in ZK, we manually
+crafted a GIF image that exploits the described memory vul-
+nerability. We passed this image and a program module that
+invokes FFmpeg’s video decoder to CHEESECLOTH, which
+generated a proof of a out-of-bound access. The only facts re-
+vealed about the exploitative GIF are what are implied by the
+fact that it trigger an out-of-bound access within 76K steps
+of execution.
+Preprocessing FFmpeg on public inputs There was po-
+tential to aggressively optimize FFmpeg’s proof state-
+ment, which was ultimately achieved by applying CHEESE-
+CLOTH’s constant folding transformation pass after manual
+program partitioning. The need for partitioning arose due to
+the interleaving of public and secret computation in the GIF
+modules, which executes by: (1) demultiplexing a given se-
+cret GIF file into a sequence of data packets; (2) initializing
+the state of the decoder, using public configuration settings;
+(3) executing the codec that contains the vulnerability.
+Although phase (2) computes entirely over public data, it
+would not be optimized by CHEESECLOTH’s constant fold-
+ing pass because the pass halts upon detecting computation
+that uses secret data, and thus would not optimize any pro-
+gram segment after phase (1). To address this issue, we man-
+ually partitioned the program by phase, applied CHEESE-
+CLOTH’s constant folding pass to each, and linked the result-
+ing optimized MicroRAM code. In general, our case study
+of FFmpeg motivates the further study and design of more
+aggressive constant folding passes, which might apply more
+sophisticated static program analyses (Secs. 2.3 and 3.2.4).
+6.3
+Leakage in OpenSSL
+OpenSSL [3] is a widely deployed open-source crypto-
+graphic library that contains implementations of the SSL and
+TLS protocols. OpenSSL versions 1.0.1 to 1.0.1f contained
+a devastating vulnerability dubbed Heartbleed, discovered in
+2014 [5], that could be exploited by a remote attacker to
+completely leak information stored over the protocol’s exe-
+cution, including other clients’ sensitive information and pri-
+vate keys.
+Comprehensive descriptions of SSL and OpenSSL are be-
+yond the scope of this paper; for the purposes of our work,
+it suffices to note that SSL parties support multiple requests,
+including both requests to store data from the another party
+and requests to reply to a heartbeat signal: a signal sent only
+10
+
+Program
+Code size (K instrs)
+Execution steps (K)
+Mult gates (M)
+Protocol time
+Protocol memory
+GRIT
+3
+5
+26.7
+3m 40s
+845 MB
+FFmpeg
+24
+79
+672.7
+1h 22m
+19 GB
+OpenSSL
+340
+1,300
+17,049.5
+36h 45m
+460 GB
+Table 1: Results for generating and running a ZK proof of software vulnerability for each case study.
+1
+void process_heartbeat(SSLRequest *req) {
+2
+unsigned int len = parse_heartbeat_len(req);
+3
+unsigned char *heartbeat = get_heartbeat(req);
+4
+unsigned char *response = malloc(len);
+5
+memcpy(response, heartbeat, len);
+6
+write(response, len); }
+Figure 3: Pseudocode depicting the Heartbleed vulnerability.
+to obtain a response to ensure that the other party is still re-
+sponsive.
+The heartbeat request and response is critical to the oper-
+ation of the Heartbleed vulnerability. A well-formed request
+consists of a data buffer d and a length field n < |d|. A cor-
+rect response to such a request returns the first n bytes con-
+tained in d. However, a party could potentially transmit an
+ill-formed request, in which n > |d|. The correct response to
+such an ill-formed request is to reject it.
+The implementation of OpenSSL (illustrated by the pseu-
+docode function process_heartbeat in Fig. 3) crucially
+failed to implement this aspect of the protocol and instead
+returned the n bytes of memory contiguous with the input
+buffer. process_heartbeat takes a heartbeat request from
+a client and echos the provided heartbeat string back. It does
+so by first parsing the length of the heartbeat string from
+the client’s request. The function then gets a pointer to the
+heartbeat string in the request. Next, it allocates a response
+buffer and copies len bytes from the heartbeat string into
+the response buffer, which is subsequently sent back to the
+client. Since process_heartbeat does not check the pro-
+vided heartbeat length against the actual length of the heart-
+beat string, if the claimed length is larger than the actual
+length of the provided heartbeat string, memory beyond the
+client’s request is sent back to the client. This is practically
+exploitable, and has been demonstrated to reveal sensitive in-
+memory data such as cryptographic keys and passwords.
+Using CHEESECLOTH, we proved in ZK that OpenSSL
+version 1.0.1f leaks arbitrary user information in 1.3M steps
+of execution, propagating data purely over register copies,
+loads, and stores; while the statement reveals a bound on the
+amount of computation required to perform the leak and in-
+formation about the types of instructions used to perform the
+leak (described below), it gives no direct indication of what
+validation is missing in the function for processing heartbeat
+requests, or that heartbeat requests are involved in the leak at
+all. We describe the statement proved, along with technical
+challenges and solutions, in more detail below.
+1
+int login_handler(
+2
+SSLConn *c, char *password, int len) {
+3
+...
+4
+label l = getLabel(c);
+5
+for (size_t i = 0; i < len; i++)
+6
+taintSource(password + i, l);
+7
+... }
+8
+int ssl3_write(
+9
+SSLConn *c, char *buf, int len) {
+10
+...
+11
+label l = getLabel(c);
+12
+for (size_t i = 0; i < len; i++)
+13
+taintSink(buf + i, l);
+14
+... }
+Figure
+4:
+Versions
+of
+the
+OpenSSL
+functions
+login_handler
+and ssl3_write
+that we
+augmented
+with operations that specify information sources and sinks.
+Passwords are tainted with the label of the current connec-
+tion, and leaks are detected if data written to the network has
+a label from a different connection.
+Specifying OpenSSL’s leakage A primary challenge of our
+work was to provide a scheme for identifying sensitive
+sources and sinks such that:
+1. A verifier with only an understanding of the data that a
+subject program handles should be able to inspect the
+modified program and definitively conclude that it cor-
+rectly defines information sources and sinks.
+2. Any modifications to the program to enable the defini-
+tion of sources and sinks between which information is
+leaked must not reveal additional information about the
+leak’s triggering input.
+Our mechanism for defining sensitivity of sources and
+sinks consists of the designated functions taintSource and
+taintSink (Sec. 5.2). We found that such a library served
+well for specifying information flow in in OpenSSL; psue-
+docode of C functions modified in the OpenSSL codebase
+to label sources and sinks are given in Fig. 4. The function
+login_handler, given an SSL connection c and a buffer
+password presumed to contain len bytes of sensitive infor-
+mation to be transmitted over c, labels len addresses be-
+ginning with password with the label of c. The function
+ssl3_write, given an SSL connection c and buffer buf pre-
+sumed to output len bytes, denotes sinks at the output chan-
+nel with the label of c for len addresses beginning with buf.
+The modifications to login_handler and ssl3_write
+11
+
+Program
+Mult gates without
+public-pc segments (M)
+Mult gates without
+sparsity (M)
+GRIT
+42.3 (37%)
+27.9 (4%)
+FFmpeg
+716.9 (6%)
+709.9 (5%)
+Table 2: The number of multiplication gates in the circuit
+with the different optimizations disabled, as well as the per-
+centage increase in size over the baseline numbers from
+Tab. 1.
+illustrate the utility of first-order labels that can be opera-
+tionally collected and set, as opposed to operations that set
+addresses as only high sources or low sinks, even in a setting
+in which the information belonging to only one principal is of
+interest. By using first-order labels, we we able to write small
+specification functions that only unified the labels between a
+network connection and a given buffer, and then succinctly
+modified the original program logic in contexts that readily
+provided a connection and related buffer.
+Proving OpenSSL’s leakage Once OpenSSL has been suit-
+ably modified to call the taintSource and taintSink func-
+tions, its leakage can be proved by generating a statement
+whose solution corresponds to an execution of a server run-
+ning OpenSSL that leaks sensitive data from one connec-
+tion to another connection. We have generated such a state-
+ment where the server first responds to a public login request
+where the password is marked as sensitive. The server then
+handles a secret malicious heartbeat request that returns the
+password from the previous request’s connection.
+Using CHEESECLOTH we prove OpenSSL’s leakage by
+validating the previously described execution which is de-
+rived from one of its originally disclosed exploits. The leak-
+age is detected through the source and sink annotations ac-
+cording to our proposed must-leak labeling scheme (Sec. 5.2)
+and the verifier only learns an upper bound on the length of
+the malicious request. We found that the labeling scheme en-
+abled leakage to be proved much more efficiently, reducing
+the overall circuit size by 30.6% over the two trace approach.
+CHEESECLOTH proved the vulnerability of OpenSSL in ap-
+proximately 37 hours, using 460 GB of protocol communica-
+tion.
+6.4
+Optimizations
+Tab. 2 contains the improvements yielded by our key opti-
+mizations (Sec. 4). We ran the GRIT and FFmpeg case stud-
+ies with each optimization disabled and report on the num-
+ber of multiplication gates in the resulting ZK circuit. In
+addition, we provide the percentage improvement over the
+baseline numbers from Tab. 1. The public-pc optimization
+reduced gate size by 37% in the shorter GRIT execution and
+6% for FFmpeg. While this is an improvement, these results
+indicate there is still room for improvement in our analysis
+that determines the number of public segments to generate
+for longer executions. The sparsity optimization with s = 2
+offers modest improvements of 4%–5% in gate size.
+7
+Related Work
+Recent work has provided the first, exciting steps toward
+proofs of vulnerability in ZK. BubbleRAM [29] is an ef-
+ficient framework for proving vulnerability, leverage novel
+protocols for converting between computations in arithmetic
+and Boolean fields, efficiently handling both read-only and
+read-write memory, and the Stack protocol [30] for prov-
+ing satisfaction of circuits with explicit disjunctions. Al-
+though our current statement compiler partially overlaps with
+BubbleRAM because it implements an older scheme for mod-
+eling RAM computations [10], most of our paper’s key con-
+tributions, namely simplifying unrolled computations using
+partial evaluation and the novel scheme for generating state-
+ments of application leakage, are largely independent of the
+contributions of [29,30], and we believe that the approaches
+could be composed. In particular, Stack was evaluated on
+code snippets representative of a practical CVE of up to 50
+LoC; due to its efficient support of disjunctions, it could scale
+to prove that one of many more such snippets is vulnera-
+ble, but it likely strongly benefit from CHEESECLOTH’s pro-
+gram optimizations if any particular code segment increased
+in size.
+Reverie [27] is a framework for proving exploits in mi-
+croprocessor code, consisting of a circuit generator that com-
+piles a given program to an arithmetic circuit and an instan-
+tiation [36] of the “MPC in the Head” protocol [33]. The
+compiler generates statements from exploits that have been
+formalized as executing a designated instruction that signals
+an error condition (i.e., violations of reachability properties,
+formalized directly in the program’s control flow); the eval-
+uation of Reverie demonstrates that it can be used to prove
+Capture the Flag (CTF) exploits that require up to 51K cy-
+cles on an MSP430 microprocessor. The core contributions
+of Reverie are largely complementary to those of CHEESE-
+CLOTH, which could potentially be adapted to efficiently
+compile vulnerability statements about programs in interme-
+diate languages to control reachability properties.
+Recent work on static program analysis in ZK [21] has
+presented techniques for proving over-approximations of all
+program executions without revealing further details of the
+program, and instantiates the framework on an abstract do-
+main for information flow based on taint tracking. The static
+analysis itself is designed to prove that a program may leak
+information: thus, it cannot yield results that directly imply
+that a program must leak, although in many cases it could
+provide evidence that could strongly inform an analysts be-
+lieft that a program may in fact leak.
+Our MicroRAM machine is inspired by TinyRAM [10] but
+departs from their design in sevaral important ways discussed
+12
+
+in Sec. 3.1. There are also some key differences in scope and
+capabilities. TinyRAM is designed to express correctness
+of any nondeterministic computations while MicroRAM fo-
+cuses on vulnerable programs. For example, SNARKs for C
+[10] approach cannot encode proofs of memory-safety vul-
+nerability in ZK directly. Instead, they encode knowledge
+of the existence of a complete, concrete vulnerability trace,
+which includes copies of exact values in all local variables
+and the values in memory at each point in the trace and the
+bug must be evident in the execution’s return value. Our ap-
+proach encodes memory vulnerabilities directly, resulting in
+a significantly more succinct witnesses to vulnerability. In
+particular we can disregard the trace after the bug is found
+and we don’t rely on the programs return value.
+Furthermore, TinyRAM approach does not scale to proofs
+of vulnerabilities in practical programs and has only been
+evaluated on programs with less than 1,200 low-level instruc-
+tions [10]. In contrast, the optimizations proposed in this
+work enables us to support programs with more than 340,000
+lines of low-level code (Tab. 1). Beyond scalability, Micro-
+RAM supports a much broader subset of the C language, in-
+cluding most of the standard C library.
+Pantry and Buffet [17, 52] represent computation as arith-
+metic constraints; a solution to the constraints is a valid trace
+of the computation. After implementing the memory consis-
+tency approach of TinyRAM, they report results orders of
+magnitude better than TinyRAM. Buffet supports all features
+in the C language, with the exceptions of goto statements
+and function pointers. To translate computation into a con-
+straint system, Pantry and Buffet must unroll loops to pub-
+licly revealed bound (although the original work does not
+explicitly discuss encoding recursive functions, we hypoth-
+esize that they would be encoded similarly, using bounded
+function inlining). The constraint system must include every
+branch of conditionals and every iteration of every loop (mul-
+tiplicatively with nested loops) which could lead to blowups
+in the constraint system, howeverthe authors suggest that this
+would only happen in degenerated cases and would not be
+common in practice. A variant Pantry/Buffet that uses zero-
+knowledge techniques to keep the state private with the same
+efficiency benefits. When presenting our approach, we com-
+pare facts about private inputs that it reveals to those revealed
+from public loop bounds (Sec. 3.2.3).
+vRAM [53] has achieved further efficiency with a inge-
+nious universal preprocessing that allows the parties to use
+a smaller circuit tailored to verifying the specific program
+on the chosen inputs. Unfortunately, such tailored circuits
+can reveal significant information about the input provided.
+Our public-pc optimization (Sec. 4.1) attempts to balance the
+gains of a tailored circuit and the privacy requirements of the
+prover.
+8
+Conclusion
+Due to a sustainted successes in the development of ZK
+protocols, recent techniques have reached the cusp of prov-
+ing knowledge of realistic vulnerabilities and proving sub-
+tle exploits in low-level code. This paper describes how
+a host of core techniques from compiler design—namely,
+conservative instruction profiling and under-approximating
+information-flow tainting—can be implemented in an opti-
+mizing proof-statement generator to produce proofs of vul-
+nerability in commodity software that can be triggered only
+be using a considerable amount of time and space.
+Our practical experience has produced a zero-knowledge
+proof of memory unsafety in FFmpeg and a proof of leakakge
+in OpenSSL that directly used the Heartbleed exploit as a wit-
+ness and demonstrates that zero knowledge proofs of vulner-
+ability in critical application software are now practical.
+Availability and Ethical Considerations
+We are in process of open sourcing the implementation
+of CHEESECLOTH for publication and artifact evaluation.
+CHEESECLOTH aids in responsible disclosure by produc-
+ing zero-knowledge proofs of the existence of vulnerabili-
+ties while keeping the vulnerabilities and exploits secret. All
+vulnerabilities used in our evaluation have been previously
+disclosed publicly, and fixes are widely deployed. Thus, the
+work presented in this paper does not constitute an unethical
+disclosure of potentially harmful information.
+Acknowledgments
+This material is based upon work supported by the Defense
+Advanced Research Projects Agency (DARPA) under Con-
+tract No. HR001120C0085. Any opinions, findings and con-
+clusions or recommendations expressed in this material are
+those of the author(s) and do not necessarily reflect the
+views of the Defense Advanced Research Projects Agency
+(DARPA). Approved for Public Release, Distribution Unlim-
+ited.
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+

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+ACCELERATED PATHS AND UNRUH EFFECT II: FINITE TIME
+DETECTOR RESPONSE IN (ANTI) DE SITTER SPACETIME AND
+HUYGEN’S PRINCIPLE.
+A PREPRINT
+Shahnewaz Ahmed,a,b,c, Mir Mehedi Faruk,d,e,f, and Muktadir Rahmang
+aSchool of Data and Sciences, BRAC University, 66 Mohakhali, Dhaka 1212. Bangladesh
+bPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
+cDepartment of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
+dDepartment of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada
+eInstitute for Theoretical Physics, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, The Netherlands.
+fDelta Institute for Theoretical Physics, Science Park 904, PO Box 94485, 1090 GL Amsterdam, The Netherlands.
+gDepartment of Physics, University of Nevada, Reno 1664 N Virginia St, Reno, NV 89557.
+January 23, 2023
+ABSTRACT
+We study the finite time response of an Unruh-DeWitt particle detector described by a qubit
+(two-level system) moving with uniform constant acceleration in maximally symmetric spacetimes.
+The D dimensional massless fermionic response function in de Sitter (dS) background is found to
+be identical to that of a detector linearly coupled to a massless scalar field in 2D dimensional dS
+background. Furthermore, we visit the status of Huygen’s principle in the Unruh radiation observed
+by the detector.
+A uniformly accelerating observer of constant acceleration a moving in Minkowski or maximally
+symmetric spacetime sees the vacuum for an inertial observer as a thermal state of temperature
+T =
+ω
+2π. Here, ω is given by[1, 2, 3, 4, 5, 6],
+ω =
+�
+�
+�
+√
+a2 + k2,
+dS, Λ > 0
+√
+a2 − k2
+AdS, Λ < 0
+a
+Minkowski, Λ = 0
+(1)
+Here the cosmological constant Λ is related to D dimension spacetime curvature, k through |Λ| =
+k2
+2 (D−2)(D−3). In recent times, Unruh radiation and its close analogue Hawking radiation has been
+extensively studied with the tool of the Unruh-Dewitt (UDW) particle detector. The UDW detector
+has applications connecting other branches of physics including but not limited to understanding
+harvesting entanglement[7, 8, 9, 10], QCD[11], complexity[12], cosmology[13, 14, 15], as well as
+in application-oriented research directions such as condensed matter systems[16, 17] like anyons[18]
+and constructing heat engines[19, 20] using UDW detector. The accelerated UDW detector shows a
+response when coupled to matter field. In one of the pioneering work on the topics related to detector
+physics was by Takagi[21] where interesting features of detector response function were elaborated.
+A complete story on fermionic response function to accelerated detectors in flat spacetime was
+developed recently in [22]. It was noted in that [22] in D-dimensional Minkowski spacetime, the
+arXiv:2301.08717v1  [hep-th]  20 Jan 2023
+
+A PREPRINT - JANUARY 23, 2023
+response of the accelerated UDW detector coupled to massless Dirac field proportional to that of a
+detector linearly coupled to a massless scalar field in 2D dimension. This observation was quite
+interesting as it helped us measure the Unruh radiation observed by the detector when coupled to
+the fermionic matter field. But it is a natural question to ask if a similar conclusion also arises when
+the fermionic field is coupled to curved spacetime. In our previous article[23] we explained how
+the same mechanism works in AdS background instead. Another significant observation made by
+several authors[24, 25] is the apparent statistics inversion in the Unruh radiation in odd dimensional
+spacetime. In odd-dimensional spacetime, we notice that the thermal radiation measured by a linear
+UDW particle detector coming from scalar field can maintain an anti periodic relation. Our previous
+article explained how non-linearity affects the statistics inversion in AdS spacetime. Of course,
+when the curvature of the spacetime was approaching zero, we reproduced the known results of
+detector response in flat spacetime[6, 23] but the similar setup in the dS background still needs to
+be discussed. In this article, we analyze the effects of non-linearity and develop the calculation
+of the fermionic response function in the dS background. In our earlier work and many other
+studies relating to the UDW detector, we investigated the response function where the detector is
+"turned on" for an infinite amount of time [6, 21, 23]. In practice it is impossible to turn on the
+detector for infinite time[20]; therefore, the finite time response has been rigorously investigated in
+the context of flat space time [26]. We start our manuscript by analysing the finite time response
+of accelerated an UDW detector in AdS spacetime coupled to real scalar fields in a non-linear
+way. In section 2, we first elaborate on the dS case for the real scalar fields and then finish the
+calculation for fermionic fields. We prove here the response function of the uniformly accelerated
+UDW detector coupled to a massless Dirac field in dS spacetime of dimension is equivalent to the
+response function of the detector linearly coupled to a massless scalar field in 2D dimensional dS
+spacetime. We have generalised the result in the case of non trivial gravitational background AdS
+spacetime in part-1[23] and dS spacetime in this article. Finally, we summarise the cases when
+Huygen’s principle is maintained or violated by the Unruh radiation observed by the accelerated
+detectors in maximally symmetric spacetime. Throughout the whole article, we chose ℏ = 1, c = 1
+and Boltzmann constant kB = 1 in our calculation.
+1
+Finite time response of UDW detector: Scalar field
+We first consider a real scalar field Φ in D dimensional (A)dS spacetime which is conformally
+coupled to gravitational background. We are are considering the AdS metric in Poincare coordinates
+but we can ofcourse choose any other coordinates system such as global coordinates. Similarly we
+will follow the flat slicing for De Sitter background. The AdS metric in Poincare coordinate,
+ds2 =
+1
+k2z2(dt2 − dx2
+1 − dx2
+2 − ... − dx2
+D−2 − dz2).
+(2)
+The dS metric in flat slicing is written as
+ds2 =
+1
+k2η2(dη2 − dx2
+1 − dx2
+2 − ... − dx2
+D−2 − dx2
+D−1).
+(3)
+Depending upon what we would like to have as our our gravitational background we pick AdS or
+dS we choose eq.(2) or eq. (3). The total action of our system of interest is,
+S = S0 + Sint + Sdetector,
+(4)
+2
+
+A PREPRINT - JANUARY 23, 2023
+The matter field action is simply-
+S0 = 1
+2
+�
+dDx
+�
+|g|
+�
+gµν▽µΦ▽νΦ + ζRΦ2�
+.
+(5)
+When scalars are conformally coupled to gravity one can specify[27],
+ζ =
+D − 2
+4(D − 1).
+(6)
+The interaction part of the Hamiltonian is simply,
+HI(τ) = λχT (ˆσ−(τ) + ˆσ+(τ))Φ(z(τ)),
+(7)
+where, λ is the strength of coupling, χT is a switching function which controls the time of interaction
+with the field, and n is any positive integer. T represents how long the detector is on. We are
+going to refer it as switching time. It is well known that any sudden jump in the switching function
+may cause divergence[20] for finite time interaction. Therefore we choose a Lorentzian switching
+function (a smooth function),
+χT (τ) =
+(T /2)2
+τ 2 + (T /2)2
+(8)
+One can simply set T → ∞, or χT = 1 in order to obtain the usual detector response (where it
+is assumed that the detector can interact with the matter fields for infinite time). The detector is
+thought as a two-level quantum system defined along a worldline x(τ). The detector Hamiltonian
+is,
+ˆHD = Ω
+2
+�
+ˆσ+ˆσ− − ˆσ−ˆσ+�
+,
+(9)
+We are thinking the UDW detector as two level system. There are two states |g⟩ and |e⟩ and Ω is the
+energy gap between these two states. The ˆσ+ and ˆσ− are the well known SU(2) ladder operators.
+|e⟩ = ˆσ+ |g⟩ ,
+(10)
+ˆσ− |g⟩ = 0
+(11)
+From this Hamiltonian we can easily see the ground and excited states of the detector are |g⟩ and
+|e⟩, respectively.
+ˆHD |e⟩ = Ω
+2 |e⟩
+(12)
+ˆHD |g⟩ = −Ω
+2 |g⟩
+(13)
+Point to note that ˆHD generates time translations with respect to the detector’s proper time τ. We
+are assuming that the detector follows a timelike trajectory x(τ) which parametrized by proper
+time τ in D dimensional spacetime. Any real scalar quantum field Φ(x) can be written as a mode
+expansion,
+Φ(x) =
+�
+dlk
+�
+fk(x)ˆbk + f ∗
+k(x)ˆb†
+k
+�
+,
+(14)
+where {uk(x)} is assumed to be a normalized basis of solutions to the Klein-Gordon equation.
+The functional form is fixed from the gravitational background. the annihilation and creation
+operators maintaining the usual commutation relations are ˆbk,ˆb†
+k, respectively. In principle the
+creation/annihilation operators help can be used construct a Hilbert space representation for the
+3
+
+A PREPRINT - JANUARY 23, 2023
+quantum field, defined in terms of the vacuum state |0⟩. The most interesting quantity to us is the
+probability amplitude related to the transition from the initial state |g, 0⟩ to a state |e, ϕ⟩. Here |ϕ⟩
+denotes an arbitrary final state of the field. The amplitude can be found following[28],
+Ag→e(ϕ) = ⟨e, ϕ| UI |g, 0⟩ =
+∞
+�
+n=0
+⟨e, ϕ| U (n)
+I
+|g, 0⟩
+(15)
+=
+�
+n odd
+λn(−i)n
+n!
+�
+dτ1 · · · dτn χ(τ1) · · · χ(τn) ⟨ϕ| T
+�
+ˆφ(τ1) · · · ˆφ(τn)
+�
+|0⟩ eiΩ(τ1−τ2+···+τn),
+Here UI the time evolution operator is given in the usual way,
+UI
+= T exp
+�
+−i
+�
+dτHI(τ)
+�
+= �∞
+n=0
+(−i)n
+n!
+�
+dτ1 · · · dτnT
+�
+HI(τ1) · · · HI(τn)
+�
+�
+��
+�
+U (n)
+I
+= �∞
+n=0 U (n)
+I
+(16)
+One can determine the transition probability to arbitrary order by tracing over the field final states
+|ϕ⟩,
+Pg→e =
+�
+Dϕ |Ag→e(ϕ)|2
+=
+�
+n,m odd
+λn+m(−i)n−m
+n!m!
+�
+dτ ′
+1 . . . dτ ′
+m dτ1 · · · dτn χ(τ ′
+1) · · · χ(τ ′
+m)χ(τ1) · · · χ(τn)
+× ⟨0| T
+�
+Φ(τ ′
+1) · · · Φ(τ ′
+m)
+�†
+T
+�
+Φ(τ1) · Φ(τn)
+�
+|0⟩ e−iΩ(τ ′
+1−···+τ ′
+m)eiΩ(τ1−···+τn),
+(17)
+In this series the lowest order term is of second order in the coupling constant λ. It is expressed as,
+P(2)
+g→e = λ2
+�
+dτdτ ′χ(τ)χ(τ ′) ⟨0| Φ(τ)Φ(τ ′) |0⟩ eiΩ(τ−τ ′) = λ2
+�
+dτdτ ′χ(τ)χ(τ ′)W (2)
+D (x(τ), x′(τ ′))eiΩ(τ−τ ′).
+(18)
+Here, W (2)
+D (x(τ), x′(τ ′)) is the D dimensional two point correlator (Wightman function). The exact
+functional form of the Wightman function again depends upon the background gravity.
+We can consider more general interaction Hamiltonian,
+HI = λ χT (τ)m(τ) OΦ[x(τ)] ,
+(19)
+Here, m(τ) the monopole operator,
+m(τ) = eiΩτ |e⟩ ⟨g| + e−iΩτ |g⟩ ⟨e| =
+�
+0
+e+iΩτ
+e−iΩτ
+0
+�
+.
+(20)
+This actually takes the ground state of the detector to the excited state and vice versa. In other words
+we can physically describe the procedure as "a click" in response to the presence of the field. Now
+the the operator OΦ outlines how the matter field is coupled to the detector. Instead of usual linear
+coupling we take a more general coupling1,
+OΦ[x(τ)] = Φn[x(τ)]
+(21)
+1normal ordering is assumed
+4
+
+A PREPRINT - JANUARY 23, 2023
+If we use the interaction Hamiltonian (21) then we re-express eq. (18),
+P(2)
+g→e = λ2
+�
+dτdτ ′χ(τ)χ(τ ′)W (2n)
+D
+(x(τ), x′(τ ′))eiΩ(τ−τ ′)
+(22)
+Here, W (2n)
+D
+(τ − τ ′) = ⟨0| : Φn(x(τ)) :: Φn(x(τ ′)) : |0⟩ is the 2n-point correlator. The detector
+response function of the UDW detector is directly proportional to the probability for the detector
+to transition from ground state to excited state. Using Lorentzian switching function eq. (8) the
+response function F(2n)(Ω, T ) will be,
+F(n)(Ω, T ) = πT 3
+4
+� ∞
+−∞
+d(∆τ) W (n)
+D (∆τ)
+∆τ 2 + T 2 e−iΩ∆τ
+(23)
+The 2n-point function W (2n)
+D
+(x, x′) is related to the the Wightman function in the following
+interesting but simple way by Wick’s theorem [29],
+W (2n)
+D
+(x, x′) = (n!)
+�
+W (2)
+D (x, x′)
+�n .
+(24)
+1.1
+Finite time response of scalar fields: AdS spacetime
+For conformally coupled scalars the Wightman function in D > 2 dimensional AdS spcatime can
+be obtained in the following form with suitable boundary condition[6],
+W (2)
+AdSD(x, x′) = ⟨0| Φ(x(τ))Φ(x(τ ′)) |0⟩ = CD
+�
+1
+(v − 1)D/2−1 −
+1
+(v + 1)D/2−1
+�
+.
+(25)
+Here, ν is the conformal invariant defined as,
+v = z2 + z′2 + (x − x′)2 − (t − t′ − iϵ)2
+2zz′
+.
+(26)
+CD is a constant,
+CD = kD−2Γ(D/2 − 1)
+2(2π)D/2
+,
+(27)
+In this article we mainly focus on Unruh effect is a widely studied phenomena which basically
+states an accelerating observer (with constant linear acceleration a) will observe a thermal bath with
+temperature T. If the observer were in flat spacetime, the temperature is given by the following
+formula T =
+ℏa
+2πckB . In the usual literature[30] (studies mostly done in flat spacetime) the path
+which was chosen for the accelerating observer had linear uniform acceleration a. The accelerating
+observer (detector) can take a circular path with constant velocity v, will end up having constant
+acceleration a. Of course the resultant radiation (detector response) due to this type of non-linear
+motion will not be quite thermal radiation as the correlators will not obey the KMS relation [31].
+We are interested in those accelerated paths which corresponds to Wightman function maintaining
+valid KMS relation.
+1.1.1
+Super critical accelerated paths in AdS
+We are considering the supercritical paths (a > k) as only these paths results in non zero response
+function for the detectors[6] in uniform linear acceleration. In our recent article[23] we successfully
+showed that using GEMS (Global Embedding Minkowski Spacetimes) approach that one can
+construct a path with constant acceleration by considering the path as an intersection between a
+5
+
+A PREPRINT - JANUARY 23, 2023
+flat plane of dimension M and D dimensional AdS hypersurface embedded in D + 1 dimensional
+flat spacetime. Here, M(M < D + 1). We also proved that for any uniform linear supercritical
+trajectories would have the same conformal invariant v as a function of proper time.
+v(τ, τ ′) = a2
+ω2 − k2
+ω2 cosh(ω(τ − τ ′) − iϵ).
+(28)
+Here ω =
+√
+a2 − k2. The example of super critical path (with constant linear acceleration) in z − t
+plane is given in ref. [23],
+t(τ) = a
+ωeωτ , z(τ) = eωτ , x1 = x2 = x3 = . . . = xD−2 = 0.
+(29)
+Another supercritical path with constant acceleration a in the x1 − t plane is given by the following
+manner[23],
+z(τ) = z0 , x1(τ) = z0k
+ω cosh(ωτ) , t(τ) = z0k
+ω sinh(ωτ) , x2 = x3 = . . . = xD−2 = 0.(30)
+z0 is a constant and τ is the proper time. We could also define the (30) path in the xi − t direction.
+However we have already showed in ref.[23] that how all uniform accelerating paths with constant
+acceleration are related by AdS isometries. Any supercritical path will result in eq. (28). Therefore,
+following eq. (25), the two point function for uniform acceleration (in any supercrtical path)
+becomes,
+GAdSD(∆τ)
+=
+ωD−2Γ( D
+2 − 1)
+(4π)
+D
+2
+�
+1
+iD−2 sinhD−2( ω∆τ
+2
+− iϵ)
+−
+1
+(sinh
+�
+A + ( ω∆τ
+2
+− iϵ)
+�
+)
+D
+2 −1(sinh
+�
+A − ( ω∆τ
+2
+− iϵ)
+�
+)
+D
+2 −1
+�
+.
+(31)
+Here, sinh A = ω/k.
+ρ- plane
+×
+×
+×i ϵ
+i ϵ + A
+i ϵ - A
+×
+×
+×
+i ϵ + i π
+i ϵ + A + i π
+i ϵ - A + i π
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+·
+×
+×
+×
+×
+× iω /2
+-iω /2
+i ϵ - i π
+i ϵ + A - i π
+i ϵ - A - i π
+×
+×
+×
+i ϵ - i π r
+i ϵ + A - i π r
+i ϵ - A - i π r
+∞
+-∞
+-i∞
+Figure 1: Contour for evaluating ID,n,α.
+6
+
+A PREPRINT - JANUARY 23, 2023
+In our previous article we demonstrated the following relation about the two-point correlator[23],
+W (2n)
+AdSD(∆τ + 2π˙i
+ω ) = (−1)nDW (2n)
+AdSD(∆τ).
+(32)
+F (n)
+AdS4(Ω, T ) vs. Ω
+Figure 2: Plot of the finite-time response of the UDW particle detector in AdS Space-time against energy Ω. (From left
+to right) 1st, 2nd and 3rd columns show the plots for different values of n, with n = 1, n = 2 and n = 3. Each row
+(from top to bottom) shows the variation of the response function with changing a (fixed k = 1, T = 3), changing k
+(fixed a = 4, T = 3) and changing T (fixed a = 4, k = 1) respectively.
+Therefore ω =
+√
+a2 − k2 is the temperature[23]. We rewrite the 2n-point function G(n)
+AdSD in the
+following manner [23],
+G(n)
+AdSD(∆τ) = (n!)Cn
+D
+� ω
+√
+2k
+�n(D−2)
+n
+�
+α=0
+�n
+α
+�(−1)α
+ip
+GD,n,α(ρ)
+(33)
+where,
+GD,n,α(ρ) = (sinh(ρ − iϵ))−p(sinh(A + (ρ − iϵ)))−q(sinh(A − (ρ − iϵ)))−q
+(34)
+ρ = ω∆τ/2
+(35)
+p = 2(n − α)(D/2 − 1)
+(36)
+q = α(D/2 − 1).
+(37)
+So, finally the expression for the finite time response function in AdS spacetime for our interaction
+7
+
+a=2
+a=3
+a=4
+a=5
+a=6
+a=7
+k=0
+k=0.5
+k=1
+k=1.5
+k=2
+k=2.5
+k=3
+T=1.5
+T=2
+T=2.5
+T=3
+ T=3.5A PREPRINT - JANUARY 23, 2023
+Hamiltonian becomes,
+F(n)(Ω, T )
+=
+πn!T 3Cn
+D
+4
+�
+ω
+√
+2k
+�n(D−2) � ∞
+−∞ d(∆τ) e−iΩ∆τ
+∆τ 2+T 2
+�n
+α=0
+�n
+α
+� (−1)α
+ip GD,n,α(ρ)
+(38)
+=
+πn!T 3Cn
+D
+4
+�
+ω
+√
+2k
+�n(D−2) �n
+α=0
+�n
+α
+� (−1)α
+ip
+� ω
+2
+� � ∞
+−∞
+dρ
+e−i(2Ω/ω)ρ
+ρ2 + (ωT /2)2GD,n,α(ρ)
+�
+��
+�
+FD,n,α
+(39)
+=
+ωπn!T 3Cn
+D
+8
+�
+ω
+√
+2k
+�n(D−2) �n
+α=0
+�n
+α
+� (−1)α
+ip FD,n,α
+(41)
+where,
+FD,n,α =
+� ∞
+−∞
+dρ
+e−i(2Ω/ω)ρ
+ρ2 + (ωT /2)2GD,n,α(ρ).
+(42)
+F (n)
+AdS4(Ω, T ) vs. a
+Figure 3: Plot of the Finite-Time response of the UDW Particle detector in AdS Space-time against acceleration a.
+(From left to right) 1st, 2nd and 3rd columns show the plots for different values of n, with n = 1, n = 2 and n = 3,
+respectively. Each row (from top to bottom) shows the variation of the response function with changing Ω (fixed k = 2,
+T = 3), changing k (Ω = 1, T = 3) and changing T (Ω = 1, k = 2) respectively.
+8
+
+Q=1
+Q=2
+Q=3
+Q=4
+0.015
+0.10
+0.003
+0.08
+k=0 
+0.010
+k=0.5
+0.06
+0.002
+k=1.0
+ k=1.5
+0.005
+0.001
+0.02
+T=5
+T=10
+T=15A PREPRINT - JANUARY 23, 2023
+Next, we evaluate FD,n,α by computing the contour integral using semi-circle contour containing
+the lower half of complex ρ plane (shown in Fig. 1). Thus we obtain,
+FD,n,α(T ) = −2πi ×
+�
+sum of the residues at ρ = −iπr, ±A − iπr ( where r = 1, 2, ...)
+and ρ = −iωT /2 of
+GD,n,α(ρ)
+ρ2 + (ωT /2)2
+�
+= −2πi ×
+�
+lim
+ρ→−iωT /2
+e−i(2Ω/ω)ρ
+sinhq (A + ρ) sinhq (A − ρ) sinhp (ρ)
+�
+ρ −
+� iωT
+2
+��+
+∞
+�
+r=1
+�
+lim
+ρ→−iπr−A
+η(q − 1)
+Γ(q)
+�
+1
+cosh(ρ + A)
+d
+dρ
+�q−1
+e−i(2Ω/ω)ρ
+cosh (A + ρ) sinhq (A − ρ) sinhp (ρ)
+�
+ρ2 +
+� ωT
+2
+�2�+
+lim
+ρ→−iπr+A
+η(q − 1)
+Γ(q)
+�
+−1
+cosh(A − ρ)
+d
+dρ
+�q−1
+−e−i(2Ω/ω)ρ
+sinhq (A + ρ) cosh (A − ρ) sinhp (ρ)
+�
+ρ2 +
+� ωT
+2
+�2�+
+lim
+ρ→−iπr
+η(p − 1)
+Γ(p)
+�
+1
+cosh ρ
+d
+dρ
+�p−1
+e−i(2Ω/ω)ρ
+sinhq (A + ρ) sinhq (A − ρ) cosh (ρ)
+�
+ρ2 +
+� ωT
+2
+�2�
+��
+(43)
+Here, η(p) is Heaviside step function and Γ(p) is gamma function. In our previous case we were
+able to analytically solve the four dimensional response function in AdS when we chose the infinite
+switching time, i.e. T = ∞. However it was not possible to calculate the detector response for
+finite switching time. We have evaluated the response function numerically in finite switching time.
+Finally we have plotted the response function in different ranges of parameter space. In figure 2,
+we plot the detector response as a function of energy gap of the two level UdW detector. In the
+three different columns of figure 2, we have chosen three values of n. We can see that when the
+detector energy gap increases the response goes to zero. In the first row we have fixed the curvature
+and switching time while varying the energy. We can clearly see the response weakens with greater
+value of n. But if we increase the acceleration of the detector the temperature of the radiation also
+increases. Therefore the first row of Fig 2 manifests response function with respect to Ω takes
+greater value when acceleration is increased. Similar trends are noticed when we look at the other
+two plots of figure 1. It is very interesting to see as the curvature of AdS spacetime goes to zero the
+detector response rises. Complete opposite trends are noticed in dS spacetime. But in both cases we
+have better response function if we can "turn on" the detector for longer time.
+As expected from previous discussion if we can accelerate the detectors more and more, we should
+be able have best response from the detector. We also see it in figure 3. In the first row of figure 3
+we can notice when energy gap rises the response weakens. Even if the detector is accelerated with
+higher value it will be difficult to excite the detector if the energy gap is too much. This problem
+persists more if we have non-linear coupling between the matter field and the detector. The trend of
+response function changing with curvature is very interesting. As the curvature increases more and
+more it becomes problematic to excite the detector. However, we should point out AdS spacetime is
+a constant curvature spacetime. So when we plot for different values of k, we mean that we are
+comparing the results the results of response function for different AdS spacetime with different
+curvatures. We do not mean that we are analysing the detector spectrum where the gravitational
+background has varying curvature.
+9
+
+A PREPRINT - JANUARY 23, 2023
+F (n)
+AdS4(Ω, T ) vs. k
+Figure 4: Plot of the Finite-Time response of the UDW Particle Detector in AdS Space-time against the curvature of
+AdS (k) for massless scalar fields. (From left to right) 1st, 2nd and 3rd columns show the plots for the n = 1, n = 2
+and n = 3 coupling to the scalar field. Each row (from top to bottom) shows the variation of the response function with
+changing a (T = 3, Ω = 1), changing Ω (a = 4, T = 3) and changing T (a = 4, Ω = 1) respectively.
+1.2
+Scalar field in dS space
+We now we use the same setup of two level detector as before but now we consider that the
+background spacetime is de Sitter spacetime. The metric of de Sitter spacetime can be expressed by
+so called flat slicing as below-
+ds2 =
+1
+k2η2(dη2 − dx2
+1 − dx2
+2 − ... − dx2
+D−1).
+(44)
+Now, we are going to discuss about the real scalar field Φ that is conformally coupled to dS
+gravitational background. We have the same matter field action as well as the same interaction
+Hamiltonian as in previous section (eq. (19)). Just as before we need to know the two-point
+correlator (Wightman function) in order to evaluate the detector response. The Wightman function
+for "Euclidean" vacuum |0⟩ can easily be obtained for conformally coupled real scalar field [32, 31],
+W (2)
+dSD(x, x′) = ⟨0| Φ(x(η))Φ(x(η′)) |0⟩ = KDv1−D/2
+(45)
+where,
+KD = kD−2Γ(D/2 − 1)
+2(2π)D/2
+.
+(46)
+10
+
+n=1
+n=2
+0.08
+n=3
+0.25
+0.10
+a=4.5
+0.20
+a=5
+0
+0.06
+a=5.5
+0.15
+a=6
+0.06
+0.04
+a=6.5
+0.10
+a=7
+0
+0.02
+0.05
+0.02
+2
+Q=1
+0.0025
+Q=2
+=U
+0.0020
+ Q=4
+0.0015
+0.0010
+0.0005
+T=1.0
+T=1.5
+T=2.0
+T=2.5
+T=3.0
+ T=3.5A PREPRINT - JANUARY 23, 2023
+Here ν is the conformal invariant.
+ν = (⃗x − ⃗x′)2 − (η − η′ − iϵ)2
+2ηη′
+.
+(47)
+In order to examine Unruh radiation through the detectors need to move through a constant
+accelerated path in dS background. An example of accelerating path in dS spacetime with constant
+linear acceleration (see Appendix 5), we can choose the following,
+η(τ) = τ0eωτ , x1(τ) = a
+ωτ0eωτ , x2 = x3 = . . . = xD−1 = 0,
+(48)
+where ω =
+√
+a2 + k2. Plugging η and xi from eq. (48) to eq. (47), the conformal invariant ν takes
+the following form in this case,
+ν = ( a
+ωτ0eωτ − a
+ωτ0eωτ ′)2 − (τ0eωτ − τ0eωτ ′)2
+2τ 2
+0 eωτeωτ ′
+= 1
+2
+� a2
+ω2 − 1
+� �eωτ − eωτ ′
+eω(τ+τ ′)/2
+�2
+= − H2
+2ω2
+�
+eω∆τ/2 − e−ω∆τ/2�2
+= −2H2
+ω2 sinh2(ω∆τ/2)
+(49)
+Following eq. (45), the two point function for uniformly accelerating paths becomes,
+GdSD(∆τ)
+=
+ωD−2Γ( D
+2 − 1)
+(4π)
+D
+2
+1
+iD−2 sinhD−2(ω∆τ/2 − iϵ).
+(50)
+We can define the transition probability rate or detector’s response function (per unit time)2 for
+interaction Lagrangian (7) of scalars[27],
+F(n)
+dSD =
+� ∞
+−∞
+d∆τe−iE∆τW (2n)
+dSD (∆τ).
+(51)
+Here, W (2n)
+dSD (τ − τ ′) = ⟨0| : Φn(x(τ)) :: Φn(x(τ ′)) : |0⟩ is the 2n correlator. The 2n-point function
+W (2n)
+dSD is related to the the Wightman function in the following way by Wick’s theorem [29],
+W (2n)
+dSD (x, x′) = (n!)
+�
+W (2)
+dSD (x, x′)
+�n
+.
+(52)
+So, the 2n correlator becomes,
+W (2n)
+dSD (∆τ) = (n!)Kn
+D
+�
+ω
+√
+2H
+�n(D−2)�
+1
+iD−2 sinhD−2( ω∆τ
+2
+− iϵ)
+�n
+.
+(53)
+Now the KMS condition can be easily checked using the equation (53).
+W (2n)
+dSD (∆τ + 2πi
+ω )
+=
+(n!)Kn
+D
+�
+ω
+√
+2H
+�n(D−2)�
+1
+iD−2 sinhD−2( ω
+2
+�
+∆τ + 2πi
+ω
+�
+− iϵ)
+�n
+=
+(−1)nDW (2n)
+dSD (∆τ).
+(54)
+2In this scenario, we are considering the detector can be switched on for infinite time.
+11
+
+A PREPRINT - JANUARY 23, 2023
+F (n)
+dS4(Ω, T ) vs. Ω
+Figure 5: Plot of the finite-time response of the UDW particle detector in dS space-time against energy Ω. (From left
+to right) 1st, 2nd and 3rd columns show the plots for different values of n, with n = 1, n = 2 and n = 3. Each row
+(from top to bottom) shows the variation of the response function with changing a (fixed k = 3, T = 3), changing k
+(fixed a = 3, T = 3) and changing T (fixed a = 3, k = 3) respectively.
+This behavior is similar to the AdS space for nonlinear coupling [33] with a major difference with
+radiation temperature being ω =
+√
+a2 + k2[3]. We can obtain the Unruh-Dewitt detector response
+function by taking α → 0 in equation (37) in [33].
+F(n)
+dSD =
+�
+n! Kn
+D
+�
+ω
+√
+2H
+�n(D−2)(−1)n(D−2)+1
+in(D−2)
+2
+ωID,n
+�
+1
+e2πE/ω − (−1)n(D−2).
+(55)
+where,
+ID,n = 2πi ×
+1
+Γ(n(D − 2)) lim
+ρ→0
+��
+1
+cosh ρ
+d
+dρ
+�n(D−2)−1 e−i 2E
+ω ρ
+cosh (ρ)
+�
+.
+(56)
+Finally, to calculate the finite-time Unruh-DeWitt detector response function for dS spacetime, we
+12
+
+a=0
+a=1
+a=2
+a=3
+a=4
+a=5
+a=6
+K=U
+k=
+k=2
+K=3
+K=4
+k=5
+=6
+T=1
+T=2
+T=3
+T=4
+T=5
+T=6A PREPRINT - JANUARY 23, 2023
+F (n)
+dS4(Ω, T ) vs. a
+Figure 6: Plot of the Finite-Time response of the UDW Particle detector in dS Space-time against acceleration a. (From
+left to right) 1st, 2nd and 3rd columns show the plots for different values of n, with n = 1, n = 2 and n = 3. Each row
+(from top to bottom) shows the variation of the response function with changing Ω (fixed k = 3, T = 3), changing k
+(Ω = 1, T = 3) and changing T (Ω = 1, k = 3) respectively.
+plug in the 2n-point correlator G(n)
+dSD from eq. (53) to eq. (51) which gives,
+F(n)
+dSD(Ω, T ) = πT 3
+4
+� ∞
+−∞
+d(∆τ) G(n)
+dSD(∆τ)
+∆τ 2 + T 2 e−iΩ∆τ
+= πT 3n!
+4
+�Γ(D/2 − 1)
+(4π)D/2
+�n �ω
+i
+�n(D−2) � ∞
+−∞
+d(∆τ)
+e−iΩ∆τ
+∆τ 2 + T 2
+1
+sinhn(D−2)( ω∆τ
+2
+− iϵ)
+= πT 3n!
+4
+�Γ(D/2 − 1)
+(4π)D/2
+�n �ω
+i
+�n(D−2) �ω
+2
+� � ∞
+−∞
+dρ
+e−2iΩρ/ω
+ρ2 + (ωT /2)2
+1
+sinhn(D−2)(ρ − iϵ)
+�
+��
+�
+FD,n
+= πT 3n!
+4
+�Γ(D/2 − 1)
+(4π)D/2
+�n �ω
+i
+�n(D−2) �ω
+2
+�
+· FD,n
+(57)
+where,
+FD,n =
+� ∞
+−∞
+dρ
+e−2iΩρ/ω
+ρ2 + (ωT /2)2
+1
+sinhn(D−2)(ρ − iϵ)
+(58)
+Finally, we evaluate the finite time response function F(n)
+dSD numerically from (57) and plot it with
+respect to energy gap, acceleration, and curvature as before. Just like the case of AdS, we can also
+13
+
+Q=1
+=U 
+Q=3
+Q=4A PREPRINT - JANUARY 23, 2023
+F (n)
+dS4(Ω, T ) vs. k
+Figure 7: Plot of the Finite-Time response of the UDW Particle Detector in dS Space-time against the curvature of
+space-time (k) for massless scalar fields. (From left to right) 1st, 2nd and 3rd columns show the plots for different
+values of n, with n = 1, n = 2 and n = 3. Each row (from top to bottom) shows the variation of the response function
+with changing Ω (a = 3, T = 3), changing a (T = 3, Ω = 1), and changing T (a = 3, Ω = 1) respectively.
+see that as the energy gap increases the response function decreases (Fig. 5). In the first row of
+figure 5 we can see if the value of acceleration is higher the response function.
+2
+Finite time response of UDW detector: Dirac fields.
+2.1
+Dirac fields in AdS
+In the similar fashion we can analyse the response function for fermions in AdS spacetime minimally
+coupled to background gravity. The fermionic matter field action is-
+S0 =
+�
+dDx
+�
+|g|¯Ψi /DΨ.
+(59)
+We can consider usual interaction Hamiltonian[20],
+HInt = λ χT (τ)m(τ) OΨ[x(τ)] ,
+(60)
+Here, the operator OΨ[x(τ)] is the normal ordered bispinor,
+OΨ[x(τ)] =: ¯Ψ[x(τ)]Ψ[x(τ)] :
+(61)
+14
+
+Q=1
+Ω=2
+Q=3
+ Q2=4A PREPRINT - JANUARY 23, 2023
+Using Lorentzian switching function eq. (8) the response function for interaction Hamiltonian (60)
+takes the following form,
+F(n)(Ω, T ) = πT 3
+4
+� ∞
+−∞
+d(∆τ) S(4)
+D (∆τ)
+∆τ 2 + T 2e−iΩ∆τ.
+(62)
+Here, the four point function S(4)
+D ,
+S(4)
+D (x(τ), x(τ ′))
+=
+⟨0| :
+�
+Ψa(x(τ)))Ψa(x(τ))
+�
+::
+�
+Ψb(x(τ ′)))Ψb(x(τ ′))
+�
+: |0⟩
+=
+Tr[S+(x, x′)S−(x′, x)].
+(63)
+As discussed in [23] for fermions in AdS spacetime,
+S(2)
+D (∆τ) = N (Γ(D/2))2
+Γ(D − 1) GAdS2D(∆τ)
+(64)
+From eq. (100) we can then conclude that detector response function for fermions can be related to
+response function for the scalars,
+JAdSD(∆τ) = N (Γ(D/2))2
+Γ(D − 1) F(1)
+AdS2D(∆τ).
+(65)
+In the next section we work on fermions in dS spacetime and we work out relations similar to eq.
+(64). Therefore, we further demonstrate that eq. (65) holds for maximally symmetric spacetime.
+The proof is similar to the case of AdS but we explicitly demonstrate it for the sake of completeness.
+2.2
+Dirac fields in dS
+In order to study Dirac field in de Sitter spacetime we choose a local Lorentz frame (vielbein) which
+is defined as, ea
+µ = δa
+µ/(Hτ) such that gµν = ea
+µeb
+νηab mimics the de-Sitter metric. Here Latin letters
+a, b corresponds to local orthonormal flat coordinates and Greek letters µ, ν signifies the de Sitter
+coordinates. Both of them takes value from 0 to D − 1. Also ηab = diag(+1, −1, . . . , −1) is the
+local flat metric. The vielbeins follow the usual orthonormal relations. Now, the curved space Γ
+matrices and the covariant derivatives are defined as,
+Γµ = eµ
+aγa,
+Dµ = ∂µ + 1
+2ωbc
+µ Ωbc,
+(66)
+where γa are gamma matrices in flat spacetime. And commutator between γ matrices are identified
+as Ωbc.
+Ωbc = 1
+4[γb, γc]
+(67)
+and the spin connections ωbc
+µ are noted as,
+ωab
+µ = eaλ�
+∂µeb
+λ −
+�
+α
+µ λ
+�
+eb
+α
+�
+(68)
+and
+�
+α
+µ λ
+�
+are the Christoffel symbols related to dS spacetime metric eq. (1). Here Γµ and γa
+maintain the well-known Clifford algebra,
+{Γµ, Γν}
+= 2gµνIN×N
+{γb, γc}
+= 2ηbcIN×N.
+(69)
+15
+
+A PREPRINT - JANUARY 23, 2023
+SAdS4(Ω, T )
+Figure 8: Plot of the Finite-Time response of the UDW Particle Detector in AdS Space-time coupled to a fermionic
+Field. Each row (from top to bottom) shows the plot of the response function against Ω (varying a[T = 3, k = 1],
+varying k[T = 3, a = 4] and varying T [a = 4, k = 1]), against k (varying a[Ω = 1, T = 3], varying Ω[a = 4, T = 3]
+and varying T [a = 4, Ω = 1]), and against a (varying Ω[k = 2, T = 3], varying k[Ω = 1, T = 3] and varying
+T [k = 2, Ω = 1]) respectively.
+with,
+N =
+�
+2
+D
+2
+D is even
+2
+D−1
+2
+D is odd.
+(70)
+In dS spacetime, the Dirac operator takes the form,
+/D = ΓµDµ ≡ eµ
+aγa�
+∂µ + 1
+2ωbc
+µ Ωbc
+�
+= kη
+�
+γa∂a − D − 1
+2η
+γ0
+�
+.
+(71)
+To derive this relation we used
+�
+α
+µ λ
+�
+=
+1
+η
+�
+gα0gµλ − δα
+λδ0
+µ − δα
+µδ0
+λ
+�
+,
+(72)
+ωab
+µ
+=
+gβ0
+η
+�
+eb
+µea
+β − ea
+µeb
+β
+�
+.
+(73)
+In dS spacetime minimally coupled Dirac fermions with mass m to background gravity will have
+the action,
+S0 =
+�
+dDx
+�
+|g|(Ψi /DΨ − mΨΨ).
+(74)
+16
+
+k=0
+a=2
+k=0.5
+a=3
+ k=1
+a=4
+k=1.5
+a=5
+ k=2 
+a=6
+k=2.5
+a=7
+k=3 
+12
+ a=4.5 
+Q=1
+ a=5
+Q=2
+10
+a=5.5
+Q=3
+a=6
+Q=4
+a=6.5
+a=7
+Q=1
+K=0
+Q=2
+k=0.5
+Q=3
+k=1
+Q=4
+k=1.5A PREPRINT - JANUARY 23, 2023
+We can split the Ψ field in two parts, namely positive and negative frequency modes.
+Ψ(x) = Ψ+(x) + Ψ−(x).
+(75)
+These are the solution of massive Dirac equation derived from equation (74)
+i /DΨ − mΨ = 0 .
+(76)
+We will first look into the positive energy mode solutions ψ(+) of (76). These solutions are
+proportional to eipx, where x = (x1, . . . , xD−1) and p = (p1, . . . , pD−1), px = plxl, and the
+summation runs over l = 1, . . . , D − 1. We now decompose the positive energy modes into upper
+and lower components,
+ψ(+) =
+�
+ψ+(η)
+ψ−(η)
+�
+eipx.
+(77)
+This can be done by using the following explicit definition Gamma matrix representation,
+γ0 =
+�
+I(N/2)×(N/2)
+0(N/2)×(N/2)
+0(N/2)×(N/2)
+−I(N/2)×(N/2)
+�
+, γa =
+�
+0(N/2)×(N/2)
+σa
+−σa
+0(N/2)×(N/2)
+�
+,
+(78)
+with a = 1, . . . , D − 1. The definitions of I(N/2)×(N/2) and 0(N/2)×(N/2) can be found in [23].
+σaσb + σbσa
+=
+2δabI(N/2)×(N/2),
+σa†
+=
+σa.
+The Dirac equation is then reduced to subsequent form for positive energy modes,
+�
+∂0 − D − 1
+2η
+± im
+kη ψ±
+�
+− iplσlψ∓ = 0.
+(79)
+Using equation (79) we can deduce two different second order differential equations for the upper
+and lower components:
+�
+η2∂2
+0 − (D − 1)η∂0 + p2η2 + (D − 1)2
+4
++ D − 1
+2
++ m2
+H2 ∓ im
+k
+�
+ψ± = 0.
+(80)
+Making the following substitution,
+ψ±(η) = ηD/2χ±(η),
+(81)
+equation (80) is reduced to the following form,
+�
+η2∂2
+0 + η∂0 + (pη)2 −
+�im
+k ± 1
+2
+�2�
+χ± = 0.
+(82)
+Now we write the solutions of (82) as
+χ±(η)
+=
+C±H(2)
+im
+k ± 1
+2(pη)
+(83)
+where H(2)
+ν (x) are Hankel function of second kind of order ν. We only considered Hankel function
+of second kind as the solution for equation (82) because for positive energy solution in Bunch-Davies
+vacuum we demand ψ(+) ∝ e−ipη [34] . The coefficients C+ and C− are not independent of each
+other. We can find the relationship C− = −ipbσbC+/p by inserting the solution matrices (83) and
+(81) into the equation (79). Moreover, we require additional quantum numbers apart from p to
+specify all the solutions. In order to do that we need to fix the spinor C+. Here we take orthonormal
+basis for spinors by choosing C+ = C(+)
+β
+w(σ), where C(+)
+β
+is a normalization constant and w(σ),
+17
+
+A PREPRINT - JANUARY 23, 2023
+σ = 1, . . . , N/2, are one-column matrices of N/2 rows, with elements w(σ)
+l
+= δlσ. Combining with
+the negative energy solutions this set β = (p, σ) will form a complete set of quantum numbers. As
+a result, the positive-energy mode functions for Bunch-Davies vacuum takes the following form,
+ψ(+)
+β
+= C(+)
+β
+ηD/2eipx
+�
+�
+w(σ)k(2)
+im
+k + 1
+2(pη)
+− ipbσb
+p w(σ)k(2)
+im
+k − 1
+2(pη)
+�
+� .
+(84)
+The coefficient C(+)
+β
+in (84) is set on from the normalization condition (using inner product defined
+over constant time hypersurface) [35]
+⟨ψ(+)
+β , ψ(+)
+β′ ⟩ =
+�
+dD−1x
+�
+|g|
+g00ψ(+)†
+β
+ψ(+)
+β′
+= δ(p − p′)δσσ′.
+(85)
+After evaluating the inner product we find the normalization constant
+C(+)
+β
+=
+√pk
+D−1
+2 e−iφ/2
+�
+8(2π)D−2 e
+mπ
+2k .
+(86)
+where φ represents an arbitrary phase. The negative-energy mode functions φ(−) can be imposing
+the condition that φ(−) ∝ e−ipx+ipη. By following the same procedure illustrated above we obtain
+the negative energy solution:
+ψ(−)
+β
+=
+√pk
+D−1
+2 eiφ/2
+�
+8(2π)D−2 e− mπ
+2k ηD/2e−ipx
+�
+�
+w(σ)H(1)
+im
+k + 1
+2(pη)
+ipbσb
+p w(σ)H(1)
+im
+k − 1
+2(pη)
+�
+� .
+(87)
+In the above equation H(1)
+ν (x) are Hankel function of first kind of order ν. Therefore we have
+successfully evaluated the complete set of solutions for the Dirac equation (76). Now we can write
+artibitary spinor solution Ψ(x) in the operator form.
+Ψ(x)
+=
+N/2
+�
+σ=1
+�
+dp
+�
+bσ(p)ψ(+)
+σ (p, x) + d†
+σ(p)ψ(−)
+σ (p, x)
+�
+(88)
+Ψ(x)
+=
+N/2
+�
+σ=1
+�
+dp
+�
+b†
+σ(p, λ)ψ(+)
+σ (p, x) + dσ(p)ψ(−)
+σ (p, x)
+�
+,
+(89)
+where
+bσ(p) |0⟩
+=
+dσ(p) |0⟩ = 0,
+(90)
+ψ
+=
+ψ†γ0,
+(91)
+{bσ(p), b†
+σ′(p′)}
+=
+δ(p − p′)δσσ′,
+(92)
+{dσ(p), d†
+σ′(p′)}
+=
+δ(p − p′)δσσ′.
+(93)
+18
+
+A PREPRINT - JANUARY 23, 2023
+Now we can have an explicit form of the Wightman functions of the fermionic field,
+S+(x, x′)
+=
+⟨0| Ψ(x)Ψ(x′) |0⟩ =
+�
+σ
+�
+dp ψ(+)
+σ (p, x)ψ(+)
+σ (p, x′)
+=
+�
+η′
+η
+�
+i
+�
+/D + Γ0
+2η
+�
++ m
+� �
+P+GdSD(x, x′, im
+k − 1
+2) + P−GdSD(x, x′, im
+k + 1
+2)
+�
+,
+(94)
+S−(x, x′)
+=
+⟨0| Ψ(x′)Ψ(x) |0⟩ =
+�
+σ
+�
+dp ψ(−)
+σ (p, x)ψ(−)
+σ (p, x′)
+=
+−
+�
+η′
+η
+�
+i
+�
+/D + Γ0
+2η
+�
++ m
+� �
+P+GdSD(x′, x, im
+k − 1
+2) + P−GdSD(x′, x, im
+k + 1
+2)
+�
+(95)
+where P ± = (IN×N ± γ0)/2 and
+GdSD(x, x′, im
+k ± 1
+2) =
+�
+dp(ηη′)
+D−1
+2 HD−2
+8(2π)D−2
+eip(x−x′)H(2)
+im
+k ± 1
+2(pη) H(1)
+m
+k ± 1
+2(pη′).
+(96)
+Now the Wightman function for massless fermions in dS background,
+S±(x, x′) = ±i
+�
+η′
+η
+�
+/D + Γ0
+2η
+�
+GdSD(x, x′)
+(97)
+and GdSD is as usual from equations (45-47) ,
+GdSD(x, x′) = GdSD(x′, x, 1
+2) = GdSD(x, x′, −1
+2) = HD−2Γ(D/2 − 1)
+2(2π)D/2
+v1−D/2.
+(98)
+From equation (97) we can further deduce that
+S±(x, x′) = ±iHηab(xa − x′a)γb
+√ηη′v
+�D − 2
+2
+�
+GdSD(x, x′)
+(99)
+The detector is moving in with constant linear acceleration a following (48) as before. In that case
+we know the detector response function of fermions (per unit time) for interaction Lagrangian is
+given by[27],
+JdSD =
+� ∞
+−∞
+d∆τe−iE∆τS(2)
+D (∆τ)
+(100)
+19
+
+A PREPRINT - JANUARY 23, 2023
+SdS4(Ω, T )
+Figure 9: Plot of the Finite-Time response of the UDW Particle Detector in dS Space-time coupled to a fermionic
+Field. Each row (from top to bottom) shows the plot of the response function against Ω (varying a[T = 3, k = 3],
+k[T = 3, a = 3], T [a = 3, k = 3]), against a (varying Ω[k = 3, T = 3], k[Ω = 1, T = 3], T [k = 3, Ω = 1]), and
+against k (varying a[Ω = 1, T = 3], Ω[a = 3, T = 3], T [a = 3, Ω = 1]) respectively.
+where,
+S(2)
+D (x(τ), x(τ ′))
+=
+⟨0| :
+�
+Ψa(x(τ)))Ψa(x(τ))
+�
+::
+�
+Ψb(x(τ ′)))Ψb(x(τ ′))
+�
+: |0⟩
+=
+Tr[S+(x, x′)S−(x′, x)]
+(101)
+=
+k2ηab(xa − x′a)ηcd(xc − x′c)
+ηη′ν2
+Tr
+�
+γbγd� �D − 2
+2
+�2
+GdSD(x, x′) GdSD(x′, x)
+=
+Nk2 (D/2 − 1)2 ηabηcdηbd(xa − x′a)(x′c − xc)
+ηη′ν2
+H2D−4Γ(D/2 − 1)2
+4(2π)D
+ν2−D
+=
+N [(D/2 − 1)Γ(D/2 − 1)]2
+Γ(D − 1)
+�k2D−2Γ(D − 1)
+2(2π)D
+�
+ν−D
+�ηac(xa − x′a)(x′c − xc)
+2ηη′
+�
+=
+N Γ(D/2)2
+Γ(D − 1)
+�k2D−2Γ(D − 1)
+2(2π)D
+�
+ν1−D
+=
+N (Γ(D/2))2
+Γ(D − 1) GdS2D(x(τ), x(τ ′)).
+(102)
+is 4-points correlator of fermionic field. Here we are taking the trace over spinor index a, b, and, we
+have used the identity,
+Tr
+�
+γbγd�
+= Nηbd
+(103)
+20
+
+Q=1
+2=2
+=U
+2=4
+Q=1
+2=2
+=U
+Q=4A PREPRINT - JANUARY 23, 2023
+So the eq.102 dictates that in any of the path S(2)
+D (∆τ) takes the following form,
+S(2)
+D (∆τ) = N (Γ(D/2))2
+Γ(D − 1) GdS2D(∆τ)
+(104)
+From eq. (100) we can then conclude that detector response function for fermions can be related to
+response function for the scalars,
+JdSD = N (Γ(D/2))2
+Γ(D − 1) F(1)
+dS2D.
+(105)
+Thus we have proved the following statement.
+The response function of an UDW detector(with uniform linear acceleration) quadratically
+coupled to a massless Dirac field in (A)dS vacuum in D ≥ 2 spacetime dimensions exactly
+equals to the response function of a UDW detector which is linearly coupled to a massless scalar
+field in 2D dimensional (A)dS vacuum times dimensional dependent numeric factor. Here,
+the fermionic field is minimally coupled to background while the scalar field is conformally
+coupled to the background.
+Upon establishing the above statement for fermionic response in maximally symmetric spacetime
+we plot the response function in figure 8 and 9 with respect to different variables such as energy gap
+Ω, acceleration a and curvature k. We can also see the similar pattern in (A)dS fermionic response
+function that the response increases with increasing acceleration a but decreases as with increasing
+energy gap Ω. The response decreases with the increment curvature k in AdS and the opposite
+pattern is noticed in dS background. Also because of the fact 4 dimensional fermionic response
+function is related to higher (eight) dimensional scalar response function we find out, accelerated
+UDW detectors respond better when coupled to fermionic fields compared to bosonic fields.
+3
+Huygen’s principle, detector and Unruh radiation
+Huygen’s principle is a well studied phenomenon specially in quantum field theory. It is a natural
+question to ponder whether the accelerated detectors observing the thermal radiation maintains the
+Huygen’s principle[25]. The observed radiation from massless scalars by UDW detectors do not
+maintain the Huygen’s principle in flat spacetime in three (odd) dimensions[25]. However this
+statement is well understood for accelerated UDW detectors in flat spacetime with linear coupling.
+But in this section we discuss the status of Huygen’s principle for scalar theories where accelerated
+UDW detectors moving in the maximally symmetric curved spacetime with non linear interaction
+coupling(21)[24]. The Huygen’s principle has several different equivalent definitions but we can
+work on with the following one [36, 37, 38]-
+i) The theory maintains the Huygen’s principle if the causal propagator Gc has support
+only on the lightcone.
+ii) The theory violates the Huygen’s principle if they are non vanishing elsewhere.
+To understand better the state of Huygen’s principle for the detected Unruh radiation we
+need to first fix the coupling between the detector and the matter field (23). For the usual linearly
+coupled (n = 1) detector, the response function simple depends upon the Wightman function.
+Concentrating on linear coupling, the causal propagator for conformally coupled scalar theory can
+21
+
+A PREPRINT - JANUARY 23, 2023
+Figure 10: (A) Support of the propagators of a massless scalar in even dimensions for odd or even coupling associated
+with (21). This also depicts support of the propagators of a massless scalar in odd dimensions with even coupling. (B)
+Support of the propagators of a massless scalar in odd dimensions for odd coupling associated with (21)
+defined as,
+Gc(x, x′) = W (2)
+D (x, x′) − W (2)
+D (x′, x) = ⟨0| [Φ(x), Φ(x′)] |0⟩
+(106)
+In flat spacetime, the origin of obeying (or violating) the Huygen’s principle can be explained for
+linearly coupled detector. In case of the linear coupling the detector response function is nothing
+but the Fourier transform of the Wightman function W (2)
+D (x, x′), which is proportional to L1− d
+2 for
+a bosonic field. Here L is the square distance between the two points x and x′. Now when the two
+point function is analytically continued to complex τ there is a brunch cut for timelike distance when
+we are in odd dimensions. This brunch cut becomes a simple pole in even dimensions. Therefore
+when we compute the response function in odd dimensions using (22) the linear detector finds a
+brunch cut in the integral expression and therefore reports a Fermi-Dirac distribution. The support
+of the propagator of scalar fields for linear detector[25] can be exactly (23) analysed with figure
+10A. For even dimensions, the support of Gc is only on the lightcone while in odd dimension the
+support of Gc is on the entire timelike region. In the case of conformally coupled scalar fields over
+(A)dS spacetime, we can follow the same argument as before. For example, the two point correlator
+of conformally coupled scalars in dS can be simply related to flat space correlator W (2)
+M (x, x′) using
+the followings relation[35, 39],
+W (2)
+dS (x, x′) = (k2η2)
+d−2
+4 W (2)
+M (x, x′)(k2η′2)
+d−2
+4
+(107)
+Therefore the pole structure for conformally coupled theories are similar to those of flat spacetime.
+In even dimensional flat spacetime the causal correlator is given by[38],
+Gc = [Φ(t, ⃗x), Φ(t + ∆t, ⃗x + ∆⃗x)] =
+i
+4π∆⃗x[δ(∆⃗t + ∆⃗x) − δ(∆⃗t − ∆⃗x)]
+(108)
+In similar fashion with odd dimensional spacetime D = d + 13, it is given by
+Gc = [Φ(t, ⃗x), Φ(t + ∆t, ⃗x + ∆⃗x)] = Γ( d−1
+2 )
+4π
+d+1
+2
+1
+L
+d−1
+2
+(109)
+3even dimensional space d.
+22
+
+XA PREPRINT - JANUARY 23, 2023
+We can see from eq. (108) that in even dimensional Minkowski spacetime the support of Gc is
+exactly on the lightcone while in odd dimensional spacetime the support of Gc is also inside the
+lightcone. Exactly similar result will hold for conformally coupled scalar theory living on (A)dS
+background through (107). We now go ahead and generalize the results with coupling n ≥ 1. The
+causal propagator can be written for any coupling n in this fashion,
+Gc(x, x′) = W (2n)(x, x′) − W (2n)(x′, x)
+(110)
+The 2n point correlators are related to two point correlators using (52). And thus the pole structure
+of 2n point correlator can be easily understood using this relation. In odd dimension the brunch cut
+in Wightman function results a brunch cut in 2n point correlator for any odd coupling n. However
+when we choose the even coupling the brunch cut turns into simple pole for the 2n point correlator.
+Huygen’s principle for scalar fields are therefore obeyed as well as no statistics inversion is noticed
+by the Unruh radiation in odd dimension only when we choose the even coupling. On the contrary
+the Huygen’s principle is violated in odd dimensions with odd coupling and statistics inversion also
+happens.
+In case of even coupling, the pole structure of the 2n-point correlator function are also
+quite interesting. Through (52) focusing in even dimensions, the Wightman function has no brunch
+cut in even dimensions. Therefore for any even coupling the 2n-point correlators will not have any
+surprise brunch cut in even dimensions. So, the Huygen’s principle is going to be satisfied trivially.
+However, in odd dimensions there is a brunch cut in Wightman function. But in the 2n-point
+correlator, when we use (52) it immediately suggests the brunch cut turns to a simple pole for any
+even n. As a result for even n, the Unruh radiation of scalars in flat space as well as in conformally
+coupled maximally symmetric scalar solutions the Huygen’s principle is always maintained in
+any dimensions. It is not possible to violate Huygen’s principle if we choose coupling with even
+n. The support of the scalar solutions are surprisingly always on the lightcone for even coupling.
+Figure 10(A) accurately depicts the status Huygen’s Principle in scalar Unruh radiation with odd
+coupling and odd dimesnions, where the support is just on the light cone. Interesting to see this is
+the exact situation when the statistics inversion happens through (54). In odd dimensions scalar
+theory propagator under consideration is anti-periodic in β = 2π
+ω when we choose odd coupling. In
+any other scenario the 2n point propagator is periodic in β where the Hugen’s principle is perfectly
+maintained in Unruh radiation.
+Let us now focus our discussion on the Huygen’s principle for fermionic theory with inter-
+action Hamiltonian (61). This is the most commonly used interaction Hamiltonian between
+fermionic theory and detector[22].
+Now the definition of Huygen principle (written before
+eq. (106)) remains the same for fermionic theory as well but the definition of Gc is given by
+Anti-Commuatator instead of commuatator algebra as in (106)4,
+Gc(x, x′) = ⟨0| {χ(x), χ(x′)} |0⟩
+(111)
+where,
+χ(x) =: Ψa(x)Ψa(x) :
+(112)
+The study of support for fermionic correlator on lightcone is a bit troublesome specially in the
+curved spacetime because of the gamma matrices. For the interaction Hamiltonian given by (61), the
+4where the trace over spin index is assumed
+23
+
+A PREPRINT - JANUARY 23, 2023
+Figure 11: Support of the four point propagators of a massless fermions in any dimensions for interaction Hamiltonian
+(60). This is the clear indication that Huygen’s principle is always maintained for fermions.
+detector response function is dependent upon the four point fermionic correlator as seen explicitly
+from (62). In ref.[25], the author focused on the two point function of fermionic fields to understand
+the status of Huygen’s principle of the Unruh radiation detected by the Unruh-DeWitt detectors
+instead of the four point function. As a result, it was concluded in ref.[25] that the Unruh radiation
+detected for fermions maintain Huygen’s principle in odd dimensions while violate it in even
+dimensions. Because the pole and brunch cut structure of four point fermionic correlator is quite
+different to the two point correlator. In case of flat space[22], AdS[23] as well as in dS ((105)), one
+can easily see that four point fermionic correlators in D dimensions is explicitly given by the two
+point scalar correlators in 2D dimensions. Therefore to understand the status of Huygen’s principle
+for Unruh radiation of the fermionic theory (60) in odd or even D dimension, we can instead think
+of the massless scalars in 2D dimensions. The scalar Wightman function when conformally coupled
+to the background (A)dS gravity solutions has no brunch cut in even dimensions. As a consequence,
+the 4-point fermionic propagator, to which the detector is sensitive always have support only on the
+light cone. It is quite surprising to see that scalar theory fails to maintain the Huygen principle in
+odd dimensions with usual linear coupling while the fermionic theory always maintains the Huygen
+principle with the usual interaction Hamiltonian(60). This result is true for matter fields in flat
+spacetime as well as conformally coupled to maximally symmetric spacetimes. The reason behind
+the conclusion being different to Huygen’s principle of fermionic Unruh radiation in ref.[25] is
+because they performed their analysis with two point function. But for the interaction Hamiltonian
+(60)5, one should actually analyse the four point fermionic correlator. Because the detector response
+is dependent on four point fermionic correlator rather than the fermionic Wightman function (see
+(62)), the correlator under consderation maintains a periodic condition with periodicity β = 2π
+ω and
+the Huygen’s principle is always maintained the irrespective of dimensionality of spacetime.
+4
+Discussion and future works
+In this article we have completed the computation of finite time response of accelerated UDW
+detectors in maximally symmetric spacetime. The behaviour of the response function with dif-
+ferent parameters are systemtically analysed in figure (2) - (9). We also concluded the analysis
+5see 2.15b no eq. of [25], where they use the same interaction Hamiltonian.
+24
+
+A PREPRINT - JANUARY 23, 2023
+for fermionic response function in maximally symmetric background (see the boxed statement
+after (105)). The result is quite powerful which also allows us to determine the status of Huygen’s
+principle of the fermionic Unruh radiation detected by UDW detector moving in maximally sym-
+metric spacetime. It is quite intriguing to note that the Huygen’s principle is always maintained
+in fermionic Unruh radiation which is minimally coupled to background as opposed to minimally
+coupled scalar[15]. We are currently working on to see if such results of fermionic response also
+holds when the UDW detectors are moving in other interesting curved spacetime solutions such as
+blackholes. As an application of finite time response we would also like to construct Unruh Otto
+engines[20] with the help of UDW detectors moving in maximally symmetric backgrounds. The
+variation of response function in dS and AdS space with respect to curvature makes the situation
+quite interesting and we are focusing on explicit conditions to extract required conditions for
+completing Otto cycle to have positive work output. The recent claim that with entangled qubits one
+can build up more efficient[40, 41] Otto engine is quite exciting and we are exploring the possibility
+to generalise it with maximally symemtric spacetimes. In our current manuscript, we have worked
+with scalar theory which is conformally coupled to background gravity but we are in a process
+of computing the finite time response function of UDW detectors for minimally coupled scalar
+theory[42].
+5
+Appendix: Constant Acceleration Path in dS spacetime
+Here, we show that the path considered in eq. 48 is a constant acceleration path. The components
+of the acceleration can be written as,
+aµ = d2xµ
+dτ 2 + 2Γµ
+αβ
+�dxα
+dτ
+� �dxβ
+dτ
+�
+(113)
+where, Γµ
+αβ are Christoffel symbols of the first kind. Writing the components out explicitly for the
+path in eq. 48, we have,
+a(0)
+=
+d2η
+dτ 2 + 2Γ0
+αβ
+�dxα
+dτ
+� �dxβ
+dτ
+�
+=
+d2η
+dτ 2 − 1
+τ
+�dη
+dτ
+�2
+− 1
+τ
+�dx1
+dτ
+�2
+=
+τ0ω2eωτ −
+1
+τ0eωτ (τ0ωeωτ)2 −
+1
+τ0eωτ
+�aτ0
+ω ωeωτ�2
+=
+−a2τ0eωτ
+(114)
+a(1)
+=
+d2x1
+dτ 2 + 2Γ1
+αβ
+�dxα
+dτ
+� �dxβ
+dτ
+�
+=
+d2x1
+dτ 2 − 2
+τ
+dx1
+dτ
+dη
+dτ
+=
+aτ0
+ω ω2eωτ −
+2
+τ0eωτ
+�aτ0
+ω ωeωτ�
+(τ0ωeωτ)
+=
+−aτ0ωeωτ
+(115)
+a(2)
+=
+a(3) = ... = a(D−1) = 0
+(116)
+25
+
+A PREPRINT - JANUARY 23, 2023
+So, the magnitude of the acceleration a becomes,
+|a|2 = −aµaµ
+= −g00(a0)2 − g11(a1)2
+= −
+1
+H2τ 2a4τ 2
+0 e2ωτ +
+1
+H2τ 2a2τ 2
+0 ω2e2ωτ
+= −
+1
+H2τ 2
+0 e2ωτ a2τ 2
+0 e2ωτ(a2 − ω2)
+= a2
+(117)
+Hence, |a| = a and the acceleration along this path is uniform.
+6
+Acknowledgments
+MMF’s research is supported by NSERC and in part by the Delta Institute of Theoretical Physics.
+The authours would like to thank Sowmitra Das and Onirban Islam for the discussions.
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+

VdAzT4oBgHgl3EQfJ_tt/content/tmp_files/2301.01089v1.pdf.txt

@@ -0,0 +1,1277 @@
+xDeepInt: a hybrid architecture for modeling the vector-wise
+and bit-wise feature interactions
+Yachen Yan
+[email protected]
+Credit Karma
+San Francisco, California
+Liubo Li
+[email protected]
+Credit Karma
+San Francisco, California
+ABSTRACT
+Learning feature interactions is the key to success for the large-
+scale CTR prediction and recommendation. In practice, handcrafted
+feature engineering usually requires exhaustive searching. In order
+to reduce the high cost of human efforts in feature engineering,
+researchers propose several deep neural networks (DNN)-based ap-
+proaches to learn the feature interactions in an end-to-end fashion.
+However, existing methods either do not learn both vector-wise
+interactions and bit-wise interactions simultaneously, or fail to
+combine them in a controllable manner. In this paper, we propose
+a new model, xDeepInt, based on a novel network architecture
+called polynomial interaction network (PIN) which learns higher-
+order vector-wise interactions recursively. By integrating subspace-
+crossing mechanism, we enable xDeepInt to balance the mixture of
+vector-wise and bit-wise feature interactions at a bounded order.
+Based on the network architecture, we customize a combined opti-
+mization strategy to conduct feature selection and interaction se-
+lection. We implement the proposed model and evaluate the model
+performance on three real-world datasets. Our experiment results
+demonstrate the efficacy and effectiveness of xDeepInt over state-
+of-the-art models. We open-source the TensorFlow implementation
+of xDeepInt: https://github.com/yanyachen/xDeepInt.
+CCS CONCEPTS
+• Computing methodologies; • Machine learning; • Machine
+learning approaches; • Neural networks;
+KEYWORDS
+CTR prediction, Recommendation System, Explicit Feature Interac-
+tion, Deep Neural Network
+ACM Reference Format:
+Yachen Yan and Liubo Li. 2020. xDeepInt: a hybrid architecture for modeling
+the vector-wise and bit-wise feature interactions. In San Diego 2020: ACM
+SIGKDD Conference on Knowledge Discovery and Data Mining, August 22–27,
+2020, San Diego, CA. ACM, New York, NY, USA, 9 pages. https://doi.org/10.
+1145/nnnnnnn.nnnnnnn
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from [email protected].
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+© 2020 Association for Computing Machinery.
+ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+1
+INTRODUCTION
+Click-through rate (CTR) prediction model [23] is an essential com-
+ponent for the large-scale recommendation system, online adver-
+tising and search ranking [7, 12, 19, 30]. In online marketplace
+scenario, accurately estimating CTR will enable the recommen-
+dation system to show users the items they prefer to view and
+explore, which has a direct impact on both short-term revenue and
+long-term user experience.
+The input features (e.g., user id, item id, item category, site do-
+main) of CTR prediction model are usually in a multi-field categori-
+cal format [31] and transformed via field-aware one-hot encoding
+and multi-hot encoding [34]. The representation of each field is a
+sparse binary vector. The corresponding cardinality of each field
+determines the dimension of the sparse vector. The concatenation
+of these sparse vectors naturally generates high-dimensional and
+sparse feature representations.
+In CTR prediction model, exploring useful feature interactions
+plays a crucial role in improving model performance[7, 8, 16, 22, 28].
+Traditionally, data scientists search and build hand-crafted fea-
+ture interactions to enhance model performance based on domain
+knowledge. In practice, feature interactions of high-quality require
+expensive cost of time and human workload [12]. Furthermore, it
+is infeasible to manually extract all possible feature interactions
+given a large number of features and high cardinality [7]. Therefore,
+learning low-order and high-order feature interactions automati-
+cally and efficiently in a high-dimensional and sparse feature space
+becomes an essential problem for improving CTR prediction model
+performance, in both academic and industrial communities.
+Deep learning models have achieved great success in recom-
+mender systems due to its great feature learning ability. Several
+deep learning architecture has been proposed from both academia
+and industry (e.g.,[7, 8, 13, 16, 20, 21, 24, 25, 28]). However, All the
+existing models utilize DNNs as building block for learning high-
+order implicit bit-wise feature interactions, without bounded order.
+When modeling explicit feature interactions, the exiting approaches
+only capture lower order explicit interactions efficiently. Learning
+higher order typically requires higher computational cost.
+In this paper, we propose a efficient neural network-based model
+called xDeepInt to learn the combination of vector-wise and bit-
+wise multiplicative feature interactions explicitly. Motivated by
+polynomial regression, we design a novel Polynomial Interaction
+Network layers to capture bounded degree vector-wise interactions
+explicitly. In order to learn the bit-wise and vector-wise interactions
+simultaneously in a controllable manner, we combine PIN with a
+subspace-crossing mechanism, which gives a significant boost to
+our model performance and brings more flexibility. The degree
+arXiv:2301.01089v1  [cs.LG]  3 Jan 2023
+
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+Yachen and Liubo
+of bit-wise interactions grows with the number of subspace. In
+summary, we make the following contributions in this paper:
+• We design a novel neural network architecture named xDeepInt
+that models the vector-wise interactions and bit-wise in-
+teractions explicitly and simultaneously, dispensing with
+jointly-trained DNN and nonlinear activation functions. The
+proposed model is lightweight. But it yields superior per-
+formance than many existing models with more complex
+structure.
+• Motivated by higher-order polynomial logistic regression,
+we design a Polynomial-Interaction-Network (PIN) layer
+which learns higher-order explicit feature interactions recur-
+sively. The degrees of interactions are controlled by tuning
+the number of PIN layers. An analysis is conducted to demon-
+strate the polynomial approximation properties of PIN.
+• We introduce a subspace-crossing mechanism for modeling
+bit-wise interactions across different fields inside PIN layer.
+The combination of PIN layer and the subspace-crossing
+mechanism allows us to control the the degree of bit-wise in-
+teractions. As the number of subspaces increases, our model
+can dynamically learn more fine-grained bit-wise feature
+interactions.
+• We design an optimization strategy which is in harmony
+with the architecture of the proposed model. We apply Group
+Lasso FTRL to the embedding table, which shrinks the entire
+rows to zero and achieves the feature selection. To optimize
+weights in PIN layers, we apply FTRL directly. The sparsity
+in weights results in selection of feature interactions.
+• We conduct a comprehensive experiment on three real-world
+datasets. The results demonstrate that xDeepInt outperforms
+existing state-of-the-art models under extreme high-dimensional
+and sparse settings. We also conduct a sensitivity analysis
+on hyper-parameter settings of xDeepInt and ablation study
+on integration of DNN.
+2
+RELATED WORK
+Deep learning based models have been applied for CTR prediction
+problem in the industry since deep neural networks become domi-
+nant in learning the useful feature representation of the mixed-type
+input data and fitting model in an end-to-end fashion[30]. This
+merit can reduce efforts in hand-crafted feature design and auto-
+matically learn the feature interactions.
+2.1
+Modeling Implicit Interaction
+Most of the DNN-based methods map the high-dimensional sparse
+categorical features and continuous features onto a low dimensional
+latent space in the initial step. Without designing specific model
+architecture, DNN-based method learns the high-order implicit fea-
+ture interactions by feeding the stacked embedded feature vectors
+into a deep feed-forward neural network.
+Deep Crossing Network [24] utilizes residual layers in the feed-
+forward structure to learn higher-order interactions with improved
+stability. Some hybrid network architectures, including Wide &
+Deep Network (WDL) [7], Product-based Neural Network (PNN) [20,
+21], Deep & Cross Network (DCN) [28], Deep Factorization Machine
+(DeepFM) [8] and eXtreme Deep Factorization Machine (xDeepFM) [16]
+employ feed-forward neural network as their deep component to
+learn higher-order implicit interactions. The complement of the
+implicit higher-order interaction improves the performance of the
+network that only models the explicit interactions [2].
+However, this type of approach detects all feature interactions at
+the bit-wise level [16] implicitly, without efficiency. And the degree
+of the interactions are not bounded.
+2.2
+Modeling Explicit Interaction
+Deep & Cross Network (DCN) [28] explores the feature interactions
+at the bit-wise level in an explicit fashion. Specifically, each cross
+layer of DCN constructs all cross terms to exploits the bit-wise
+interactions. The number of recursive cross layers controls the
+degree of bit-wise feature interactions.
+Some recent models explicitly learn the vector-wise feature in-
+teractions using a specific form of the vector product. Deep Fac-
+torization Machine (DeepFM) [8] combines factorization machine
+layer and feed-forward neural network through joint learning fea-
+ture embedding. Factorization machine layer models the pairwise
+vector-wise interaction between feature 𝑖 and feature 𝑗 by the inner
+product of ⟨𝑥𝑖,𝑥𝑗⟩ = �𝑘
+𝑡=1 𝑥𝑖𝑡𝑥𝑗𝑡. Then, the vector-wise interac-
+tions are concatenated with the output units of the feed-forward
+neural network. Product Neural Network (PNN) [20, 21] introduces
+the inner product layer and the outer product layer to learn ex-
+plicit vector-wise interactions and bit-wise interactions respectively.
+xDeepFM [16] learns the explicit vector-wise interaction by using
+Compressed Interaction Network (CIN) which has an RNN-like
+architecture and learns all possible vector-wise interactions us-
+ing Hadamard product. The convolutional filters and the pooling
+mechanism are used to extract information. FiBiNET [13] utilizes
+Squeeze-and-Excitation network to dynamically learn the impor-
+tance of features and models the feature interactions via bilinear
+function.
+In the recent research of the sequencing model, the architecture
+of the Transformer [26] has been widely used to understand the
+associations between relevant features. With different layers of the
+multi-head self-attentive neural networks, AutoInt [25] can learn
+different orders of feature combinations of input features. Residual
+connections [9, 27] are added to carry through different degrees of
+feature interaction.
+The aforementioned approaches learn explicit feature interac-
+tions by using outer product, kernel product or multi-head self-
+attention, which require expensive computational cost.
+3
+MODEL
+In this section, we give an overview of the architecture of xDeepInt.
+First, we introduce input and embedding layer, which map con-
+tinuous features and high-dimensional categorical features onto a
+dense vector. Second, we present the Polynomial Interaction Net-
+work(PIN) which utilizes iterative interaction layers with residual
+connections [9, 27] to explicitly learn the vector-wise interactions.
+Third, we implement the subspace-crossing mechanism to model
+bit-wise interaction. The number subspaces controls the degree of
+mixture of bit-wise and vector-wise interactions.
+
+Research Track Paper
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+Embedding Layer
+Categorical 
+Feature
+Bucktized
+ Numeric Feature
+1st PIN Layer
+2nd PIN Layer
+...
+l-th PIN Layer
+Input Feature Map
+Output Layer
+Figure 1: The architecture of unrolled Polynomial Interac-
+tion Network with residual connections
+3.1
+Embedding Layer
+In large-scale recommendation system, inputs include both con-
+tinuous features and categorical features. Categorical features are
+often directly encoded by one-hot encoding, which results in an
+excessively high-dimensional and sparse feature space. Suppose we
+have 𝐹 fields. In our feature preprocessing step, we bucketize all
+the continuous features to equal frequency bins, then embed the
+bucketized continuous features and categorical features to same
+latent space 𝑅𝐾,
+x𝑒
+𝑓 = x𝑜
+𝑓 V𝑓 ,
+where x𝑒
+𝑓 = [𝑥𝑓 ,1,𝑥𝑓 ,2, · · · ,𝑥𝑓 ,𝐾], 𝑥𝑓 ,𝑘 is the 𝑘-th bit of the 𝑓 -th
+field of the embedding feature map, V𝑓 is an embedding matrix for
+field 𝑓 , and x𝑜
+𝑓 is a one-hot vector. Lastly, we stack 𝐹 embedding
+vectors and obtain an 𝐹-by-𝐾 input feature map 𝑋0:
+𝑋0 =
+
+x𝑒
+1
+x𝑒
+2...
+x𝑒
+𝐹
+
+3.2
+Polynomial Interaction Network
+Consider a 𝑙-th order polynomial with 𝑓 variables of the following
+form:
+Π𝑙
+𝑗=1(
+𝑓∑︁
+𝑖=1
+𝑎𝑖𝑗𝑥𝑖 + 𝑏𝑗).
+(1)
+This polynomial contains all possible multiplicative combinations
+of 𝑥𝑖’s with order less than or equal to 𝑙 and has an iterative form:
+𝐹 (𝑙−1) (𝑥1, · · · ,𝑥𝑓 )(
+𝑓∑︁
+𝑖=1
+𝑎𝑖𝑙𝑥𝑖 + 𝑏𝑙)
+(2)
+where 𝐹 (𝑙−1) (𝑥1, · · · ,𝑥𝑓 ) = Π𝑙−1
+𝑗=1(�𝑓
+𝑖=1 𝑎𝑖𝑗𝑥𝑖 + 𝑏𝑗). Motivated by
+the iterative form, we propose polynomial interaction network
+defined by the following formula:
+𝑋𝑙 = 𝑓 (𝑊𝑙−1,𝑋𝑙−1,𝑋0) + 𝑋𝑙−1
+= 𝑋𝑙−1 ◦ (𝑊𝑙−1𝑋0) + 𝑋𝑙−1
+= 𝑋𝑙−1 ◦ [𝑊𝑙−1𝑋0 + 1]
+(3)
+where ◦ denotes the Hadamard product. For instance, [𝑎𝑖,𝑗]𝑚×𝑛 ◦
+[𝑏𝑖,𝑗]𝑚×𝑛 = [𝑎𝑖,𝑗𝑏𝑖,𝑗]𝑚×𝑛. 𝑊𝑙−1 ∈ 𝑅𝐹×𝐹 and 1 ∈ 𝑅𝐹×𝐾 with all
+entries are equal to one. 𝑋𝑙−1,𝑋𝑙 ∈ 𝑅𝐹×𝐾 are the output matrices
+of (𝑙-1)-th and 𝑙-th interaction layer. Like (1), the 𝑙-th PIN layer’s
+output is the weighted sum of all vector-wise feature interactions
+of order less than or equal to 𝑙.
+The architecture of the polynomial interaction network is moti-
+vated by the following aspects.
+First, the polynomial interaction network has a recursive struc-
+ture. The outputs of the current layer are built upon the previ-
+ous layer’s outputs and the first order feature map, ensuring that
+higher-order feature interactions are based on lower-order feature
+interactions from previous layers.
+Second, we use the Hadamard product to model the explicit
+vector-wise interaction, which brings us more flexibility in crossing
+the bits of each dimension in shared latent space and preserves
+more information of each degree of feature interactions.
+Third, we build a field aggregation layer 𝐴𝑔𝑔(𝑙) (𝑋) = 𝑊𝑙𝑋 which
+combines the feature map at the vector-wise level using a linear
+transformation 𝑊𝑙. Each vector of the field aggregation feature
+map can be viewed as a combinatorial feature vector constructed
+by the weighted sum of the input feature map. Then we take the
+Hadamard product of the output of the previous layer and field
+aggregation feature map for this layer. This operation allows us
+
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+Yachen and Liubo
+.
+Figure 2: Details of PIN layer
+The aggregation layer is defined by 𝐴𝑔𝑔(𝑙−1) (𝑋0) = 𝑊𝑙−1 · 𝑋0. The PIN
+takes the Hadamard product of the aggregated 1st-order feature map and
+the output of the previous PIN layer to generate the higher order
+vector-wise interaction.
+to explore all possible 𝑙-th order polynomial feature interactions
+based on existing (𝑙-1)-th order feature interactions.
+Last, we utilize residual connections [9, 27] in the polynomial
+interaction network, allowing a different degree of vector-wise
+polynomial feature interactions to be combined, including the first
+feature map. Since the polynomial interaction layer’s outputs con-
+tain all degree of feature interactions, the skipped connection en-
+able next polynomial interaction layer to focus on searching useful
+higher-order feature interactions while complementing lower-order
+feature interactions. As the number of layer increases, the degree
+of the polynomial feature interactions increases. The recurrent ar-
+chitecture of the proposed polynomial interaction network enables
+to bound the degree of polynomial feature interactions.
+3.3
+Subspace-crossing Mechanism
+The Polynomial Interaction Network (PIN) models the vector-wise
+interactions. However, PIN does not learn the bit-wise interaction
+in the shared latent embedding space. In order to cross the bits of
+different embedding dimensions, we propose the subspace-crossing
+mechanism which allows xDeepInt to learn the bit-wise interactions.
+Suppose we split the embedding space into ℎ sub-spaces, the input
+feature map 𝑋0 is then represented by ℎ sub-matrices as follow:
+𝑋0 = [𝑋0,1,𝑋0,2, · · · ,𝑋0,ℎ]
+(4)
+where 𝑋0,𝑖 ∈ 𝑅𝐹×𝐾/ℎ and 𝑖 = 1, 2 · · · ,ℎ. Next, we stack all sub-
+matrices at the field dimension and construct a stacked input feature
+map 𝑋
+′
+0 ∈ 𝑅(𝐹∗ℎ)×(𝐾/ℎ).
+𝑋
+′
+0 =
+
+𝑋0,1
+𝑋0,2
+...
+𝑋0,ℎ
+
+,
+(5)
+where 𝑋0,𝑗 ∈ 𝑅𝐹×(𝐾/ℎ) and ℎ denotes the number of sub-spaces.
+By splitting the embedding vector of each field to ℎ sub-vectors
+and stacking them together, we can align bits of different embed-
+ding dimension and create the vector-wise interactions on stacked
+sub-embeddings. Accordingly, we feed 𝑋
+′
+0 into the Polynomial In-
+Figure 3: Subspace-crossing mechanism
+teraction Network (PIN):
+𝑋
+′
+1 = 𝑋
+′
+0 ◦ [𝑊
+′
+0𝑋
+′
+0 + 1]
+...
+𝑋
+′
+𝑙 = 𝑋
+′
+𝑙−1 ◦ [𝑊
+′
+𝑙−1𝑋
+′
+0 + 1]
+(6)
+where𝑊
+′
+𝑙 ∈ 𝑅(𝐹∗ℎ)×(𝐹∗ℎ), 1 ∈ 𝑅(𝐹∗ℎ)×(𝐾/ℎ) and𝑋
+′
+𝑙 ∈ 𝑅(𝐹∗ℎ)×(𝐾/ℎ).
+The field aggregation of feature map and the multiplicative inter-
+actions building by Hadamard product are both of the vector-wise
+level in vanilla PIN layers. The subspace-crossing mechanism en-
+hanced PIN takes theℎ aligned subspaces as input, so that encourag-
+ing the PIN to capture the explicit bit-wise interaction by crossing
+features of the difference subspaces. The number of subspaces ℎ
+controls the complexity of bit-wise interactions. Larger ℎ helps the
+model to learn more complex feature interactions.
+3.4
+Output Layer
+The output of Polynomial Interaction Network is a feature map
+that consists of different degree of feature interactions, including
+raw input feature map reserved by residual connections and higher-
+order feature interactions learned by PIN. For the final prediction,
+we merely use formula as follows:
+ˆ𝑦 = 𝜎 �(𝑊𝑜𝑢𝑡𝑋𝑙 + 𝑏1𝑇 )1�
+(7)
+where 𝜎 is the sigmoid function, 𝑊𝑜𝑢𝑡 ∈ 𝑅1×𝐹 is a feature map
+aggregation vector that linearly combines all the features in the
+feature map, 1 ∈ 𝑅𝐾 and 𝑏 ∈ 𝑅 is the bias.
+3.5
+Optimization and Regularization
+For optimization, we use Group Lasso Follow The Regularized
+Leader (G-FTRL) [? ] as the optimizer for the embedding layers for
+feature selection, and Follow The Regularized Leader (FTRL) [19]
+as the optimizer for the PIN layers for interaction selection.
+Group lasso FTRL regularizes the entire embedding vector of
+insignificant features in each field to exactly zero, which essentially
+conducts feature selection and brings more training efficiency for
+industrial settings. The group lasso regularization is applied prior
+to the subspace splitting mechanism such that feature selection is
+consistent between each subspaces.
+FTRL regularizes the single elements of weight kernel in PIN lay-
+ers to exactly zero, which excludes insignificant feature interactions
+and regularizes the complexity of the model.
+
+Research Track Paper
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+Group Lasso 
+FTRL
+FTRL
+Embedding Table
+Weight
+Figure 4: Group Lasso FTRL v.s. FTRL
+The Group Lasso FTRL regularizes the embedding table with group-wise
+sparsity. FTRL regularizes the weight kernels of PIN layer with
+element-wise sparsity.
+This optimization strategy takes advantages of the properties
+of the different optimizers and achieves row-wise sparsity and
+element-wise sparsity at embedding table and weight kernel respec-
+tively. Therefore, it improves generalization ability and efficiency
+for both training and serving. It also plays an important role in
+model compression.
+3.6
+Training
+The loss function we use is Log Loss,
+LogLoss = − 1
+𝑁
+�𝑁
+𝑖=1(𝑦𝑖𝑙𝑜𝑔( ˆ𝑦𝑖) + (1 − 𝑦𝑖)𝑙𝑜𝑔(1 − ˆ𝑦𝑖))
+(8)
+where 𝑦𝑖 is the true label and ˆ𝑦𝑖 is the estimated click through rate.
+𝑁 is the total number of training examples.
+3.7
+Difference Between PIN and DCN
+PIN and DCN both have an iterative form. However, the two net-
+work architectures are quite different in extracting feature interac-
+tions.
+𝑥𝑙 = 𝑓𝐷𝐶𝑁 (𝑥𝑙−1,𝑤𝑙−1,𝑏𝑙−1)
+= 𝑥0𝑥𝑇
+𝑙−1𝑤𝑙−1 + 𝑏𝑙−1 + 𝑥𝑙−1
+(9)
+For DCN, feature map are flattened and concatenated as a single
+vector. All higher order bit-wise interactions are firstly constructed
+by the term 𝑥0𝑥𝑇
+𝑙−1, and then aggregated by a linear regression for
+next layer. This structure results in a special format of the output.
+As discussed in [16], the output the DCN layer is a scalar multiple
+of 𝑥0. [16] also pointed out the downsides: 1) the output of DCN
+is in a special form, with each hidden layer is a scalar multiple of
+𝑥0 and thus limits expressive power; 2) interactions only come in a
+bit-wise fashion.
+PIN constructs vector-wise feature interaction using Hadamard
+product, which preserve the information at vector-wise level. In
+order to allow different fields to cross at vector level, PIN firstly
+aggregates the input feature map by a linear transformation𝑊𝑙−1𝑋0
+for each iterative PIN layer and build interactions by term 𝑋𝑙−1 ◦
+𝑊𝑙−1𝑋0. Accordingly, PIN keeps the vector-wise structure of feature
+interactions and does not limit the output to a scalar multiple of 𝑋0.
+Moreover, each PIN layer is directly connected with input feature
+map 𝑋0, which improves model trainability. We also prove PIN’s
+polynomial approximation property in later section.
+3.8
+xDeepInt Analysis
+In this section, we analyze polynomial approximation property of
+the proposed xDeepInt model. We consider an xDeepInt model with
+𝑙 PIN layers, a subspace crossing mechanism with ℎ subspaces and
+𝐹 input feature with the same embedding size 𝐾.
+3.8.1
+Polynomial Approximation. In order to understand how PIN
+exploits the vector-wise interactions, we examine the polynomial
+approximation properties of PIN. Let X(0) ∈ 𝑅𝐹×𝐾 be the embedded
+feature vector with x𝑖 = [𝑥𝑖1, · · · ,𝑥𝑖𝐾] being the 𝑖-th row. x(𝑙)
+𝑖
+=
+[𝑥 (𝑙)
+𝑖1 , · · · ,𝑥 (𝑙)
+𝑖𝐾 ] is the 𝑖-th row of the output of 𝑙-th layer. Then, 𝑥 (𝑙)
+𝑖𝑘
+has the following explicit form:
+𝑥 (𝑙)
+𝑖𝑘 = 𝑥 (0)
+𝑖𝑘
+𝑙−1
+�
+𝑟=0
+(
+𝐹
+∑︁
+𝑖=1
+𝑤 (𝑟)
+𝑖𝑗 𝑥 (0)
+𝑗𝑘 + 1), for 𝑘 = 1, · · · , 𝐾
+(10)
+where 𝑊 (𝑟) = [𝑤 (𝑟)
+𝑖𝑗 ]1≤𝑖,𝑗 ≤𝐹 is the weight matrix at 𝑟-th PIN layer.
+The product �𝑙−1
+𝑟=0(�𝐹
+𝑖=1 𝑤 (𝑟)
+𝑖𝑗 𝑥 (0)
+𝑗𝑘 + 1) is the weighted sum of all
+possible crossed terms of the embbeded input at the 𝑘-th bit having
+order less than or equal to 𝑙 − 1. Thus, 𝑥 (𝑙)
+𝑖𝑘 is the weighted sum of
+all crossed terms that contains 𝑥 (0)
+𝑖𝑘 and has the order less than or
+equal to 𝑙.
+For the bit-wise interaction modeled by subspace-crossing mech-
+anism, we consider the case where the number of subspaces equals
+to the embedding size 𝐾. In this extreme case, each row of 𝑊 ′
+0𝑋 ′
+0
+is a weighted sum of all bits in all fields. This design allows the
+combination of embedded features at different bits. To be more
+explicit, we consider the stacked input feature map with ℎ = 𝐾
+𝑋
+′
+0 =
+
+𝑋0,1
+𝑋0,2
+...
+𝑋0,𝐾
+
+,
+(11)
+where 𝑋0,𝑖 =
+
+𝑥 (0)
+𝑖,1
+𝑥 (0)
+𝑖,2
+...
+𝑥 (0)
+𝑖,𝐹
+
+∈ 𝑅𝐹 .The weight matrix 𝑊 ′
+0 is given by
+𝑊 ′
+0 =
+
+𝑤 (0)
+1,1
+𝑤 (0)
+1,2
+· · ·
+𝑤 (0)
+1,𝐾
+𝑤 (0)
+1,𝐾+1
+· · ·
+𝑤 (0)
+1,𝐹𝐾
+...
+...
+...
+...
+...
+...
+...
+𝑤 (0)
+𝐹𝐾,1
+𝑤 (0)
+𝐹𝐾,2
+· · ·
+𝑤 (0)
+𝐹𝐾,𝐾
+𝑤 (0)
+𝐹𝐾,𝐾+1
+· · ·
+𝑤 (0)
+𝐹𝐾,𝐹𝐾
+
+∈ 𝑅𝐹𝐾×𝐹𝐾 .
+(12)
+
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+Yachen and Liubo
+Thus, the 𝑘-th row of 𝑊 ′
+0𝑋 ′
+0 + 1 has the following form:
+𝐹
+∑︁
+𝑖=1
+𝐾
+∑︁
+𝑗=1
+𝑤 (0)
+𝑘,(𝑖−1)𝐾+𝑗𝑥𝑖,𝑗 + 1.
+(13)
+This is a linear combination of bits in all fields, which allows the
+PIN exploit all crossing of the feature map at bit-wise level. For
+example, the [(𝑖 − 1)𝐾 + 𝑗]-th row of 𝑋 ′
+𝑙 is given by
+𝑥 (𝑙)
+𝑖,𝑗 = 𝑥 (0)
+𝑖,𝑗
+𝑙−1
+�
+𝑟=0
+𝐹
+∑︁
+𝑖=1
+𝐾
+∑︁
+𝑗=1
+𝑤 (𝑟)
+𝑘,(𝑖−1)𝐾+𝑗𝑥𝑖,𝑗 + 1
+(14)
+𝑥 (𝑙)
+𝑖,𝑗 is the weighted sum of all crossed terms that contains 𝑥 (0)
+𝑖,𝑗 and
+has the order less than or equal to 𝑙.
+3.8.2
+Time Complexity. The cost of computing feature maps𝑊𝑙−1𝑋
+at 𝑙-th PIN layer is O(ℎ𝐹2𝐾). For a 𝐿-layers xDeepInt model, the
+total cost of feature maps is O(𝐿ℎ𝐹2𝐾). The additional cost is from
+the Hadamard product and residual connection, which is O(𝐿𝐹𝐾).
+In practice, ℎ is not too large. Hence, the total time complexity
+mainly relies on the number of fields 𝐹 and embedding size 𝐾. For
+an 𝐿 layers DNN with each layer has 𝐷𝑘 hidden nodes, the time
+complexity is O(𝐹𝐾 ×𝐷1×𝐷2+�𝐿−1
+𝑘=2 𝐷𝑘−1𝐷𝑘𝐷𝑘+1). The time com-
+plexity of xDeepInt relies on the number of subspaces. Therefore,
+xDeepInt has higher time complexity than DNN when modelling
+higher degrees of bit-wise interactions.
+3.8.3
+Space Complexity. The embedding layer contains �𝐹
+𝑓 =1 𝐾 ×
+𝐶𝑓 parameters, where 𝐶𝑓 is the cardinality of 𝑓 -th field. The output
+layer aggregates the feature map at the last PIN layer. Hence, the
+output layers require 𝐹 + 1 parameters. The subspace crossing
+mechanism needs ℎ2 × 𝐹2 parameters at each PIN layer, which
+is the exact size of the weight matrix 𝑊 ′𝑟 with 0 ≤ 𝑟 ≤ 𝑙 − 1.
+There are (𝐾/ℎ) × 𝑘′ + ℎ2 × 𝐹2 × 𝑙 parameters in 𝑙 PIN layers.
+Usually, we will pick a small ℎ to control the model complexity and
+𝑘′ is comparable to 𝐾. Accordingly, the overall complexity of the
+xDeepInt is approximate 𝑂(ℎ2 × 𝐹2 × 𝑙), which is affected by the
+embedding size 𝐾 heavily. A plain 𝐿 layers DNN with each layer has
+𝐷𝑘 hidden nodes requires 𝐹𝐾 ×𝐷1 +�𝐿
+𝑘=2 𝐷𝑘−1𝐷𝑘 parameters. The
+complexity mainly depends on the embedding size and the number
+of hidden nodes at each layer. To reduce the space complexity of
+xDeepInt, we can apply the method introduced in [16]. The space
+complexity of the model can be further reduced by exploiting a
+𝐿-order decomposition and replace the weight matrix 𝑊 ′𝑟 with two
+low-rank matrices.
+4
+EXPERIMENTS
+In this section, we focus on evaluating the effectiveness of our
+proposed models and answering the following questions:
+• Q1: How does our proposed xDeepInt perform in CTR pre-
+diction problem? Is it effective and efficient under extreme
+high-dimensional and sparse data settings?
+• Q2: How do different hyper-parameter settings influence
+the performance of xDeepInt?
+• Q3: Will modeling implicit higher-order feature interactions
+further improve the performance of xDeepInt?
+4.1
+Experiment Setup
+4.1.1
+Datasets. We evaluate our proposed model on three public
+real-world datasets widely used for research.
+1. Avazu.1 Avazu dataset is from kaggle competition in 2015.
+Avazu provided 10 days of click-through data. We use 21 features
+in total for modeling. All the features in this dataset are categorical
+features.
+2. Criteo.2 Criteo dataset is from Kaggle competition in 2014.
+Criteo AI Lab officially released this dataset after, for academic
+use. This dataset contains 13 numerical features and 26 categorical
+features. We discretize all the numerical features to integers by
+transformation function ⌊𝐿𝑜𝑔 �𝑉 2�⌋ and treat them as categorical
+features, which is conducted by winning team of Criteo competi-
+tion.
+3. iPinYou.3 iPinYou dataset is from iPinYou Global RTB(Real-
+Time Bidding) Bidding Algorithm Competition in 2013. We follow
+the data processing steps of [32] and consider all 16 categorical
+features.
+For all the datasets, we randomly split the examples into three
+parts: 70% is for training, 10% is for validation, and 20% is for test-
+ing. We also remove each categorical features’ infrequent levels
+appearing less than 20 times to reduce sparsity issue. Note that
+we want to compare the effectiveness and efficiency on learning
+higher-order feature interactions automatically, so we do not do any
+feature engineering but only feature transformation, e.g., numerical
+feature bucketing and categorical feature frequency thresholding.
+4.1.2
+Evaluation Metrics. We consider AUC and LogLoss for eval-
+uating the performance of the models.
+LogLoss LogLoss is both our loss function and evaluation metric.
+It measures the average distance between predicted probability and
+true label of all the examples.
+AUC Area Under the ROC Curve (AUC) measures the probabil-
+ity that a randomly chosen positive example ranked higher by the
+model than a randomly chosen negative example. AUC only con-
+siders the relative order between positive and negative examples.
+A higher AUC indicates better ranking performance.
+4.1.3
+Competing Models. We compare xDeepInt with following
+models: LR(logistic regression) [18, 19], FM(factorization machine) [22],
+DNN (plain multilayer perceptron), Wide & Deep [7], DeepCross-
+ing [24], DCN (Deep & Cross Network) [28], PNN (with both in-
+ner product layer and outer product layer) [20, 21], DeepFM [8],
+xDeepFM [16], AutoInt [25] and FiBiNET [13]. Some of the models
+are state-of-the-art models for CTR prediction problem and are
+widely used in the industry.
+4.1.4
+Reproducibility. We implement all the models using Tensor-
+flow [1]. The mini-batch size is 4096, and the embedding dimension
+is 16 for all the features. For optimization, we employ Adam [15]
+with learning rate set to 0.001 for all the neural network models, and
+we apply FTRL [18, 19] with learning rate as 0.01 for both LR and
+FM. For regularization, we choose L2 regularization with 𝜆 = 0.0001
+for dense layer. Grid-search for each competing model’s hyper-
+parameters is conducted on the validation dataset. The number of
+1https://www.kaggle.com/c/avazu-ctr-prediction
+2https://www.kaggle.com/c/criteo-display-ad-challenge
+3http://contest.ipinyou.com/
+
+Research Track Paper
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+DNN, Cross, CIN, Interacting layers is from 1 to 4. The number of
+neurons ranges from 128 to 1024. All the models are trained with
+early stopping and are evaluated every 2000 training steps.
+For the hyper-parameters search of xDeepInt, The number of
+recursive feature interaction layers is from 1 to 4. For the number
+of sub-spaces ℎ, the searched values are 1, 2, 4, 8 and 16. Since our
+embedding size is 16, this range covers from complete vector-wise
+interaction to complete bit-wise interaction. We use G-FTRL opti-
+mizer for embedding table and FTRL for PIN layers with learning
+rate as 0.01.
+4.2
+Model Performance Comparison (Q1)
+Table 1: Performance Comparison of Different Algorithms
+on Criteo, Avazu and iPinYou Dataset.
+Criteo
+Avazu
+iPinYou
+Model
+AUC
+LogLoss
+AUC
+LogLoss
+AUC
+LogLoss
+LR
+0.7924
+0.4577
+0.7533
+0.3952
+0.7692
+0.005605
+FM
+0.8030
+0.4487
+0.7652
+0.3889
+0.7737
+0.005576
+DNN
+0.8051
+0.4461
+0.7627
+0.3895
+0.7732
+0.005749
+Wide&Deep
+0.8062
+0.4451
+0.7637
+0.3889
+0.7763
+0.005589
+DeepFM
+0.8069
+0.4445
+0.7665
+0.3879
+0.7749
+0.005609
+DeepCrossing
+0.8068
+0.4456
+0.7628
+0.3891
+0.7706
+0.005657
+DCN
+0.8056
+0.4457
+0.7661
+0.3880
+0.7758
+0.005682
+PNN
+0.8083
+0.4433
+0.7663
+0.3882
+0.7783
+0.005584
+xDeepFM
+0.8077
+0.4439
+0.7668
+0.3878
+0.7772
+0.005664
+AutoInt
+0.8053
+0.4462
+0.7650
+0.3883
+0.7732
+0.005758
+FiBiNET
+0.8082
+0.4439
+0.7652
+0.3886
+0.7756
+0.005679
+xDeepInt
+0.8111
+0.4408
+0.7675
+0.3872
+0.7791
+0.005565
+The overall performance of different models is listed in Table 1. We
+have the following observations in terms of model effectiveness:
+• LR is generally worse than other algorithms, which indicates
+that learning higher-order feature interactions is essential
+for CTR model performance.
+• FM brings the most significant boost in performance while
+we increase model complexity. This reveals the importance
+of learning explicit vector-wise feature interactions.
+• Models that combining vector-wise and bit-wise interactions
+together consistently outperform other models. This phe-
+nomenon indicates that both types of feature interactions are
+essential to prediction performance and compensate each
+other.
+• xDeepInt achieves the best prediction performance among
+all models. However, different datasets favor feature interac-
+tions of different degrees and bit-wise feature interactions of
+different complexity. The superior performance of our model
+could attribute to the fact that xDeepInt model the bounded
+degree of polynomial feature interactions by adjusting the
+depth of PIN and achieve different complexity of bit-wise
+feature interactions by changing the number of sub-spaces.
+4.3
+Hyper-Parameter Study (Q2)
+In order to have deeper insights of the proposed model, we conduct
+experiments on three datasets and compare several variants of
+xDeepInt on different hyper-parameter settings.
+4.3.1
+Depth of Network. The depth of PIN determines the order
+of feature interactions. Table 2 illustrates the performance change
+with respect to the number of PIN layers. When the number of
+layers set to 0, our model is equivalent to logistic regression and no
+interactions are learned. The performance of xDeepInt achieves the
+best when the number of layers is about 3 or 4. In this experiment,
+we set the number of sub-spaces as 1, to disable the bit-wise feature
+interactions.
+Table 2: Impact of hyper-parameters: number of layers
+#Layers
+0
+1
+2
+3
+4
+5
+Criteo
+AUC
+0.7921
+0.8038
+0.8050
+0.8057
+0.8063
+0.8061
+LogLoss
+0.4580
+0.4477
+0.4466
+0.4461
+0.4452
+0.4454
+Avazu
+AUC
+0.7536
+0.7654
+0.7664
+0.7675
+0.7670
+0.7662
+LogLoss
+0.3951
+0.3888
+0.3879
+0.3872
+0.3875
+0.3883
+iPinYou
+AUC
+0.7690
+0.7740
+0.7775
+0.7791
+0.7783
+0.7772
+LogLoss
+0.005604
+0.005576
+0.005569
+0.005565
+0.005580
+0.005571
+4.3.2
+Number of Sub-spaces. The subspace-crossing mechanism
+enables the proposed model to control the complexity of bit-wise in-
+teractions. Table 3 demonstrates that subspace-crossing mechanism
+boosts the performance. In this experiment, we set the number of
+PIN layers as 3, which is generally a good choice but not the best
+setting for each dataset.
+Table 3: Impact of hyper-parameters: number of sub-spaces
+#Sub-spaces
+1
+2
+4
+8
+16
+Criteo
+AUC
+0.8072
+0.8081
+0.8089
+0.8096
+0.8101
+LogLoss
+0.4445
+0.4435
+0.4425
+0.4421
+0.4418
+Avazu
+AUC
+0.7660
+0.7668
+0.7674
+0.7672
+0.7668
+LogLoss
+0.3880
+0.3877
+0.3875
+0.3878
+0.3879
+iPinYou
+AUC
+0.7772
+0.7783
+0.7788
+0.7787
+0.7784
+LogLoss
+0.005590
+0.005583
+0.005568
+0.005572
+0.005580
+4.3.3
+Activation Function. By default, we use linear activation func-
+tion on neurons of PIN layers. We also would like to explore how
+different activation function of PIN affect the performance. Table 4
+shows that the linear activation function is the most performant
+one for the PIN. We study the effect of activation function on Criteo
+dataset.
+Table 4: Impact of hyper-parameters: activation function
+AUC
+LogLoss
+linear
+0.8111
+0.4408
+tanh
+0.8100
+0.4418
+sigmoid
+0.8082
+0.4434
+softplus
+0.8080
+0.4436
+swish
+0.8100
+0.4418
+relu
+0.8098
+0.4419
+leaky relu
+0.8102
+0.4415
+elu
+0.8099
+0.4418
+selu
+0.8100
+0.4418
+4.3.4
+Optimizer. We also build our model with Adam optimizer,
+same as all the competing models, to compare with our G-FTRL
+and FTRL combined optimization strategy. Table 5 shows that our
+G-FTRL and FTRL combined optimization strategy achieves better
+
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
+Yachen and Liubo
+performance. Table 6 shows that our optimization strategy gets
+higher degree of feature sparse ratio (ratio of all zero embedding
+vectors in embedding table) and sparse ratio (ratio of zero weights in
+PIN layers), which results in lightweight model. One thing should be
+noted is that xDeepInt still achieves the best prediction performance
+among all models when using Adam optimizer, which demonstrates
+the effectiveness of xDeepInt architecture.
+Table 5: Impact of hyper-parameters: optimizer
+Dataset
+Model
+LogLoss
+AUC
+Criteo
+G-FTRL/FTRL
+0.4408
+0.8111
+Adam
+0.4415
+0.8105
+Avazu
+G-FTRL/FTRL
+0.3872
+0.7675
+Adam
+0.3873
+0.7674
+iPinYou
+G-FTRL/FTRL
+0.005565
+0.7791
+Adam
+0.005583
+0.7784
+Table 6: Analysis of model sparsity
+Dataset
+Feature Sparse ratio
+Weight Sparse ratio
+Criteo
+0.6506
+0.1030
+Avazu
+0.2193
+0.0448
+iPinYou
+0.8274
+0.0627
+4.4
+Ablation Study: Integrating Implicit
+Interactions (Q3)
+In this section, we conduct ablation study comparing the perfor-
+mance of our proposed model with and without integrating implicit
+feature interactions.
+Feed-forward neural network is widely used by various model
+architectures for learning implicit feature interactions. In this ex-
+periment, we jointly train xDeepInt with a three-layer feed-forward
+neural network and name the combined model as xDeepInt+ to
+compare with vanilla xDeepInt.
+Table 7 compares vanilla xDeepInt and xDeepInt+. We observe
+that the jointly-trained feed-forward neural network does not boost
+the performance of vanilla xDeepInt. The reason is that vanilla
+xDeepInt model has already learned bit-wise interactions through
+the subspace-crossing mechanism. Thus, feed-forward neural net-
+work does not bring in additional predictive power.
+Table 7: Ablation study comparing the performance of
+xDeepInt with and without integrating DNN
+Dataset
+Model
+LogLoss
+AUC
+Criteo
+xDeepInt
+0.4408
+0.8111
+xDeepInt+
+0.4412
+0.8107
+Avazu
+xDeepInt
+0.3872
+0.7675
+xDeepInt+
+0.3874
+0.7673
+iPinYou
+xDeepInt
+0.005565
+0.7791
+xDeepInt+
+0.005581
+0.7787
+5
+CONCLUSION
+In this paper, we design a novel network layer named polynomial
+interaction network (PIN), which learns the higher order vector-
+wise feature interactions on the embedding space. By incorporating
+PIN with the subspace-crossing mechanism, our proposed model
+xDeepInt learns bit-wise and vector-wise feature interactions of
+bounded degree simultaneously in controllable manner. We add
+residual connections to PIN layers, such that the output of each layer
+is an ensemble of the low-order and high-order interactions. The
+degree of interaction is controlled by the number of PIN layers, and
+the complexity of bit-wise interaction is controlled by the number
+of sub-spaces. Additionally, an optimization method is introduced
+to performs feature selection and interaction selection based on the
+network structure. Our experimental result demonstrates that the
+proposed xDeepInt outperforms existing state-of-art methods on
+real-world datasets. To our best knowledge, xDeepInt is the first
+neural network architecture that achieves state-of-art performance
+without integration of feed-forward neural network using non-
+linear activation functions.
+We have multiple directions of future work. First, the proposed
+model only focuses on modeling fixed-length feature vectors. In or-
+der to model historical and sequential behavior in recommendation
+systems[33, 34], We are interested in making our model architecture
+applicable to variable-length feature vectors. Second, We would like
+to extend the application of polynomial interaction layers to more
+modeling scenarios and exploit PIN’s potential on other problems.
+Third, the model is fully explainable when the subspace crossing
+mechanism is disable. The explainability of the model is another
+direction of future work.
+
+Research Track Paper
+DLP-KDD 2020, August 24, 2020, San Diego, California, USA
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+

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+arXiv:2301.05401v1  [astro-ph.HE]  13 Jan 2023
+Draft version January 16, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Long-duration Gamma-ray Burst Progenitors and Magnetar Formation
+Cui-Ying Song1 and Tong Liu2
+1Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China; [email protected]
+2Department of Astronomy, Xiamen University, Xiamen 361005, China; [email protected]
+ABSTRACT
+Millisecond magnetars produced in the center of dying massive stars are one prominent model to
+power gamma-ray bursts (GRBs). Their detailed nature, however, remains unsolved. To explore the
+effects of the initial mass, rotation, mass loss, and metallicity of the progenitor stars of 10 − 30 M⊙ on
+the formation and properties of the protomagnetars, we evolve over 150 single star models from the
+pre-main-sequence to core collapse by using the stellar evolution code MESA. We find that all of the
+fast-rotating stars become Wolf-Rayet stars. The final stellar, helium and carbon-oxygen core masses
+roughly increase with increasing initial mass, and decrease moderately with increasing initial rotation
+rate. We illustrate the effects of these intrinsic signatures on the hydrogen and helium envelopes and
+the metallicity. We then discuss the progenitors of the different types of supernovae. Furthermore, we
+find that the compactness parameter remains a nonmonotonic function of the initial mass and initial
+velocity when the effects of different metallicity and wind mass loss are considered. More importantly,
+we present the estimated period, magnetic field strength, and masses of protomagnetars in all cases.
+The typical rotational energy of these millisecond magnetars is sufficient to power long-duration GRBs.
+Keywords: Massive stars (732) - Stellar evolutionary models (2046) - Gamma-ray bursts
+(629) - Magnetars (992)
+1. INTRODUCTION
+Observations of long-duration gamma-ray bursts
+(LGRBs) associated with core-collapse supernovae
+(CCSNe, e.g., Galama et al. 1998; Bloom et al.
+1999; Hjorth et al. 2003; Stanek et al. 2003) sug-
+gest that some fraction of LGRBs originate from
+the death of massive stars (see reviews by Piran
+2004; Woosley & Bloom 2006; Kumar & Zhang
+2015; Cano et al. 2017).
+After the massive star
+collapses, a black hole (BH) hyperaccretion sys-
+tem (e.g., Woosley 1993; MacFadyen & Woosley
+1999; Liu et al. 2017) or a rapidly rotating neu-
+tron star (NS) with a strong magnetic field
+(magnetar, e.g., Usov 1992; Duncan & Thompson
+1992;
+Wheeler et al.
+2000;
+Zhang & Mészáros
+2001) might be formed.
+In the BH hyper-
+accretion scenario,
+the annihilation of neutri-
+nos and anti-neutrinos escaped from neutrino-
+dominated accretion disks (e.g., Popham et al.
+1999; Narayan et al. 2001; Chen & Beloborodov
+2007;
+Gu et al.
+2006;
+Liu et al.
+2007,
+2015;
+Zalamea & Beloborodov
+2011;
+Kawanaka et al.
+2013, see a review by Liu et al. 2017) or the
+large-scale
+magnetic
+fields
+extract
+the
+rota-
+tional
+energy
+of
+BHs,
+i.e.,
+Blandford-Znajek
+mechanism,
+(,
+e.g., Blandford & Znajek 1977;
+Lee et al. 2000; Wang et al. 2002; McKinney 2005;
+McKinney et al. 2012; Komissarov & Barkov 2009;
+Tchekhovskoy et al. 2011; Lei et al. 2013) to launch
+ultrarelativistic jets. The release energy from the
+spin-down of newborn magnetars could also ac-
+count for gamma-ray bursts (GRBs, e.g., Usov
+1992; Wheeler et al. 2000; Metzger et al. 2011,
+2017; Rowlinson et al. 2013; Kaspi & Beloborodov
+2017). The collimated jet production mechanism
+of magnetars has been studied by some literatures
+
+2
+Song et al.
+(Komissarov & Barkov
+2007;
+Bucciantini et al.
+2008, 2009, 2012; Kiuchi et al. 2012; Siegel et al.
+2014; Shankar et al. 2021). If the ultrarelativistic
+jet produced by the central engine could break
+out from the progenitor envelope and circumstel-
+lar medium and be along the line of sight, the
+observable LGRB could be triggered.
+The characteristics of LGRB progenitors are still
+unclear since no direct observational evidence has
+been presented to reveal the association of any
+progenitor and the observed LGRBs.
+The event
+rate of LGRBs is much lower than that of CC-
+SNe (e.g., Podsiadlowski et al. 2004; Izzard et al.
+2004; Guetta et al. 2005; Soderberg et al. 2006;
+Wanderman & Piran 2010), implying that only
+a small fraction of massive stars could produce
+LGRBs and CCSNe at the end of their lives.
+Soderberg et al. (2010) estimated that the ratio of
+LGRBs to Ibc supernovae (SN) is approximately
+1% based on a radio survey. Georgy et al. (2012)
+obtained this fraction from rotating stellar models
+and found that it could exceed 25%. The absence
+of hydrogen/helium lines in the spectra of SNe as-
+sociated with LGRBs suggests that their progen-
+itors have undergone violent mass loss or chem-
+ical mixing of elements.
+Wolf-Rayet (WR) stars
+with striping H/He envelopes before explosion have
+been proposed as GRB progenitors (e.g., Woosley
+1993; Woosley & Heger 2006; Crowther 2007). WR
+stars can be subdivided into three classes based
+on the exhibition of broad emission lines in spec-
+tra. These groups include N-rich WR stars (WN)
+and C-rich and O-rich WR stars (WC/WO). The
+origin of WR is an ongoing study.
+It is widely
+believed that rapidly rotating single massive stars
+could undergo quasi-chemically homogeneous evo-
+lution (e.g., Yoon & Langer 2005; Yoon et al. 2006;
+Ekström et al. 2012; Maeder & Meynet 2012), al-
+lowing the systems to lose their envelopes while still
+retaining enough angular momentum to produce
+LGRBs. Furthermore, WR stars can also be born
+in massive close binaries through tidal interactions
+with their companion stars (e.g., Cantiello et al.
+2007; Yoon et al. 2010; Eldridge et al. 2011; Langer
+2012; de Mink et al. 2013).
+Several recent studies have investigated LGRB
+progenitor evolution models and corresponding
+remnants (Heger et al. 2003; Hirschi et al. 2005;
+Obergaulinger & Aloy 2017; Aguilera-Dena et al.
+2018; Aloy & Obergaulinger 2021).
+Roy et al.
+(2020) showed the effects of mass, rotation rate,
+and metallicity of massive stars on their evolution
+into WN and subsequently into WC, which are pos-
+sible Type-Ic SNe/GRB progenitors. By studying
+the surface helium and nitrogen mass fraction evo-
+lution, they found that an O star with rotation rate
+Ω/Ωcirt ≥ 0.4 could evolve into the WN phase.
+This might denote that rapidly rotating mas-
+sive stars could satisfy the requirement of Type-
+Ic SNe/LGRB production.
+Aguilera-Dena et al.
+(2020) assumed that these successful neutrino-
+driven explosion models would produce magnetar
+and power superluminous SNe, and that failed ex-
+plosion models would lead to the BH form and
+power LGRBs. The subnormal distribution of the
+initial SN explosion energy could naturally build
+the “lower mass gap” in the mass distribution of
+compact objects (Liu et al. 2021). However, in ad-
+dition to the collapsar, the magnetar might also
+be the central engine of LGRBs.
+It is difficult
+to determine whether the central engine of GRB
+is a BH or magnetar.
+The shallow decay phase
+(Dai 2004; Zhang et al. 2006; Dall’Osso et al. 2011;
+Li et al. 2016) and “internal plateau” (Troja et al.
+2007; Lyons et al. 2010) in some GRB afterglow
+light curves have been proposed as the magnetar
+signature, but the BH hyperaccretion model can-
+not be ruled out, especially for long-lasting plateaus
+(e.g., Yi et al. 2022). Regardless of the type of cen-
+tral engine, the shallow decay phase might be re-
+lated to a precessing jet (Huang & Liu 2021). In
+this paper, we focus on the progenitors of LGRBs
+and the formation of magnetars.
+The physical conditions for dying massive stars
+to produce GRBs remain an open question. De-
+velopments to enhance theoretical models are re-
+quired to study how various physical parameters
+could replicate the formation conditions and char-
+acteristics of the GRB central engines. Conversely,
+to predict the final fate of massive stars and rem-
+
+LGRB progenitors and magnetar formation
+3
+Figure 1. (a) Hertzsprung-Russell diagram of the massive star with different initial masses Mini = 15 and 30 M⊙,
+metallicity Z = 0.01 and 0.1 Z⊙, initial rotation rates Ω/Ωcrit = 0.1 and 0.6, and scaling factors of the “Dutch” wind
+loss rate ηwind = 0.5 and 1. (b) Evolution of the central density and temperature in the models. The endpoint of the
+line is marked by the pentagram, corresponding to the time of the iron core collapse.
+nants, researchers should use observational GRB
+and SN data to constrain stellar evolution models.
+In the framework of the 1D single star evolu-
+tion scenario, we explore how initial rotation, mass,
+metallicity, and mass loss impact the final fate of
+the massive stars.
+More importantly, the initial
+signatures of the magnetars are discussed.
+This
+paper is structured as follows.
+In Section 2, we
+describe the physical ingredients of our stellar evo-
+lution models. We present our analysis of pre-SNe
+and the characteristics of the protomagnetar in Sec-
+tion 3. Conclusions and discussion are presented in
+Section 4.
+2. PROGENITOR MODELING IN MESA
+We
+use
+the
+Modules
+for
+Experiments
+in
+Stellar Astrophysics software package (MESA,
+Paxton et al. 2011, 2013, 2015, 2018, 2019, version
+21.12.1) to simulate a large set of 158 massive star
+models from the pre-main-sequence stage until the
+onset of iron core collapse. Here, we define collapse
+as the moment when the infall velocity of any point
+inside the stellar model exceeds 1,000 km s−1. We
+take the test suite 20M_pre_ms_to_core_collapse
+as our baseline model.
+Our stellar models cover
+an initial mass range between 10 and 30 M⊙
+with a step size of 1 M⊙, which are expected
+to form NSs upon collapse.
+The initial metal-
+licity is Z = 0.01 and 0.1 Z⊙, where Z is the
+mass fraction of elements heavier than helium and
+Z⊙ is the solar metallicity (e.g., Grevesse & Sauval
+1998; von Steiger & Zurbuchen 2016).
+Following
+Tout et al. (1996) and Pols et al. (1998), we use
+the initial helium fraction Y = 0.24 + 2Z and ini-
+tial hydrogen mass fraction X = 1 − Y − Z. The
+initial rotational rate Ω/Ωcrit = 0.1 − 0.8 in 0.1
+intervals, which corresponds to the initial equato-
+rial angular velocity Ω ≈ 80 − 770 km s−1. Here,
+Ωcrit = (GM/R3
+e)1/2 is the critical angular veloc-
+ity at the equator of the star, where M and Re
+denote the mass of the star and the equator ra-
+dius, respectively. For stellar winds, all models are
+evolved with the “Dutch” wind-loss scheme (e.g.,
+de Jager et al. 1988;
+Nieuwenhuijzen & de Jager
+1990;
+Nugis & Lamers
+2000;
+Vink et al.
+2001;
+
+4
+Song et al.
+Figure 2. The evolution of stellar mass and total angular momentum.
+Glebbeek et al.
+2009),
+and
+the
+scaling
+factor
+ηwind = 0.5 or 1. We treat convection using the
+MLT++ scheme of MESA. According to the Ledoux
+criterion and standard mixing length theory (MLT)
+approximation (Cox & Giuli 1968), we chose a mix-
+ing length parameter αMLT = 1.5.
+The effect
+of semiconvection (Langer 1991; Yoon et al. 2006)
+is included, and we adopt a baseline choice of
+αsc = 0.01 except after core helium depletion,
+where αsc = 0.
+Considering thermohaline mix-
+ing (Kippenhahn et al. 1980), we set αth = 2 and
+0 before and after core helium depletion, respec-
+tively.
+The hydrodynamic mixing instabilities at
+convective boundaries, called overshoot mixing, are
+treated as an exponential decay process (Herwig
+2000) beyond the Schwarzschild boundary.
+We
+adopt overshoot mixing parameters fov = 0.005
+and f0 = 0.001 (Paxton et al. 2011).
+The de-
+tailed definition and range of these parameters are
+described and discussed by Paxton et al. (2011,
+2013), Farmer et al. (2016) and on the MESA web-
+site1.
+1 https://docs.mesastar.org/en/release-r22.05.1/
+We use the approx21_cr60_plus_co56.net net-
+work to compute nuclear reactions in the stel-
+lar interior (Timmes 1999; Paxton et al. 2011).
+This approximate nuclear reaction network con-
+sists of 21 base isotopes plus
+60Cr and
+56Co,
+and it is a traditional workhorse in massive
+star models.
+Nuclear reaction rates are from
+JINA REACLIB (Cyburt et al. 2010), where type 2
+opacity tables (Iglesias & Rogers 1996) and the
+Skye (Jermyn et al. 2021) equation of state are
+adopted.
+We start by running models at the pre-main-
+sequence stage and activating rotation when near
+the zero-age-main-sequence. The implementation
+of rotation in MESA closely follows Heger et al.
+(2000) and Heger et al. (2005). Five different ro-
+tationally induced mixing processes are included:
+Solberg-H/oiland instability, secular shear insta-
+bility,
+Eddington-Sweet
+circulation,
+Goldreich-
+Schubert-Fricke, and Spruit-Tayler dynamo, which
+could lead to angular momentum redistribution
+and chemical mixing.
+It should be noted that
+the Spruit-Tayler dynamo only operates in radia-
+tive regions of the star and produces poloidal mag-
+netic fields.
+The toroidal magnetic fields gener-
+
+LGRB progenitors and magnetar formation
+5
+ated by differential winding are orders of mag-
+nitude larger than the poloidal magnetic fields
+(Heger et al. 2005). We assume that the compo-
+sitional mixing efficiency fc = 1/30 (Heger et al.
+2000).
+The stellar evolution tracks and central condi-
+tions for some of our models are shown in Figure
+1. The blue and red lines indicate initial masses
+M = 15 and 30 M⊙, respectively. The darker lines
+indicate models with a faster initial rotation speed.
+The solid, dotted, and dashed lines correspond to
+different metallicity and the scaling factor of wind
+(Z/Z⊙, ηwind)=(0.1, 1), (0.1, 0.5), (0.01, 1). The
+endpoint of lines is marked by a pentagram, cor-
+responding to the time of the iron core collapse.
+The more rapidly rotating model evolves toward
+the blue part of the Hertzsprung-Russell (H-R) di-
+agram and ends its life with a WR star. The slower
+rotating models evolve toward the red part of the
+H-R diagram and become red supergiants.
+The
+effective temperature and luminosity generally in-
+crease with increasing initial mass, rotation veloc-
+ity, and scaling factor of wind and decrease with
+increasing metallicity. Roughly, the more massive
+the stars are, the higher the central density and
+temperature.
+In Figure 2, we present the evolution of stellar
+mass and total angular momentum. More rapid ro-
+tation, lower metallicity, and a smaller scaling fac-
+tor of wind can extend the star lifetime. The final
+mass decreases with increasing rotation rate. The
+larger the initial velocity of a star is, the greater
+the angular momentum loss.
+3. RESULTS
+3.1. Pre-SN Properties
+We summarize the properties of all models at
+the end of their evolution in Table 1 of the Ap-
+pendix, listing the initial mass, metallicity, rota-
+tional rate, and corresponding velocity at the equa-
+tor, the “Dutch” wind scale factor, the initial total
+angular momentum, final mass, ages, radius, ef-
+fective temperature and luminosity, final angular
+momentum and masses of the He/CO/Fe core at
+the end of the calculation. The compactness pa-
+rameter, mass, average magnetic field strength, and
+rotation period of the protomagnetar are also pre-
+sented in the table.
+In Figure 3, we show the final stellar masses
+Mf, helium core masses MHe, carbon-oxygen core
+masses MCO, and iron core mass MFe of mas-
+sive stars at the beginning of iron core collapse
+as a function of the initial surface angular veloc-
+ity Ω. The different colors correspond to various
+initial parameters.
+The He and CO core masses
+are measured using the mass coordinate, where the
+mass fraction of hydrogen is less than 0.1 for MHe
+and the mass fraction of helium decreases below
+0.1 for MCO. Here, we define the iron core mass
+boundary as the outermost location where the Si
+mass fraction is less than 0.1.
+Remarkably, the
+radius and composition of the stellar core might
+change with different core mass definitions (e.g.,
+Sukhbold & Woosley 2014; Laplace et al. 2021). It
+is difficult to precisely define the stellar core bound-
+ary because there is a composition and density gra-
+dient at the edge. Moreover, the stellar core mass
+and boundary are also affected by semiconvection
+and overshoot (Schootemeijer et al. 2019).
+It is obvious from Figure 3 (a) and (b) that MHe
+values approximate Mf values when Ω/Ωcrit ≥ 0.4.
+For the more massive and lower metallicity stars,
+the corresponding values Ω/Ωcrit ≥ 0.3. In other
+words, stars almost entirely lose their hydrogen en-
+velopes and become naked He cores.
+Note that
+Mf, MHe, MCO and MFe show little change when
+Ω/Ωcrit ≥ 0.4. The values of Mf, MHe, and MCO
+are positively correlated with the initial mass Mini
+and negatively correlated with the initial metallic-
+ity Z and scaling factor of wind loss ηwind when
+the initial surface angular velocity is low. This is
+similar to models without rotation. For high initial
+surface angular velocity, rotation plays an impor-
+tant role in enhanced stellar mass loss according
+to the prescription, i.e.,
+˙M ∝ 1/(1 − Ω/Ωcrit)0.43
+(Paxton et al. 2013).
+Similar to Figure 3, the Mf, MHe, MCO, and MFe
+of massive stars at the beginning of iron core col-
+lapse as a function of initial stellar mass Mini are
+presented in Figure 4. Here, we set Ω/Ωcrit = 0.6.
+
+6
+Song et al.
+Figure 3. Final stellar mass Mf, He core mass MHe, CO core mass MCO and iron core mass MFe at the beginning
+of core collapse as a function of initial surface angular velocity Ω/Ωcrit.
+
+LGRB progenitors and magnetar formation
+7
+Figure 4. Final stellar mass Mf, He core mass MHe, CO core mass MCO and iron core mass MFe at the beginning
+of core collapse as a function of initial stellar mass Mini.
+
+8
+Song et al.
+Figure 5.
+Final hydrogen, helium, and metallicity
+mass fraction on the surface versus initial rotation rate
+for different initial mass, metallicity, and mass loss
+rates.
+Figure 6.
+Final hydrogen, helium, and metallicity
+mass fraction on the surface versus initial mass for dif-
+ferent mass loss rates and initial metallicity values at
+the initial surface angular velocity Ω = 0.6 Ωcrit.
+
+LGRB progenitors and magnetar formation
+9
+It is easy to find that Mf, MHe, and MCO increase
+as the initial mass increases, except for some fluc-
+tuations. MFe of massive stars with different initial
+Z and ηwind fluctuates with Ω and Mini, and there
+is no obvious correlation. Iron core masses in our
+models range from 1.29 to 1.92 M⊙, and most of
+them are concentrated between 1.4 − 1.6 M⊙.
+In Figure 5, we show the final hydrogen (up-
+per panel), helium (middle panel), and metallicity
+(lower panel) mass fraction on the surface versus Ω
+for different Mini, Z, and ηwind. As shown in the
+left panel of Figure 5, all of the hydrogen mass frac-
+tions on the surface of massive stars are lower than
+0.1 when Ω/Ωcrit ≥ 0.4. This suggests that rapid
+rotation can cause stars to lose their hydrogen en-
+velopes. It should be noted that the mass fractions
+of hydrogen, helium, and metallicity show slight
+changes when Ω/Ωcrit ≥ 0.4. In Figure 6, we show
+the final hydrogen, helium, and metallicity mass
+fraction values on the surface versus Mini for dif-
+ferent Z and ηwind. Here, we set Ω/Ωcrit=0.6. The
+results reveal that the surface helium mass fraction
+decreases gradually with increasing Mini, while the
+surface metallicity mass fraction displays the oppo-
+site trend.
+Massive stars without hydrogen envelopes could
+core-collapse and give rise to type Ib or Ic SNe.
+Type Ib or Ic SNe are distinguished based on the
+presence or absence of helium lines in the spec-
+trum. Sometimes, it is difficult to distinguish them
+from limited observational data.
+Most progeni-
+tor models still retain some helium after the core
+collapse. The different characteristics of the pro-
+genitor stars which lead to the difference between
+Ib or Ic SNe remain unknown (Hachinger et al.
+2012; Dessart et al. 2011, 2012, 2022). Therefore,
+these stars are usually collectively referred to as the
+Ibc SNe.
+Following Eldridge & Tout (2004) and
+Eldridge (2005), we use surface helium abundance
+Ysurface to estimate the SNe type produced by the
+progenitors. If Ysurface < 0.3, a type Ic SN occurs;
+if 0.3 < Ysurface < 0.7, we label the star as type
+Ibc due to uncertain, and if Ysurface > 0.7, a type
+Ib star results. As shown in the middle panel of
+Figures 5 and 6 and He masses in 3 and 4, most
+of our progenitor models would give rise to Ibc SN,
+and a small fraction of them would produce Ic or
+Ib SN.
+3.2. Protomagnetar signatures
+After a massive star collapses, a proto-NS might
+be born in the center. In this paper, we take the
+iron core masses as the NS baryonic masses Mb.
+The binding energy will be released by emitting
+neutrinos during the iron core collapse to the NS.
+Following Lattimer & Prakash (2001), the NS grav-
+itational mass MNS can be calculated by
+Mb − MNS
+MNS
+≈
+0.6 GMNS/c2
+1 − 0.5 GMNS/c2 .
+(1)
+The change in NS mass due to material fallback
+in accretion during the explosion is negligible, espe-
+cially for stars with zero-age-main-sequence masses
+less than 30 M⊙. The results of accurate calcula-
+tions, including fallback, exhibit good agreement
+with this simple approach (e.g., Sukhbold et al.
+2016; Ertl et al. 2020).
+Magnetic fields can be generated by differen-
+tial rotation in radiative regions (e.g., Spruit 2002;
+Petrovic et al. 2005; Heger et al. 2005). The aver-
+age strength of dynamo-generated magnetic fields
+inside the iron core is given by
+< Bφ >=
+� MFe
+0
+Bφ(m)dm
+� MFe
+0
+dm
+.
+(2)
+Assuming the conservation of magnetic flux and
+iron core contracted homogeneously to the NS with
+radius RNS = 12 km, we obtain the average surface
+magnetic field strength of NSs.
+The
+core
+compactness
+was
+defined
+as
+(O’Connor & Ott 2011)
+ξM0 =
+M0/M⊙
+R(Mbary = M0)/1000 km,
+(3)
+where M0 = 2.5 M⊙ was chosen to evaluate the
+compactness parameter (e.g., Ugliano et al. 2012;
+Sukhbold & Woosley 2014), which corresponds to
+the point where the collapse speed first reaches
+1, 000 km s−1. This has been pointed out as a rough
+
+10
+Song et al.
+Figure 7. The periods PNS of the protomagnetar as a function of initial mass Mini. All progenitor models are
+included.
+Figure 8. The mass MNS of the protomagnetar as a function of initial mass Mini. All progenitor models are included.
+indicator of whether the collapse of a nonrotat-
+ing stellar core could lead to a successful neutrino-
+driven explosion (ξ2.5 < 0.45) or form a BH (ξ2.5 >
+0.45). While this criterion is not sufficient to ac-
+curately predict the fate of rapidly spinning stars,
+it can still reveal structural features of the stel-
+lar core (e.g., Ertl et al. 2016; Müller et al. 2016;
+Aguilera-Dena et al. 2020).
+The natal spin rates of NSs can be determined
+by angular momentum within the iron core:
+PNS = 2πINS
+JFe
+.
+(4)
+
+LGRB progenitors and magnetar formation
+11
+Figure 9. The average magnetic field strength of the protomagnetar as a function of initial mass Mini. All progenitor
+models are included.
+Figure 10. Compactness parameters ξ2.5 of progenitor models as a function of initial rotation rate Ω/Ωcrit and mass
+Mini. The ξ2.5 = 0.45 line separates models that could explode (gray) and disallowed (white) regions.
+
+12
+Song et al.
+Figure 11. The total rotation energy Erot as a function of periods PNS. All models involved in the diagram eventually
+evolved into WR stars.
+
+LGRB progenitors and magnetar formation
+13
+Here, the moment of inertia of the NSs is set as INS
+= 0.35MNSR2
+NS (e.g., Lattimer & Prakash 2001).
+The periods PNS, masses MNS and average sur-
+face magnetic fields BNS of proto-NSs after core
+collapse are presented in Figures 7, 8, and 9, re-
+spectively. The left and right panels correspond to
+the “Dutch” wind scaling factor ηwind = 1 and 0.5,
+respectively. The pentagrams and solid circles rep-
+resent different metallicity Z = 0.01 and 0.1 Z⊙.
+The ratio of the angular velocity to critical angu-
+lar velocity Ω/Ωcrit is indicated by the color bar,
+which varies from 0.1 − 0.8. In Figure 7, we find
+that when ηwind = 1, the PNS of most proto-NSs
+born in lower initial metallicity (Z = 0.01 Z⊙) pro-
+genitors are larger than those of progenitors pro-
+duced by higher initial metallicity (Z = 0.1 Z⊙).
+However, this phenomenon disappears when ηwind
+= 0.5. The periods PNS range from 1.0 − 2.8 ms.
+The rotation rate of NSs does not increase mono-
+tonically with the initial velocity of the predeces-
+sor star, which may be related to the fact that the
+angular momentum loss during star evolution is in-
+fluenced by several factors. As shown in Figure 8,
+the masses MNS span from 1.18 M⊙ to 1.68 M⊙,
+with a median value of 1.4 M⊙.
+Similar to Figure 7, we can find in Figure 9
+that the average surface magnetic fields BNS of
+most proto-NSs born from lower initial metallicity
+(Z = 0.01 Z⊙) progenitors are smaller than those
+of progenitors produced by higher initial metallic-
+ity (Z = 0.1 Z⊙) when ηwind = 1, but not for ηwind
+= 0.5. Even after taking into account the effects
+of initial velocity, stellar wind loss, and metallicity,
+progenitor stars with larger initial masses are still
+more likely to form fast-spinning NSs. The more
+massive NSs are more likely to have faster rotation
+rates and stronger magnetic field strengths.
+Re-
+gardless, these NSs are definitely classified as mil-
+lisecond magnetars.
+The compactness parameter ξ2.5 is calculated ac-
+cording to Equation (3). As shown in Figure 10,
+ξ2.5 varies with the initial rotation rate Ω/Ωcrit and
+initial mass Mini of progenitors. The ξ2.5 = 0.45
+line separates models that could explode, which are
+driven by neutrino (gray) and disallowed (white)
+regions. It is easy to find that most of our progeni-
+tor models could explode successfully. The models
+with Z = 0.01 Z⊙ and ηwind = 1 appear to have
+rather low compactness. The values of the com-
+pactness parameter are less affected by the initial
+velocity of stars, especially for ηwind = 1. For mod-
+els with the same initial velocity, the compactness
+parameter varies with the initial mass in a non-
+monotonic way. It is consistent with previous stud-
+ies (Müller et al. 2016; Aguilera-Dena et al. 2020).
+The compactness parameter is positively correlated
+with the mass of the NSs in all cases.
+To explore the ability of newborn magnetars to
+power GRBs, the total rotation energy Erot and
+PNS of magnetars born from different values of ini-
+tial mass, metallicity, wind loss rate, and initial
+rotation rate are shown in Figure 11. All of the
+progenitor stars are WR stars that have lost their
+hydrogen envelope. Erot ranges from ∼ 5 × 1051
+to ∼ 2 × 1052 ergs, and PNS ranges from 1.2 to
+2.5 ms.
+We conclude that stars with larger ini-
+tial masses tend to produce rapidly rotating mag-
+netars with more rotational energy, even when dif-
+ferent initial rotation rates and stellar wind losses
+are taken into account.
+This phenomenon is es-
+pecially evident for stars with Z = 0.01Z⊙. Con-
+sidering that the envelopes are easily disrupted by
+jets in most cases, the above results indicate that
+our models could meet the energy requirements
+of most of the observed LGRBs, including some
+collapsar-origin SGRBs (e.g. Zhang et al. 2009;
+Levesque et al. 2010; Thöne et al. 2011; Xin et al.
+2011). In other words, we present theoretical evi-
+dence on magnetar-origin GRBs by simulating the
+evolution of fast-rotating, low-metallicity, and mas-
+sive stars.
+4. CONCLUSIONS AND DISCUSSION
+We use MESA code to simulate a grid of fast-
+rotating (Ω/Ωcrit = 0.1−0.8 with 0.1 interval), low
+metallicity (Z = 0.1 and 0.01 Z⊙), and massive
+star (10 − 30 M⊙ with mass interval 1 M⊙) mod-
+els with different “Dutch” wind-loss scaling factors
+(ηwind = 0.5 and 1) from the pre-main-sequence
+stage until core collapse.
+The effects of the ini-
+tial rotation, mass, metallicity, and mass loss on
+
+14
+Song et al.
+the formation and characteristics of the GRB pro-
+genitor and protomagnetar are explored. The fi-
+nal stellar, helium and carbon-oxygen core masses
+roughly increase with increasing initial mass and
+decrease moderately with increasing initial rota-
+tion rate. We also discuss the progenitors of the
+different types of supernovae. Most of our progen-
+itor models would give rise to type Ibc SNe, and
+a small fraction of them will produce type Ic or Ib
+SNe. Furthermore, we find that the compactness
+parameter remains a nonmonotonic function of the
+initial mass and initial velocity when the effects of
+different metallicity and wind mass loss are consid-
+ered. All of our stellar models evolved to WR stars
+at a high initial rotation rate Ω/Ωcrit ≥ 0.4. The
+period of a protomagnetar ranges from 1.0 − 2.8
+ms with a median value of 1.63 ms, and the mass
+ranges from 1.18 − 1.68 M⊙ with a median value
+of 1.41 M⊙.
+The average surface magnetic field
+strength is concentrated on the order of ∼ 1014 G.
+The Erot can be up to about 2 × 1052 ergs. The
+rotational energy, magnetic field strength, and pe-
+riod of the protomagnetar are satisfied to power the
+typical LGRBs.
+Observation and theory demonstrated that GRB
+rates and properties vary with redshift due to the
+different distributions of metallicity, star forma-
+tion rate, and initial mass distributions of massive
+stars at different redshifts (e.g., Yonetoku et al.
+2004; Hirschi et al. 2005; Langer & Norman 2006;
+Madau & Dickinson 2014; Taggart & Perley 2021).
+The evolution of LGRB progenitors is also influ-
+enced by redshift (Yoon et al. 2006; Fryer et al.
+2022). However, the analysis of the metallicity of
+GRB host galaxies reveals that GRBs favor subso-
+lar metallicity, even considering the redshift effect
+(Levan et al. 2016).
+Therefore, we only consider
+two subsolar metallicity values Z = 0.1 and 0.01
+Z⊙ in our stellar models.
+It is unclear how much the evolutionary path-
+ways of single and binary stars contributed to the
+production of type Ibc SN and GRB progenitors
+(e.g., Langer 2012). Here, we do not consider the
+interaction of binary stars and focus on exploring
+the effects of different parameters on single star
+evolution and the formation of LGRB central en-
+gines. The evolution of binary stars is more com-
+plicated than that of a single star.
+In addition
+to some usual uncertainties in single star evolu-
+tion, as we mentioned above, the evolution path
+and outcome of a binary star system also depend
+on the initial mass ratio, orbital eccentricity, ma-
+terial, and angular momentum exchange. Binary
+channels might also be important for the forma-
+tion of GRB progenitors (e.g., Fryer et al. 1999;
+Fryer & Heger 2005; Eldridge et al. 2011).
+In a
+close binary system, the mass gainer can be spun up
+to break up the rotation velocity through mass and
+angular momentum transport (e.g., Cantiello et al.
+2007).
+Therefore,
+the binary progenitors for
+LGRBs do not require a fast initial rotation veloc-
+ity to strip the stellar hydrogen envelope and retain
+sufficient angular momentum to produce a GRB
+jet (e.g., Podsiadlowski et al. 2010; Chrimes et al.
+2020). Moreover, Laplace et al. (2021) studied the
+systematic differences in the core density and com-
+position structure of solar metallicity nonrotating
+single and donor stars in binary systems with an
+initial mass range of 11 − 21 M⊙.
+They con-
+cluded that single stars systematically possessed
+more massive He cores than binary-striped stars
+with the same initial mass. Lloyd-Ronning (2022)
+proposed that radio-loud GRBs are produced by in-
+teracting binary systems, while radio-quiet GRBs
+originate from the collapse of single stars.
+The material fallback process during an SN ex-
+plosion can affect the final properties of compact
+remnants (Zhang et al. 2008; Fryer et al. 2012;
+Ugliano et al. 2012; Janka 2013; Müller et al. 2019;
+Ertl et al. 2020). First, the mass of NS could grow
+by accretion of fallback material.
+In this paper,
+we take the iron core mass as the baryonic mass
+of NS, then convert it to NS gravitational mass
+through an analytical, radius-dependent approach
+(Lattimer & Prakash 2001) and do not consider de-
+tailed SNe explosion physics.
+The mass of our
+NSs ranges from 1.18 M⊙ to 1.68 M⊙, with a me-
+dian value of 1.41 M⊙. This result exhibits good
+agreement with the previous simulation, includ-
+ing fallback (e.g., Ugliano et al. 2012; Müller et al.
+
+LGRB progenitors and magnetar formation
+15
+2016; Sukhbold et al. 2016).
+However, if a large
+amount of matter could fall on the newly born NS,
+a black hole might form.
+Second, recent three-
+dimensional simulations of nonrotating progenitors
+indicate that the NS can be spun up to millisecond
+periods by fallback (Chan et al. 2020). Here, we
+estimate the rotation rate of the newly born NS by
+the conservation of the total angular momentum in
+the iron core, neglecting the fallback process.
+There is no firm conclusion regarding the up-
+per and lower limits of NS mass born in na-
+ture (Özel & Freire 2016).
+It is difficult to form
+NSs with masses below 1.2 M⊙ through iron-core
+collapse SN explosions, although low-mass NSs
+(≈ 0.93 M⊙) have been observed.
+These low-
+mass NSs may be born in an electron-capture SNe
+explosion with 8−10 M⊙ progenitors (Lattimer
+2012). The well-measured massive pulsar masses,
+such as J1614+2230 (1.97±0.04 M⊙), J0348+0432
+(2.01±0.04 M⊙), J2215+5135 (2.27±0.17 M⊙),
+and J0740+6620 (2.14±0.1 M⊙), indicate that
+the
+maximum
+NS
+mass
+can
+be
+at
+least
+2
+M⊙ (Demorest et al. 2010; Antoniadis et al. 2013;
+Linares et al. 2018; Cromartie et al. 2020).
+The
+observation of a compact object in a binary merger
+GW190814 (Abbott et al. 2020) could be an indi-
+cation of the most massive NS mass reaching 2.6 −
+2.9 M⊙ (Godzieba et al. 2021). The NS mass dis-
+tributions in binary systems have been investigated
+by many authors (Thorsett & Chakrabarty 1999;
+Fryer et al. 2012).
+Woosley (2019), Ertl et al.
+(2020), and Woosley et al. (2020) calculated the
+evolution and explosion of a grid of nonrotating he-
+lium stars and studied the remnant mass distribu-
+tions. They simplify binary star evolution and as-
+sume that the entire hydrogen envelope is promptly
+removed by binary interaction at the moment of he-
+lium core ignition. Their results show that the me-
+dian NS masses in binary systems are in the range
+of 1.32 − 1.37 M⊙ and vary slightly with mass loss
+and metallicity (Woosley et al. 2020). They set the
+lightest and heaviest NS gravitational mass at 1.24
+and 2.3 M⊙, respectively.
+Previous observations
+have shown overwhelming evidence that the mass
+distribution of NSs is bimodal (e.g., Özel et al.
+2012; Antoniadis et al. 2016; Alsing et al. 2018),
+with two peaks at 1.3 M⊙ and 1.5−1.7 M⊙.
+The distribution of the compactness parameter
+for nonrotating stars has been studied by some au-
+thors (O’Connor & Ott 2011; Ugliano et al. 2012;
+Sukhbold & Woosley
+2014;
+Müller et al.
+2016).
+The approximate critical value between black
+hole and NS formation varies in different stud-
+ies.
+Müller et al. (2016) proposed a semianalytic
+method to determine explodability and found that
+ξ2.5 = 0.278 is the best value for discriminating suc-
+cessful or failed explosions.
+Aguilera-Dena et al.
+(2020) calculated the compactness of 42 rotating
+massive star models with initial equatorial rotation
+velocities of 600 km s−1 and Z = 0.02 Z⊙. They
+show that semianalytic exploding models are com-
+patible with the BH and NS thresholds η2.5 = 0.45.
+They suggested that the compactness is a non-
+monotonic function of the initial mass for rapidly
+rotating stars. In this paper, we explore the effects
+of the initial rotation rate, initial mass, metallicity
+and mass loss on compactness. The results show
+that the compactness parameter remains a non-
+monotonic function of the initial mass and initial
+velocity when the effects of different metallicity and
+wind mass loss are considered. In the future, we
+will test the accuracy of the compactness param-
+eter as a criterion for rotating massive star explo-
+sions by simulating the process of SN explosion. We
+will also investigate the effects of the initial mass,
+metallicity and rotational rate on the SN explosion
+energy, ejecta mass and remnant characteristics.
+Software:
+MESA
+(Paxton et al.
+2011,
+2013,
+2015,
+2018, 2019),
+MatPlotLib
+(Hunter
+2007), NASA ADS, py_mesa_reader, NumPy
+(van der Walt et al.
+2011),
+and
+Pandas
+(Reback et al. 2022).
+
+16
+Song et al.
+ACKNOWLEDGMENTS
+We appreciate Prof. Alexander Heger and Prof.
+Bernhard Müller for their constructive suggestions
+and comments, and thank Prof.
+Dong Lai, Dr.
+Tuan Yi, Dr. Zhenyu Zhu, and Shuai Zha for help-
+ful discussion. The computations in this paper were
+run on the π2.0 cluster supported by the Center
+for High Performance Computing at Shanghai Jiao
+Tong University. This work was supported by the
+National Natural Science Foundation of China un-
+der grants 12103033, 12173031, and 12221003.
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+LGRB progenitors and magnetar formation
+21
+APPENDIX
+The properties of the star models at the end of their evolution are shown in Table 1.
+
+22
+Song et al.
+Table 1. Properties of the models at the end of their evolution.
+Progenitor Parameters
+Stellar Properties at Core Collapse
+Protomagnetar
+Mini
+Z
+Ω
+Vequ
+ηwind
+Jini
+Mf
+Ages
+Rf
+log Teff logL
+JHe
+MHe
+JCO
+MCO
+JFe
+MFe ξ2.5 MNS
+BNS
+PNS
+(M⊙) (Z⊙) (Ωcrit) (km s−1)
+(1052ergs s) (M⊙) (Myr) (R⊙)
+(K)
+(L⊙) ( 1050 ergs s) (M⊙) (1049 ergs s) (M⊙) (1048 ergs s) (M⊙)
+(M⊙) (1014 G) (ms)
+10
+0.01
+0.1
+89
+0.5
+0.31
+9.97
+28.31 558.77
+3.59
+4.81
+0.61
+3.51
+1.15
+1.98
+4.19
+1.47 0.01 1.33
+1.69
+2
+10
+0.01
+0.3
+265
+0.5
+0.91
+9.97
+38.49
+29.79
+4.28
+5.04
+1.41
+5.38
+4.54
+3.43
+5.15
+1.47 0.11 1.33
+2.04
+1.62
+10
+0.01
+0.5
+428
+0.5
+1.4
+5.87
+43.95
+0.92
+5.12
+5.36
+3.17
+5.87
+3.62
+3.44
+4.96
+1.61 0.23 1.44
+1.5
+1.83
+10
+0.01
+0.6
+502
+0.5
+1.58
+5.98
+44.93
+0.68
+5.15
+5.24
+3.66
+5.98
+8.64
+4.18
+5.63
+1.51
+0.2
+1.36
+1.68
+1.52
+10
+0.01
+0.6
+502
+1
+1.58
+4.28
+45.02
+0.95
+5.04
+5.05
+0.76
+4.28
+2.48
+2.87
+4.05
+1.5
+0.1
+1.35
+2.46
+2.1
+10
+0.01
+0.7
+571
+0.5
+1.71
+5.93
+45.77
+0.68
+5.15
+5.23
+3.61
+5.93
+8.43
+4.13
+5.18
+1.59 0.21 1.42
+2.07
+1.73
+10
+0.01
+0.8
+593
+0.5
+1.56
+5.82
+44.95
+0.7
+5.16
+5.27
+3.13
+5.82
+6.2
+4
+4.42
+1.59 0.24 1.43
+1.67
+2.03
+10
+0.1
+0.1
+82
+0.5
+0.31
+9.73
+28.48 657.59
+3.55
+4.81
+0.27
+3.47
+0.82
+2.02
+2.99
+1.47 0.02 1.33
+0.8
+2.8
+10
+0.1
+0.6
+461
+1
+1.58
+7.35
+44.7
+1.73
+5.03
+5.55
+1.87
+7.2
+8.51
+5.21
+4.52
+1.63 0.26 1.46
+1.53
+2.03
+10
+0.1
+0.6
+461
+0.5
+1.58
+5.85
+45.62
+0.78
+5.11
+5.17
+2.87
+5.85
+5.55
+4.12
+5.73
+1.53 0.21 1.37
+2.95
+1.51
+11
+0.01
+0.6
+510
+0.5
+1.9
+6.35
+38.36
+0.68
+5.25
+5.61
+3.74
+6.35
+8.9
+4.48
+5.77
+1.64 0.26 1.46
+2
+1.6
+11
+0.01
+0.6
+510
+1
+1.9
+4.5
+38.43
+0.81
+5.09
+5.11
+0.81
+4.5
+2.93
+3.07
+4.18
+1.54 0.18 1.39
+2.2
+2.09
+11
+0.1
+0.6
+469
+0.5
+1.9
+6.28
+38.65
+0.81
+5.11
+5.21
+2.57
+6.28
+8.06
+4.42
+4.97
+1.53 0.13 1.37
+1.8
+1.74
+11
+0.1
+0.6
+469
+1
+1.9
+7.91
+38.02
+1.51
+5.01
+5.34
+2.33
+7.81
+11
+5.7
+4.8
+1.55 0.16 1.39
+1.72
+1.83
+12
+0.01
+0.6
+518
+1
+2.24
+4.86
+33.26
+0.62
+5.17
+5.21
+0.94
+4.86
+3.87
+3.44
+3.82
+1.59 0.24 1.43
+1.42
+2.35
+12
+0.1
+0.1
+85
+0.5
+0.45
+11.72 20.79 688.92
+3.57
+4.92
+0.74
+4.1
+1.96
+2.43
+3.43
+1.43 0.04
+1.3
+1.04
+2.38
+12
+0.1
+0.6
+476
+0.5
+2.26
+8.07
+33.12
+0.7
+5.18
+5.35
+4.22
+8.07
+18.1
+5.95
+3.38
+1.51 0.16 1.36
+1.05
+2.53
+12
+0.1
+0.6
+476
+1
+2.26
+8.19
+33
+0.8
+5.15
+5.37
+2.56
+8.19
+13.3
+6.07
+5
+1.53 0.24 1.37
+4.31
+1.73
+13
+0.01
+0.6
+525
+0.5
+2.61
+7.16
+29.2
+0.67
+5.18
+5.3
+4.03
+7.16
+13.5
+5.18
+5.69
+1.47 0.17 1.33
+3.62
+1.47
+13
+0.01
+0.6
+525
+1
+2.61
+5.02
+29.34
+0.87
+5.17
+5.52
+0.96
+5.02
+4.23
+3.58
+4.39
+1.57 0.19 1.41
+1.45
+2.03
+13
+0.1
+0.6
+483
+0.5
+2.64
+7.41
+29.33
+0.7
+5.25
+5.65
+3.39
+7.41
+13.8
+5.46
+5.73
+1.5
+0.14 1.36
+1.97
+1.49
+13
+0.1
+0.6
+483
+1
+2.64
+8.68
+29.06
+0.91
+5.13
+5.38
+2.74
+8.68
+14.9
+6.56
+4.86
+1.51 0.13 1.36
+3.61
+1.76
+14
+0.01
+0.6
+531
+1
+3
+5.32
+26.14
+0.61
+5.18
+5.25
+1.05
+5.32
+5
+3.86
+4.03
+1.59 0.21 1.42
+1.8
+2.23
+14
+0.1
+0.6
+490
+1
+3.05
+9.28
+25.9
+1.03
+5.12
+5.45
+3.59
+9.28
+18.7
+7.02
+4.08
+1.48 0.11 1.34
+1.14
+2.06
+14
+0.1
+0.6
+490
+0.5
+3.05
+7.87
+26.12
+0.61
+5.21
+5.36
+3.6
+7.87
+15.4
+5.77
+6.43
+1.61 0.23 1.44
+2.5
+1.41
+15
+0.01
+0.2
+190
+1
+1.33
+14.95
+17.2
+570.08
+3.68
+5.18
+1.82
+6.06
+6.65
+4.07
+6.31
+1.64 0.26 1.46
+2.41
+1.46
+15
+0.01
+0.4
+372
+0.5
+2.52
+8.09
+22.48
+0.58
+5.22
+5.37
+4.23
+8.09
+17.2
+6
+5.7
+1.51
+0.2
+1.36
+2.39
+1.51
+15
+0.01
+0.4
+372
+1
+2.52
+5.68
+22.57
+0.59
+5.18
+5.22
+1.13
+5.68
+5.26
+4.18
+4.47
+1.63 0.22 1.46
+1.41
+2.05
+15
+0.01
+0.5
+458
+1
+3.02
+5.55
+23.07
+0.6
+5.17
+5.18
+1.1
+5.55
+4.5
+4.05
+4.38
+1.54 0.19 1.38
+1.52
+1.99
+15
+0.01
+0.6
+537
+0.5
+3.42
+7.79
+23.48
+0.58
+5.22
+5.37
+4
+7.79
+15.5
+5.75
+5.01
+1.55
+0.2
+1.39
+2.21
+1.75
+Table 1 continued
+
+LGRB progenitors and magnetar formation
+23
+Table 1 (continued)
+Progenitor Parameters
+Stellar Properties at Core Collapse
+Protomagnetar
+Mini
+Z
+Ω
+Vequ
+ηwind
+Jini
+Mf
+Ages
+Rf
+log Teff logL
+JHe
+MHe
+JCO
+MCO
+JFe
+MFe ξ2.5 MNS
+BNS
+PNS
+(M⊙) (Z⊙) (Ωcrit) (km s−1)
+(1052ergs s) (M⊙) (Myr) (R⊙)
+(K)
+(L⊙) ( 1050 ergs s) (M⊙) (1049 ergs s) (M⊙) (1048 ergs s) (M⊙)
+(M⊙) (1014 G) (ms)
+15
+0.01
+0.6
+537
+1
+3.42
+5.5
+23.56
+0.64
+5.15
+5.18
+1.08
+5.5
+4.97
+4.02
+3.76
+1.58 0.19 1.41
+0.81
+2.37
+15
+0.01
+0.7
+612
+0.5
+3.72
+7.75
+23.88
+0.54
+5.24
+5.38
+3.98
+7.75
+16.1
+5.69
+5.99
+1.58 0.23 1.41
+2.4
+1.49
+15
+0.01
+0.7
+612
+1
+3.72
+5.34
+23.94
+0.64
+5.16
+5.19
+1.01
+5.34
+4.84
+3.85
+4.19
+1.58 0.18 1.42
+0.99
+2.13
+15
+0.01
+0.8
+658
+0.5
+3.69
+7.95
+23.83
+0.6
+5.21
+5.36
+4.52
+7.95
+16.9
+5.8
+6.64
+1.61 0.22 1.44
+1.96
+1.37
+15
+0.01
+0.8
+658
+1
+3.69
+5.44
+23.91
+0.58
+5.19
+5.25
+1.07
+5.44
+4.78
+3.98
+4.48
+1.63 0.24 1.46
+1.41
+2.05
+15
+0.1
+0.1
+88
+1
+0.69
+12.7
+14.51 719.67
+3.59
+5.04
+1.19
+4.89
+3.93
+3.09
+4.13
+1.53 0.15 1.37
+2.7
+2.09
+15
+0.1
+0.1
+88
+0.5
+0.69
+13.52 15.55 748.97
+3.61
+5.13
+1.46
+5.66
+5.27
+3.77
+4.59
+1.53 0.19 1.38
+1.56
+1.89
+15
+0.1
+0.2
+175
+1
+1.36
+14.63 16.44 697.36
+3.61
+5.1
+0.98
+5.39
+3.38
+3.49
+4.51
+1.57 0.16 1.41
+1.83
+1.97
+15
+0.1
+0.2
+175
+0.5
+1.36
+14.49 16.71 750.76
+3.62
+5.17
+1.81
+5.94
+6.18
+3.99
+5.48
+1.59 0.22 1.43
+1.73
+1.64
+15
+0.1
+0.3
+261
+0.5
+1.99
+14.76 19.35 670.84
+3.66
+5.24
+1.55
+6.94
+5.85
+4.74
+3.44
+1.29 0.38 1.18
+0.04
+2.16
+15
+0.1
+0.4
+343
+0.5
+2.57
+12.01 22.24
+1.15
+5.16
+5.71
+8.89
+12.01
+42.4
+9.31
+7.74
+1.69 0.35
+1.5
+2.88
+1.22
+15
+0.1
+0.4
+343
+1
+2.57
+10.19 22.24
+0.7
+5.22
+5.51
+3.34
+10.19
+19.4
+7.99
+5.7
+1.79 0.47 1.59
+1.95
+1.75
+15
+0.1
+0.5
+422
+0.5
+3.08
+7.78
+23.11
+0.61
+5.21
+5.35
+3.02
+7.78
+14
+5.73
+6.03
+1.55 0.21
+1.4
+3.1
+1.46
+15
+0.1
+0.6
+496
+0.5
+3.48
+7.88
+23.55
+0.67
+5.19
+5.35
+3.71
+7.88
+14.7
+5.74
+5.56
+1.58
+0.2
+1.42
+2.08
+1.61
+15
+0.1
+0.6
+495
+1
+3.48
+9.78
+23.33
+0.7
+5.21
+5.47
+3.94
+9.78
+21.7
+7.55
+6.49
+1.64 0.26 1.47
+1.5
+1.42
+15
+0.1
+0.7
+565
+1
+3.79
+9.21
+23.69
+0.81
+5.16
+5.39
+3.12
+9.21
+18.3
+7.15
+5.24
+1.55 0.15
+1.4
+2.22
+1.68
+15
+0.1
+0.8
+622
+1
+3.9
+9.43
+23.79
+0.79
+5.17
+5.45
+3.37
+9.43
+19.4
+7.3
+6.66
+1.56 0.23
+1.4
+3.37
+1.32
+15
+0.1
+0.8
+623
+0.5
+3.9
+7.49
+24.06
+0.66
+5.18
+5.33
+3.32
+7.49
+13.5
+5.52
+5.07
+1.57 0.15 1.41
+3.91
+1.76
+16
+0.01
+0.6
+543
+1
+3.86
+5.8
+21.45
+0.58
+5.19
+5.24
+1.19
+5.8
+5.1
+4.23
+4.32
+1.62 0.23 1.45
+1.01
+2.12
+16
+0.1
+0.6
+501
+0.5
+3.94
+8.37
+21.42
+0.64
+5.21
+5.4
+4.14
+8.37
+18.4
+6.13
+5.31
+1.54 0.17 1.38
+2.73
+1.64
+16
+0.1
+0.6
+501
+1
+3.94
+10.22 21.17
+0.68
+5.22
+5.5
+3.84
+10.22
+22.5
+8.01
+6.34
+1.83
+0.6
+1.62
+2.53
+1.61
+17
+0.01
+0.6
+549
+1
+4.32
+5.86
+19.73
+0.65
+5.19
+5.32
+1.16
+5.86
+5.44
+4.28
+4.35
+1.56 0.16
+1.4
+1.44
+2.03
+17
+0.1
+0.6
+507
+1
+4.42
+10.34 19.44
+0.78
+5.18
+5.45
+3.78
+10.34
+22.7
+8.03
+6.38
+1.63 0.24 1.46
+2.36
+1.44
+17
+0.1
+0.6
+507
+0.5
+4.42
+8.8
+19.61
+0.73
+5.17
+5.37
+4.12
+8.8
+17.8
+6.53
+5.27
+1.54 0.14 1.39
+4.83
+1.66
+18
+0.01
+0.6
+554
+0.5
+4.81
+9.1
+18.14
+0.61
+5.22
+5.41
+4.92
+9.1
+21.9
+6.71
+7.87
+1.66 0.27 1.48
+3.13
+1.19
+18
+0.01
+0.6
+554
+1
+4.81
+6.03
+18.22
+0.56
+5.2
+5.26
+1.18
+6.03
+5.9
+4.45
+4.17
+1.61 0.23 1.44
+1.11
+2.18
+18
+0.1
+0.6
+512
+0.5
+4.93
+9.46
+18.14
+0.78
+5.17
+5.4
+4.83
+9.46
+22
+7.08
+5.67
+1.45
+0.1
+1.31
+3.25
+1.46
+19
+0.01
+0.6
+559
+0.5
+5.32
+9.59
+16.87
+0.59
+5.24
+5.46
+5.31
+9.59
+23.9
+7.13
+3.97
+1.5
+0.18 1.35
+1.32
+2.14
+19
+0.01
+0.6
+559
+1
+5.32
+6.45
+16.93
+0.59
+5.19
+5.25
+1.38
+6.45
+7.04
+4.77
+4.43
+1.48 0.16 1.34
+2.16
+1.9
+19
+0.1
+0.6
+516
+0.5
+5.46
+9.34
+16.87
+0.75
+5.2
+5.51
+4.33
+9.34
+21
+7.03
+5.5
+1.46
+0.1
+1.32
+2.51
+1.51
+19
+0.1
+0.6
+516
+1
+5.46
+10.21 16.71
+0.64
+5.23
+5.49
+3.64
+10.21
+19.3
+7.55
+5.7
+1.69 0.35
+1.5
+1.86
+1.66
+20
+0.01
+0.1
+100
+1
+1.14
+19.63 10.95 657.71
+3.7
+5.41
+3.7
+8.71
+18
+6.43
+6.71
+1.64 0.28 1.46
+1.92
+1.37
+Table 1 continued
+
+24
+Song et al.
+Table 1 (continued)
+Progenitor Parameters
+Stellar Properties at Core Collapse
+Protomagnetar
+Mini
+Z
+Ω
+Vequ
+ηwind
+Jini
+Mf
+Ages
+Rf
+log Teff logL
+JHe
+MHe
+JCO
+MCO
+JFe
+MFe ξ2.5 MNS
+BNS
+PNS
+(M⊙) (Z⊙) (Ωcrit) (km s−1)
+(1052ergs s) (M⊙) (Myr) (R⊙)
+(K)
+(L⊙) ( 1050 ergs s) (M⊙) (1049 ergs s) (M⊙) (1048 ergs s) (M⊙)
+(M⊙) (1014 G) (ms)
+20
+0.01
+0.2
+199
+1
+2.26
+19.88 12.02 429.99
+3.78
+5.33
+3.54
+8.86
+15.9
+6.44
+7.41
+1.84 0.59 1.62
+2.49
+1.38
+20
+0.01
+0.3
+296
+0.5
+3.32
+18.01 14.58
+2.24
+5.06
+5.89
+27.3
+18.01
+42.2
+8.93
+10.4
+1.86 0.61 1.64
+5.26
+1
+20
+0.01
+0.3
+296
+1
+3.32
+9.63
+14.74
+0.64
+5.21
+5.41
+3.11
+9.63
+16.4
+7.2
+4.96
+1.52 0.12 1.37
+1.83
+1.74
+20
+0.01
+0.4
+390
+0.5
+4.29
+10.27 15.15
+0.67
+5.21
+5.44
+5.79
+10.27
+27.9
+7.77
+6.08
+1.51 0.13 1.36
+3.04
+1.41
+20
+0.01
+0.4
+390
+1
+4.29
+6.74
+15.21
+0.57
+5.2
+5.28
+1.41
+6.74
+7.4
+5.01
+4.11
+1.5
+0.18 1.35
+1.59
+2.07
+20
+0.01
+0.5
+480
+0.5
+5.15
+9.72
+15.48
+0.74
+5.18
+5.43
+4.84
+9.72
+22.8
+7.22
+4.87
+1.5
+0.11 1.35
+1.46
+1.75
+20
+0.01
+0.5
+480
+1
+5.15
+6.58
+15.56
+0.58
+5.2
+5.27
+1.4
+6.58
+7.45
+4.88
+4.87
+1.53 0.17 1.37
+1.49
+1.78
+20
+0.01
+0.6
+564
+0.5
+5.84
+9.91
+15.76
+0.67
+5.21
+5.45
+5.45
+9.91
+25.4
+7.38
+5.45
+1.55 0.14 1.39
+2.49
+1.61
+20
+0.01
+0.6
+564
+1
+5.84
+6.65
+15.83
+0.55
+5.21
+5.28
+1.47
+6.65
+7.17
+4.91
+4.82
+1.57
+0.2
+1.41
+2.65
+1.84
+20
+0.01
+0.7
+644
+0.5
+6.36
+9.82
+16
+0.55
+5.26
+5.47
+5.39
+9.82
+25
+7.35
+7.15
+1.61 0.22 1.45
+3.25
+1.27
+20
+0.01
+0.7
+644
+1
+6.36
+6.48
+16.07
+0.53
+5.21
+5.27
+1.38
+6.48
+7.12
+4.84
+5.11
+1.57 0.21 1.41
+1.87
+1.73
+20
+0.01
+0.8
+716
+0.5
+6.67
+9.76
+16.14
+0.5
+5.28
+5.46
+5.24
+9.76
+25.6
+7.49
+6.55
+1.61 0.23 1.44
+2.94
+1.39
+20
+0.01
+0.8
+716
+1
+6.67
+6.34
+16.22
+0.55
+5.23
+5.35
+1.29
+6.34
+6.59
+4.71
+4.24
+1.61 0.23 1.44
+1.6
+2.15
+20
+0.1
+0.1
+92
+1
+1.18
+15
+10.45 854.89
+3.62
+5.31
+2.77
+7.45
+11.6
+5.32
+5.95
+1.63 0.26 1.46
+2.51
+1.54
+20
+0.1
+0.3
+273
+0.5
+3.41
+16.4
+14.47
+16.9
+4.62
+5.87
+9.18
+15.61
+53.6
+12.55
+4.73
+1.85
+0.6
+1.63
+0.53
+2.17
+20
+0.1
+0.3
+264
+1
+3.41
+18.73 13.36 934.99
+3.67
+5.56
+3.21
+10.37
+14.5
+7.72
+4.92
+1.47 0.16 1.33
+2.29
+1.7
+20
+0.1
+0.4
+360
+0.5
+4.42
+11.66 15.09
+0.45
+5.34
+5.64
+6.45
+11.66
+32.3
+8.81
+7.46
+1.6
+0.26 1.44
+2.08
+1.21
+20
+0.1
+0.4
+360
+1
+4.42
+12.62 14.87
+0.83
+5.28
+5.92
+4.75
+12.62
+30.2
+10.04
+6.59
+1.58 0.22 1.41
+3.52
+1.35
+20
+0.1
+0.5
+443
+0.5
+5.3
+9.79
+15.47
+0.75
+5.18
+5.41
+4.19
+9.79
+20.7
+7.32
+5.2
+1.49 0.11 1.34
+4.68
+1.63
+20
+0.1
+0.5
+443
+1
+5.29
+11.91 15.28
+0.54
+5.29
+5.58
+4.65
+11.91
+27.6
+9.25
+5.42
+1.58 0.27 1.41
+3.61
+1.64
+20
+0.1
+0.6
+521
+0.5
+6.01
+9.42
+15.76
+0.73
+5.18
+5.4
+3.79
+9.42
+19.4
+7.11
+4.76
+1.49 0.11 1.34
+2.6
+1.78
+20
+0.1
+0.6
+521
+1
+6.01
+11.6
+15.58
+0.59
+5.27
+5.56
+4.62
+11.6
+29
+9.18
+5.98
+1.65
+0.3
+1.48
+2.74
+1.56
+20
+0.1
+0.7
+596
+0.5
+6.55
+9.33
+16
+0.77
+5.17
+5.39
+4.2
+9.33
+20.4
+7.05
+5.35
+1.47 0.09 1.33
+2.66
+1.56
+20
+0.1
+0.8
+664
+1
+6.89
+11.76 15.96
+0.51
+5.32
+5.65
+5.41
+11.76
+33.5
+9.52
+6.65
+1.76 0.44 1.56
+4.44
+1.48
+20
+0.1
+0.8
+665
+0.5
+6.88
+9.09
+16.14
+0.5
+5.27
+5.44
+4.28
+9.09
+20.4
+6.86
+6.32
+1.68
+0.3
+1.49
+2.5
+1.49
+21
+0.01
+0.6
+568
+0.5
+6.4
+10.31 14.82
+0.76
+5.18
+5.45
+5.73
+10.31
+26.3
+7.68
+5.77
+1.44
+0.1
+1.31
+3.69
+1.43
+21
+0.01
+0.6
+568
+1
+6.4
+6.56
+14.9
+0.54
+5.22
+5.28
+1.35
+6.56
+7.33
+4.89
+4.21
+1.54 0.21 1.39
+1.29
+2.07
+21
+0.1
+0.6
+525
+0.5
+6.58
+9.7
+14.81
+0.71
+5.19
+5.42
+3.83
+9.7
+19.9
+7.29
+5.15
+1.53 0.13 1.38
+3.02
+1.68
+21
+0.1
+0.6
+525
+1
+6.58
+11.82 14.67
+0.59
+5.27
+5.57
+4.68
+11.82
+28.7
+9.3
+6.63
+1.55 0.23 1.39
+3.54
+1.32
+22
+0.01
+0.6
+572
+0.5
+6.97
+10.43 13.99
+0.66
+5.22
+5.48
+5.31
+10.43
+25.5
+7.72
+7.62
+1.66 0.19 1.48
+3.02
+1.22
+22
+0.01
+0.6
+572
+1
+6.97
+6.93
+14.05
+0.59
+5.2
+5.29
+1.49
+6.93
+7.83
+5.15
+4.73
+1.56 0.17
+1.4
+1.37
+1.87
+22
+0.1
+0.6
+530
+0.5
+7.18
+9.88
+13.99
+0.67
+5.21
+5.43
+3.84
+9.88
+19.6
+7.41
+5.96
+1.51 0.16 1.36
+2.6
+1.44
+Table 1 continued
+
+LGRB progenitors and magnetar formation
+25
+Table 1 (continued)
+Progenitor Parameters
+Stellar Properties at Core Collapse
+Protomagnetar
+Mini
+Z
+Ω
+Vequ
+ηwind
+Jini
+Mf
+Ages
+Rf
+log Teff logL
+JHe
+MHe
+JCO
+MCO
+JFe
+MFe ξ2.5 MNS
+BNS
+PNS
+(M⊙) (Z⊙) (Ωcrit) (km s−1)
+(1052ergs s) (M⊙) (Myr) (R⊙)
+(K)
+(L⊙) ( 1050 ergs s) (M⊙) (1049 ergs s) (M⊙) (1048 ergs s) (M⊙)
+(M⊙) (1014 G) (ms)
+23
+0.01
+0.6
+577
+0.5
+7.56
+11.02 13.24
+0.57
+5.26
+5.53
+5.92
+11.02
+28.3
+8.19
+8.15
+1.82
+0.5
+1.6
+2.28
+1.24
+23
+0.01
+0.6
+577
+1
+7.56
+6.98
+13.31
+0.52
+5.23
+5.32
+1.48
+6.98
+8.08
+5.21
+4.69
+1.57 0.21 1.41
+2.76
+1.89
+23
+0.1
+0.6
+533
+0.5
+7.8
+10.03 13.25
+0.63
+5.22
+5.44
+4.12
+10.03
+21.5
+7.59
+6.08
+1.57
+0.2
+1.41
+2.23
+1.46
+23
+0.1
+0.6
+533
+1
+7.8
+12.45 13.13
+0.54
+5.3
+5.61
+5.27
+12.45
+30.4
+9.57
+6.74
+1.6
+0.27 1.43
+4.88
+1.34
+24
+0.01
+0.6
+581
+0.5
+8.16
+11.41 12.57
+0.5
+5.3
+5.56
+6.18
+11.41
+29.4
+8.48
+7.58
+1.7
+0.36 1.51
+2.86
+1.26
+24
+0.1
+0.6
+537
+1
+8.44
+12.59 12.48
+0.75
+5.23
+5.61
+5.78
+12.59
+31.2
+9.49
+4.87
+1.5
+0.14 1.36
+1.66
+1.75
+24
+0.1
+0.6
+537
+0.5
+8.44
+10.92 12.57
+0.59
+5.24
+5.48
+4.82
+10.92
+24.8
+8.23
+7.28
+1.65 0.29 1.47
+6.77
+1.27
+25
+0.01
+0.1
+103
+0.5
+1.7
+24.93
+8.65
+110.83
+4.11
+5.48
+3.71
+9.73
+17.4
+7.08
+7.03
+1.92 0.66 1.69
+2.26
+1.51
+25
+0.01
+0.2
+205
+1
+3.38
+24.72
+9.92
+109.61
+4.15
+5.64
+7.79
+13.45
+43
+10.49
+10
+1.8
+0.43
+1.6
+5.7
+1
+25
+0.01
+0.3
+306
+1
+4.96
+9.38
+11.32
+0.64
+5.21
+5.39
+2.46
+9.38
+13.7
+7.06
+5.33
+1.51 0.12 1.36
+2.27
+1.6
+25
+0.01
+0.4
+404
+0.5
+6.43
+11.74 11.56
+0.48
+5.32
+5.58
+5.81
+11.74
+29
+8.8
+6.72
+1.73 0.38 1.54
+2.01
+1.44
+25
+0.01
+0.4
+404
+1
+6.43
+7.63
+11.62
+0.51
+5.25
+5.35
+1.66
+7.63
+9.37
+5.72
+4.09
+1.57 0.21 1.41
+1.21
+2.17
+25
+0.01
+0.5
+497
+0.5
+7.72
+11.83 11.79
+0.5
+5.31
+5.61
+6.28
+11.83
+29.8
+8.75
+7.02
+1.68 0.33
+1.5
+2.09
+1.34
+25
+0.01
+0.5
+497
+1
+7.72
+7.67
+11.85
+0.53
+5.24
+5.36
+1.73
+7.67
+9.59
+5.75
+4.39
+1.52
+0.2
+1.37
+2.12
+1.97
+25
+0.01
+0.6
+585
+0.5
+8.79
+11.45 11.99
+0.51
+5.3
+5.57
+5.52
+11.45
+27.1
+8.53
+7.09
+1.76 0.45 1.56
+2.41
+1.39
+25
+0.01
+0.6
+585
+1
+8.79
+7.46
+12.07
+0.49
+5.26
+5.39
+1.65
+7.46
+9.09
+5.58
+4.8
+1.55 0.23
+1.4
+1.27
+1.83
+25
+0.01
+0.7
+669
+0.5
+9.6
+11.86 12.17
+0.49
+5.31
+5.59
+6.91
+11.86
+33.5
+8.86
+7.37
+1.66 0.28 1.48
+3.03
+1.27
+25
+0.01
+0.7
+669
+1
+9.6
+7.42
+12.24
+0.5
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+747
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+27
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+2.23
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+25
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+747
+1
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+0.51
+5.24
+5.34
+1.7
+7.52
+9.4
+5.63
+4.5
+1.58 0.21 1.41
+1.45
+1.98
+25
+0.1
+0.1
+95
+1
+1.76
+13.86
+8.96
+1271
+3.61
+5.62
+6.4
+11.93
+34
+9.29
+8.47
+1.72 0.36 1.53
+4.09
+1.14
+25
+0.1
+0.2
+190
+1
+3.49
+23.86
+9.46
+955.17
+3.7
+5.71
+4.36
+11
+21.5
+8.28
+5.63
+1.82 0.57 1.61
+0.46
+1.8
+25
+0.1
+0.2
+190
+0.5
+3.49
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+9.58 1085.14
+3.65
+5.62
+5.08
+12.42
+26.9
+9.68
+6.87
+1.79 0.45 1.58
+2.48
+1.45
+25
+0.1
+0.3
+283
+1
+5.13
+19.98 10.77 1501.66
+3.62
+5.81
+7.04
+16.04
+32.4
+12.02
+6.66
+1.76 0.39 1.56
+3.08
+1.47
+25
+0.1
+0.4
+372
+0.5
+6.65
+13.2
+11.54
+0.63
+5.27
+5.65
+7.34
+13.2
+39.1
+10.19
+5.43
+1.53 0.14 1.38
+2.11
+1.6
+25
+0.1
+0.4
+372
+1
+6.65
+12.87 11.39
+0.55
+5.3
+5.63
+5.54
+12.87
+53.7
+12.8
+4.58
+1.53 0.16 1.38
+3.13
+1.9
+25
+0.1
+0.5
+458
+1
+7.99
+12.69 11.68
+0.71
+5.24
+5.61
+5.75
+12.69
+31.7
+9.63
+5.2
+1.53 0.13 1.38
+2.75
+1.67
+25
+0.1
+0.6
+540
+1
+9.1
+12.51 11.91
+0.71
+5.23
+5.6
+5.81
+12.51
+31.6
+9.47
+6.83
+1.57 0.13 1.41
+6.02
+1.3
+25
+0.1
+0.7
+619
+0.5
+9.95
+11.21 12.15
+0.51
+5.32
+5.66
+5.71
+11.21
+27.2
+8.3
+6.95
+1.77
+0.5
+1.57
+2.08
+1.42
+25
+0.1
+0.7
+620
+1
+9.95
+11.94 12.06
+0.65
+5.28
+5.7
+5.16
+11.94
+27.2
+9.04
+7.14
+1.63
+0.2
+1.46
+4.28
+1.29
+25
+0.1
+0.8
+693
+0.5
+10.5
+10.79 12.26
+0.65
+5.23
+5.5
+5.36
+10.79
+24.7
+7.88
+7.6
+1.81 0.57
+1.6
+2.56
+1.33
+26
+0.01
+0.6
+588
+0.5
+9.44
+11.66 11.48
+0.5
+5.3
+5.57
+5.46
+11.66
+26.7
+8.66
+6.68
+1.68 0.34
+1.5
+1.9
+1.41
+Table 1 continued
+
+26
+Song et al.
+Table 1 (continued)
+Progenitor Parameters
+Stellar Properties at Core Collapse
+Protomagnetar
+Mini
+Z
+Ω
+Vequ
+ηwind
+Jini
+Mf
+Ages
+Rf
+log Teff logL
+JHe
+MHe
+JCO
+MCO
+JFe
+MFe ξ2.5 MNS
+BNS
+PNS
+(M⊙) (Z⊙) (Ωcrit) (km s−1)
+(1052ergs s) (M⊙) (Myr) (R⊙)
+(K)
+(L⊙) ( 1050 ergs s) (M⊙) (1049 ergs s) (M⊙) (1048 ergs s) (M⊙)
+(M⊙) (1014 G) (ms)
+26
+0.01
+0.6
+588
+1
+9.44
+7.61
+11.54
+0.55
+5.23
+5.33
+1.66
+7.61
+9.36
+5.72
+3.73
+1.56 0.18
+1.4
+1.56
+2.37
+26
+0.1
+0.6
+544
+1
+9.77
+12.34 11.37
+0.6
+5.27
+5.58
+5.47
+12.34
+52.9
+12.27
+5.83
+1.48 0.13 1.34
+3.71
+1.45
+27
+0.01
+0.6
+592
+1
+10.1
+7.82
+11.07
+0.52
+5.25
+5.37
+1.73
+7.82
+9.71
+5.87
+4.08
+1.55 0.19 1.39
+1.25
+2.15
+27
+0.1
+0.6
+547
+1
+10.5
+12.41 10.91
+0.63
+5.25
+5.57
+5.2
+12.41
+51
+12.38
+5
+1.47 0.14 1.33
+2.84
+1.68
+27
+0.1
+0.6
+548
+0.5
+10.5
+11.82 10.97
+0.52
+5.3
+5.58
+5.27
+11.82
+27.1
+8.88
+6.37
+1.58 0.23 1.42
+2.53
+1.4
+28
+0.01
+0.6
+595
+0.5
+10.8
+12.95 10.57
+0.53
+5.31
+5.64
+7.26
+12.95
+69.8
+12.88
+4.92
+1.55 0.22 1.39
+3.88
+1.78
+28
+0.01
+0.6
+595
+1
+10.8
+8.22
+10.64
+0.53
+5.24
+5.38
+1.93
+8.22
+10.9
+6.18
+4.48
+1.55 0.19
+1.4
+1.96
+1.96
+28
+0.1
+0.6
+550
+0.5
+11.2
+11.69 10.55
+0.5
+5.3
+5.57
+4.96
+11.69
+26.4
+8.86
+6.86
+1.57 0.23 1.41
+4.52
+1.29
+28
+0.1
+0.6
+550
+1
+11.2
+12.28 10.48
+0.59
+5.27
+5.58
+5.29
+12.28
+51.7
+12.24
+4.2
+1.51 0.15 1.36
+1.62
+2.04
+29
+0.01
+0.6
+599
+1
+11.5
+8.33
+10.24
+0.52
+5.25
+5.38
+1.93
+8.33
+10.9
+6.27
+4.38
+1.53 0.19 1.37
+1.09
+1.97
+29
+0.1
+0.6
+554
+1
+11.9
+12.27 10.13
+0.53
+5.35
+5.78
+5.35
+12.27
+52.5
+12.23
+6.45
+1.61 0.24 1.44
+3.95
+1.4
+29
+0.1
+0.6
+554
+0.5
+11.9
+11.97 10.17
+0.48
+5.32
+5.6
+4.93
+11.97
+27.2
+9.19
+5.74
+1.57 0.24 1.41
+2.32
+1.55
+30
+0.01
+0.3
+314
+0.5
+6.86
+14.2
+9.33
+0.62
+5.29
+5.69
+7.23
+14.2
+38.4
+10.86
+5.94
+1.51 0.14 1.36
+4.28
+1.44
+30
+0.01
+0.3
+314
+1
+6.86
+13.59
+9.27
+0.67
+5.26
+5.64
+5.15
+13.59
+50.2
+13.53
+5.54
+1.51 0.14 1.36
+3.93
+1.55
+30
+0.01
+0.4
+415
+0.5
+8.9
+13.92
+9.48
+0.7
+5.25
+5.66
+7.46
+13.92
+71.8
+13.85
+5.56
+1.59 0.14 1.43
+2.12
+1.62
+30
+0.01
+0.4
+415
+1
+8.9
+8.94
+9.53
+0.48
+5.28
+5.44
+2.15
+8.94
+12.2
+6.77
+4.85
+1.64 0.24 1.46
+1.43
+1.9
+30
+0.01
+0.5
+511
+0.5
+10.7
+13.76
+9.65
+0.64
+5.28
+5.67
+7.72
+13.76
+38.5
+10.4
+6.12
+1.56 0.15
+1.4
+3.97
+1.44
+30
+0.01
+0.5
+511
+1
+10.7
+8.57
+9.72
+0.63
+5.2
+5.36
+1.92
+8.57
+10.7
+6.42
+4.11
+1.48 0.14 1.33
+1.56
+2.04
+30
+0.01
+0.6
+602
+0.5
+12.2
+13.72
+9.82
+0.69
+5.25
+5.65
+7.76
+13.72
+74.4
+13.64
+6.28
+1.53 0.14 1.38
+5.7
+1.38
+30
+0.01
+0.6
+602
+1
+12.2
+8.5
+9.88
+0.6
+5.21
+5.36
+2.02
+8.5
+11.4
+6.38
+5.17
+1.53 0.17 1.38
+1.31
+1.68
+30
+0.01
+0.7
+690
+0.5
+13.4
+13.55
+9.96
+0.7
+5.25
+5.64
+7.52
+13.55
+71.7
+13.47
+4.88
+1.49 0.12 1.35
+5.03
+1.74
+30
+0.01
+0.7
+690
+1
+13.4
+8.59
+10.02
+0.63
+5.2
+5.36
+2.07
+8.59
+11.8
+6.48
+5.19
+1.55 0.15 1.39
+1.59
+1.69
+30
+0.01
+0.8
+773
+0.5
+14.1
+13.55 10.05
+0.7
+5.25
+5.64
+7.5
+13.55
+72.3
+13.48
+5.49
+1.49 0.13 1.35
+2.56
+1.55
+30
+0.01
+0.8
+773
+1
+14.1
+8.37
+10.11
+0.61
+5.21
+5.35
+1.97
+8.37
+11
+6.29
+3.99
+1.5
+0.15 1.35
+6.3
+2.13
+30
+0.1
+0.1
+98
+1
+2.44
+27.05
+7.46 1310.05
+3.63
+5.72
+8.38
+14.26
+46.6
+11.34
+7.26
+1.77 0.52 1.57
+3.91
+1.36
+30
+0.1
+0.2
+194
+0.5
+4.82
+29.15
+7.99 1028.65
+3.67
+5.67
+5.93
+13.64
+32.7
+10.56
+5.85
+1.82 0.57 1.61
+0.59
+1.73
+30
+0.1
+0.5
+471
+0.5
+11.1
+12.83
+9.65
+0.62
+5.27
+5.61
+5.44
+12.83
+30.8
+9.93
+5.34
+1.53 0.14 1.38
+3.92
+1.63
+30
+0.1
+0.5
+471
+1
+11.1
+12.67
+9.6
+0.62
+5.26
+5.58
+5.22
+12.67
+51.7
+12.65
+5.45
+1.52 0.22 1.37
+4.03
+1.58
+30
+0.1
+0.6
+556
+1
+12.7
+12.91
+9.76
+0.63
+5.26
+5.6
+5.74
+12.91
+56.4
+12.88
+6.4
+1.61 0.15 1.44
+2.92
+1.42
+30
+0.1
+0.6
+557
+0.5
+12.7
+12.35
+9.81
+0.51
+5.31
+5.61
+5.32
+12.35
+28.3
+9.36
+6.43
+1.62 0.25 1.45
+3.01
+1.42
+30
+0.1
+0.7
+639
+0.5
+13.9
+12.39
+9.95
+0.54
+5.3
+5.62
+6.22
+12.39
+30.3
+9.16
+5.27
+1.56 0.22
+1.4
+1.82
+1.67
+
+LGRB progenitors and magnetar formation
+27
+Notes.
+The columns are the initial mass Mini
+(Column 1), the initial metallicity Z (Column 2),
+the initial rotation rate Ω (Column 3) and corre-
+sponding velocity at the equator Vequ (Column 4),
+the “Dutch” wind scale factor ηwind (Column 5),
+the initial total angular momentum Jini (Column
+6), final mass Mf (Column 7), final star ages (Col-
+umn 8), final radius Rf (Column 9), final effective
+temperature log Teff (Column 10) and luminosity
+log L (Column 11) in logarithmic form, final angu-
+lar momentum in He core JHe (Column 12), He core
+mass at the end of calculation MHe (Column 13),
+final angular momentum in CO core JCO (Column
+14), CO core mass at the end of calculation MCO
+(Column 15), final angular momentum in Fe core
+JFe (Column 16), Fe core mass at the end calcula-
+tion MFe (Column 17), the compactness parameter
+(Column 18), the NS mass (Column 19), the av-
+erage surface magnetic field strength (Column 20),
+and NS rotation period (Column 21).
+

YNA0T4oBgHgl3EQfFf_E/content/tmp_files/2301.02034v1.pdf.txt

@@ -0,0 +1,1557 @@
+Draft version January 6, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Multi-stage reconnection powering a solar coronal jet
+David M. Long
+,1 Lakshmi Pradeep Chitta
+,2 Deborah Baker
+,1 Iain G. Hannah
+,3 Nawin Ngampoopun
+,1
+David Berghmans
+,4 Andrei N. Zhukov
+,4, 5 and Luca Teriaca
+2
+1University College London, Mullard Space Science Laboratory, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK
+2Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077, G¨ottingen, Germany
+3School of Physics & Astronomy, University of Glasgow, University Avenue, Glasgow G12 8QQ, UK
+4Solar-Terrestrial Centre of Excellence – SIDC, Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, Belgium
+5Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia
+(Received January 6, 2023; Revised January 6, 2023; Accepted January 6, 2023)
+Submitted to ApJ
+ABSTRACT
+Coronal jets are short-lived eruptive features commonly observed in polar coronal holes and are
+thought to play a key role in the transfer of mass and energy into the solar corona. We describe
+unique contemporaneous observations of a coronal blowout jet seen by the Extreme Ultraviolet Imager
+onboard the Solar Orbiter spacecraft (SO/EUI) and the Atmospheric Imaging Assembly onboard
+the Solar Dynamics Observatory (SDO/AIA). The coronal jet erupted from the south polar coronal
+hole, and was observed with high spatial and temporal resolution by both instruments. This enabled
+identification of the different stages of a breakout reconnection process producing the observed jet. We
+find bulk plasma flow kinematics of ∼100–200 km s−1 across the lifetime of its observed propagation,
+with a distinct kink in the jet where it impacted and was subsequently guided by a nearby polar plume.
+We also identify a faint faster feature ahead of the bulk plasma motion propagating with a velocity of
+∼715 km s−1 which we attribute to untwisting of newly reconnected field lines during the eruption. A
+Differential Emission Measure (DEM) analysis using the SDO/AIA observations revealed a very weak
+jet signal, indicating that the erupting material was likely much cooler than the coronal passbands
+used to derive the DEM. This is consistent with the very bright appearance of the jet in the Lyman-α
+passband observed by SO/EUI. The DEM was used to estimate the radiative thermal energy of the
+source region of the coronal jet, finding a value of ∼ 2 × 1024 ergs, comparable to the energy of a
+nanoflare.
+1. INTRODUCTION
+Coronal jets are collimated ejections of plasma from
+the solar atmosphere that could escape into the helio-
+sphere.
+Typically observed using extreme ultraviolet
+(EUV) (e.g., Alexander & Fletcher 1999; Nistic`o et al.
+2009; Chandrashekhar et al. 2014) or X-ray (e.g., Shi-
+bata et al. 1992; Shimojo et al. 1996; Cirtain et al. 2007;
+Sterling et al. 2015) observations, they are better ob-
+served in coronal holes, due to the lower background
+intensity making them easier to identify (e.g., Savcheva
+et al. 2007). They are the subject of detailed investi-
+Corresponding author: David M. Long
+[email protected]
+gation, as they offer a mechanism for transferring mass
+and energy to the outer corona and solar wind (cf. Shen
+2021).
+They also represent an opportunity to study
+smaller scale evolution of the mechanisms involved in
+larger solar eruptions, with recent observations (e.g.,
+Nistic`o et al. 2009; Sterling et al. 2015) and simulations
+(e.g., Wyper et al. 2017, 2018) suggesting that coronal
+jets can be interpreted as being produced by the erup-
+tion of mini-filaments via breakout reconnection. A de-
+tailed overview of observations and modelling of coronal
+jets can be found in the recent reviews by Raouafi et al.
+(2016) and Shen (2021).
+Coronal jets are typically identified as being divided
+into two distinct types, “standard” and “blowout” jets
+(cf. Moore et al. 2010), based on morphological be-
+haviour. Standard jets follow the model originally pro-
+arXiv:2301.02034v1  [astro-ph.SR]  5 Jan 2023
+
+ID2
+Long et al.
+a; Ly−α, t =  06:04:52
+150
+225
+300
+ 
+−1800
+−1750
+−1700
+−1650
+−1600
+Y (Arcsec)
+b; Ly−α, t =  06:04:52 − 06:03:52
+150
+225
+300
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+c; 174Å, t =  06:04:52
+150
+225
+300
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+d; 174Å, t =  06:04:52 − 06:03:52
+150
+225
+300
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+e; 304Å, t =  06:04:53
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+−1020
+−1000
+−980
+−960
+−940
+Y (arcsec)
+f; 304Å, t =  06:04:53 − 06:03:57
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+g; 171Å, t =  06:04:57
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+h; 171Å, t =  06:04:57 − 06:03:57
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+i; 193Å, t =  06:04:52
+20
+40
+60
+80
+100
+120
+X (arcsec)
+−1020
+−1000
+−980
+−960
+−940
+Y (Arcsec)
+j; 193Å, t =  06:04:52 − 06:03:57
+20
+40
+60
+80
+100
+120
+X (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+k; 211Å, t =  06:04:57
+20
+40
+60
+80
+100
+120
+X (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+l; 211Å, t =  06:04:57 − 06:03:57
+20
+40
+60
+80
+100
+120
+X (arcsec)
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 1. Jet eruption in a polar coronal hole. The coronal jet (indicated by the white arrow) observed by EUI/HRI (top row)
+and SDO/AIA (middle and bottom rows) using a combination of intensity and running difference images for each passband.
+Top row shows the EUI/HRI 174 ˚A and Lyman-alpha passbands, middle row shows the SDO/AIA 304 ˚A and 171 ˚A passbands,
+and the bottom row shows the SDO/AIA 193 ˚A and 211 ˚A passbands. In each case, intensity images have been enhanced using
+the Multiscale Gaussian Normalisation technique of Morgan & Druckm¨uller (2014), while the times of the images used to make
+the running difference images are given in the panel title. Note that all times indicate the time at Earth. An animated version of
+this figure is available as movie 0.mp4, with a duration of 13 s which shows the temporal evolution of the erupting jet observed
+by both Solar Orbiter EUI and SDO/AIA.
+posed by Shibata et al. (1992), with a narrow spire that
+remains thin throughout the lifetime of the jet, a rela-
+tively dim source region, and no emission in the cooler
+304 ˚A passband. In contrast, blowout jets initially be-
+have as standard jets, with the spire subsequently broad-
+ening to match the width of the footpoints. Blowout jets
+have also been observed to produce emission in the 304 ˚A
+passband, with Moore et al. (2010) suggesting that re-
+connection as a result of an emerging bipole which drives
+the jet could release the overlying magnetic field and en-
+able the observed surge (i.e., the eruption of cooler mate-
+rial). This cooler material has traditionally been identi-
+fied using the 304 ˚A passband, but Alexander & Fletcher
+(1999) presented observations of a coronal jet made us-
+ing the Lyman-α 1216 ˚A passband from the Transition
+Region And Coronal Explorer (TRACE; Handy et al.
+1999).
+In this case, the jet produced bright emission
+in the Lyman-α passband, and could be tracked at a
+cadence of ∼60 s.
+The multiple coronal extreme ultraviolet (EUV) pass-
+bands provided by the Atmospheric Imaging Assembly
+(AIA; Lemen et al. 2012) onboard the Solar Dynamics
+Observatory (SDO; Pesnell et al. 2012) has enabled the
+development of multiple techniques to estimate plasma
+
+EUI jet eruption
+3
+parameters such as density and temperature using dif-
+ferential emission measure (DEM; see, e.g., the codes
+developed by Aschwanden et al. 2013; Hannah & Kon-
+tar 2012, 2013; Plowman et al. 2013; Cheung et al. 2015;
+Pickering & Morgan 2019).
+This has enabled many
+plasma diagnostic studies of different solar phenomena,
+including coronal jets (e.g., Chen et al. 2013; Zhang &
+Ji 2014; Joshi et al. 2020). More recently, DEM anal-
+ysis has also been used to probe the radiative thermal
+energy released during solar nanoflares, with Purkhart
+& Veronig (2022) finding an energy range of 1024 –
+1029 erg for 30 SDO/AIA image series between 2011
+and 2018.
+The high spatial and temporal resolution
+provided by SDO/AIA has also enabled a detailed in-
+vestigation of the evolution of coronal jets (cf. Morton
+et al. 2012b,a; Chen et al. 2012). The High Resolution
+Imager (HRI) component of the Extreme Ultraviolet Im-
+ager (EUI; Rochus et al. 2020) onboard the Solar Orbiter
+(M¨uller et al. 2020) spacecraft will enable much higher
+spatial and temporal resolution studies of solar phenom-
+ena in its 174 ˚A and Lyman-α passbands. Already this
+is providing new insights into very small features asso-
+ciated with polar coronal jets (e.g., Mandal et al. 2022),
+offering sub-arcsecond imaging of these features.
+In this work, we use a combination of observations
+from Solar Orbiter EUI and SDO/AIA to examine the
+initial evolution of a small scale coronal jet erupting from
+the southern polar coronal hole. The event is described
+in section 2, with an analysis of the observations pre-
+sented in section 3 before some conclusions are drawn in
+section 4.
+2. OBSERVATIONS AND DATA ANALYSIS
+The coronal jet discussed here erupted from the
+south pole of the Sun on 2021-Sept-14 beginning at
+∼05:59:02 UT as observed by Solar Orbiter EUI1. At
+the time, Solar Orbiter was located at 0.587 astronom-
+ical units from the Sun at an angle of 47.372 degrees
+behind the Earth, and was doing a calibration manoeu-
+vre, pointing at the North and South Poles, and East
+and West limbs to enable cross-calibration of the dif-
+ferent high-resolution telescopes. The erupting jet was
+observed by both the 174 ˚A and Lyman-α passbands
+with an image scale of 0.492′′ (1.028′′) per pixel in the
+174 ˚A (Lyman-α) passbands, and a temporal resolution
+of 5 s (see top row of Figure 1). For the analysis de-
+1 Note that the distances of the individual spacecraft from the Sun
+meant that phenomena were observed at different times by the
+different spacecraft, so to avoid confusion, we will use the time
+at Earth throughout this discussion.
+scribed here, we used the calibrated level-2 EUI/HRI
+data from EUI Data Release 4.2
+The jet was also well observed by SDO/AIA, with
+the eruption starting at ∼06:02:30 UT as observed near
+Earth. SDO/AIA has an image scale of 0.6′′ per pixel at
+a 12 s cadence in each of the 7 EUV passbands (94, 131,
+171, 193, 211, 304, 335 ˚A), with the data presented here
+processed using the standard aia prep.pro routine con-
+tained within SolarSoftWare (Freeland & Handy 1998).
+The eruption was well observed by the 304, 171, and
+193 ˚A passbands (with their temperature response func-
+tions peaking at T∼ 104.9, 105.9, and 106.2 respectively),
+and was also identifiable using the 211 ˚A passband
+(T∼ 106.25) as shown in the middle and bottom rows
+of Figure 1, with no signal apparent within the 94, 131,
+or 335 ˚A passbands (T∼ 106.85, 105.7, and 106.4 respec-
+tively).
+Note that a full description of the response
+function for each of the AIA passbands can be found
+in Boerner et al. (2012).
+3. RESULTS
+The erupting jet is shown as observed using multiple
+passbands at ∼06:04:52 UT in Figure 1, with the tem-
+poral evolution of the jet shown in more detail in the
+associated animation movie 0.mp4. The jet has a clear
+“λ” style morphology (see e.g., Raouafi et al. 2016, for
+details), with each passband also showing an apparent
+kink in the jet following its initial eruption as indicated
+by the white arrows. Inspection of the temporal evo-
+lution of the jet eruption using the larger field of view
+provided by SDO/AIA suggests that this kinking is due
+to the erupting jet interacting with a nearby very faint
+polar plume, which deflects the initially laterally prop-
+agating jet. As observed by EUI/HRI, the initially nar-
+row jet reverses its propagation in the direction parallel
+to the solar limb and starts to expand. This is apparent
+in both the 174 ˚A and Lyman-α passbands, with the
+higher resolution of the 174 ˚A passband also suggesting
+a slight untwisting of the jet as it initially erupts.
+This overall behaviour of the jet is also seen in the
+SDO/AIA passbands shown in Figure 1, with the differ-
+ent viewpoint and additional passbands providing addi-
+tional information. From this viewpoint, the clear non-
+radial motion is consistent with its interpretation as a
+“λ” style jet. The jet also appears to be guided radially
+outward from the Sun following the kinked evolution (al-
+though the kinking is less pronounced here), with little
+to none of the spreading observed by EUI/HRI. This
+lack of spread in the jet in AIA images could be due to
+2 https://doi.org/10.24414/s5da-7e78
+
+4
+Long et al.
+the line-of-sight effects. While the jet can be identified
+in the 193 ˚A and 211 ˚A passbands (observing plasma
+at ∼1–2 MK), it is clearest in the 171 ˚A and particu-
+larly the 304 ˚A passbands, suggesting that the feature
+is likely composed of cooler plasma, which is consistent
+with the clear identification of the jet in the EUI/HRI
+Lyman-alpha passband.
+3.1. Initial evolution
+The very high spatial and temporal cadence of the
+EUI/HRI 174 ˚A passband enables a detailed analysis of
+the initial stages of the jet eruption, as shown in Fig-
+ure 2. Prior to the onset of the eruption, overarching
+loops above the coronal bright point can be identified,
+as shown in Figure 2a.
+Within ∼60 s, the reconnec-
+tion process has begun, with the eruption of the mini-
+filament identifiable in Figure 2b.
+This mini-filament
+then interacts with the overlying loops to drive breakout
+reconnection as shown in the image 5 s later (Figure 2c).
+This can be seen as rapid disconnection and opening of
+the overarching loops.
+The result of bi-directional flows induced by this
+breakout reconnection can be seen in the evolution of
+small plasmoid-like blobs draining from the reconnection
+site down the legs of the overarching loops to the solar
+surface (as indicated by the unidirectional white arrows
+in Figure 2d-e). Meanwhile the mini-filament continues
+to erupt, with its spine indicated by the bi-directional
+arrow in panels e-j of Figure 2. However, as it erupts,
+its legs start to stretch (Figure 2g) and subsequently
+break, producing bright side lobes (Figure 2h-i). This
+enables the eruption of more plasmoid material into the
+erupting jet (indicated by the red arrow in Figure 2h &
+i and then by the white arrow in Figure 2j-l) which then
+evolves out into the solar atmosphere.
+In total, this process takes ∼140 s from the onset of
+the eruption to the erupting plasma escaping out into
+the outer corona, with the very high temporal and spa-
+tial resolution of EUI/HRI enabling a detailed analysis
+of the different reconnection stages of the initiation pro-
+cess. While the same process was also observed by the
+Lyman-α passband, the lower spatial resolution and very
+high intensity of the plasma in this passband made it dif-
+ficult to make a direct comparison of this small-scale be-
+haviour. The difference between the two passbands can
+be seen in panels a & c of Figure 1, with the small-scale
+loop features identifiable in the 174 ˚A image (panel c)
+not as apparent in the Lyman-α image (panel a).
+3.2. Jet kinematics
+Once the plasma from the jet had been ejected via
+the breakout reconnection process, the next step was to
+quantify its evolution. A series of distance-time stack
+plots were used to further examine the temporal evolu-
+tion of the jet as shown in Figures 3 and 4 respectively.
+The initial evolution of the jet close to the origin was ex-
+amined using observations from EUI/HRI as shown in
+Figure 3. The top row of Figure 3 shows individual snap-
+shots of the initial stages of the erupting jet as observed
+by EUI/HRI 174 ˚A (panel a) and EUI/HRI Lyman-α
+(panel b) at 06:03:48 UT. Panels c & d then show the
+distance-time stack plot along the white box indicated
+in panels a & b. The white fiducial lines here indicate
+that the jet had an initial velocity of ∼166 km s−1 as
+observed by both EUI/HRIEUV and EUI/HRILYA.
+Following this initial evolution, the longer term evo-
+lution of the erupting jet was tracked using the larger
+field of view of SDO/AIA (see Figure 4). Here, the top
+row shows the evolution of the erupting jet, with the
+bottom row showing distance-time stackplots along the
+white region defined in panel b for the 171 ˚A (left col-
+umn) and 193 ˚A (right column) passbands. The bright
+jet feature was then identified in the stackplots as the
+positions or points of the maximum intensity associated
+with the jet at each time-step (as shown with symbols
+in panel c for the 171 ˚A passband). These points were
+then fitted using a linear fit to derive the kinematics.
+This reveals a slight deceleration in the evolution of the
+jet as observed by both passbands, from ∼200 km s−1
+close to the source to ∼195 km s−1 further out following
+the interaction with the nearby faint streamer.
+Although care has been taken to try and ensure that
+the same feature was identified using the datasets from
+the different spacecraft, there may be several reasons
+why the derived kinematics differ slightly. The EUI/HRI
+observations are taken very close to the source of the
+erupting jet while the SDO/AIA observations cover a
+much longer distance and therefore timeframe. The an-
+gle of the evolution of the jet with respect to the plane
+of sky will also affect the kinematics derived using each
+instrument.
+While both SDO/AIA passbands in Figure 4 show a
+feature propagating outward from the Sun with a fitted
+velocity of ∼200 km s−1, it is also possible to identify
+a very faint feature in the 193 ˚A passband with a ve-
+locity of ∼715 km s−1 propagating ahead of the jet.
+This feature is not very clear in the SDO/AIA obser-
+vations presented here, and could not be easily identi-
+fied in coronagraph images from the LASCO C2 corona-
+graph onboard the SOHO spacecraft, making it difficult
+to quantify. However, this feature is consistent with the
+previous observations by Cirtain et al. (2007) of two ve-
+locities associated with x-ray jets from a polar coronal
+hole. Pariat et al. (2015) subsequently suggested that
+
+EUI jet eruption
+5
+2021-09-14T06:06:13
+2021-09-14T06:06:03
+2021-09-14T06:05:58
+2021-09-14T06:05:53
+2021-09-14T06:04:53
+2021-09-14T06:04:18
+2021-09-14T06:03:58
+2021-09-14T06:03:38
+2021-09-14T06:03:28
+2021-09-14T06:02:03
+2021-09-14T06:02:58
+2021-09-14T06:03:03
+overarching loops 
+(coronal bright point)
+Mini-filament eruption
+Breakout reconnection
+a)
+b)
+c)
+d)
+e)
+f)
+g)
+h)
+i)
+j)
+k)
+l)
+Figure 2. The initial evolution of the jet eruption observed in the 174 ˚A passband by EUI/HRI. Panel a shows the initial
+overarching loops prior to the onset of the eruption. Within ∼60 s, (panel b), the mini-filament starts to erupt, interacting with
+the overarching loops and driving breakout reconnection (panel c). Bright plasma blobs produced in this reconnection process
+then start to drain down the leg of the loops (white arrow in panels d-f), with the spine of the jet indicated by the double-ended
+white arrow in panels e-j. The reconnection opens magnetic field lines, enabling the eruption of the jet plasma (indicated by
+the red arrow in panels h-i and subsequently by the white arrow in panels j-l).
+
+6
+Long et al.
+t = 06:03:47
+0
+20
+40
+60
+80
+Pixels
+0
+20
+40
+60
+80
+Pixels
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+(a)
+06:00
+06:04
+06:08
+06:12
+Start Time (14−Sep−21 05:57:55)
+0
+5
+10
+15
+06:00
+06:04
+06:08
+06:12
+Start Time (14−Sep−21 05:57:55)
+0
+5
+10
+15
+165 km/s
+(c)
+t = 06:03:47
+0
+10
+20
+30
+40
+Pixels
+ 
+ 
+ 
+ 
+ 
+ 
+0
+10
+20
+30
+40
+ 
+ 
+ 
+ 
+ 
+(b)
+06:00
+06:04
+06:08
+06:12
+Start Time (14−Sep−21 05:57:55)
+0
+5
+10
+15
+06:00
+06:04
+06:08
+06:12
+Start Time (14−Sep−21 05:57:55)
+0
+5
+10
+15
+168 km/s
+Distance (Mm)
+(d)
+Figure 3. The initial stages of the jet eruption observed
+in the 174 ˚A and Lyman-α passbands by EUI/HRI. Panels
+a & b show the initial stages of the eruption in the 174 ˚A
+and Lyman-α passbands respectively, with the white dotted
+box in panels a & b used to produce the distance-time stack
+plots shown in panels c & d. A fiducial line is overlaid on
+the leading edge of the bright jet feature in panels c & d to
+guide the eye and to illustrate the propagation speed of the
+jet.
+an inclined solar jet could produce a standard jet fol-
+lowed by a helical jet due to the untwisting of newly
+reconnected magnetic field lines as the eruption evolves.
+This standard jet could then be observed as a wave trav-
+elling at a phase speed close to the local Alfv´en speed
+(∼800 km s−1 as measured in X-ray observations by
+Cirtain et al. 2007), with the helical (or blowout) jet ob-
+served as a bulk plasma flow travelling at a fraction of
+the phase speed. The faint feature observed here using
+EUV observations can therefore be interpreted as the
+standard jet wave simulated by Pariat et al. (2015) and
+observed in X-rays by Cirtain et al. (2007).
+3.3. Differential Emission Measure analysis
+The evolution of the jet plasma was examined in more
+detail using a Differential Emission Measure (DEM) ap-
+proach in order to quantify how the temperature and
+density of the jet plasma evolved with time. The DEM
+φ(T) is defined as,
+φ(T) = n2
+e(T) dh
+dT ,
+(1)
+where ne is the electron number density and h is the
+line of sight depth.
+Computing DEMs from observa-
+tions is an ill-posed problem, and multiple techniques
+have been described to try and solve it using observa-
+tions from SDO/AIA (see, e.g., Long et al. 2021, for
+more details). In this case, we used the Regularised In-
+version technique developed by Hannah & Kontar (2012,
+2013) to derive the differential emission measure of the
+field of view shown in Figure 1. A selection of the re-
+sulting DEMs can be seen in Figure 5 for temperatures
+of 105.9 – 106.4 K in bins of 100.1 K. Note that above
+this temperature it was not possible to identify any sig-
+nature of the erupting jet feature. The left two columns
+in Figure 5 show the DEM images in each temperature
+bin at t= 06 : 05 : 23 UT, with the right two columns
+showing the temporal evolution of the DEM in each tem-
+perature bin for the region shown in Figure 4b used to
+calculate the kinematics of the jet eruption as observed
+by SDO/AIA. Although very faint, a signature of the
+jet eruption can be identified in the different DEM tem-
+perature bins shown here, with the feature identified by
+arrows in each DEM stackplot to help guide the eye.
+This faint feature matches the feature observed in Fig-
+ure 4 propagating at a velocity of ∼200 km s−1, with
+no evidence of the faster feature identified in the 193 ˚A
+shown in Figure 4d.
+To try and further understand the plasma evolution
+of the jet, the EM-weighted temperature and density of
+the field of view was estimated using the approach of
+Vanninathan et al. (2015); Long et al. (2019). The EM-
+weighted electron number density can be defined as,
+ne =
+��
+φ(T)dT
+h
+,
+(2)
+where h is the plasma scale height, while the DEM-
+weighted temperature can defined as,
+T =
+��
+φ(T)TdT
+�
+φ(T)dt ,
+(3)
+(see, e.g., Cheng et al. 2012). The resulting tempera-
+ture and number density stackplots along the jet region
+highlighted in Figure 4b are shown in panels a and b of
+Figure 6 respectively. The erupting jet can be identi-
+fied (and is highlighted using arrows to help guide the
+
+EUI jet eruption
+7
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+(a) AIA 171 Å
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+(b) AIA 193 Å
+50 Mm
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:39)
+0
+50
+100
+150
+Distance along the jet (Mm)
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:39)
+0
+50
+100
+150
+Distance along the jet (Mm)
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+195 km s
+−1
+200 km s
+−1
+(c) AIA 171 Å
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:46)
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:46)
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+��1
+(d) AIA 193 Å
+195 km s
+−1
+200 km s
+−1
+715 km s
+−1
+(d) AIA 193 Å
+Figure 4. The longer term evolution of the jet eruption as observed using the 171 ˚A and 193 ˚A passbands on SDO/AIA. Top
+row shows the field of view used to track the jet evolution, with the bottom row showing the temporal evolution along the region
+highlighted in white in panel b for the 171 ˚A (left) and 193 ˚A (right) passbands.
+eye), but is again very faint in both number density and
+temperature plots. A profile of both temperature and
+density along the white dashed line shown in Figure 6a &
+b is shown in black in panels c & d for temperature and
+number density respectively. In both cases the temporal
+evolution is very noisy, and a heavily smoothed line has
+been added in red to highlight the slight increase above
+the background corresponding to the passage of the jet
+feature.
+3.4. Helical Morphology of the Jet and Link to
+Switchbacks
+An apparent helical (or kink, twist) morphology of the
+jet material can be seen in panels a–d of Figure 1. Such
+morphologies in jets are not uncommon, as noted by e.g.,
+Shimojo et al. (1996); Wang et al. (1998); Veselovsky
+et al. (1999); Jiang et al. (2007); Moore et al. (2015).
+They suggest the presence of helical or kinked magnetic
+field lines, similar to structures observed in erupting
+prominences. We note that the jet is also visible in the
+Lyman-α HRI channel (Figure 1a, b), indicating that
+a part of the erupting material is at chromospheric or
+transition region temperatures (see e.g., Canfield et al.
+1996; Sterling et al. 2015). As noted above, helical field
+
+8
+Long et al.
+Log T = 5.9−6.0
+ 
+ 
+ 
+ 
+ 
+−1200
+−1100
+−1000
+−900
+Solar Y
+Log T = 5.9−6.0
+ 
+ 
+ 
+ 
+ 
+0
+50
+100
+150
+Solar Y
+Log T = 5.9−6.0
+ 
+ 
+ 
+ 
+ 
+0
+50
+100
+150
+Solar Y
+Log T = 6.0−6.1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Log T = 6.0−6.1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Log T = 6.0−6.1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Log T = 6.1−6.2
+ 
+ 
+ 
+ 
+ 
+−1200
+−1100
+−1000
+−900
+Solar Y
+Log T = 6.1−6.2
+ 
+ 
+ 
+ 
+ 
+0
+50
+100
+150
+Solar Y
+Log T = 6.1−6.2
+ 
+ 
+ 
+ 
+ 
+0
+50
+100
+150
+Solar Y
+Log T = 6.2−6.3
+−100
+0
+100
+200
+Solar X
+ 
+ 
+ 
+ 
+ 
+Log T = 6.2−6.3
+06:00
+06:05
+06:10
+06:15
+Tstart = 14−Sep−21 06:00:11
+ 
+ 
+ 
+ 
+ 
+Log T = 6.2−6.3
+06:00
+06:05
+06:10
+06:15
+Tstart = 14−Sep−21 06:00:11
+ 
+ 
+ 
+ 
+ 
+Log T = 6.3−6.4
+−100
+0
+100
+200
+Solar X
+−1200
+−1100
+−1000
+−900
+Solar Y
+Log T = 6.3−6.4
+06:00
+06:05
+06:10
+06:15
+Tstart = 14−Sep−21 06:00:11
+0
+50
+100
+150
+Solar Y
+Log T = 6.3−6.4
+06:00
+06:05
+06:10
+06:15
+Tstart = 14−Sep−21 06:00:11
+0
+50
+100
+150
+Solar Y
+T = 06:05:23 UT
+18.0
+19.0
+20.0
+21.0
+DEM [cm−5 K−1]
+Figure 5. Images (left two columns) and stackplots (right two columns) for different DEM temperature bins (as defined in the
+panel titles) showing the evolution of the jet in each temperature bin. The propagating jet feature has been highlighted using
+black arrows in the stackplots to help guide the eye.
+lines in jets are usually interpreted as the result of in-
+terchange reconnection of large-scale open coronal hole
+field lines with small-scale closed field lines at the coro-
+nal base (Pariat et al. 2009).
+The kinked magnetic field lines in jets may be linked
+to the magnetic field switchbacks that are frequently
+observed in the near-Sun solar wind by the Parker So-
+lar Probe mission (PSP; Kasper et al. 2019; Bale et al.
+2019).
+Switchbacks represent significant deviations of
+the magnetic field from the nominal Parker spiral, which
+in extreme cases may result in the inversion of the ra-
+dial component of the magnetic field. The link of coro-
+nal jets with switchbacks was suggested by Sterling &
+Moore (2020) who argued that slightly kinked magnetic
+field lines resulting from the interchange reconnection
+may evolve to switchbacks as they propagate in the he-
+liosphere.
+The size of the observed kinked jet in the transverse
+(i.e. perpendicular to the radial) direction can be esti-
+mated from Figure 1c to be around 6×103 km. At this
+moment, the jet is located at 1.03 R⊙ from the centre
+of the Sun.
+Assuming the radial expansion of struc-
+tures propagating outwards, the transverse size of the
+corresponding feature at 35.7 R⊙ (the distance of the
+first perihelion of Parker Solar Probe) would be around
+7×106 km. This is much larger than the transverse scale
+of switchbacks at this distance from the Sun, which is
+reported to be around 104 km (Horbury et al. 2020).
+However, Figure 1c shows that the jet does not look like
+a single kinked feature, but rather has a developed fine
+structure with many individual sub-features at scales
+down to a few HRI pixels, oriented at different directions
+with respect to the radial. If a kink has the minimal re-
+solved transverse size of two HRI pixels (i.e.
+around
+
+EUI jet eruption
+9
+ 
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:41)
+0
+50
+100
+150
+Distance (Mm)
+ 
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:41)
+0
+50
+100
+150
+Distance (Mm)
+6.0
+6.1
+6.2
+a; EM weighted temperature (Log10T)
+ 
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:41)
+ 
+ 
+ 
+ 
+ 
+ 
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:41)
+ 
+ 
+ 
+ 
+ 
+8.0
+8.2
+8.4
+8.6
+b; EM weighted Density (cm−3)
+ 
+ 
+ 
+ 
+ 
+1.00
+1.05
+1.10
+1.15
+1.20
+T (MK)
+c
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:47)
+8.15
+8.20
+8.25
+8.30
+8.35
+Log10Density (cm−3)
+d
+06:04
+06:08
+06:12
+06:16
+Start Time (14−Sep−21 06:01:47)
+0.8
+1.0
+1.2
+1.4
+1.6
+Integrated emission measure (cm−5 x1027)
+e
+Figure 6. The differential emission measure of the erupting jet. Top row shows the emission measure weighted temperature
+(left) and electron number density (right) along the region of interest shown in Figure 4b used to calculate the jet kinematics.
+The faint jet feature has been highlighted using black arrows to help guide the eye. Panels c & d show a cut in temperature
+and the number density respectively along the white dashed line shown in panels a & b, with the original data shown in black
+and a heavily smoothed line shown in red to help guide the eye. Panel e shows the evolution in integrated emission measure in
+the blue square at the source of the jet eruption shown in Figure 1e.
+400 km at the time of our observations), then its ra-
+dial expansion would lead to the switchback transverse
+size of around 5×105 km at 35.7 R⊙. This size is still
+an order of magnitude larger than the maximal switch-
+back size (not necessarily in the transverse direction) of
+7×104 km inferred by Horbury et al. (2020). This means
+that the origin of individual switchbacks in the corona
+cannot be resolved by HRI, even if non-radial structures
+at the resolved scales of the jet may suggest the presence
+of helical fields at even smaller scales. The situation will
+improve only slightly at the closest perihelion of Solar
+Orbiter around 0.284 au, as the linear spatial resolution
+of HRI observations will be only a factor two better than
+that of the observations taken at 0.587 au reported here.
+Another important scale is that of switchback patches,
+which contain multiple smaller-scale switchbacks (Bale
+et al. 2021).
+Each switchback patch typically corre-
+sponds to the supergranulation scales of around 3◦–5◦
+in heliographic longitude (Bale et al. 2021), which is
+a few times larger than the transverse size of our jet
+(around 0.5◦). The jet reported in this study therefore
+corresponds to an intermediate scale situated between
+the scales of individual switchbacks and the switchback
+patches.
+4. DISCUSSION & CONCLUSIONS
+As part of its commissioning activities taking obser-
+vations of the southern polar coronal hole on 2021-Sept-
+14, Solar Orbiter/EUI observed a small coronal hole
+jet eruption with very high spatial and temporal res-
+olution.
+Despite the difference in spacecraft position,
+the jet was also well observed by SDO/AIA, enabling
+a multi-viewpoint, multi-thermal analysis of the erup-
+tion.
+In both cases, the jet was observed to initially
+
+10
+Long et al.
+erupt almost laterally, with a clear kink seen in the jet
+evolution as it impacted a nearby polar plume observed
+by both EUI/HRI and SDO/AIA (see movie 0.mp4 as-
+sociated with Figure 1). The jet was then observed by
+SDO/AIA to evolve radially away from the Sun, while
+the jet appeared to reverse its propagation in the direc-
+tion parallel to the solar limb while propagating outward
+from the Sun as observed by EUI/HRI.
+The very high spatial and temporal resolution pro-
+vided by EUI/HRI in the 174 ˚A passband enabled a
+detailed analysis of the initiation of the jet.
+As out-
+lined in Figure 2, the jet formed as a result of breakout
+reconnection between the erupting mini-filament and a
+small overlying loop system. This resulted in the further
+stretching and reconnection along the erupting filament,
+with small plasmoid-like blobs visible flowing away from
+the site of the reconnection down the legs of the loop
+system. This reconnection opened the overlying mag-
+netic field which then enabled the eruption of the jet
+itself. Although the process was also observed using the
+Lyman-α passband, the lower resolution and bright na-
+ture of the plasma in this passband meant that it was
+not possible to discern any fine structure comparable to
+that observed using the 174 ˚A passband.
+As observed in Figures 3 and 4, the jet displays slightly
+different kinematics when observed by SDO/AIA and
+EUI/HRI. Figure 3 shows the initial evolution of the
+jet observed close to the source prior to the interaction
+with the nearby polar plume as seen by EUI/HRI 174 ˚A
+and Lyman-α, with Figure 4 showing the longer term
+evolution of the jet as observed by SDO/AIA. These
+figures suggest a constant linear velocity for the jet of
+∼165-200 km s−1, with the slight differences between the
+observations from the two spacecraft most likely due to
+the di���erent angle of observation, with the jet exhibit-
+ing different kinematics when propagating out of the
+plane of sky as seen from different perspectives. The jet
+was also best observed using the cooler passbands avail-
+able from both SDO/AIA and EUI/HRI, with a very
+clear jet seen in both AIA 304 ˚A and HRI Lyman-α, al-
+though it was possible to discern a fainter feature in the
+171/174 ˚A, 193 ˚A, and 211 ˚A passbands. A very faint
+feature was also observed in the SDO/AIA 193 ˚A pass-
+band propagating away from the origin with a velocity
+of ∼715 km s−1 as shown in Figure 4. This is consis-
+tent with the previous observations in X-rays by Cirtain
+et al. (2007) and simulations of Pariat et al. (2015) of a
+wave travelling at a phase velocity close to the Alfv´en
+speed ahead of bulk plasma motion in a coronal jet, and
+implies that the jet is the result of interchange recon-
+nection between open and closed magnetic field in the
+corona, consistent both with the blowout jet interpreta-
+tion, and the unique high resolution EUI/HRI observa-
+tions in EUV of the breakout reconnection in the initial
+stages of the eruption, and the observations of an initial
+untwisting of the jet as it erupted. Although this be-
+haviour could be linked to the switchbacks detected by
+PSP, the size of the kink observed here is much larger
+than the typical switchback size measured by Horbury
+et al. (2020).
+While the jet can be clearly seen in the individual
+passbands shown in Figure 1, it was much more difficult
+to identify using the derived differential emission mea-
+sure, as can be seen in Figure 5. The jet can be identi-
+fied using the temperature bins shown in the figure, but
+no signal was observed in the temperature bins above
+∼106.4 K. A further analysis of the emission measure
+weighted temperature and number density (as shown in
+Figure 6) show that a slight increase in both tempera-
+ture and density could be identified associated with the
+passage of the jet, but it was very small in both cases.
+These observations, combined with the very clear pre-
+sentation of the jet in the 304 ˚A passband observed by
+SDO/AIA and the Lyman-α observed by EUI/HRI, in-
+dicate that the jet primarily consisted of cooler material
+which was below the threshold of the DEM calculated
+using SDO/AIA observations. This indicates that the
+jet was erupting chromospheric material, again consis-
+tent with a blowout jet interpretation.
+However, the DEM derived from SDO/AIA can also
+be used to estimate the radiative thermal energy associ-
+ated with the onset of the jet eruption. As discussed by
+Benz & Krucker (1999) and more recently by Purkhart
+& Veronig (2022), it is possible to estimate the radiative
+thermal energy Eth released during a coronal bright-
+ening using the evolution of emission measure via the
+equation,
+Eth = 3kBT
+�
+∆EMqhA,
+(4)
+where kB is Boltzmann’s constant, T is the tempera-
+ture during the peak of the EM evolution, ∆EM is the
+change in integrated emission measure relative to the
+pre-event value measured in cm−5, h is the total line of
+sight thickness of the event, A is the total area of the
+event, and q is the filling factor, representing the differ-
+ence between observed and actual volumes occupied by
+the emitting plasma. As this is difficult to accurately
+estimate, and does not measurably affect the estimated
+energy, we have assumed a filling factor q of 1 (cf. Par-
+nell & Jupp 2000; Purkhart & Veronig 2022). The re-
+gion chosen here as corresponding to the onset of the jet
+eruption is indicated by the blue square in Figure 1e,
+with the temporal evolution of the integrated emission
+measure in this region shown in Figure 6e. Note that
+due to the highly variable nature of the evolution of
+
+EUI jet eruption
+11
+the emission measure, the ∆EM was defined using the
+change between the start and peak of the smoothed red
+dashed line shown in Figure 6e. We also followed the
+lead of Purkhart & Veronig (2022), assuming a line of
+sight distance h =
+√
+A, after Benz & Krucker (1999).
+From equation 4, we derived a radiative thermal energy
+for the source region of the coronal jet of 1.5×1024 ergs,
+comparable to the energy of a nanoflare (see e.g. Chitta
+et al. 2021). However, it is worth noting that this is most
+likely an underestimation of the true radiative thermal
+energy produced by the source region. The jet in this
+case erupted from very close to the limb, but still on-
+disk as observed by SDO/AIA, and this close proximity
+to the limb leads to an increased absorption of the EUV
+radiation along the line of sight to the observer.
+These observations highlight the importance of high
+time cadence observations with very high spatial resolu-
+tion when studying the evolution of small scale features
+in the solar atmosphere. The HRI/EUV observations of
+the erupting jet presented here show the different stages
+of the breakout reconnection process which led to the
+eruption of the jet in very fine detail, and reveal un-
+twisting of the jet as it erupts, while the Lyman-α ob-
+servations show the cool nature of the erupting plasma.
+The additional point-of-view of Solar Orbiter is also very
+useful, as it highlights the interaction of the erupting
+jet with the nearby polar plume; behaviour that would
+have been missed if only observations from SDO/AIA
+were used to study the jet. However, the multiple pass-
+bands provided by SDO/AIA are vital for understanding
+the plasma diagnostics associated with the erupting jet,
+and for estimating the energy released during the erup-
+tion. The combination of observations from both SDO
+and Solar Orbiter will offer new insights into coronal
+jets, particularly as Solar Orbiter moves out of the solar
+ecliptic towards the poles.
+The authors wish to thank the anonymous referee whose
+suggestions helped to improve the paper. DML wishes
+to thank Peter Wyper and Etienne Pariat for useful dis-
+cussions which helped to codify the interpretation of the
+jet presented in this paper.
+Solar Orbiter is a space
+mission of international collaboration between ESA and
+NASA, operated by ESA. The EUI instrument was built
+by CSL, IAS, MPS, MSSL/UCL, PMOD/WRC, ROB,
+LCF/IO with funding from the Belgian Federal Science
+Policy Office (BELSPO/PRODEX PEA 4000134088);
+the Centre National d’Etudes Spatiales (CNES); the
+UK Space Agency (UKSA); the Bundesministerium f¨ur
+Wirtschaft und Energie (BMWi) through the Deutsches
+Zentrum f¨ur Luft- und Raumfahrt (DLR); and the
+Swiss Space Office (SSO). SDO data are courtesy of
+NASA/SDO and the AIA, EVE, and HMI science teams.
+D.M.L. is grateful to the Science Technology and Fa-
+cilities Council for the award of an Ernest Rutherford
+Fellowship (ST/R003246/1). L.P.C. gratefully acknowl-
+edges funding by the European Union. Views and opin-
+ions expressed are however those of the author(s) only
+and do not necessarily reflect those of the European
+Union or the European Research Council (grant agree-
+ment No 101039844). Neither the European Union nor
+the granting authority can be held responsible for them.
+D.Baker and I.G.H are funded under STFC consolidated
+grant numbers ST/S000240/1 and ST/T000422/1 re-
+spectively. N.N is supported by STFC PhD studentship
+grant ST/W507891/1. A.N.Z. thanks the Belgian Fed-
+eral Science Policy Office (BELSPO) for the provision
+of financial support in the framework of the PRODEX
+Programme of the European Space Agency (ESA) under
+contract number 4000136424.
+Facilities: SDO/AIA, Solar Orbiter EUI
+Software: SSW/IDL (Freeland & Handy 1998)
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Z9E0T4oBgHgl3EQfnQEE/content/tmp_files/2301.02508v1.pdf.txt

@@ -0,0 +1,1888 @@
+End-to-End 3D Dense Captioning with Vote2Cap-DETR
+Sijin Chen1*
+Hongyuan Zhu2
+Xin Chen3
+Yinjie Lei4
+Tao Chen1†
+Gang YU3
+1Fudan University
+2Institute for Infocomm Research, A*STAR
+3Tencent PCG
+4Sichuan University
+https://github.com/ch3cook-fdu/Vote2Cap-DETR
+Abstract
+3D dense captioning aims to generate multiple cap-
+tions localized with their associated object regions. Exist-
+ing methods follow a sophisticated “detect-then-describe”
+pipeline equipped with numerous hand-crafted components.
+However, these hand-crafted components would yield sub-
+optimal performance given cluttered object spatial and
+class distributions among different scenes.
+In this pa-
+per, we propose a simple-yet-effective transformer frame-
+work Vote2Cap-DETR based on recent popular DEtection
+TRansformer (DETR). Compared with prior arts, our
+framework has several appealing advantages:
+1) With-
+out resorting to numerous hand-crafted components, our
+method is based on a full transformer encoder-decoder ar-
+chitecture with a learnable vote query driven object de-
+coder, and a caption decoder that produces the dense cap-
+tions in a set-prediction manner. 2) In contrast to the two-
+stage scheme, our method can perform detection and cap-
+tioning in one-stage. 3) Without bells and whistles, exten-
+sive experiments on two commonly used datasets, ScanRe-
+fer and Nr3D, demonstrate that our Vote2Cap-DETR sur-
+passes current state-of-the-arts by 11.13% and 7.11% in
+[email protected], respectively. Codes will be released soon.
+1. Introduction
+3D dense captioning [11, 7, 38, 36, 18, 4] requires a sys-
+tem to localize all the objects in a 3D scene, and generate
+descriptive sentences for each object. This problem is chal-
+lenging given 1) the sparsity of point clouds and 2) the clut-
+tered distribution of objects.
+3D dense captioning can be divided into two tasks, ob-
+ject detection and object caption generation. Scan2Cap[11],
+MORE[18], and SpaCap3D[36] propose well-designed re-
+lation reasoning modules to efficiently model relations
+among object proposals.
+[42] introduces contextual in-
+*This work is accomplished when visiting the Advanced Perception
+Reasoning Lab at I2R, A*STAR.
+†Corresponding author.
+Two-stage methods
+Feature
+Encoding
+This is a gray chair. It is to the 
+right of a file cabinet.
+The rolling office chair. The chair 
+is next to the square desk.
+⋮
+Object
+Detection
+Relation
+Modelling
+Caption
+Generation
+Post
+Processing
+Vote2Cap-DETR
+Caption
+Head
+Detection
+Head
+Transformer
+Feature
+Encoding
+Decoder
+Vote Query
+This is a gray chair. It is to the 
+right of a file cabinet.
+The rolling office chair. The chair 
+is next to the square desk.
+⋮
+Figure 1. Illustration of existing two-stage 3D dense caption-
+ing method (upper) and our Vote2Cap-DETR (bottom). Exist-
+ing methods adopt a two-stage pipeline that heavily depends on a
+detector’s output. Therefore, we propose a transformer-based one-
+stage model, Vote2Cap-DETR, that frames 3D dense captioning
+as a set prediction problem.
+formation from two branches to improve the caption.
+3DJCG[4] and D3Net[7] study the correlation between 3D
+visual grounding and 3D dense captioning, and point out
+that these two tasks promote each other. Additionally, χ-
+Trans2Cap[38] discusses how to transfer knowledge from
+additional 2d information to boost 3d dense captioning.
+Among existing methods, they all adopt a two-stage
+“detect-then-describe” pipeline[11, 18, 36, 4, 7, 42] (Fig-
+ure 1). This pipeline first generates a set of object propos-
+als, then decodes each object by a caption generator with
+an explicit reasoning procedure.
+Though these methods
+have achieved remarkable performance, the “detect-then-
+describe” pipeline suffers from the following issues: 1) Be-
+cause of the serial and explicit reasoning, this task highly
+depends on the object detection performance, which limits
+the mutual promotion of detection and captioning. 2) The
+heavy reliance on hand-crafted components, e.g., radii, 3D
+operators, the definition of proposal neighbors, and post-
+processing (non-maximum suppression[25]) introduces ad-
+1
+arXiv:2301.02508v1  [cs.CV]  6 Jan 2023
+
+ditional hyper-parameters, leading to a sub-optimal perfor-
+mance given the sparse object surfaces and cluttered object
+distributions among different indoor scenes. This inspires
+us to design an one-stage 3D dense captioning system.
+To address the above issues, we propose Vote2Cap-
+DETR, a full transformer encoder-decoder architecture for
+one-stage 3D dense captioning.
+Unlike the traditional
+“detect-then-describe” pipeline, we directly feed the de-
+coder’s output into the localization head and caption head
+in parallel.
+By casting 3D dense captioning as a set-to-
+set problem, each target instance and its language annota-
+tion is matched with a query in an one-to-one correspon-
+dence manner, helping feature representation for proposals
+be more discriminative to identify each distinctive object
+in a 3D scene. Additionally, we also propose a novel vote
+query driven decoder to introduce spatial bias for better lo-
+calization of objects in a cluttered 3D scene.
+With the fully attentional design, we resolve 3D dense
+captioning with the following innovations: 1) Our method
+treats the 3D dense captioning task as a set prediction prob-
+lem. The proposed Vote2Cap-DETR directly decodes the
+features into object sets with their locations and correspond-
+ing captions by applying two parallel prediction heads. 2)
+We propose a novel vote decoder by reformulating the ob-
+ject queries in 3DETR into the format of the vote query,
+which is a composition of the embeddings of the seeds point
+and the vote transformation of the box with respect to the
+seeds. This indicates the connection between the vote query
+in Vote2Cap-DETR with the VoteNet, but with better lo-
+calization and higher training efficiencies; 3) We develop a
+novel query driven caption head, which absorbs the relation
+and attribute modeling into the self- and cross-attention, so
+that it can look into both the local and global context to bet-
+ter describe the scene. Extensive experiments on two com-
+monly used datasets, ScanRefer and Nr3D, demonstrate that
+our approach surpasses prior arts with many hand-crafted
+procedures by a large margin, which demonstrates the supe-
+riority that, full transformer architecture with sophisticated
+vote head and caption head can inspire many 3D vision and
+language tasks.
+To summarize, the main contributions of this work in-
+clude:
+• We propose a novel one-stage and fully attention
+driven architecture for 3D dense captioning as a set-
+to-set prediction problem, which achieves object local-
+ization and caption generation in parallel.
+• Extensive
+experiments
+show
+that
+our
+proposed
+Vote2Cap approach achieves a new state-of-the-art
+performance on both Nr3D[1] (45.53% [email protected]) and
+ScanRefer[11] (73.77% [email protected]).
+2. Related Work
+We briefly summarize works on 3D dense captioning,
+and DETR-based methods for image and 3D object detec-
+tion. Additionally, we also introduce some methods for im-
+age captioning, which are closely related to our work.
+3D Dense Captioning.
+3D dense captioning, a task
+that requires translating 3D scene information to a set
+of bounding boxes and natural language descriptions, is
+challenging and has raised great interest among schol-
+ars recent years.
+Scan2Cap[11] and MORE[18] build
+graph on a detector’s[29, 17] box estimations with hand-
+crafted rules to reason complex relations among objects
+in a 3D scene.
+SpaCap3D[36] build a spatiality-guided
+transformer to model spatial relations among the detector’s
+output.
+3DJCG[4] and D3Net[7] study the joint promo-
+tion of 3D dense captioning and 3D visual grounding. χ-
+Trans2Cap[38] introduces additional 2D prior to comple-
+ment information for 3D dense captioning with knowledge
+transfer. Recently, [42] shifts attention to contextual infor-
+mation for the perception of non-object information. These
+approaches have made great attempts to solve the 3D dense
+captioning problem. However, they all follow a “detect-
+then-describe” pipeline, which is heavily dependent on a de-
+tector’s performance. Our proposed Vote2Cap-DETR dif-
+fers from existing works in that, our method is a one-stage
+model that detects and generates captions in parallel, and
+treats 3D dense captioning as a set prediction problem.
+DETR: from 2D to 3D. DEtection Transformer(DETR)[5]
+is a transformer[34] based architecture that treats object
+detection as a set prediction problem, and does not re-
+quire non-maximum suppression[25] for post-processing.
+Though great results have been achieved, DETR suffers
+from slow convergence. Many follow-up works[43, 39, 14,
+23, 9, 16] put efforts on speeding up DETR’s training by in-
+troducing multi-scale features, cross attention designs, and
+label assignment techniques. Researchers also attempt to
+introduce transformer architectures to 3D object detection.
+GroupFree3D[21] learns proposal features from the whole
+point cloud through the transformer rather than grouping
+local points. 3DETR[24] analyzes the potential of the stan-
+dard transformer model, and generates proposals by uni-
+formly sampling seed points from a 3D scene. In our work,
+we extend the DETR architecture for 3D dense captioning
+that makes caption generation and box localization fully in-
+terrelated with parallel decoding. Additionally, we propose
+vote query for better performance and faster convergence.
+Image Captioning. Image captioning requires a model to
+generate sentences describing key elements in an image,
+which has become a hot topic in computer vision. Existing
+image captioning works adopt an encoder-decoder architec-
+ture, where the decoder generates sentences from visual fea-
+tures extracted by the encoder. [2, 12, 15, 27] adopt a detec-
+2
+
+tor to extract region features as visual clues for the decoder,
+while [20, 41] extract grid features directly from an image.
+Additionally, [26] generates captions with both region and
+grid visual features. Though these methods are effective
+in image captioning, they cannot be directly applied to 3D
+dense captioning, which requires both accurately localizing
+and describing a 3D object, rather than simply captioning a
+whole 2D scene image. In contrast, our proposed caption
+head sufficiently leverages the rich context information in
+3D point cloud, receives visual clues from both the object
+query and its local context, and fuses them to achieve effec-
+tive 3D dense captioning.
+3. Method
+As shown in Fig. 2, given a 3D scene, our goal is to
+localize objects of interest and generate informative natu-
+ral language descriptions for each object. The input of our
+model is a point cloud PC = [pin; fin] ∈ RN×(3+F ) rep-
+resenting an indoor 3D scene. Here, pin ∈ RN×3 is the
+absolute locations for each point, and fin ∈ RN×F is addi-
+tional input feature for each point, such as color, normal,
+height, or multiview feature introduced by [11, 6].
+The
+expected output is a set of box-caption pairs ( ˆB, ˆC) =
+{(ˆb1, ˆc1), · · · , (ˆbK, ˆcK)}, representing an estimation of K
+distinctive objects in this 3D scene.
+Specifically, our system adopts 3DETR[24] encoder as
+our scene encoder, and transformer decoder to capture both
+object-object and object-scene interactions by the attention
+mechanism. Then, we adopt two task-specific heads for ob-
+ject detection and caption generation.
+3.1. 3DETR Encoder
+Inspired by DETR[5], 3DETR[24] has made a successful
+attempt at bringing full transformer architecture to the 3D
+object detection task, which removes many hard-coded de-
+sign decisions as the popular VoteNet and PointNet++ mod-
+ules in most two-stage methods.
+In 3DETR encoder, the input PC is first tokenized with
+a set-abstraction layer[30]. Then, point tokens are fed into
+a masked transformer encoder with a set-abstraction layer
+followed by another two encoder layers. We denote the en-
+coded scene tokens as [penc; fenc] ∈ R1,024×(3+256).
+3.2. Vote Query
+Though 3DETR has achieved initial success in 3D ob-
+ject detection, it suffers from certain limitations. 3DETR
+proposes the box estimation around the query points (aka
+proposal centers) sampled from the scenes, which can make
+these boxes far away from real objects given the sparse ob-
+ject surfaces, resulting in slow convergence to capture dis-
+criminative object features with further miss detections.
+Prior works on fast convergence DETR models[23, 10,
+40] show that by injecting more structured bias to ini-
+tialize object queries, such as anchor points or content-
+aware queries, accelerates training. Therefore, we propose
+the vote query, which introduces both 3D spatial bias and
+content-related information, for faster convergence and per-
+formance improvement.
+More specifically, we reformulate the object queries in
+3DETR into the format of vote query, as a composition of
+the embedding of the reference points and vote transforma-
+tion around them. This helps to build the connection be-
+tween the object query in 3DETR and the vote set prediction
+widely studied in VoteNet.
+The detailed structure is shown in Figure 3. Here, vote
+∆pvote is predicted from encoded scene token feature fenc
+with a Feed Forward Network (FFN) FFNvote that learns
+to shift the encoded points to objects’ centers spatially:
+pvote = penc + ∆pvote = penc + FFNvote (fenc) .
+(1)
+Then, we sample 256 points pseed from penc with farthest
+point sampling, and locate each point’s offset estimation for
+pvq = pseed + ∆pvote. Finally, we gather features from
+(penc, fenc) for pvq with a set-abstraction layer[30], to for-
+mulate the vote query feature fvq ∈ R256×256. We repre-
+sent vote query as (pvq, fvq).
+Following 3DETR[24], our model adopts an eight-layer
+transformer decoder, and the i-th layer’s input query feature
+f i
+query is calculated through
+f i
+query = Layeri−1
+�
+f i−1
+query + FFN (PE (pvq))
+�
+,
+(2)
+where f 0
+query = fvq, and PE(·) is the 3D Fourier posi-
+tional encoding function[32]. Experiments in later sections
+demonstrate that: 1) Vote query injects additional spatial
+bias to object detection and boosts the detection perfor-
+mance. 2) Encoding features from the point cloud as initial
+queries accelerates convergence.
+3.3. Parallel Decoding
+We adopt two task-specific heads for simultaneous ob-
+ject detection and caption generation. The two task heads
+are agnostic to each other’s output.
+Detection Head. Detecting objects in a 3D scene requires
+box corner estimation ˆB and class estimation ˆS (containing
+“no object” class) from each object query feature. Follow-
+ing 3DETR[24], box corner estimation is reformulated into
+offset estimation from a query point to an object’s center,
+and box size estimation. All subtasks are implemented by
+FFNs. In practice, the object localization head is shared
+through different layers in the decoder, following all exist-
+ing works on DETR[5, 24, 23, 10].
+Caption Head. 3D dense captioning requires attribute de-
+tails on an object and its relation with its close surroundings.
+However, the vote query itself is agnostic to box predictions
+for the whole scene, and fails to provide adequate attribute
+3
+
+Caption
+Head
+Detection
+Heads
+Vote Query Generation
+Parallel Decoding
+Scene
+Encoder
+Tokenizer
+Feature Encoding
+Transformer
+Decoder
+𝑝𝑣𝑞, 𝑓𝑣𝑞
+𝓥𝒒
+𝓥𝒔
+Prediction Results
+This is a tan cabinet. 
+It is in the corner of 
+the room.
+This is a chair. It is 
+placed at a table .
+This is a gray chair. It 
+is to the left of another 
+chair.
+…
+…
+Notations
+𝑝𝑒𝑛𝑐, 𝑓𝑒𝑛𝑐
+3D Fourier Positional Encoding
+Query Feature
+Surrounding Contextual Information
+𝓥𝒒
+𝓥𝒔
+Figure 2. Approach. Vote2Cap-DETR is an one-stage transformer model that takes a 3D point cloud as its input, and generates a set of
+box predictions and sentences localizing and describing each object in the point cloud. The scene encoder first generates encoded scene
+tokens (penc, fenc) from the input point cloud. Then, we generate vote query (pvq, fvq) from the encoded scene tokens, which introduce
+both spatial bias pvq and content-aware feature fvq to initial object queries. The transformer decoder decodes each vote query with two
+parallel task heads for captioning and detection. We optimize Vote2Cap-DETR with a set loss.
+FPS.
+down sampling
++
+𝑝𝑠𝑒𝑒𝑑
+Δ𝑝𝑣𝑜𝑡𝑒
+Set 
+Abstraction
+𝑝𝑒𝑛𝑐, 𝑓𝑒𝑛𝑐
+Vote Query Generation
+𝑓𝑒𝑛𝑐
+𝑝𝑒𝑛𝑐
+𝑝𝑣𝑞, 𝑓𝑣𝑞
+𝑭𝑭𝑵𝒗𝒐𝒕𝒆
+Figure 3. Vote Query Generation. Vote query pvq contains spa-
+tial bias (∆pvote) to initial object queries (pseed), which are sam-
+pled from the scene with farthest point sampling (FPS) and gath-
+ered feature fvq from the point cloud for each query.
+and spatial relations for generating informative captions.
+Therefore, the main difficulty is how to leverage sufficient
+surrounding contextual information without confusing the
+caption head.
+To address the above issues, we propose Dual-Clued
+Captioner(DCC), a lightweight transformer decoder-based
+caption head, for 3D dense captioning. DCC consists of
+a stack of 2 identical transformer decoder blocks, sinusoid
+position embedding, and a linear classification head. To
+generate informative captions, DCC receives two streams
+of visual clue V = (Vq, Vs). Here, Vq is the last decoder
+layer’s output feature of a vote query, and Vs is contextual
+information surrounding the absolute location of each vote
+Linear
+Masked
+Self-Attn
+Cross-Attn
+FFN
+× 𝑳𝒄𝒂𝒑
+Caption Head
+1D
+𝓥𝒔
+𝓥𝒒
+the
+chair
+is
+pulled
+…
+Word Embed
+the
+chair
+is
+pulled
+into
+the
+table
+Figure 4.
+Dual-Clued Captioner(DCC). DCC is a lightweight
+transformer based caption head that uses vote query feature Vq
+as caption perfix to identify the described region, and contextual
+features Vs surrounding the vote query to complement with more
+surrounding information for more descriptive caption generation.
+query. When generating a caption for a proposal, we substi-
+tute the standard Start Of Seqenece(‘SOS’) prefix with Vq
+of the described query identifying the object to be described
+following [36]. Since the vote query is agnostic of actual
+neighbor object proposals because of the parallel detection
+branch, we introduce the vote query’s ks nearest local con-
+text token features as its local surroundings Vs as keys for
+cross attention. During the evaluation, we generate captions
+through beam search with a beam size of 5.
+3.4. Set prediction loss for 3D Dense Captioning
+Our proposed Vote2Cap-DETR generates a set of paired
+box-caption proposals ( ˆB, ˆC) for 3D dense captioning. It
+4
+
+requires supervision for vote query (Lvq), detection head
+(Ldet), and caption head (Lcap).
+Vote Query Loss. We borrow vote loss from VoteNet[29]
+as Lvq, to help the vote query generation module learn to
+shift points penc to an object’s center:
+Lvq = 1
+M
+M
+�
+i=1
+Ngt
+�
+j=1
+��pi
+vote − cntj
+��
+1 · I
+�
+pi
+enc ∈ Ij
+�
+. (3)
+Here, I(·) is an indicator function that equals 1 when the
+condition meets and 0 otherwise, Ngt is the number of in-
+stances in a 3D scene, M is the size of pvote, and cntj is the
+center of jth instance Ij.
+Detection Loss. Following 3DETR[24], we use the same
+Hungarian algorithm to assign each proposal with a ground
+truth label. Since 3D dense captioning is closely related to
+the object localization ability, we apply a larger weight on
+the gIoU loss component for total set loss[24]:
+Lset = α1Lgiou +α2Lcls +α3Lcenter−reg +α4Lsize−reg,
+(4)
+where α1 = 10, α2 = 1, α3 = 5, α4 = 1 are set heuristi-
+cally. The set loss Lset is applied to all ndec−layer layers in
+the decoder for better convergence.
+Caption Loss. Following the standard practice of image
+captioning, we train our caption head first with standard
+cross-entropy loss (MLE training), and then fine-tune it
+with Self-Critical Sequence Training (SCST)[31]. During
+MLE training, the model is trained to predict the (t + 1)th
+word ct+1
+i
+, given the first t words c[1:t]
+i
+and the visual clue
+V. The loss function for a T-length sentence is defined as:
+Lci =
+T
+�
+i=1
+Lci(t) = −
+T
+�
+i=1
+log ˆP
+�
+ct+1
+i
+|V, c[1:t]
+i
+�
+.
+(5)
+After the caption head is trained under word-level supervi-
+sion, we fine-tune it with SCST. During SCST, the model
+generates multiple captions ˆc1,··· ,k with a beam size of k,
+and another ˆg through greedy search as a baseline. The loss
+function for SCST is defined as:
+Lci = −
+k
+�
+i=1
+(R (ˆci) − R (ˆg)) · 1
+|ˆci| log ˆP (ˆci|V) .
+(6)
+Here, the reward function R (·) is the CIDEr metric for cap-
+tion evaluation, and the log probability of caption ˆci is nor-
+malized by caption length |ˆci|, to encourage the model to
+treat captions with different length equally important.
+Set to Set Training for 3D Dense Captioning. We pro-
+pose an easy-to-implement set-to-set training strategy for
+3D dense captioning. Given a 3D scene, we randomly sam-
+ple one sentence from the corpus for each annotated in-
+stance. Then, we assign language annotations to the cor-
+responding number of proposals in the corresponding scene
+with the same Hungarian algorithm. During training, we
+average losses for captions Lci on all annotated instances in
+a batch, to compute the caption loss Lcap. To balance losses
+for different tasks, our loss function for the whole system is
+defined as:
+L = β1Lvq + β2
+ndec−layer
+�
+i=1
+Lset + β3Lcap,
+(7)
+where β1 = 10, β2 = 1, β3 = 5 are set heuristically.
+4. Experiments
+We first present the datasets, metrics, and implementa-
+tion details for 3D dense captioning (section 4.1). Then, we
+provide comparisons with all state-of-the-art methods (sec-
+tion 4.2). We also provide studies on the effectiveness of
+different parts in our model (section 4.3). Finally, we visu-
+alize several qualitative results to address the effectiveness
+of our method (section 4.4).
+4.1. Datasets, Metrics, and Implementation Details
+Datasets.
+We report results on two commonly used
+datasets, ScanRefer [6] and Nr3D[1], both of which are
+built on 3D scenes from ScanNet[13]. ScanNet[13] con-
+tains 1,201 indoor 3D scenes for training and 312 for
+validation. ScanRefer/Nr3D contains 36,665/32,919 free-
+form language annotations describing 7,875/4,664 ob-
+jects from 562/511 3D scenes for training, and evalu-
+ates on 9,508/8,584 sentences for 2,068/1,214 objects from
+141/130 3D scenes.
+Evaluation Metrics. Following [11, 4, 18, 36], we first
+apply NMS on object proposals to drop duplicate object
+predictions. Each object proposal is a box-sentence pair
+(ˆbi, ˆci), containing box corner prediction ˆbi and generated
+sentence ˆci. Then, each instance is assigned an object pro-
+posal with the largest IoU among the remaining proposals.
+Here, we use (bi, Ci) to represent an instance’s label, where
+bi is a box corner’s label and Ci is the corpus containing all
+caption annotations for this instance. To jointly evaluate the
+model’s localization and caption generation capability, we
+adopt the m@kIoU metric[11]:
+m@kIoU = 1
+N
+N
+�
+i=1
+m (ˆci, Ci) · I
+�
+IoU
+�
+ˆbi, bi
+�
+≥ k
+�
+.
+(8)
+Here, N is the number of total annotated instances in the
+evaluation dataset, and m could be any metric for natu-
+ral language generation, such as CIDEr[35], METEOR[3],
+BLEU-4[28], and ROUGE-L[19].
+Implementation Details. We offer implementation details
+of different baselines. “w/o additional 2D” means the in-
+put PC ∈ R40,000×10 contains absolute location as well as
+5
+
+color, normal and height for 40, 000 points representing a
+3D scene. “additional 2D” means we replace color informa-
+tion with 128-dimensional multiview feature extracted by
+ENet[8] from 2D images following [11].
+We first pre-train the whole network without the cap-
+tion head, on ScanNet[13] detection dataset with ScanRe-
+fer[6] categories for 1, 080 epochs (about 163k iterations,
+34 hours), using the AdamW optimizer[22] with a learning
+rate decaying from 5 × 10−4 to 10−6 by a cosine anneal-
+ing scheduler, a weight decay of 0.1, a gradient clipping of
+0.1, and a batch size of 8 following [24]. Then, we load the
+pre-trained detector, and train our caption head with MLE
+loss for another 720 epochs (51k/46k iterations for Scan-
+Refer/Nr3D, 11/10 hours). To prevent overfitting, we fix
+the learning rate of the detector as 10−6, and set that of the
+caption head decaying from 10−4 to 10−6 using another co-
+sine annealing scheduler. Due to the high memory cost of
+SCST, we tune the caption head with a batch size of 2 and
+freeze the detector for 180 epochs (50k/46k iterations for
+ScanRefer/Nr3D, 14/11 hours) with a fixed learning rate of
+10−6. We evaluate the model every 2, 000 iterations during
+training for consistency with existing works[11, 36], and
+all experiments mentioned above are conducted on a single
+RTX3090 GPU.
+4.2. Comparison with Existing Methods
+In this section, we compare performance with exist-
+ing works on metrics C, M, B-4, R as abbreviations for
+CIDEr[35], METEOR[3], BLEU-4[28], Rouge-L[19] un-
+der IoU thresholds of 0.25, 0.5 for ScanRefer (Table 1)
+and 0.5 for Nr3D (Table 2).
+“-” indicates that neither
+the original paper nor any follow-up works provide such
+results.
+Since different supervision on the caption head
+has a huge influence on the captioning performance, we
+make separate comparisons for MLE training and SCST.
+Among all the listed methods, experiments other than
+D3Net[7] and 3DJCG[4] utilize the standard VoteNet[29]
+detector. Meanwhile, D3Net[7] adopts PointGroup[17], a
+3D instance segmentation model, for better object detec-
+tion.
+3DJCG[4] improves VoteNet’s localization perfor-
+mance with an FCOS[33] head, which predicts distance
+from a voting point to each side of a bounding box. Ad-
+ditionally, 3DJCG and D3Net focus on the joint promotion
+of 3D dense captioning and 3D visual grounding, therefore
+their reported models are trained with data from both tasks.
+Among methods listed under SCST, χ-Trans2Cap[38] com-
+bines MLE training with standard SCST in an additive man-
+ner, Scan2Cap and D3Net[7] adopt the same reward com-
+bining CIDEr score and listener losses with a weighted sum.
+It’s worth mentioning that our model adopts the standard
+SCST, whose reward function is CIDEr score.
+Table 1 reports comparisons on ScanRefer[6] validation
+dataset. Our Vote2Cap-DETR surpasses current state-of-
+the-art methods. For example, under MLE training with ad-
+ditional 2D inputs, our Vote2Cap-DETR achieves 59.32%
+[email protected] while 3DJCG[4] achieves 49.48% (9.84% [email protected]↑)
+with additional training data. Additionally, under SCST, our
+Vote2Cap-DETR achieves 70.63% [email protected], while 62.64%
+(7.99% [email protected]↑) for current state-of-the-art D3Net[7] with
+more training labels and semi-supervised training on more
+training data.
+In Table 2, we list results on the Nr3D[1] dataset with ad-
+ditional 2D input following [36]. Since Scan2Cap[11] has
+not reported results on Nr3D, we adopt the best-reported
+result from [4]. Our proposed Vote2Cap-DETR also sur-
+passes current state-of-the-art methods.
+4.3. Ablation Study
+Since 3D dense captioning concerns both localization
+and caption generation, we perform ablation studies to un-
+derstand the effectiveness of different components.
+Does the vote query improve 3DETR? We performed ab-
+lation experiments in Table 3 and Figure 5 to see if the vote
+query can improve 3DETR’s localization and convergence.
+Introducing position features pvq alone helps improve de-
+tection performance (0.97% mAP50↑). However, it (green
+line in Figure 5) converges slower in the earlier training pro-
+cedure than the 3DETR baseline (blue line in Figure 5), in-
+ferring the vote query generation module is not well learned
+to predict accurate spatial offset estimations at early train-
+ing epochs. Introducing additional content feature fvq in
+vote query features results in another boost in both detec-
+tion performance (2.98% mAP50↑) and training speed (red
+line in Figure 5). The overall localization performance of
+Vote2Cap-DETR is about 7.2% mAP higher than the popu-
+lar VoteNet.
+0
+20000
+40000
+60000
+80000
+100000
+120000
+140000
+160000
+Training Iterations
+0
+10
+20
+30
+40
+50
+Validation [email protected]
+(pquery, f0
+query) = (pvq, fvq)
+(pquery, f0
+query) = (pseed, 0)
+(pquery, f0
+query) = (pvq, 0)
+Figure 5. Vote query and convergence. We take out convergence
+study on a different combination of content feature fvq and posi-
+tion pvq in vote query. The baseline model (pquery, f 0
+query) =
+(pseed, 0) downgrades to 3DETR. Introducing pvq boosts perfor-
+mance but decelerates training since FFNvote requires time to
+converge, and fvq accelerates training.
+Does 3D context feature help captioning? Since the per-
+6
+
+Method
+Ldes
+w/o additional 2D input
+w/ additional 2D input
+IoU = 0.25
+IoU = 0.50
+IoU = 0.25
+IoU = 0.50
+C↑
+B-4↑
+M↑
+R↑
+C↑
+B-4↑
+M↑
+R↑
+C↑
+B-4↑
+M↑
+R↑
+C↑
+B-4↑
+M↑
+R↑
+Scan2Cap[11]
+MLE
+53.73
+34.25
+26.14
+54.95
+35.20
+22.36
+21.44
+43.57
+56.82
+34.18
+26.29
+55.27
+39.08
+23.32
+21.97
+44.78
+MORE[18]
+58.89
+35.41
+26.36
+55.41
+38.98
+23.01
+21.65
+44.33
+62.91
+36.25
+26.75
+56.33
+40.94
+22.93
+21.66
+44.42
+SpaCap3d[36]
+58.06
+35.30
+26.16
+55.03
+42.76
+25.38
+22.84
+45.66
+63.30
+36.46
+26.71
+55.71
+44.02
+25.26
+22.33
+45.36
+3DJCG[4]
+60.86
+39.67
+27.45
+59.02
+47.68
+31.53
+24.28
+51.80
+64.70
+40.17
+27.66
+59.23
+49.48
+31.03
+24.22
+50.80
+D3Net[7]
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+46.07
+30.29
+24.35
+51.67
+Ours
+71.45
+39.34
+28.25
+59.33
+61.81
+34.46
+26.22
+54.40
+72.79
+39.17
+28.06
+59.23
+59.32
+32.42
+25.28
+52.53
+χ-Trans2Cap[38]
+SCST
+58.81
+34.17
+25.81
+54.10
+41.52
+23.83
+21.90
+44.97
+61.83
+35.65
+26.61
+54.70
+43.87
+25.05
+22.46
+45.28
+Scan2Cap[11]
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+48.38
+26.09
+22.15
+44.74
+D3Net[7]
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+62.64
+35.68
+25.72
+53.90
+Ours
+84.15
+42.51
+28.47
+59.26
+73.77
+38.21
+26.64
+54.71
+86.28
+42.64
+28.27
+59.07
+70.63
+35.69
+25.51
+52.28
+Table 1.
+Evaluating Vote2Cap-DETR on ScanRefer[6]. We compare Vote2Cap-DETR with all published state-of-the-art 3D dense
+caption methods on the ScanRefer dataset. Though our method does not depend on hand-crafted NMS[25] to drop overlapped boxes,
+we follow the standard evaluation protocol from [11] for fair comparison and provide evaluation without NMS in Table 6. Our proposed
+Vote2Cap-DETR achieves new state-of-the-art under both MLE training and SCST.
+Method
+Ldes
+[email protected]↑
+[email protected]↑
+[email protected]↑
+[email protected]↑
+Scan2Cap[11]
+MLE
+27.47
+17.24
+21.80
+49.06
+SpaCap3d[36]
+33.71
+19.92
+22.61
+50.50
+D3Net[7]
+33.85
+20.70
+23.13
+53.38
+3DJCG[4]
+38.06
+22.82
+23.77
+52.99
+Ours
+43.84
+26.68
+25.41
+54.43
+χ-Tran2Cap[38]
+SCST
+33.62
+19.29
+22.27
+50.00
+D3Net[7]
+38.42
+22.22
+24.74
+54.37
+Ours
+45.53
+26.88
+25.43
+54.76
+Table 2.
+Evaluating Vote2Cap-DETR on Nr3D[1]. Likewise,
+we perform the standard evaluation on the Nr3D dataset, and our
+proposed Vote2Cap-DETR surpasses prior arts.
+pquery
+f 0
+query
+IoU=0.25
+IoU=0.50
+1st layer IoU=0.50
+mAP↑
+AR↑
+mAP↑
+AR↑
+mAP↑
+AR↑
+VoteNet Baseline
+63.42
+82.18
+44.96
+60.65
+-
+-
+pseed
+0
+67.25
+84.91
+48.18
+64.98
+34.80
+55.06
+pvq
+0
+67.33
+85.60
+49.15
+66.38
+30.23
+58.44
+pvq
+fvq
+69.61
+87.20
+52.13
+69.12
+46.53
+66.51
+Table 3. Vote query and performance. We provide quantitative
+results for Figure 5. Introducing pvq as query positions improves
+detection, and gathering fvq from content further boosts perfor-
+mance.
+formance of 3D dense captioning is affected by both lo-
+calization and caption capability, we freeze all parameters
+other than the caption head, and train with 3D only input
+and standard cross entropy loss (MLE training) for a fair
+evaluation. We use object-centric decoder[36] as our base-
+line, which is a decoder that generates captions with object
+feature as a caption’s prefix. In Table 4, “-” refers to the
+object-centric decoder baseline, “global” means naively in-
+cluding all context tokens extracted from the scene encoder
+in the decoder, “local” is our proposed caption head that
+includes a vote query’s ks (ks = 128 empirically) nearest
+context tokens extracted from the scene encoder.
+With the object feature as a caption’s prefix, caption
+generation performance benefits from introducing addi-
+tional contextual information. Additionally, compared with
+naively introducing contextual information from the whole
+scene, introducing local information could be more benefi-
+cial. This demonstrates our motivation that close surround-
+ings matter when describing an object.
+key
+IoU=0.25
+IoU=0.5
+C↑
+B-4↑
+M↑
+R↑
+C↑
+B-4↑
+M↑
+R↑
+-
+68.62
+38.61
+27.67
+58.47
+60.15
+34.02
+25.80
+53.82
+global
+70.05
+39.23
+27.84
+58.44
+61.20
+34.66
+25.93
+53.79
+local
+70.42
+39.98
+27.99
+58.89
+61.39
+35.24
+26.02
+54.12
+Table 4.
+Different keys for caption generation. We provide
+a comparison on different keys used in caption generation. Intro-
+ducing contextual information relates to more informative captions
+generated. Since 3D dense captioning is more object-centric, in-
+troducing vote queries’ local contextual feature is a better choice.
+Do Set-to-Set Training benefit dense captioning? To ana-
+lyze effectiveness of set-to-set training, we follow the train-
+ing procedure that utilize a smaller learning rate for all pa-
+rameters other than the caption head, and freeze these pa-
+rameters during SCST. We name the baseline training strat-
+egy as “Sentence Training”, which traverses through all
+sentence annotations in the dataset and is widely adopted
+in various works[11, 36].
+As is shown in Figure 7, our
+proposed “Set-to-Set” training achieves comparable results
+with the traditional “Sentence Training” during MLE train-
+ing, and converges faster because of a bigger batch size on
+the caption head, which also benefits SCST.
+Training
+Ldes
+[email protected]↑
+[email protected]↑
+[email protected]↑
+[email protected]↑
+Sentence
+MLE
+61.21
+35.35
+26.12
+54.52
+Set-to-Set
+61.81
+34.46
+26.22
+54.40
+Sentence
+SCST
+71.39
+37.57
+26.01
+54.28
+Set-to-Set
+73.77
+38.21
+26.64
+54.71
+Table 5.
+Set to Set training and performance. We compare
+our proposed set-to-set training with traditional “Sentence Train-
+ing”, which traverses through all sentence annotations. We achieve
+comparable performance with MLE training, and 2.38% [email protected]
+improvement with SCST.
+Is Vote2Cap-DETR robust to NMS? Similar to other
+DETR works, the set loss will encourage the model to pro-
+duce compact predictions.
+We compare performance on
+7
+
+3DJCG: This is a rectangular 
+whiteboard. It is on the wall.
+SpaCap3D: The whiteboard is 
+affixed to the wall. It is to the right 
+of the window.
+Ours: The tv is on the wall. It is to 
+the right of the table.
+GT: This is a big black tv. It is 
+above a thin table.
+scene0011_00
+3DJCG: This is a brown table. It 
+is in the middle of the room.
+SpaCap3D: This is a wooden
+table. It is in the center of the 
+room. 
+Ours: This is a wooden table. It is 
+in the corner of the room. 
+GT: This is a small table with a 
+wood look. It is the table closest 
+to the front of the room in the 
+upper left corner.
+scene0015_00
+3DJCG: The is a small brown
+cabinet. It is to the right of the 
+desk.
+SpaCap3D: The cabinet is below 
+the desk. It is to the left of the 
+chair.
+Ours: This is a white cabinet. It is 
+to the right of the table.
+GT: A white cabinet is sitting on 
+the floor next to the wall. It is to 
+the left of the couch.
+scene0025_00
+3DJCG: This is a brown table. It 
+is in front of the couch.
+SpaCap3D: This is a wooden 
+coffee table. It is in front of the 
+couch.
+Ours: This is a brown ottoman. It 
+is to the right of the chair.
+GT: This is a brown ottoman. It is 
+in front of a couch.
+scene0050_00
+Figure 6. Qualitative Comparisons. We compare qualitative results with two state-of-the-art “detect-then-describe” methods, 3DJCG[4]
+and SpaCap3D[36]. We underline phrases describing spatial locations, and mark correct attribute words in green and wrong descriptions
+in red. Our method produces tight bounding boxes close to ground truth annotations and produce accurate descriptions of object attributes,
+classes and spatial relationship.
+0
+10000
+20000
+30000
+40000
+50000
+Training Iterations
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Validation [email protected]
+MLE Training on Set-to-Set
+MLE Training on Sentence
+SCST on Set-to-Set
+SCST on Sentence
+Figure 7.
+Set-to-Set training and convergence. Convergence
+speed analysis of two different training strategies with MLE train-
+ing as well as SCST. Set-to-Set training enables a larger batch size
+for the caption head, which accelerates convergence on 3D dense
+captioning.
+both 3D dense caption ([email protected]) and detection (mAP50,
+AR50) in Table 6.
+Since the m@kIoU metric (Eq.
+8)
+does not contain any penalties on redundant predictions,
+getting rid of NMS[25] results in performance growth on
+[email protected]. Absence of NMS restricts the detection precision
+performance (mAP50) of SpaCap3D (14.47% mAP50 ↓)
+and 3DJCG (17.55% mAP50 ↓), however that of Vote2Cap-
+DETR remains stable.
+Models
+w/ NMS
+w/o NMS
+[email protected]↑
+mAP50↑
+AR50↑
+[email protected]↑
+mAP50↑
+AR50↑
+SpaCap3D
+43.93
+37.77
+53.96
+51.35
+23.30
+64.14
+3DJCG
+50.22
+47.58
+62.12
+54.94
+30.03
+68.69
+Vote2Cap-DETR
+70.63
+52.79
+66.09
+71.57
+52.82
+67.80
+Table 6. Effect of NMS. We analyze whether the absence of NMS
+affects the 3D dense captioning performance ([email protected]) as well as
+detection performance (mAP50, AR50).
+4.4. Qualitative Results
+We compare qualitative results with two state-of-the-art
+models, SpaCap3D[36] and 3DJCG[4] in Figure6. One can
+see that our method produces tight bounding boxes close to
+the ground-truth. Moreover, our method can produce ac-
+curate descriptions of object attributes, classes, and spatial
+relationships.
+5. Conclusion.
+In this work, we present Vote2Cap-DETR, a trans-
+former based one-stage approach, for 3D dense caption-
+ing. The proposed Vote2Cap-DETR adopts a fully trans-
+former encoder-decoder architecture that decodes a set of
+vote queries to box predictions and captions in parallel. We
+show that by introducing spatial bias and content-aware fea-
+tures, vote query boosts both convergence and detection
+performance. Additionally, we develop a novel lightweight
+query-driven caption head for informative caption genera-
+tion. Experiments on two widely used datasets for 3D dense
+8
+
+captioning validates that our propose one-stage Vote2Cap-
+DETR model surpasses prior works with heavy dependence
+on hand-crafted components by a large margin.
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+puter vision and pattern recognition, pages 15465–15474,
+2021. 3
+[42] Yufeng Zhong, Long Xu, Jiebo Luo, and Lin Ma. Contextual
+modeling for 3d dense captioning on point clouds.
+arXiv
+preprint arXiv:2210.03925, 2022. 1, 2
+[43] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang
+Wang, and Jifeng Dai. Deformable detr: Deformable trans-
+formers for end-to-end object detection.
+arXiv preprint
+arXiv:2010.04159, 2020. 2
+10
+
+Appendix
+In our supplementary material, we first propose a non-transformer baseline for our method that builds on VoteNet[29] in
+section A. Then, we provide additional experimental details in section B. Finally, we provide several qualitative studies in
+section C. It’s also worth mentioning that our proposed Vote2Cap-DETR sets a new state-of-the-art on the Scan2Cap online
+test benchmark (Figure 8).
+A. VoteNet baseline with set-to-set training
+In this section, we perform ablation study by replacing our Vote2Cap-DETR’s components (SceneEncoder, Vote Query,
+Transformer Decoder) with VoteNet to study the behavior of non-transformer architecture’s behavior. In Table 7, we observe
+that without delicate hand-crafted relation modelling modules, the VoteNet baseline surpasses 3DJCG[4] by 3.48 in [email protected]
+↑ and 6.23 in [email protected] ↑ and achieves comparable results on other metrics with MLE training. The results demonstrate the
+novel caption head and set-to-set training can also improve non-transformer architecture’s dense captioning performance. On
+the other hand, the VoteNet baseline still falls short in terms of our Vote2Cap-DETR, which demonstrates that Vote Query
+can help learn more discriminate features in an end-to-end manner for end tasks without resorting to many hand-crafted
+components as in VoteNet.
+Method
+IoU = 0.25
+IoU = 0.5
+C↑
+B-4↑
+M↑
+R↑
+C↑
+B-4↑
+M↑
+R↑
+3DJCG[4]
+64.70
+40.17
+27.66
+59.23
+49.48
+31.03
+24.22
+50.80
+Ours(VoteNet)
+70.93
+39.92
+28.09
+58.88
+52.96
+30.59
+24.40
+50.10
+Ours(Full)
+72.79
+39.17
+28.06
+59.23
+59.32
+32.42
+25.28
+52.53
+Table 7.
+VoteNet baseline with set-to-set training. We replace Vote2Cap-DETR’s components with VoteNet. One can see that non-
+transformer VoteNet architecture also benefits from our novel caption head and set to set training. Although there are still performance
+gaps with our Vote2Cap-DETR architecture.
+B. Experiments
+We provide evaluations on the Scan2Cap online test benchmark (section B.1) as well as additional experimental details
+(section B.2 & B.3) in this section.
+B.1. Scan2Cap Test Benchmark
+Our proposed Vote2Cap-DETR achieves a new state-of-the-art for all metrics on the Scan2Cap online test benchmark
+(Figure 8, https://kaldir.vc.in.tum.de/scanrefer benchmark/benchmark captioning).
+B.2. Per-Class mAP Results
+We list per class mAP results for VoteNet[29], 3DETR[24], and our proposed Vote2Cap-DETR on ScanNet scenes[13]
+under an IoU threshold of 0.5 in Table 8. The overall performance is listed in the main paper.
+Method
+cabinet
+bed
+chair
+sofa
+table
+door
+window
+bookshelf
+picture
+counter
+desk
+curtain
+refrigerator
+shower curtain
+toilet
+sink
+bathtub
+others
+VoteNet[29]
+21.41
+78.41
+78.47
+74.44
+55.42
+34.68
+14.91
+29.80
+9.04
+16.57
+51.12
+34.62
+40.12
+45.82
+89.93
+37.23
+83.41
+13.79
+3DETR[24]
+26.30
+75.78
+82.19
+59.15
+62.25
+39.16
+21.47
+33.14
+16.45
+34.41
+49.68
+38.34
+42.83
+33.33
+88.68
+52.62
+82.41
+29.06
+Vote2Cap-DETR
+31.98
+81.48
+85.80
+64.37
+65.20
+41.19
+28.47
+39.81
+22.94
+39.02
+54.46
+36.66
+40.19
+56.10
+87.97
+44.38
+85.12
+33.28
+Table 8. Per-class AP under IoU threshold of 0.5 on ScanNet scenes.
+B.3. Implementation Details
+Our proposed Vote2Cap-DETR first goes through the feature encoding module, then we generate vote queries from the
+encoded feature as object queries, and we decode the vote queries to bounding boxes and captions in the end.
+11
+
+Figure 8. Scan2Cap[11] test benchmark. Our proposed Vote2Cap-DETR achieves a new state-of-the-art for all metrics on the Scan2Cap
+online test benchmark.
+Feature Encoding directly operates on the input point cloud PC to 1,024 tokens with a feature size of 256. We first
+tokenizes the input point cloud PC = [pin; fin] ∈ R40,000×(3+din) to point tokens [ptoken; ftoken] ∈ R2,048×(3+256) with
+a set-abstraction layer[30] with hidden sizes of [3 + din, 64, 128, 256]. Then, our scene encoder encodes point tokens
+[ptoken; ftoken] ∈ R2,048×(3+256) to [penc; fenc] ∈ R1,024×(3+256). We adopt the same encoder as 3DETR-m[24], which
+contains a three-layer transformer encoder with a set-abstraction layer between the first two layers. Each encoder layer has
+a feature size of 256 and Feed Forward Network (FFN) with a hidden size of 128. The first encoder layer operates on 2,048
+points, while the last two operates on the 1,024 points downsampled by the set-abstraction layer. Additionally, three binary
+attention masks are applied to each encoder layer with a radius of [0.16, 0.64, 1.44] respectively to force the interactions of
+points in a given radius.
+Vote Query Generator generates 256 object queries [pvq; fvq] ∈ R256×(3+256) from the encoded points [penc; fenc] ∈
+R1,024×(3+256). It contains an FFN FFNvote with a hidden size of 256 to generate offset estimation and feature projection
+with respective to fenc. It also use a set abstraction layer to gather feature fvq ∈ R256×256 from encoded scene feature for
+pvq ∈ R256×3 as described in the main paper.
+Parallel Decoding aims to decode the vote queries [pvq; fvq] to corresponding box estimations and captions. The trans-
+former decoder consists of eight identical transformer decoder layers with four heads for both self-attention and cross-
+attention. It operates on vote queries [pvq; fvq] and encoded feature [penc; fenc] for the final query feature [pvq, fout] ∈
+R256×(3+256). Follow the transformer decoder are two parallel heads, the detection head and the caption head. The detection
+head generates center offset estimation ([−0.5, 0.5]3) from vote queries’ absolute location pvq, normalized size estimation
+12
+
+Scan2Cap Benchmark
+This table lists the benchmark results for the Scan2Cap Dense Captioning Benchmark scenario.
+Captioning F1-Score
+Dense Captioning
+Object Detection
+[email protected]
+[email protected]
+[email protected]
+[email protected]
+DCmAP
+[email protected]
+Method
+Info
+vote2cap-detr
+0.3128 1
+0.1778 1
+0.2842 1
+0.1316 1
+0.1825 1
+0.4454 1
+CFM
+0.2360 2
+0.1417 2
+0.2253 2
+0.1034 2
+0.1379 5
+0.3008 5
+CM3D-Trans+
+0.2348 3
+0.1383 3
+0.2250 4
+0.1030 3
+0.1398 4
+0.2966 7
+Yufeng Zhong, Long Xu, Jiebo Luo, Lin Ma: Contextual Modeling for 3D Dense Captioning on Point Clouds.
+Forest-xyz
+0.2266 4
+0.1363 4
+0.2250 3
+0.1027 4
+0.1161 10
+0.2825 10
+D3Net - Speaker
+P
+0.2088 5
+0.1335 6
+0.2237 5
+0.1022 5
+0.1481 3
+0.4198 2
+Dave Zhenyu Chen, Qirui Wu, Matthias Niessner, Angel X. Chang: D3Net: A Unified Speaker-Listener Architecture for 3D Dense Captioning and Visual Grounding. 17th European Conference on Computer Vision
+(ECCV), 2022
+3DJCG(Captioning)
+P
+0.1918 6
+0.1350 5
+0.2207 6
+0.1013 6
+0.1506 2
+0.3867 3
+Daigang Cai, Lichen Zhao, Jing Zhangt, Lu Sheng, Dong Xu: 3DJCG: A Unified Framework for Joint Dense Captioning and Visual Grounding on 3D Point Clouds. CVPR2022 Oral
+REMAN
+0.1662 7
+0.1070 7
+0.1790 7
+0.0815 7
+0.1235 8
+0.2927 9
+NOAH
+0.1382 8
+0.0901 8
+0.1598 8
+0.0747 8
+0.1359 6
+0.2977 6
+SpaCap3D
+P
+0.1359 9
+0.0883 9
+0.1591 g
+0.0738 g
+0.1182 9
+0.3275 4
+X-Trans2Cap
+P
+0.1274 10
+0.0808 11
+0.1392 11
+0.0653 11
+0.1244 7
+0.2795 11
+Yuan, Zhihao and Yan, Xu and Liao, Yinghong and Guo, Yao and Li, Guanbin and Cui, Shuguang and Li, Zhen: X-Trans2Cap: Cross-Modal Knowledge Transfer Using Transformer for 3D Dense Captioning. CVPR
+2022
+MORE-xyz
+P
+0.1239 11
+0.0796 12
+0.1362 12
+0.0631 12
+0.1116 12
+0.2648 12
+Yang Jiao, Shaoxiang Chen, Zequn Jie, Jingjing Chen, Lin Ma, Yu-Gang Jiang: MoRE: Multi_oRder RElation Mining for Dense Captioning in 3D Scenes. ECCV 2022
+SUN+
+0.1148 12
+0.0846 10
+0.1564 10
+0.0711 10
+0.1143 11
+0.2958 8
+Scan2Cap
+P
+0.0849 13
+0.0576 13
+0.1073 13
+0.0492 13
+0.0970 13
+0.2481 13
+Dave Zhenyu Chen, Ali Gholami, Matthias Niener and Angel X. Chang: Scan2Cap: Context-aware Dense Captioning in RGB-D Scans. CVPR 2021([0, 1]3), and semantic class estimation from fout using separate FFN heads with a hidden size of 256. Note that we do
+not estimate the rotation angles since ScanNet[13] does not contain any rotated boxes. Our proposed caption head, DCC,
+generates captions with respect to final query features fout as Vq and pvq’s surrounding contextual features Vs. DCC is a
+two layer transformer decoder with four heads for multi-head attentions, as well as a feature size of 256, a sinusoid position
+encoding, and a vocabulary of 3,433 for ScanRefer[6] and 2,937 for Nr3D[1].
+C. Qualitative Results
+Qualitative results on Nr3D. We showcase qualitative results on 3D dense captioning on the Nr3D[1] dataset in Figure 9.
+Our proposed Vote2Cap-DETR is also able to generate tight bounding boxes as well as accurate descriptions for each object
+in a 3D scene.
+Visualization results of vote queries. We visualize the vote queries’ position pvq in our Vote2Cap-DETR and seed
+queries’ position pseed of 3DETR in Figure 10. Most of the vote queries focus on objects in a 3D scene, while pseed is mostly
+distributed in background areas.
+Visualization of detection results. We visualize several detection results in Figure 11. Our proposed Vote2Cap-DETR is
+able to generate accurate box predictions for a 3D scene.
+13
+
+scene0025_00
+scene0081_00
+scene0187_00
+scene0207_00
+scene0300_00
+scene0307_00
+scene0527_00
+scene0645_00
+SpaCap3D: The keyboard
+closest to the door.
+Ours: The monitor closest 
+to the door.
+GT: The monitor closest to 
+the door.
+SpaCap3D: The table with 
+the plant on it.
+Ours: The round table in 
+the corner of the room.
+GT: Round table near the 
+end of the couch.
+SpaCap3D: The table that 
+is not in the middle of the 
+room.
+Ours: The table in the 
+middle of the room.
+GT: The center-most box. 
+In-between the two chairs.
+SpaCap3D: The door that 
+is open. 
+Ours: The door to the left 
+of the stove.
+GT: White door nearest to 
+stove.
+SpaCap3D: The bag on 
+the table.
+Ours: The monitor closest 
+to the window.
+GT: The computer monitor 
+the is closest to the 
+windows.
+SpaCap3D: The bookshelf 
+that is not next to the tv.
+Ours: It is the bookshelf 
+closest to the door.
+GT: The largest shelf by 
+the door.
+SpaCap3D: The mirror
+that is not above the sink.
+Ours: Facing the sink, the 
+cabinet on the right. 
+GT: Upper cabinet above 
+the right sink.
+SpaCap3D: The ottoman 
+closest to the door.
+Ours: The ottoman closest 
+to the couch. 
+GT: The ottoman closest 
+to the couch.
+Figure 9. Visualization of 3D dense captioning on Nr3D[1]. We visualize several results generated by our proposed Vote2Cap-DETR
+comparing with SpaCap3D[36] on the Nr3D[37] dataset. Our proposed method generates tight bounding box as well as accurate descrip-
+tions.
+14
+
+𝑃𝐶
+𝑝𝑠𝑒𝑒𝑑
+𝑝𝑣𝑞
+scene0011_00
+scene0015_00
+scene0064_00
+scene0131_00
+scene0169_00
+scene0378_00
+Figure 10. Visualization of vote queries. We visualize absolute position of different object queries, pseed used in 3DETR (marked in
+blue) and pvq used in our proposed Vote2Cap-DETR (marked in red) with the input point cloud PC. Most of the vote queries focus on
+objects in a 3D scene (as red arrows pointed out), while pseed is mostly distributed in background areas (as blue arrows pointed out).
+15
+
+GT
+VoteNet
+3DETR
+Vote2Cap-DETR
+scene0088_00
+scene0144_00
+scene0169_00
+scene0257_00
+scene0342_00
+scene0354_00
+Figure 11.
+Visualization of detection performance. We visualize detection results of VoteNet[29], 3DETR[24], and our proposed
+Vote2Cap-DETR. Our proposed Vote2Cap-DETR is able to generate accurate localization results.
+16
+
+E11

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+arXiv:2301.02963v1  [physics.app-ph]  8 Jan 2023
+Diffusive Pseudo-Conformal Mapping: Anisotropy-Free
+Transformation Thermal Media with Perfect Interface Matching
+Gaole Dai,1, ∗ Fubao Yang,2 Jun Wang,3, 4 Liujun Xu,5, † and Jiping Huang2, ‡
+1School of Sciences, Nantong University, Nantong 226019, China
+2Department of Physics, State Key Laboratory of Surface Physics,
+and Key Laboratory of Micro and Nano Photonic Structures (MOE),
+Fudan University, Shanghai 200433, China
+3School of Physics, East China University of
+Science and Technology, Shanghai 200237, China
+4Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou 325001, China
+5Graduate School of China Academy of Engineering Physics, Beijing 100193, China
+(Dated: January 10, 2023)
+Abstract
+Transformation media provide a fundamental paradigm for field regulation, but their tricky
+anisotropy challenges fabrication. Though optical conformal mapping has been utilized to eliminate
+anisotropy, two key factors still hinder its development in thermotics, i.e., the distinct diffusion
+nature and inevitable interface mismatching. Here, we put forth the concept of diffusive pseudo-
+conformal mapping, overcoming the inherent difference between diffusion and waves and achieving
+perfect interface matching. The proposed mapping directly leads to heat guiding and expanding
+functions with anisotropy-free transformation thermal media, whose feasibility is confirmed by
+experiments or simulations. Besides diverse applications, we provide a unified perspective for two
+distinct types of prevailing bilayer cloaks by uncovering their profound ties with pseudo-conformal
+mapping. These results greatly simplify the preparation of transformation thermotics and have
+implications for regulating other diffusion and wave phenomena.
+1
+
+Heat control is essential for every aspect of human life, such as energy utilization, chip
+cooling, and infrared detection. The past decade has witnessed the development of transfor-
+mation theory [1, 2] in heat conduction, a diffusion process that intrinsically differs from wave
+dynamics [3, 4]. Transformation thermotics indicates that anisotropic and inhomogeneous
+thermal parameters in physical space can mimic heat transfer in curved space, ensured by the
+form-invariance of heat equations under coordinate transformations. However, anisotropy is
+a considerable restriction on practical realization because natural materials usually exhibit
+isotropic thermal properties. Though alternative schemes like diffusive scattering cancel-
+lation were proposed to avoid anisotropy [5–8], they generally apply to specific scenarios
+such as thermal invisibility. Therefore, removing the intrinsic anisotropy of transformation
+thermal media is still a big challenge.
+Conformal transformation optics [1, 9] can eliminate anisotropy with two-dimensional
+(2D) optical conformal mapping that locally preserves the angles and orientations of
+curves [10].
+Diverse wave phenomena have been realized with anisotropy-free transfor-
+mation refractive index [11–19]. However, two critical problems challenge the application
+of optical conformal mapping in thermotics.
+On the one hand, diffusion and waves are
+fundamentally different in governing equations (i.e., the Laplace equation vs. the Helmholtz
+equation) and key parameters (i.e., thermal conductivity vs. refractive index). On the other
+hand, matching the interface heat flux between the functional device and background is still
+tricky due to the lack of an intuitive impedance matching criteria, which is essential for
+accurate and robust heat manipulation. Thus, directly utilizing optical conformal mapping
+for precise diffusion control is impracticable.
+Here, we propose the concept of diffusive pseudo-conformal mapping with angle preser-
+vation for certain families of curves representing thermal fields. Fortunately, “pseudo” does
+not affect the crucial advantage of “conformal” in removing anisotropy and contributes to
+perfect interface matching. The proposed mapping yields precise heat manipulation with
+anisotropy-free transformation thermal media, and we take heat guiding and expanding as
+two typical examples, with experimental or simulated confirmation. Moreover, we reveal
+the geometric origin of bilayer cloaks previously designed by scattering cancellation from
+the perspective of pseudo-conformal mapping. These results feature scalability in handling
+complex heat transfer [20–22] and provide a unified geometric perspective towards various
+thermal functions with isotropic materials.
+2
+
+For clarity, we first establish diffusive conformal mapping and illustrate its restrictions.
+A 2D conformal mapping f : z0 �→ z (z0 = x0 + iy0 and z = f(z0) = x + iy) is holomorphic,
+satisfying the Cauchy–Riemann equations [10],
+∂f
+∂z0
+= 1
+2
+� ∂f
+∂x0
++ i ∂f
+∂y0
+�
+= 0,
+(1)
+where z0 is the complex conjugate of z0. We consider form-invariant heat conduction equa-
+tions in physical space (position denoted by z) and virtual space (position denoted by z0),
+∇ · (κ∇T(z)) − Q = 0,
+(2a)
+∇0 · (κ0∇0T0(z0)) − Q0 = 0,
+(2b)
+where T, κ, and Q are the temperature, thermal conductivity (rank-2 tensor), and internal
+source power in physical space, respectively. Their counterparts with a subscript “0” denote
+corresponding parameters in virtual space.
+The transformation rules to ensure T(z) =
+T0(f −1(z0)) are [23, 24]
+κ(z) = Jfκ0(z0)JT
+f / det Jf,
+(3a)
+Q(z) = Q0(z0)/ det Jf,
+(3b)
+where Jf is the Jacobian matrix of f, and JT
+f is the transpose of Jf.
+Due to the holo-
+morphicity of f, JfJT
+f / det Jf is an identity matrix [9], so Eq. (3a) can be simplified as
+κ(z) = κ0(z0(z)). With the familiar paradigm of transformation theory, virtual space is
+isotropic and homogeneous, so κ and κ0 are identical constant scalars. This result is con-
+sistent with the conventional technique using conformal mapping to solve heat equations in
+irregular geometric domains [25]. Different from the engineered gradient index in conformal
+transformation optics, such a frozen degree of freedom (i.e., κ = κ0) restricts the function
+design for thermal manipulation. Besides, transformation media are often in contact with
+the background that undergoes trivial mapping without changing material properties. How-
+ever, conformal mapping usually cannot ensure the continuity of boundary conditions due
+to the strict constraint of Eq. (1), i.e., conformality for all curves, as shown in the examples
+below. In fact, in most cases, we do not even have the option to consider interface matching
+due to the uniqueness of conformal mapping between two given domains. Therefore, we
+3
+
+must develop diffusive pseudo-conformal mapping further and design thermal guiding and
+expanding functions.
+Thermal guiding can bend the heat flux by an arbitrary angle. Its virtual space and
+physical space are shown in Figs. 1(a) and 1(c), respectively. In the virtual space, we apply
+a thermal bias along the y0-axis. The inlet (hot source) is put on the bottom horizontal
+boundary B0C0 and the outlet (cold source) is on the top boundary F0E0. The vertical
+boundaries F0A0B0 and E0D0C0 are thermally insulated. In the physical space, we expect
+the flux is rotated by angle ϕ when flowing out of the guide without changing its magnitude.
+In Fig. 1(a), we plot the grid lines of the Cartesian coordinates. They are just the heat
+flux streamlines (constant-x0 curves) and the isotherms (constant-y0 curves). The upper
+rectangle (with a height W) undergoes a composition of rotation and translation to the
+background in Fig. 1(c) without changing its thermal conductivity. The lower rectangle
+(with a height L) is transformed into the partial annulus (the guide) in Fig. 1(c). Notably,
+an identity transformation should happen between the inlets B0C0 and BC.
+Also, the
+mappings for the background and the guide should have the same effect at their interface
+AD (on the constant-azimuth line equal to ϕ in Fig. 1(c), which is mapped from A0D0 in
+Fig. 1(a)). The other boundaries, FAB and EDC, are still insulated in the physical space.
+Although there exists a conformal mapping between the guide and its preimage according
+to Riemann mapping theorem [10], it generally cannot transform B0C0 (or A0D0) into BC
+(or AD), let alone achieve the interface effect as we want (see our discussion based on the
+theory of quasiconformal mapping in Supplemental Material, Note I [26]).
+To construct the guide, a two-step approach is implemented. A non-conformal mapping
+is first used [Figs. 1(a) and 1(b)]: x1 = ln x0 and y1 = ϕy0/L, from the virtual space to an
+intermediate named the auxiliary space (position denoted by z1 = x1 + iy1). The second
+step to the physical space [Figs. 1(b) and 1(c)] is conformal: z = exp(z1), which can lead to
+the waveguide in optics [9]. The composition transformation is also non-conformal and the
+thermal conductivities in the auxiliary space (denoted by κ1) and the physical space (satisfy-
+ing κ(z) = κ1(z1)) should be diagonally anisotropic, writing κ = κ0diag
+�
+∂x1
+∂x0/ ∂y1
+∂y0, ∂y1
+∂y0/ ∂x1
+∂x0
+�
+in the Cartesian coordinate system.
+However, if κ0 itself is diagonally anisotropic, e.g.,
+κ0 ∼ diag
+�
+∂y1
+∂y0/ ∂x1
+∂x0, ∂x1
+∂x0/ ∂y1
+∂y0
+�
+, anisotropy can be eliminated in the other two spaces.
+In
+addition, since only κy0y0
+0
+contributes to heat flux, we can take κ0 = κy0y0
+0
+without changing
+the results in any space. In this way, thermal conductivities in all spaces are isotropic and
+4
+
+we have
+κ(z) = κ1(z1(z)) = κ0
+ϕ
+L
+�
+x2 + y2.
+(4)
+We also plot the grid lines in Figs. 1(b) and 1(c) mapped from their counterparts in
+Fig. 1(a). The three sets of grid lines are all orthogonal since they happen to be streamlines
+and isotherms in their spaces. Intuitively, we call the mapping in the first step “pseudo-
+conformal” for maintaining orthogonality of certain curves. By doing a pseudo-conformal
+mapping first and then a conformal one, the composition is still pseudo-conformal. The
+streamlines in Figs. 1(a) and 1(c), which are both evenly distributed, so the heat flux mag-
+nitude is not changed. Also, the flux in the guide does turn an angle ϕ as we expected.
+To confirm our design, wo perform an experiment with the following parameters: a = 0.5,
+L = 3W = ϕ/0.65, ϕ = π/6 and κ0 = 400 W m−1 K−1 (e.g., copper). Here, the unit length
+is 40 cm.
+The thermal conductivity profile based on Eq. (4) is shown in Fig. 2(a) and
+the background has the same value as κ0.
+The inhomogeneous κ is approximated by a
+composite of copper and air holes [Fig. 2(b)]. We plot six straight lines, including the inlet
+(Line 1) and outlet (Line 6) in Fig. 2(b). Line 4 is the interface of the guide and background.
+Lines 2 and 3 divide the guide into thirds, and Line 5 is in the middle of the background.
+Ideally, the temperature difference relative to the outlet on each line (isotherm) is marked
+in Fig. 2(b). The total thermal bias is ∆T, which is experimentally realized by water baths
+heating or cooling the sample. Fig. 2(c) shows the observed temperatures. Each of the six
+lines is roughly isothermal, and we plot horizontal lines corresponding to their mean value
+(excluding edge extremes).
+Notably, the temperature differences between Lines 4-6 are
+almost equal. We can conclude that the sample can bend the flux and keep its homogeneity.
+More details about the experimental setup and data are presented in Supplemental Material,
+Note II [26].
+Next, we design a thermal expander that can convert the heat flux emitted by the point
+source into parallel flows. In the virtual space [Fig. 3(a)], we consider the lower half-plane.
+The isotherms form concentric semi-circles with the point source at the center, and the heat
+flux magnitude varies with azimuth. To determine the temperature distribution, its value
+on a certain isotherm should be given, e.g., 290 K on {z0 : |z0| = 1, y ⩽ 0} via an external
+source. The x0-axis is thermally insulated except for the point source. By doing a conformal
+mapping z = ii+z0
+i−z0 used in the optical counterpart [9], i.e., a M¨obius transformation, the
+unit lower half-disk becomes the unit upper half-disk in the physical space [Fig. 3(b)]. The
+5
+
+point source is now at z = i and its power becomes Q = Q0/2 to ensure T(z) = T0(z0(z)).
+The diameter on the x-axis {z : |x| ⩽ 1, y = 0} is an isotherm (and also the external source)
+mapped from {z0 : |z0| = 1, y ⩽ 0} and the heat flux on it is parallel (along the negative
+y-axis). However, flux magnitude has a varying value depending on which azimuth it is
+mapped from. For practical applications, we want a homogenized flux so it can keep parallel
+in a rectangular extension attached to {z : |x| ⩽ 1, y = 0} when the external source is
+moved to the bottom of the extension. This homogenization can be realized by z = ii+z1(z0)
+i−z1(z0).
+Here, Arg[z1] = 2 arctan
+�
+Arg[z0]−2π
+Arg[z0]−π
+�
++ 2π is pseudo-conformal from the virtual space to the
+auxiliary space, and Arg denotes the argument (see Supplemental Material, Note III for how
+to construct this mapping [26]). This approach leads to the same boundary conditions of
+external source and insulation as the conformal one. Now, the thermal conductivity in the
+expander is
+κ(x, y) = κ1
+�
+1 +
+�
+(x2 + y2 − 1)2
+x4 + 2x2 (y2 + 1) + (y2 − 1)2
+�−1
+.
+(5)
+Since κ is still isotropic, the heat flux on the x-axis can keep parallel. In Fig. 3(c), we plot
+the thermal fields in the new physical space. We can see that the intersections of streamlines
+with the x-axis are now uniformly distributed, which is different from the case in Fig. 3(b).
+This is an intuitive representation of homogenization. We further show the heat flux on
+the x-axis produced by the two mappings in Fig. 3(d). The theoretical results agree with
+the finite-element numerical ones from COMSOL Multiphysics (https://www.comsol.com/).
+In Supplemental Material, Note IV [26], we confirm the performance of our design when
+considering an extension. Eq.(5) can also be written in a compact form using transposed
+bipolar coordinates (See Supplemental Material, Note V [26]).
+Besides functional design in different scenarios, our approach can also reveal the under-
+lying ties between material parameters and coordinate transformations for some previous
+works based on alternative schemes of transformation theory. Here, for example, we provide
+a geometric insight of bilayer cloaks [5–8]. Transformation-based annular cloak (shell cloak)
+depends on a non-homeomorphic mapping to generate its inner boundary, e.g., transforming
+a point or a line into a closed curve [1, 2]. Due to the geometric symmetry, we consider
+the upper half of an annular cloak, i.e., a carpet cloak. The virtual and physical spaces in
+Fig. 4 show a transformation from the upper half-disk D+ with a radius R2 to the upper
+half-annulus A+ = {z : R1 ⩽ |z| ⩽ R2, y ⩾ 0} [Fig. 4(c)]. Here A+ is actually the outer
+6
+
+layer of a bilayer cloak. For simplicity, we take R1 = 1 (in meters). The area outside D+
+or A+ is the background. A cloak means the area enclosed by A+ is expected to have no
+disturbance to the background. We still use a two-step pseudo-conformal mapping. First,
+D+ is compressed/expanded into an half-ellipse E+ = {z1 : 0 ⩽ x2
+1/a2 + y2
+1/b2 ⩽ 1, y1 ⩾ 0}
+[Fig. 4(b)] by x1 = R2
+a x0 and y1 = R2
+b y0. Here we take a2 = R2
+2 + R2
+1 and b2 = R2
+2 − R2
+1 so
+the foci of the half-ellipse are (±2, 0). This non-conformal mapping is angle-preserving for
+the Descartes grid lines in Fig. 4(a). Second, conformally map E+ to A+:
+
+
+
+
+
+
+
+z = z1 −
+�
+z2
+1 − 4
+2
+,
+if x1 < 0
+z = z1 +
+�
+z2
+1 − 4
+2
+,
+if x1 ⩾ 0.
+(6)
+This mapping is one branch of the inverse Zhukovsky transform [35] (See Supplemental
+Material, Note VI for detailed explanation [26]). In particular, the upper boundary of A+
+(i.e., {z : |z| = R2, y ⩾ 0}) finally undergoes an identity transformation as well as the
+background. If the thermal bias is applied along the x0-axis, the grid lines in Fig. 4(a)
+represent horizontal streamlines and vertical isotherms and only κx0x0
+0
+contributes to heat
+flux. Taking κ0 = κx0x0
+0
+, we find the thermal conductivity in A+ is
+κ = R2
+2 + R2
+1
+R2
+2 − R2
+1
+κ0,
+(7)
+which is just the cloaking condition for the outer layer of a bilayer cloak [5]. In addition,
+the lower boundary of A+ including {z : |z| = R1, y ⩾ 0} and {z : R1 ⩽ |x| ⩽ R2, y = 0}
+(denoted by Γ as a whole) are mapped from a streamline in the virtual space. The heat flux
+should have no normal component on Γ, which can be ensured by the insulation condition.
+This corresponds to the cloaking condition for the inner layer of a bilayer cloak, i.e., zero-
+value thermal conductivity and arbitrary shape as along as it covers {z : |z| = R1, y ⩾ 0}.
+If the thermal bias is instead along the y0-axis, our mapping is still angle-preserving for
+the vertical streamlines and horizontal isotherms in Fig. 4(a). Take κ0 = κy0y0
+0
+and we have
+κ = R2
+2 − R2
+1
+R2
+2 + R2
+1
+κ0.
+(8)
+The lower boundary Γ is mapped from an isotherm, leading to another bilayer cloak made
+of zero-index materials [7]. We call the previous cloak “normal bilayer” to distinguish them.
+The term “zero-index” refers to the constant-temperature condition on the inner layer that
+can be replaced by an effectively infinite thermal conductivity [7, 8]. Figs. 4(d) and 4(e)
+7
+
+numerically confirm the performance of the two cloaks. In addition to the invisibility effect,
+we can see that the patterns of isotherms and streamlines in the two cloaks have a duality
+relationship by swapping the family of curves of streamlines and isotherms. Further, this
+mapping can be performed on the entire complex plane to build the shell cloak. Our method
+can also be used to design invisibility devices with other geometries, such as confocally
+elliptical cloaks (See Supplemental Material, Note II for detailed discussions [26]).
+In summary, we propose the concept of diffusive pseudo-conformal mapping to simulta-
+neously achieve precise heat flux regulation and maintain the material isotropy in transfor-
+mation thermotics. By preserving the angles of certain families of curves (e.g., isotherms
+and streamlines), our approach can circumvent anisotropy and perfectly match the interface
+heat flux between the transformation media and background. We demonstrate our theory
+by designing feasible and robust thermal devices that can bend or parallelize heat flux at our
+will. Also, we revisit scattering-cancellation-based bilayer cloaks from a unified perspective
+of pseudo-conformal mapping. In addition to reobtaining the parameters without inversely
+solving heat equations, the intrinsic geometric relationship between different types of bilayer
+cloaks is also revealed. The idea of diffusive pseudo-conformal mapping can be further devel-
+oped for controlling transient heat conduction [24, 38], multithermotics [39, 40], and other
+diffusion physics [41–44]. The consideration of perfect interface matching also benefits the
+design of transformation wave media, such as avoiding impedance mismatch that plagues
+conformal invisibility devices [45].
+We acknowledge financial support from the National Natural Science Foundation of China
+(Grants No. 11725521, No. 12035004, No. 12147169, and No. 12205101) and the Science
+and Technology Commission of Shanghai Municipality (Grant No. 20JC1414700).
+∗ [email protected]
+† [email protected]
+‡ [email protected]
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+10
+
+FIG. 1. Transformation for a heat flux guide (gray and white meshes) including the background
+(brown and white meshes). (a), (b) and (c) are the virtual space, the auxiliary space (only showing
+the transformation of the guide) and the physical space, respectively.
+11
+
+■口口口口口口口口口口口0口=yo
+Z= ez1FIG. 2. (a) Normalized thermal conductivity (κ/κ0) profile of the guide. The white curves are
+isolines. (b) Sample structure made of copper and air holes for experimental setup. (c) Measured
+temperatures showing the data on the red lines plotted in (b). The arc length means the distance
+from left endpoint.
+12
+
+40
+Heatfluxguide(p=元/6)
+Samplestructure
+Copper
+Airhole
+(390W
+W/2
+30
+3
+5
+(cm)
+20
+Line
+A
+Line
+azi:元/6
+10
+Line3
+azi:/9
+Line2
+Normalized
+azi:元/18
+conductivity
+0
+0.65
+30
+35
+40
+45
+50
+55
+60
+30
+35
+40
+45
+50
+55
+60
+x (cm)
+x (cm)
+Line1
+Line3
+Line5
+320
+Line2
+Line4
+Line6
++316.53K
+315
+310
++308.76K
+305
++302.24K
+300
+296.95K
+295
+294.50K
+291.98K
+290
+Observedtemperatures inexperiments
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Arclength(cm)FIG. 3. Thermal expander. (a) The virtual space with the point source at the origin. Concentric
+semicircles are isotherms. Gray curves with arrows represent streamlines, which are also constant-
+azimuth lines ranging from 1.1π to 1.9π (0.1π interval from left to right). (b) The physical space
+for a conformal mapping. (c) The physical space for a pseudo-conformal mapping. The thermal
+fields in (a)–(c) are illustrated based on finite element numerical results. (d) The magnitude of
+the normalized heat flux on the x-axis for the two physical spaces. The solid lines are theoretical
+results while the scatter charts with markers are numerical results. Here, we take κ0 = 400 W m−1
+K−1 and Q0 = 3000 W m−3.
+13
+
+口0.0
+1.0
+(m)
+w)
+0.5
+0.5
+-1.0
+0.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Xo (m)
+x(m)
+Normalizedheatflux
+1.3
+1.0
+(w)
+1.0
+0.5
+Simulation(C)
+Simulation(PC)
+Theory (C)
+0.0
+0.7
+Theory(PC)
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+x (m)
+x (m)FIG. 4. Carpet cloaks. (a)–(c) show the geometric transformation to construct such a cloak. (a),
+(b), and (c) are the virtual space, auxiliary space and the physical space, respectively. (d) is the
+computed temperature profile of a normal bilayer cloak with an insulating inner layer. We place
+the hot source (300 K) and the cold source (200 K) on the left and right boundaries, respectively,
+generating a thermal bias along the x-axis. (e) is the computed temperature profile of a zero-index
+cloak with an constant-temperature inner layer (realized by an external source). We illustrate white
+curves for isotherms and gray curves with arrows for streamlines. Here, we take R2 = 2R1 = 2 m
+and κ0 = 400 W m−1 K−1. The entire simulation domain is limited to a 6 m × 3 m rectangle.
+14
+
+口口口口3
+(w) x
+2
+R
+-3
+-2
+-1
+0
+1
+2
+3
+x (m)
+3
+2
+(w)
+Z1+Vz-
+0
+-3
+-1
+0
+1
+2
+(w) x

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+arXiv:2301.08326v1  [physics.ed-ph]  19 Jan 2023
+A low-cost confocal microscope for the undergraduate lab
+A. Reguilon, W. Bethard, and E. Brekke∗
+Department of Physics, St. Norbert College, De Pere, WI 54115
+Abstract
+We demonstrate a simple and cost-efficient scanning confocal microscope setup for use in ad-
+vanced instructional physics laboratories. The setup is constructed from readily available com-
+mercial products, and the implementation of a 3D-printed flexure stage allows for further cost
+reduction and pedagogical opportunity. Experiments exploring the thickness of a microscope slide
+and the surface of solid objects with height variation are presented as foundational components of
+undergraduate laboratory projects, and demonstrate the capabilities of a confocal microscope. This
+system allows observation of key components of a confocal microscope, including depth perception
+and data acquisition via transverse scanning, making it an excellent pedagogical resource.
+1
+
+I.
+INTRODUCTION
+The design and use of optical instruments is an area of broad interest, appealing par-
+ticularly to students at the intersection of biological fields with physics and engineering.
+With the large number of undergraduate physicists interested in careers in biomedical fields,
+a confocal microscope provides an excellent opportunity to see how physics principles re-
+late to state-of-the-art equipment used for image formation. The properties, advantages,
+and applications of confocal micrsocopy have been carefully investigated in the biological
+community.1–3
+While many undergraduate optics courses spend significant time on the ideas of optical
+instruments and traditional microscopes, confocal microscopy is an important application
+that is often absent from the student experience. There has been a variety of designs for
+homebuilt confocal microscopes in the lab,4–9 but these designs are often more focused on
+the production of an image than on student understanding of the physics underlying the
+operation of the instrument. Previous versions were often either too expensive, or left essen-
+tial elements hidden in commercial components, making them nonideal for undergraduate
+pedagogy.
+In this paper we present a simple scanning confocal microscope experiment, ideal for
+the undergraduate optics or advanced instructional lab environment. Rather than being
+designed to minimize its size or acquire images on par with commercial confocal micro-
+scopes, our instructional setup is designed to clearly demonstrate the physics involved in
+the image acquisition method and to make the optical components easy to manipulate. Our
+setup provides an excellent system to illustrate scanning confocal microscopy and also gives
+experience with the calibration of a data acquisition process.
+Our system is designed to be versatile, with potential to develop student experimental
+skills in the areas optical systems, electronics, and 3D printing. Projects can be adapted to
+suit the context of an advanced lab setting, an optics or electronics course, an independent
+student project, or a senior thesis. In the following sections, we outline an experimental
+process that can be used as a whole or broken up into parts, depending on the pedagogical
+purposes of the instructor. In the first experiment, a microscope slide is used to illustrate the
+ability of a confocal setup to acquire depth information in a sample. A second experiment
+involves using this setup to investigate height variations of a solid material. Finally, either of
+2
+
+these experiments can incorporate a 3D printed translation stage,10 which allows integration
+of programming and electronic control.
+II.
+DESIGN
+100 mm lens
+500 mm lens
+150 mm lens
+Objective lens
+Mirror
+Beam 
+Splitter
+650 nm 
+laser
+Photodiode
+Sample on 
+Translation Stage
+Iris
+Iris
+X
+Y
+FIG. 1. The experimental setup for the confocal microscope. A generic visible laser is used, and
+expanded to fill a microscope objective lens. The objective lens focuses the light onto a sample,
+which is mounted on a translation stage at the focus of the objective. The light reflected from the
+sample is separated by a beam splitter, and focused onto an image plane, where an iris limits light
+to that which came from the focal plane of the objective. This intensity is then monitored by a
+photodiode.
+Our confocal microscope experimental setup is shown in Fig. 1. The essential elements
+of this system are readily available commercial components. We use a simple 650 nm laser
+diode, with a optical power of 5 mW. This laser is expanded with a two-lens beam expander
+to a beam waist of 5 mm waist to fill a commercial microscope objective lens. The beam
+3
+
+expander is constructed so that the distance between the lenses is the sum of the focal
+lengths, which increases the beam size by a factor of f2/f1. An iris is placed at the focus
+of the beam expander to make the beam more circular. Once the beam is collimated, the
+distance between the expander and the objective, including the presence of a steering mirror,
+is chosen for convenience. The sample is placed on a translation stage, which can be either
+a commercially purchased or a 3D-printed stage. The objective focuses the light onto the
+sample, where a portion is reflected back through the objective and a beamsplitter redirects
+the light towards a photodetector.
+TABLE I. The parts required for construction of the confocal microscope.
+Part
+Specs
+Supplier and part number Price (US$)
+Laser Diode
+650 nm, 5 mW
+Adafuit 1054
+10.00
+Diode Holder
+Thorlabs RA90
+10.30
+Feet (10)
+1′′ x 2.3′′ x 3/8′′
+Thorlabs BA1S (5 PACK)
+48.20
+Lens Holder (7)
+Ø1′′ Optics, 8-32 Tap
+Thorlabs LMR1
+109.83
+Mirror Mount
+Kinematic Mount for Ø1′′
+Thorlabs KM100
+39.86
+Mirror
+Ø1′′ 400 - 750 nm
+Thorlabs BB1-E02
+77.35
+Objective Lens
+10X infinite conjugate
+Amscope PL10X-INF-V300
+66.99
+Thread Adaptor
+SM1 to RMS
+Thorlabs SM1A3
+18.50
+Posts (10)
+2′′
+Thorlabs TR2-P5
+50.52
+Post Holders (10)
+2′′
+Thorlabs PH2-P5
+81.28
+150 mm Lens
+Ø1′′, AR Coating: 650–1050 nm
+Thorlabs LB1437-B
+35.50
+500 mm Lens
+Ø1′′, AR Coating: 650–1050 nm
+Thorlabs LA1908-B
+33.01
+100 mm Lens
+Ø1′′, AR Coating: 650–1050 nm
+Thorlabs LA1509-B
+34.39
+Beamsplitter
+Economy Ø1′′ 50:50
+Thorlabs EBS1
+35.78
+Ring Actuated Iris (2)
+(Ø0.8 - Ø12 mm)
+Thorlabs SM1D12D
+145.86
+Photodetector
+350 - 1100 nm
+Thorlabs DET36A2
+134.20
+Ultralight Breadboard
+24′′ x 24′′ x 0.98′′
+Thorlabs PBG2424F
+842.93
+3-Axis RollerBlock
+Long-Travel Bearing Stage
+Thorlabs RB13M
+1,566.15
+Total
+3,340.65
+4
+
+If the sample is at the focal point of the objective, the light coming back through the
+system will be collimated, and a final imaging lens will focus it through an iris before the
+photodetector. Hence, if the sample is at exactly the focal length away from the objective,
+the image plane will be at the iris location, and a large portion of the light will make it to
+the photodetector. However, if the sample’s surface is not at the focal plane, the amount of
+light at the photodetector will be greatly reduced.
+Essential to the operation of the confocal microscope is the fact that the iris is in the
+image plane at the light focus.
+Thus, it effectively images only the part of the sample
+at the focal plane of the objective lens. Unlike traditional microscopes, this allows depth
+information in a sample.
+The apparatus described here is a cost-effective means for demonstrating these ideas in
+an undergraduate setting. A full list of the parts needed can be seen in Table I, where it
+is assumed power supplies and an oscilloscope are available. The inclusion of the mirror
+and lens focal lengths were chosen by convenience with an available breadboard, and can
+be modified as desired. The total cost of the equipment comes to US $3,340.65. However,
+the largest cost is the 3-axis translation stage, which can be replaced with a 3D-printed
+flexure stage,10,11 reducing the cost to under US $2,000 and opening up new opportunities
+for development.
+This setup can be used to examine several simple objects, which provide excellent insight
+into the confocal imaging process. A deeper understanding of depth information from the
+microscope can come from examining standard microscope slides, especially those with con-
+cave sample openings. The slide can be mounted on the translation stage in a number of
+ways; in our experiment, it was clipped to a 3D-printed stand, such that the laser passed
+through the slide with nothing behind it. The system can also be used to examine sur-
+face variation of solid objects. We have found that 3D-printed objects with known height
+variation works well, but a variety of other objects can be used.
+III.
+RESULTS AND ANALYSIS
+In order to demonstrate the capabilities of the confocal microscope, two experiments are
+outlined here. The first involves observing the thickness of semitransparent materials, with
+commercial microscope slides offering an excellent starting point. Using a microscope slide
+5
+
+to demonstrate how the confocal microscope works offers students the chance to determine
+depth information on a sample.
+As a starting point, a microscope slide with a concave
+sample opening can be used. In this case two reflections occur, one off of the slide’s front
+surface and the other off of its back surface.
+As a result, adjusting the position of the
+sample longitudinally along the direction of laser propagation allows the observation of
+two successive peaks, as each of the slide’s front surface and back surface is in the focal
+plane of the objective. An example of the data observed as the slide platform is adjusted
+longitudinally (in the x-direction) is shown in Fig. 2. Opening the iris at the image plane
+makes clear the way in which the confocal microscope causes a specific image plane for
+different depths in the material. Without this iris, depth information is lost, as can be seen
+in Fig. 2.
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+Photodiode Signal (V)
+5.5
+5.0
+4.5
+4.0
+3.5
+3.0
+2.5
+2.0
+X axis position (mm)
+ Slide Edge
+ Slide Center
+ Without Iris
+FIG. 2. As the distance between the objective lens and the slide is varied, intensity peaks are seen
+when either the front or the back surface of the slide is at the focal plane of the objective. This is
+shown both at the flat edge of the slide, and at the curved center where the slide is not as thick.
+The shifting of the second peak reveals the changing thickness of the slide. Removing the iris in
+the imaging plane reveals the necessity of limiting the light from a single image plane for depth
+perception.
+By using a microscope slide with a concave sample opening, the thickness of the glass
+6
+
+can be observed as a function of the position along the slide. As a starting point, this can
+be done by moving the slide a small distance transversely (in the y-direction), then scanning
+longitudinally (in the x-direction) to observe the two peaks, stepping through positions. An
+example of the data obtained with this two axis scan technique is shown in Fig. 3.
+5.0
+4.8
+4.6
+4.4
+Position of highest reflection (mm)
+10
+8
+6
+4
+2
+0
+Distance along slide (mm)
+ Two Axis Scan
+ One Axis Scan
+FIG. 3. The distance from the objective where peak reflected light is observed as a function of
+distance along the slide. The region where the slide is flat is visible, before the concave opening
+begins at 4 mm. Then the changing thickness where the concave sample opening begins can be
+observed. The two-axis scan involves moving the translation stage transversely a small amount
+before scanning longitudinally to find the maximum. The one-axis method involves only scanning
+transversely, using a known calibration of the photodiode voltage with depth.
+This setup can also be used to demonstrate the usefulness of a scanning confocal micro-
+scope by acquiring the same depth information, while only moving the slide transversely to
+the laser beam. This makes the image collection process much faster, but requires calibrat-
+ing the system. As the slide is shifted transversely in the area where the depth varies, the
+different depths will cause different voltages in the photodiode. To correlate these voltage
+changes to height changes, a fit for the voltage as a function of height in the region scanned
+is found from the longitudinal scan, such as shown in Fig. 4.
+The shape of the intensity peak is caused by the changing beam waist as it propagates
+7
+
+2.5
+2.0
+1.5
+1.0
+0.5
+Voltage (V)
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Distance (mm)
+ Linear Fit
+Range: 0.066 mm
+2.5
+2.0
+1.5
+1.0
+0.5
+Voltage (V)
+1.4
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+Distance (mm)
+Range: 0.21 mm
+Lorentzian Fit
+(a)
+(b)
+FIG. 4.
+The intensity peak obtained as the slide is translated longitudinally can be used to provide
+a voltage vs. height calibration for the sample. An appropriate fitting function can be chosen to
+fit the peak over a particular height variation region on a sample. a) A linear fit to a portion of
+the peak, providing a simple calibration valid over 0.066 mm around the sample. b) A Lorentzian
+fit valid over 0.21 mm range around the sample.
+through the image plane, with the intensity limited by the iris size.
+This is a complex
+dependence, but can be roughly fit using a Lorentzian over much of the peak. For a more
+precise calibration, it is desired to find a function that fits the intensity peak very well
+over a particular range of heights of the sample. Using this calibration, voltage changes
+when scanning transversely can be used to calculate height changes in the sample. The
+simplest calibration is to use a portion of the graph that can be well approximated as
+linear, as shown in Fig. 4(a). However, this linear fit is only valid over a limited height
+variation, less than 100 µm. Over a larger height variation, an exponential or polynomial
+fit may also work well, where we illustrate a Lorentzian fit in Fig. 4(b). Allowing students
+to determine this calibration technique provides additional insight and experience in data
+analysis methods. Taking the data again for the slide thickness using a transverse scan
+calibrated with the Lorentzian fit is shown in Fig. 3, and agrees with that taken when
+collecting points individually with motion in both dimensions. Further, this process allows
+students to understand how data can be collected and analyzed quickly in this ‘scanning’
+configuration of a confocal microscope.
+The ability of the confocal microscope to determine height variations of a solid sample
+8
+
+represents another key category of experiment that can be undertaken. To illustrate this
+point, stereolithography (SLA) 3D-printed objects of known height variation are used. SLA
+3D printing, like fused deposition modeling (FDM) 3D printing, creates a model layer by
+layer.
+Instead of extruding filament, a laser polymerizes resin at a specific point.
+This
+process allows for a quantifiable amount of material to be deposited with a known volumetric
+resolution. This property was exploited to create steps, with a known thickness, on the
+surface of a 3D-printed block small enough to demonstrate the accuracy of the system.
+Variations on this experiment could be done with a variety of objects, as long as the object
+has height variation on the scale of 100s of microns.
+With these solid objects, there are no longer multiple intensity peaks from different
+depths of the sample. Instead, the location of the surface can be seen by the maximum
+of the intensity peak.
+Here, the surface height variation can be investigated either by
+successively stepping transversely and longitudinally, or by scanning only transversely using
+a calibrated intensity vs. height graph, as discussed above. For our experiment two different
+surface variations were printed, one that increased 50 µm in height for each 200 µm shift
+in position, and another that increased 100 µm in height for each 200 µm shift in position.
+Figure 5 shows the measured surface location as a function of the distance along the sample
+for each of these materials using the two-axis scan technique, and illustrates the ability
+of the system to determine solid surface height variations. The precision of 3D printers
+can limit the resolution of features created using this method, but it is expected that the
+cost of high-resolution 3D printers will continue to decrease, allowing more intricate surface
+patterns to be investigated. Additional solid objects such as coins can also be examined, but
+the high reflectivity and angled surfaces can cause complex behavior in the signal intensity,
+and extra care is required.
+IV.
+INCORPORATION OF 3D-PRINTED TRANSLATION STAGE
+3D printing has developed into an extremely valuable tool for the undergraduate lab, and
+has been incorporated into a number of experimental designs.12–14 Due to recent advances
+and open-source development, it is now possible to print and program excellent open-source
+translation stages,10,11 and 3D printing has been incorporated into a variety of microscope
+designs.15–20 In the confocal microscopy setup presented here, 3D printing presents the pos-
+9
+
+4.8
+4.6
+4.4
+4.2
+4.0
+3.8
+Material Height (mm)
+10.5
+10.0
+9.5
+9.0
+Distance along printed sample (mm)
+ 100 micron step
+ 50 micron step
+FIG. 5.
+The height of a solid 3D-printed object, as measured with the confocal microscope. This
+object was designed and printed to show the capabilities of the microscope, and a number of similar
+objects can be used.
+sibility of replacing a commercial translation stage system with a homebuilt option. This
+replacement serves two main purposes: First, it can reduce the cost of the necessary equip-
+ment by a significant fraction. Second, it allows meaningful student experience in design,
+3D printing, electronics, and programming. This allows the confocal microscope project to
+be quite general, appealing to students interested in a wide range of engineering or data
+acquisition fields.
+One option that incorporates 3-axis translation and excellent precision is the OpenFlexure
+Block stage.11 This setup is well documented elsewhere, and here was incorporated with
+little modification to the design presented. This stage was specified to have a step size in
+the x direction of 12.4 ± 0.2 nm and translate 2 mm on any axis. While this translation
+distance is not large enough to scan over large portions of the slide, it does allow the same
+experimentation on slide thickness described above, with the data taken again as observed in
+Fig. 2. Having both the data for slide thickness taken from the commercial translations stage
+and the OpenFluxure Block stage allows an additional means of verifying the calibration for
+10
+
+distance per step on the block stage. Using these data, our calibration was determined to be
+12.1 ± 0.4 nm, consistent with previous measurements.11 The use of 3D-printed translation
+stages makes the scanning confocal microscope even more accessible, and would combine
+well with projects commonly undertaken in an undergraduate lab environment.
+V.
+CONCLUSIONS AND FUTURE DIRECTIONS
+The design for a homebuilt scanning confocal microscope presented here is ideal for an un-
+dergraduate instructional laboratory setting. Through the measurement of microscope slide
+thicknesses and surface variations, undergraduates develop a better sense of the methods of
+scanning confocal microscopy and the information it provides. This design provides a robust
+setup with understandable mechanisms at a low price that enables easy implementation.
+Our setup is intended for pedagogical purposes rather than quality imaging competitive
+with commercial confocal microscopes, but can be extended for more complete imaging. If
+desired, the OpenFlexure stage can be automated, allowing for expansion of this project
+into acquiring scanned three dimensional images. This is especially appealing in the context
+of an independent project or senior thesis looking to incorporate additional programming.
+An investigation of the resolution of the system2 can also be employed as an extension to
+the experiments presented here. Investigating the limiting resolution would require materials
+with much finer features, as with this wavelength and numerical aperture lens the theoretical
+lateral resolution is expected to be near 1 µm, and the axial resolution on order 15 µm. Here
+several factors could be investigated to determine maximum resolution, including beam size,
+the minimal size and location of the iris, and the numerical aperture of the objective lens
+used.
+The confocal microscope system described in this paper can easily be expanded, either to
+improve its quality or examine further applications. Enhancements include the addition of an
+additional beamsplitter and camera, the use of achromatic doublets to decrease aberrations,
+improving the laser quality through fiber-coupling before use, or a galvo-scanning mirror in
+place of the x-y translation stage. In addition, the experiments outlined here can easily be
+extended to simple biological samples or fluorescence microscopy, allowing further insight
+into the way the optical system is commonly applied in biomedical fields.
+11
+
+ACKNOWLEDGMENTS
+We wish to acknowledge the contributions of Joseph Coonen and the programming insight
+of Michael Olson.
+∗ [email protected]
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+Curr. Protoc. Cytom. 92, e68 (2020).
+3 J. C. Erie, J. W. McLaren,
+and S. V. Patel, “Confocal Microscopy in Ophthalmology,”
+Am. J. Ophthalmol. 148, 639 (2009).
+4 P. Xi, B. Rajwa, J.T. Jones, and J.P. Robinson, “The design and construction of a cost-efficient
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+5 J. Hsu, S. Dhingra, and B. D’Urso, “Design and construction of a cost-efficient Arduino-based
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+6 S. Arunkarthick, M. M. Bijeesh, A. S. Vetcha, N. Rastogi, P. Nandakumar, and G. K. Varier,
+“Design and construction of a confocal laser scanning microscope for biomolecular imaging,”
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+7 C. M.. Jennings, J. B. King, and S. H. Parekh, “Low-cost, minimalist line-scanning confocal
+microscopy,” Opt. Letters 47, 4191 (2022).
+8 P. K.. Shakhi,
+M. M. Bijeesh,
+G. K. Varier,
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+Gong,
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+Kulkarni,
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+Zhu,
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+D.
+Nguyen,
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+Bio. Opt. Express 10, 3497 (2019).
+10 J. P. Sharkey, C. C. W. Foo, A. Kabla, J. J. Baumberg,
+and R. W. Bowman, “A one-piece
+printed flexure stage for open-source microscopy,” Rev. of Sci. Inst. 87, 025104 (2016).
+11 Q. Meng, K. Harrington, J. Stirling, and R. Bowman, “The OpenFlexure Block Stage: sub-100
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+nm fibre alignment with a monolithic plastic flexure stage,” Optics Express 28, 4763 (2020).
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+and T. Bixby, “Optical measurements on a budget: A 3D printed ellipsometer,”
+Am. J. Phys. 90, 445 (2022).
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+housing,” Am. J. Phys. 88, 1170 (2020).
+14 B. Schmidt, M. Pacholok,
+D. King
+and J. Kariuki, “Application of 3D Printers to
+Fabricate Low-Cost Electrode Components for Undergraduate Experiments and Research,”
+J. Chem.Educ. 99, 1160 (2022).
+15 T. Matsui and D. Fujiwara, “Optical sectioning robotic microscopy for everyone: the structured
+illumination microscope with the OpenFlexure stages,” Optics Express 30, 23208 (2022).
+16 J. T. Collins, J. Knapper, J. Sterling, J. Mduda, C. Mkindi, V. Mayagaya, G. A. Mwakajinga,
+P. T. Nyakyi, V. L. Sanga, D. Carbery, L. White, S. Dale, Z. J. Lim, J. J. Baumberg, P. Cicuta,
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+13
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='08326v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='ed-ph]  19 Jan 2023  A low-cost confocal microscope for the undergraduate lab  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Reguilon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Bethard, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Brekke∗  Department of Physics, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Norbert College, De Pere, WI 54115  Abstract  We demonstrate a simple and cost-efficient scanning confocal microscope setup for use in ad-  vanced instructional physics laboratories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The setup is constructed from readily available com-  mercial products, and the implementation of a 3D-printed flexure stage allows for further cost  reduction and pedagogical opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Experiments exploring the thickness of a microscope slide  and the surface of solid objects with height variation are presented as foundational components of  undergraduate laboratory projects, and demonstrate the capabilities of a confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This  system allows observation of key components of a confocal microscope, including depth perception  and data acquisition via transverse scanning, making it an excellent pedagogical resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  1  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  INTRODUCTION  The design and use of optical instruments is an area of broad interest, appealing par-  ticularly to students at the intersection of biological fields with physics and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  With the large number of undergraduate physicists interested in careers in biomedical fields,  a confocal microscope provides an excellent opportunity to see how physics principles re-  late to state-of-the-art equipment used for image formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The properties, advantages,  and applications of confocal micrsocopy have been carefully investigated in the biological  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='1–3  While many undergraduate optics courses spend significant time on the ideas of optical  instruments and traditional microscopes, confocal microscopy is an important application  that is often absent from the student experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' There has been a variety of designs for  homebuilt confocal microscopes in the lab,4–9 but these designs are often more focused on  the production of an image than on student understanding of the physics underlying the  operation of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Previous versions were often either too expensive, or left essen-  tial elements hidden in commercial components, making them nonideal for undergraduate  pedagogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  In this paper we present a simple scanning confocal microscope experiment, ideal for  the undergraduate optics or advanced instructional lab environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Rather than being  designed to minimize its size or acquire images on par with commercial confocal micro-  scopes, our instructional setup is designed to clearly demonstrate the physics involved in  the image acquisition method and to make the optical components easy to manipulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Our  setup provides an excellent system to illustrate scanning confocal microscopy and also gives  experience with the calibration of a data acquisition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Our system is designed to be versatile, with potential to develop student experimental  skills in the areas optical systems, electronics, and 3D printing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Projects can be adapted to  suit the context of an advanced lab setting, an optics or electronics course, an independent  student project, or a senior thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' In the following sections, we outline an experimental  process that can be used as a whole or broken up into parts, depending on the pedagogical  purposes of the instructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' In the first experiment, a microscope slide is used to illustrate the  ability of a confocal setup to acquire depth information in a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' A second experiment  involves using this setup to investigate height variations of a solid material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Finally, either of  2  these experiments can incorporate a 3D printed translation stage,10 which allows integration  of programming and electronic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  DESIGN  100 mm lens  500 mm lens  150 mm lens  Objective lens  Mirror  Beam  Splitter  650 nm  laser  Photodiode  Sample on  Translation Stage  Iris  Iris  X  Y  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The experimental setup for the confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' A generic visible laser is used, and  expanded to fill a microscope objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The objective lens focuses the light onto a sample,  which is mounted on a translation stage at the focus of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The light reflected from the  sample is separated by a beam splitter, and focused onto an image plane, where an iris limits light  to that which came from the focal plane of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This intensity is then monitored by a  photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Our confocal microscope experimental setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The essential elements  of this system are readily available commercial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' We use a simple 650 nm laser  diode, with a optical power of 5 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This laser is expanded with a two-lens beam expander  to a beam waist of 5 mm waist to fill a commercial microscope objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The beam  3  expander is constructed so that the distance between the lenses is the sum of the focal  lengths, which increases the beam size by a factor of f2/f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' An iris is placed at the focus  of the beam expander to make the beam more circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Once the beam is collimated, the  distance between the expander and the objective, including the presence of a steering mirror,  is chosen for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The sample is placed on a translation stage, which can be either  a commercially purchased or a 3D-printed stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The objective focuses the light onto the  sample, where a portion is reflected back through the objective and a beamsplitter redirects  the light towards a photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The parts required for construction of the confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Part  Specs  Supplier and part number Price (US$)  Laser Diode  650 nm, 5 mW  Adafuit 1054  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='00  Diode Holder  Thorlabs RA90  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='30  Feet (10)  1′′ x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='3′′ x 3/8′′  Thorlabs BA1S (5 PACK)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='20  Lens Holder (7)  Ø1′′ Optics, 8-32 Tap  Thorlabs LMR1  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='83  Mirror Mount  Kinematic Mount for Ø1′′  Thorlabs KM100  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='86  Mirror  Ø1′′ 400 - 750 nm  Thorlabs BB1-E02  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='35  Objective Lens  10X infinite conjugate  Amscope PL10X-INF-V300  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='99  Thread Adaptor  SM1 to RMS  Thorlabs SM1A3  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='50  Posts (10)  2′′  Thorlabs TR2-P5  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='52  Post Holders (10)  2′′  Thorlabs PH2-P5  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='28  150 mm Lens  Ø1′′, AR Coating: 650–1050 nm  Thorlabs LB1437-B  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='50  500 mm Lens  Ø1′′, AR Coating: 650–1050 nm  Thorlabs LA1908-B  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='01  100 mm Lens  Ø1′′, AR Coating: 650–1050 nm  Thorlabs LA1509-B  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='39  Beamsplitter  Economy Ø1′′ 50:50  Thorlabs EBS1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='78  Ring Actuated Iris (2)  (Ø0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='8 - Ø12 mm)  Thorlabs SM1D12D  145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='86  Photodetector  350 - 1100 nm  Thorlabs DET36A2  134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='20  Ultralight Breadboard  24′′ x 24′′ x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='98′′  Thorlabs PBG2424F  842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='93  3-Axis RollerBlock  Long-Travel Bearing Stage  Thorlabs RB13M  1,566.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='15  Total  3,340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='65  4  If the sample is at the focal point of the objective, the light coming back through the  system will be collimated, and a final imaging lens will focus it through an iris before the  photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Hence, if the sample is at exactly the focal length away from the objective,  the image plane will be at the iris location, and a large portion of the light will make it to  the photodetector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' However, if the sample’s surface is not at the focal plane, the amount of  light at the photodetector will be greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Essential to the operation of the confocal microscope is the fact that the iris is in the  image plane at the light focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Thus, it effectively images only the part of the sample  at the focal plane of the objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Unlike traditional microscopes, this allows depth  information in a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  The apparatus described here is a cost-effective means for demonstrating these ideas in  an undergraduate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' A full list of the parts needed can be seen in Table I, where it  is assumed power supplies and an oscilloscope are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The inclusion of the mirror  and lens focal lengths were chosen by convenience with an available breadboard, and can  be modified as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The total cost of the equipment comes to US $3,340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' However,  the largest cost is the 3-axis translation stage, which can be replaced with a 3D-printed  flexure stage,10,11 reducing the cost to under US $2,000 and opening up new opportunities  for development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  This setup can be used to examine several simple objects, which provide excellent insight  into the confocal imaging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' A deeper understanding of depth information from the  microscope can come from examining standard microscope slides, especially those with con-  cave sample openings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The slide can be mounted on the translation stage in a number of  ways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' in our experiment, it was clipped to a 3D-printed stand, such that the laser passed  through the slide with nothing behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The system can also be used to examine sur-  face variation of solid objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' We have found that 3D-printed objects with known height  variation works well, but a variety of other objects can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  RESULTS AND ANALYSIS  In order to demonstrate the capabilities of the confocal microscope, two experiments are  outlined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The first involves observing the thickness of semitransparent materials, with  commercial microscope slides offering an excellent starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Using a microscope slide  5  to demonstrate how the confocal microscope works offers students the chance to determine  depth information on a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  As a starting point, a microscope slide with a concave  sample opening can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' In this case two reflections occur, one off of the slide’s front  surface and the other off of its back surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  As a result, adjusting the position of the  sample longitudinally along the direction of laser propagation allows the observation of  two successive peaks, as each of the slide’s front surface and back surface is in the focal  plane of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' An example of the data observed as the slide platform is adjusted  longitudinally (in the x-direction) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Opening the iris at the image plane  makes clear the way in which the confocal microscope causes a specific image plane for  different depths in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Without this iris, depth information is lost, as can be seen  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  Photodiode Signal (V)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  X axis position (mm)  Slide Edge  Slide Center  Without Iris  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' As the distance between the objective lens and the slide is varied, intensity peaks are seen  when either the front or the back surface of the slide is at the focal plane of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This is  shown both at the flat edge of the slide, and at the curved center where the slide is not as thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  The shifting of the second peak reveals the changing thickness of the slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Removing the iris in  the imaging plane reveals the necessity of limiting the light from a single image plane for depth  perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  By using a microscope slide with a concave sample opening, the thickness of the glass  6  can be observed as a function of the position along the slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' As a starting point, this can  be done by moving the slide a small distance transversely (in the y-direction), then scanning  longitudinally (in the x-direction) to observe the two peaks, stepping through positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' An  example of the data obtained with this two axis scan technique is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4  Position of highest reflection (mm)  10  8  6  4  2  0  Distance along slide (mm)  Two Axis Scan  One Axis Scan  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The distance from the objective where peak reflected light is observed as a function of  distance along the slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The region where the slide is flat is visible, before the concave opening  begins at 4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Then the changing thickness where the concave sample opening begins can be  observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The two-axis scan involves moving the translation stage transversely a small amount  before scanning longitudinally to find the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The one-axis method involves only scanning  transversely, using a known calibration of the photodiode voltage with depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  This setup can also be used to demonstrate the usefulness of a scanning confocal micro-  scope by acquiring the same depth information, while only moving the slide transversely to  the laser beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This makes the image collection process much faster, but requires calibrat-  ing the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' As the slide is shifted transversely in the area where the depth varies, the  di��erent depths will cause different voltages in the photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' To correlate these voltage  changes to height changes, a fit for the voltage as a function of height in the region scanned  is found from the longitudinal scan, such as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  The shape of the intensity peak is caused by the changing beam waist as it propagates  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  Voltage (V)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  Distance (mm)  Linear Fit  Range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='066 mm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  Voltage (V)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  Distance (mm)  Range: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='21 mm  Lorentzian Fit  (a)  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  The intensity peak obtained as the slide is translated longitudinally can be used to provide  a voltage vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' height calibration for the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' An appropriate fitting function can be chosen to  fit the peak over a particular height variation region on a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' a) A linear fit to a portion of  the peak, providing a simple calibration valid over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='066 mm around the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' b) A Lorentzian  fit valid over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='21 mm range around the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  through the image plane, with the intensity limited by the iris size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  This is a complex  dependence, but can be roughly fit using a Lorentzian over much of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' For a more  precise calibration, it is desired to find a function that fits the intensity peak very well  over a particular range of heights of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Using this calibration, voltage changes  when scanning transversely can be used to calculate height changes in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The  simplest calibration is to use a portion of the graph that can be well approximated as  linear, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' However, this linear fit is only valid over a limited height  variation, less than 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Over a larger height variation, an exponential or polynomial  fit may also work well, where we illustrate a Lorentzian fit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Allowing students  to determine this calibration technique provides additional insight and experience in data  analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Taking the data again for the slide thickness using a transverse scan  calibrated with the Lorentzian fit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 3, and agrees with that taken when  collecting points individually with motion in both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Further, this process allows  students to understand how data can be collected and analyzed quickly in this ‘scanning’  configuration of a confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  The ability of the confocal microscope to determine height variations of a solid sample  8  represents another key category of experiment that can be undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' To illustrate this  point, stereolithography (SLA) 3D-printed objects of known height variation are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' SLA  3D printing, like fused deposition modeling (FDM) 3D printing, creates a model layer by  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Instead of extruding filament, a laser polymerizes resin at a specific point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  This  process allows for a quantifiable amount of material to be deposited with a known volumetric  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This property was exploited to create steps, with a known thickness, on the  surface of a 3D-printed block small enough to demonstrate the accuracy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Variations on this experiment could be done with a variety of objects, as long as the object  has height variation on the scale of 100s of microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  With these solid objects, there are no longer multiple intensity peaks from different  depths of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Instead, the location of the surface can be seen by the maximum  of the intensity peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Here, the surface height variation can be investigated either by  successively stepping transversely and longitudinally, or by scanning only transversely using  a calibrated intensity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' height graph, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' For our experiment two different  surface variations were printed, one that increased 50 µm in height for each 200 µm shift  in position, and another that increased 100 µm in height for each 200 µm shift in position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Figure 5 shows the measured surface location as a function of the distance along the sample  for each of these materials using the two-axis scan technique, and illustrates the ability  of the system to determine solid surface height variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' The precision of 3D printers  can limit the resolution of features created using this method, but it is expected that the  cost of high-resolution 3D printers will continue to decrease, allowing more intricate surface  patterns to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Additional solid objects such as coins can also be examined, but  the high reflectivity and angled surfaces can cause complex behavior in the signal intensity,  and extra care is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  INCORPORATION OF 3D-PRINTED TRANSLATION STAGE  3D printing has developed into an extremely valuable tool for the undergraduate lab, and  has been incorporated into a number of experimental designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='12–14 Due to recent advances  and open-source development, it is now possible to print and program excellent open-source  translation stages,10,11 and 3D printing has been incorporated into a variety of microscope  designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='15–20 In the confocal microscopy setup presented here, 3D printing presents the pos-  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='8  Material Height (mm)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='5  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='0  Distance along printed sample (mm)  100 micron step  50 micron step  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  The height of a solid 3D-printed object, as measured with the confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This  object was designed and printed to show the capabilities of the microscope, and a number of similar  objects can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  sibility of replacing a commercial translation stage system with a homebuilt option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This  replacement serves two main purposes: First, it can reduce the cost of the necessary equip-  ment by a significant fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Second, it allows meaningful student experience in design,  3D printing, electronics, and programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This allows the confocal microscope project to  be quite general, appealing to students interested in a wide range of engineering or data  acquisition fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  One option that incorporates 3-axis translation and excellent precision is the OpenFlexure  Block stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='11 This setup is well documented elsewhere, and here was incorporated with  little modification to the design presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This stage was specified to have a step size in  the x direction of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='2 nm and translate 2 mm on any axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' While this translation  distance is not large enough to scan over large portions of the slide, it does allow the same  experimentation on slide thickness described above, with the data taken again as observed in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Having both the data for slide thickness taken from the commercial translations stage  and the OpenFluxure Block stage allows an additional means of verifying the calibration for  10  distance per step on the block stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Using these data, our calibration was determined to be  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='4 nm, consistent with previous measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='11 The use of 3D-printed translation  stages makes the scanning confocal microscope even more accessible, and would combine  well with projects commonly undertaken in an undergraduate lab environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  CONCLUSIONS AND FUTURE DIRECTIONS  The design for a homebuilt scanning confocal microscope presented here is ideal for an un-  dergraduate instructional laboratory setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Through the measurement of microscope slide  thicknesses and surface variations, undergraduates develop a better sense of the methods of  scanning confocal microscopy and the information it provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This design provides a robust  setup with understandable mechanisms at a low price that enables easy implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  Our setup is intended for pedagogical purposes rather than quality imaging competitive  with commercial confocal microscopes, but can be extended for more complete imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' If  desired, the OpenFlexure stage can be automated, allowing for expansion of this project  into acquiring scanned three dimensional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' This is especially appealing in the context  of an independent project or senior thesis looking to incorporate additional programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  An investigation of the resolution of the system2 can also be employed as an extension to  the experiments presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Investigating the limiting resolution would require materials  with much finer features, as with this wavelength and numerical aperture lens the theoretical  lateral resolution is expected to be near 1 µm, and the axial resolution on order 15 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Here  several factors could be investigated to determine maximum resolution, including beam size,  the minimal size and location of the iris, and the numerical aperture of the objective lens  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  The confocal microscope system described in this paper can easily be expanded, either to  improve its quality or examine further applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' Enhancements include the addition of an  additional beamsplitter and camera, the use of achromatic doublets to decrease aberrations,  improving the laser quality through fiber-coupling before use, or a galvo-scanning mirror in  place of the x-y translation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content=' In addition, the experiments outlined here can easily be  extended to simple biological samples or fluorescence microscopy, allowing further insight  into the way the optical system is commonly applied in biomedical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  11  ACKNOWLEDGMENTS  We wish to acknowledge the contributions of Joseph Coonen and the programming insight  of Michael Olson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='  ∗ erik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='brekke@snc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
+page_content='edu  1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
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+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE_T4oBgHgl3EQf0Bwx/content/2301.08326v1.pdf'}
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dNAzT4oBgHgl3EQfnv3m/content/tmp_files/2301.01587v1.pdf.txt

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+Condensed Matter Physics, 2022, Vol. 25, No. 4, 43704: 1–11
+DOI: 10.5488/CMP.25.43704
+http://www.icmp.lviv.ua/journal
+Temperature dependence of dielectric permittivity in
+incommensurately modulated phase of ammonium
+fluoroberyllate
+B. I. Horon
+1,2∗, O. S. Kushnir
+2, P. A. Shchepanskyi
+1, V. Yo. Stadnyk
+1
+1 General Physics Department, Ivan Franko National University of Lviv, 23 Drahomanov Street, 79005 Lviv,
+Ukraine
+2 Optoelectronics and Information Technologies Department, Ivan Franko National University of Lviv,
+107 gen. Tarnavskyi Street, 79013 Lviv, Ukraine
+Received July 15, 2022, in final form September 28, 2022
+We study the temperature dependence of dielectric permittivity along the polar axis for ferroelectric ammonium
+fluoroberyllate (AFB) crystal in the vicinity of its phase transition points. The experimental data within incom-
+mensurately modulated phase of AFB is compared with the predictions of phenomenological models known
+from the literature: the Curie-Weiss (CW) law, the generalized Curie-Weiss (GCW) law, and the models by Le-
+vanyuk and Sannikov (LS) and by Prelovšek, Levstik and Filipič (PLF) suggested for improper ferroelectrics. It is
+shown that the LS approach describes the temperature behavior of the dielectric permittivity for the AFB crystal
+better than the CW, GWC and PLF models. The main physical reasons of this situation are elucidated.
+Key words: phase transitions, incommensurate phases, improper ferroelectrics, dielectric permittivity,
+ammonium fluoroberyllate
+1. Introduction
+Ammonium fluoroberyllate (NH4)2BeF4 (or AFB) is an improper ferroelectric crystal that belongs
+to a large A2BX4 family. It undergoes two phase transitions (PTs) approximately at the temperatures
+𝑇C ≈ 177 K and 𝑇i ≈ 183 K [1–6], which separate a low-temperature ferroelectric phase, an intermediate
+incommensurate phase and a high-temperature paraelectric phase. Although the AFB crystals have been
+thoroughly studied during decades (see, e.g., [6–11]), some problems of their PTs and critical phenomena
+still remain a matter of dispute.
+In particular, AFB reveals an intriguing temperature dependence of its dielectric permittivity: unlike
+the optical birefringence and many other characteristics, dielectric anomaly at 𝑇i is in fact absent, while
+the 𝑇C point is marked by only a weak dielectric peak [2–5, 12]. The dielectric properties of AFB have
+been the main subject of theoretical studies by Levanyuk and Sannikov [3] and by Prelovšek, Levstik
+and Filipič [5] (abbreviated respectively as LS and PLF), which are both based upon the hypothesis of
+improper ferroelectricity in AFB. In spite of this fact, the final expressions obtained in [3, 5] turn out to
+be different in many respects.
+The other notable fact is that there has been no study where an experimental temperature dependence
+of the dielectric permittivity for the AFB crystals would be simultaneously compared with different
+theoretical formulae in order to estimate the advantages and shortcomings of the latter. The only exception,
+our recent work [13], represents a short technical report based upon contemporary methods of nonlinear
+fitting and statistical techniques (see the works [14, 15]). Although a number of weak methodical points
+∗Corresponding author: [email protected]
+This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution
+of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
+43704-1
+arXiv:2301.01587v1  [cond-mat.mtrl-sci]  4 Jan 2023
+
+B. I. Horon, O. S. Kushnir, P. A. Shchepanskyi, V. Yo. Stadnyk
+−5
+0
+5
+10
+15
+20
+T
+−
+T
+C
+, K
+10
+30
+50
+ε
+T
+i
+Curie-W
+eiss
+power law
+a
+−5
+0
+5
+10
+15
+20
+T
+−
+T
+C
+, K
+10
+30
+50
+ε
+T
+i
+LS
+PLF
+b
+Figure 1. (Colour online) Experimental temperature dependence 𝜀(𝑇) of dielectric permittivity for the
+AFB crystals (circles) and its fitting (lines) with the theoretical models (1), (2) (panel a) and (3), (4)
+(panel b) within the incommensurate phase.
+associated with fitting [16–18] are omitted in this work, no physical reasoning and data interpretation
+have been made there.
+In the present study we compare all of the available phenomenological approaches which can, in
+principle, be applied to describe the dielectric properties of the AFB crystals and explain why the LS
+theory [3] exceeds the performance of other approaches and perfectly fits the experimental data for
+dielectric permittivity.
+2. Experimental data and short description of theoretical models
+A single crystal of AFB for our studies was grown from aqueous solution of a stoichiometric mixture
+of NH4 and BeF2, using a standard method of slow cooling. The dielectric permittivity was measured
+along the polar axis with an automated capacitive apparatus (the temperature region 170–200 K, the
+tolerance of temperature measurement ∼0.1 K, and the working frequency 1 kHz). Figure 1 displays the
+experimental temperature dependence of the dielectric permittivity for the AFB crystals. As seen from
+figure 1, no anomaly is visible at 𝑇i, in compliance with the main bulk of experimental data known from
+the literature.
+Note also that the maximum dielectric permittivity detected by us (𝜀max ≈ 55) correlates well with the
+data obtained in the earlier measurements for the improper AFB crystals (𝜀max ≈ 35–160 [1–5, 12, 19]).
+This is contrary to the proper ferroelectrics where the values 𝜀max ∼ 103 − 105 are often detected
+(see [10]). Such a small 𝜀max peak can indeed be successfully interpreted using the idea that the dielectric
+anomaly in improper ferroelectrics is a secondary effect while a true order parameter has the symmetry
+different from that of spontaneous electric polarization. For the same or somewhat different reasons,
+weak dielectric anomalies are also typical of ferroelastics [20], multiferroics [21] and ferroelectrics with
+noticeable amounts of structural defects [14, 15, 22].
+Now we proceed to phenomenological consideration of the dielectric properties of AFB. Since the
+both LS [3] and PLF [5] models have not dealt with the 𝜀(𝑇) function within the ferroelectric phase, we
+analyze the dielectric data only within the incommensurate and paraelectric phases. Another important
+point is a so-called ‘background’ dielectric permittivity 𝜀b which can be independent of the PTs. The
+problem of the background versus the PT-driven anomaly is familiar in examining the specific heat
+of ferroics, since the appropriate anomaly is often comparable with the lattice contributions (see, e.g.,
+[23, 24]). However, this is not so in the field of dielectric studies of proper ferroelectrics where huge
+anomalous peaks are mostly observed, so that neglecting the background would not hinder the possibility
+of obtaining highly accurate fitting data. As a consequence, even a constant background term 𝜀b has
+rarely been considered in the dielectric studies of ferroelectrics, not to mention a temperature dependent
+background 𝜀b(𝑇). Still, a few relevant exceptions are known from the literature [14, 25]. Of course,
+consideration of 𝜀b can become very important when improper ferroelectrics like AFB are addressed.
+43704-2
+
+Temperature dependence of dielectric permittivity in incommensurately modulated phase of ammonium fluoroberyllate
+Finally, the both LS and PLF models [3][5] treat the dielectric function as temperature-independent
+within the paraelectric phase and this is consistent with our both experimental results and the whole bulk
+of the literature data (especially with that obtained in a broad temperature range [20]). Therefore, we
+restrict ourselves to the simplest assumption 𝜀b = const.
+Next, the dielectric permittivity within the incommensurate phase can, in principle, be described by
+one of the following theoretical models.
+Model (1). A canonical Curie-Weiss law with a constant background 𝜀b and a Curie constant 𝐶CW:
+𝜀(𝑇) = 𝜀b + 𝐶CW
+𝑇 − 𝑇C
+.
+(2.1)
+Model (2). A power law representing generalization of the Curie-Weiss formula (2.1), with the
+exponent 𝛾 > 1:
+𝜀(𝑇) = 𝜀b +
+𝐶𝛾
+(𝑇 − 𝑇C)𝛾 .
+(2.2)
+Model (3). The LS model [3] for the incommensurate phase which in fact states that
+𝜀(𝑇) = 𝜀b + 𝜀2
+b𝐴 𝑡 6 + 𝑡
+4 − 𝑡 ,
+(2.3)
+where 𝐴 is a constant and 𝑡 implies the reduced temperature:
+𝑡 = 𝑇i − 𝑇
+𝑇i − 𝜃 ,
+with 𝜃 (𝑇C < 𝜃 < 𝑇i) being an instability point for the order parameter. It can be defined in terms of the
+distance Δ𝑇 from the 𝑇C point (𝜃 = 𝑇C + Δ𝑇). Note that Δ𝑇 can be expressed in terms of the free-energy
+expansion [3] (see section 4).
+According to formula (2.3), the 𝜀(𝑇) function diverges at 𝑡 = 4 and tends to 𝜀 = 𝜀b at 𝑇 = 𝑇i. Since
+the model predicts the same constant value 𝜀 = 𝜀b in the paraelectric phase, the dielectric function
+is continuous at the incommensurate–paraelectric PT, while the slope of 𝜀(𝑇) curve suffers an abrupt
+change at 𝑇i. Although the authors [3] themselves have defined the applicability limits of formula (2.3)
+as a narrow temperature region below the 𝑇i point (e.g., as a region of small 𝑡 in terms adopted in this
+work), we have checked the model (3) in the overall range between the temperatures 𝑇C and 𝑇i.
+Model (4). The PLF model [5] for the incommensurate phase:
+𝜀(𝑇) = 𝜀b + 𝜀b
+𝑐
+�
+𝐸(𝜏)
+(1 − 𝜏2)𝐾(𝜏) − 1
+�
+,
+(2.4)
+where 𝑐 is a constant, while 𝐾(𝜏) and 𝐸(𝜏) denote the complete elliptic integrals of the first and second
+kinds, respectively. The authors [5] have not linked the elliptic modulus 𝜏 with the PT parameters.
+Nonetheless, the relation 𝜏 = 𝑇i−𝑇
+𝑇i−𝑇C can be postulated due to the properties 𝜏 → 0 and 𝜏 → 1 holding
+respectively at 𝑇 → 𝑇i and 𝑇 → 𝑇C [5]. According to the PLF approach [5], formula (2.4) can be assumed
+to be applicable within the entire incommensurate phase. A criticality at 𝑇C in formula (2.4) is due to the
+behaviour of the terms (1 − 𝜏2) and 𝐾(𝜏) at 𝜏 → 1. Since the equality 𝐾(𝜏) = 𝐸(𝜏) takes place at 𝜏 = 0,
+we have 𝜀 = 𝜀b at 𝑇 = 𝑇i. Finally, the 𝑇i point, which corresponds to the anomaly found experimentally
+in the specific heat, is hardly detectable in the dielectric permittivity. Similarly to the LS model, the
+only track of this PT is a change in the 𝜀(𝑇) slope, which can be detected in a smoothed temperature
+dependence d𝜀(𝑇)/d𝑇 (not shown in figure 1).
+It is worthwhile that, contrary to the models (1) and (2), the temperature-independent background 𝜀b
+is introduced into the formulae (2.3) and (2.4) directly from the expansion of free energy Φ (in fact, via
+the relation Φ ∼ 𝑃2/(2𝜀b), with 𝑃 being the electric polarization [3] — see section 4). Notice also that,
+in case of AFB, we have evidently different experimental background levels found in the paraelectric and
+ferroelectric phases (see figure 1). Finally, theoretical considerations [3] testify that it is unnecessary to
+retain any temperature-dependent 𝜀b(𝑇) terms.
+43704-3
+
+B. I. Horon, O. S. Kushnir, P. A. Shchepanskyi, V. Yo. Stadnyk
+3. Fitting the results and their discussion
+Now we fit our experimental data 𝜀(𝑇) using the phenomenological models (1)–(4), determine the
+best model and explain in detail the practical advantages and disadvantages of those models. Procedures of
+nonlinear fitting are implemented according to a standard Levenberg–Marquardt algorithm. A goodness-
+of-fit is evaluated with 𝜒2 and Wald–Wolfowitz statistical tests. Finally, the error margins for the model
+parameters are found with a bootstrap technique, using 2000 synthetic datasets. The appropriate details
+are elucidated elsewhere [13]. Figure 1 illustrates the fitting results and table 1 displays a short account
+of the main model parameters.
+Table 1. Some of fitting parameters of the 𝜀(𝑇) dependence corresponding to phenomenological mod-
+els (1)–(4).
+Model
+Results
+Parameter
+Value
+(1)
+𝐶CW
+0.179
+(2)
+𝐶𝛾
+7.392
+𝛾
+0.326
+(3)
+𝐴
+0.0094
+Δ𝑇, K
+4.01
+(4)
+𝑐
+30.373
+The Curie-Weiss law underestimates the experimental 𝜀(𝑇) curve at the temperatures more or less
+distant from the PT (figure 1a) and so fails to appropriately fit the dielectric permittivity. Moreover, the
+Curie-Weiss fit reveals too large Z-score (see table 2). The PLF model (4) has the characteristics similar to
+those of the Curie-Weiss law (see figure 1b and table 2). The other pattern takes place for the model (2),
+which corresponds to generalized power-law for the temperature dependence 𝜀(𝑇). Here, the fitting
+function overestimates most of the experimental data point values and fails to catch the background (see
+figure 1a), whereas the Z-score is just as large as that for the other models mentioned above. Moreover,
+the model provides a 𝛾 value noticeably less than unity (see table 1), which is a physically shallow result.
+Hence, the generalized power law for the dielectric permittivity of AFB is also insufficient.
+Table 2. Results of 𝜒2 and Wald–Wolfowitz tests for phenomenological models (1)–(4).
+Model
+Statistical Tests
+Parameter
+Value
+(1)
+𝜒2
+5252.33
+Reduced 𝜒2
+165.22
+Z-score
+−3.45
+(2)
+𝜒2
+1525.25
+Reduced 𝜒2
+47.66
+Z-score
+−4.85
+Correlation(𝐶𝛾, 𝛾)
+−0.87
+(3)
+𝜒2
+320.87
+Reduced 𝜒2
+9.72
+Z-score
+0.34
+Correlation(𝐴, Δ𝑇)
+−0.95
+(4)
+𝜒2
+2189.92
+Reduced 𝜒2
+72.99
+Z-score
+−4.39
+43704-4
+
+Temperature dependence of dielectric permittivity in incommensurately modulated phase of ammonium fluoroberyllate
+0
+1
+2
+3
+4
+5
+T
+−
+T
+C
+, K
+0
+10
+20
+30
+1
+ε
+−
+ε
+b
+PLF
+LS
+CW
+a
+10
+−2
+10
+−1
+10
+0
+T
+−
+T
+C
+, K
+10
+−1
+10
+0
+10
+1
+ε
+−
+ε
+b
+PLF
+LS
+CW
+b
+Figure 2. Temperature dependences of reciprocal dielectric permittivity (a) and log-log plots of dielectric
+permittivity (b). Circles correspond to experimental data and lines correspond to different theoretical
+models: Curie-Weiss (CW), LS and PLF (see the legend). In the both cases, the background term 𝜀𝑏 is
+extracted from the experimental data.
+On the contrary, the theoretical curve referred to the LS model (3) fits fairly well the experimental
+data, and the appropriate statistical tests provide quite satisfactory results (see figure 1b, table 1 and
+table 2). Moreover, it becomes evident that the model (3) can in fact be applied in the entire temperature
+range under study, contrary to the cautions of the authors [3]. Finally, the term 𝜀b = 7.12 found from
+the LS fitting (not shown in table 1) turns out to be very close to the experimental dielectric background
+averaged over the paraelectric phase. In other words, the LS phenomenology obviously exceeds the
+performance of the other models. For completeness, we list the PT points derived with the model (3),
+which are not displayed in table 1 for the sake of brevity: 𝑇C = 177.64 K (found from the dielectric peak),
+𝑇i = 183.19 K and 𝜃 = 181.65 K. Finally, the confidence intervals for the model parameters 𝐴 and Δ𝑇
+are given respectively by −0.0053–0.0272 and 3.668–4.366 (cf. with the data of table 1).
+Now, we wish to clarify more scrupulously why, contrary to the model (3), the models (1), (2) and (4)
+agree worse with the experimental results for the AFB crystals. For this purpose, we display both the
+experimental and theoretical data 𝜀(𝑇) either in the ‘Curie-Weiss’ coordinates (𝜀−𝜀b)−1 vs. (𝑇 −𝑇C) (see
+figure 2a) or on the double logarithmic scale log(𝜀 − 𝜀b) vs. log(𝑇 − 𝑇C) (see figure 2b). To prevent the
+overloading of these figures, we do not show the data obtained with the generalized power law (2.2). The
+results for this model differ only by insignificant details from those illustrated in figure 2a and figure 2b
+for the models (1) and (4).
+Although the data of figure 2 can hardly be used for a thorough quantitative interpretation (see the
+discussion in section 2 and [16–18]), it illustrates well the main practical tendencies for the above models.
+The Curie-Weiss law for the dielectric permittivity implies a straight line in the (𝜀 − 𝜀b)−1 vs. (𝑇 − 𝑇C)
+coordinates and a straight line with the slope −1 on the log-log scale (see the dot lines in figure 2a and
+figure 2b). A close examination of the behavior of theoretical PLF function 2.4 involving the elliptical
+integrals testifies that there exists a temperature region where the model (4) can be approximately reduced
+to the inverse power law, i.e., the Curie-Weiss relation. Namely, this is an intermediate region above the PT
+temperature 𝑇C given by 𝑇 −𝑇C ≈ 10−1–100 K (see a dashed line in figure 2a and, especially, in figure 2b).
+It corresponds to moderately large reduced temperatures 𝜏 (𝜏 ≈ 0.82–0.98). Note that formula (2.4) was
+used in the work [26] for interpretation of the dielectric properties of incommensurate Rb2ZnCl4 crystals,
+and the authors [26] actually confirmed that, in the region of intermediate relative temperatures (𝑇 −𝑇C),
+the relation (2.4) yields the results very close to the Curie-Weiss law.
+At the temperatures more distant from 𝑇C (i.e., in the region defined by the inequality 𝑇 − 𝑇C > 1 K,
+or at 𝜏 ≈ 0–0.82), one can observe severe deviations of the PLF model from the inverse power law (see
+figure 2a). Finally, the predictions of the PLF model become progressively different from those of the
+Curie-Weiss law also in the region given by 𝑇 − 𝑇C < 0.1 K, i.e. at 𝜏 > 0.98 (see figure 2b). Eventually,
+both the Curie-Weiss and the PLF models do not claim to consider the fluctuation corrections in the
+critical region since they correspond to the mean-field theory, which is inapplicable in the closest vicinity
+of the PT points.
+43704-5
+
+B. I. Horon, O. S. Kushnir, P. A. Shchepanskyi, V. Yo. Stadnyk
+Now that we have ascertained the main formal differences among the theoretical models, we are in
+a position to compare better the experiment with all the theoretical predictions. As seen from figure 2a,
+the experimental dependence 𝜀(𝑇) without the background deviates significantly from the Curie-Weiss
+law and, moreover, from any other inverse power law. The latter fact is also evident from figure 2b, since
+the slope of the experimental curve on the log-log scale changes continuously. Of course, the PLF model
+predicting nearly inverse power law would fail in describing such data. On the contrary, the LS model
+(see dash-dot lines in figure 2) is governed by a combination of terms linear in temperature, which enter
+both the numerator and the denominator of formula (2.3). This mathematical structure provides a gradual
+change in the slope of the LS curve in both figure 2a and figure 2b and so satisfactorily describes the
+experimental data.
+4. Comparison of LS and PLF models for the dielectric permittivity
+The fact that the LS model describes better the dielectric permittivity of the AFB crystals than
+the PLF model is unexpected and even counter-intuitive. Indeed, the LS approach looks simpler than
+the PLF model [5] which was developed later, the authors [5] may have been familiar with the LS
+results [3] and, moreover, they supposed their model to be applicable in a wider temperature region of the
+incommensurate phase, compared with the LS model. To understand better these problems, we elucidate
+in brief the main differences in the physical assumptions underlying the two models. To do that, we
+outline the main points of derivation of formulae (2.3) and (2.4).
+The authors [3, 5] started from the same free-energy expansion in the framework of the mean-field
+theory. In polar coordinates, it can be written as follows:
+Φ = 𝛼
+2 𝜌2 + 𝛽1
+4 𝜌4 + 𝛽2
+4 𝜌4 cos 4𝜑 − 𝜎𝜌2 d𝜑
+d𝑧 + 𝛿
+2
+��d𝜌
+d𝑧
+�2
++ 𝜌2
+�d𝜑
+d𝑧
+�2�
+− 𝐸𝑃 + 𝜅
+2𝑃2 + 𝑎𝜌2𝑃 cos 2𝜑.
+(4.1)
+Here, 𝛼 = 𝛼′(𝑇 − 𝜃), 𝜌 and 𝜑 are respectively the amplitude and the phase of the order parameter,
+𝐸 is the electric field, 𝑃 is the polarization, 𝜃 is the temperature point of structural instability, and
+𝛼′, 𝛽1, 𝛽2, 𝜎, 𝛿, 𝜅 and 𝑎 denote temperature-independent constants (see also section 2). The most
+fundamental fact is the availability in formula (4.1) of a Lifshitz invariant proportional to 𝜎, which is
+symmetry-allowed for incommensurately modulated media. Due to nonzero 𝜎 term in (4.1), the initial
+high-temperature phase does not lose its stability under the condition 𝛼 = 0 (i.e., at 𝑇 = 𝜃). This occurs
+with respect to inhomogeneous displacements at some other value 𝛼 = 𝛼0 = 𝜎2/𝛿 corresponding to a
+higher paraelectric–incommensurate temperature 𝑇i.
+Note that the 𝛽2 term in formula (4.1), which is associated with spatial anisotropy of the order
+parameter, notably complicates the problem of finding a steady-state solution for the free energy. Since
+the above approach is focused first of all on some (small but not too small) vicinity of 𝑇i, as is always
+the case with the mean-field theory, strongly anisotropic higher-order terms in the order parameter are
+omitted in (4.1). They drive a lock-in commensurate PT at 𝑇C rather than the incommensurate PT at 𝑇i
+(see, e.g., the work [26]).
+Probably, the main difference in the LS and PLF models [3, 5] is the approaches adopted by the
+authors to solve the system of Lagrange–Euler equations
+𝜕Φ
+𝜕𝜌 − d
+d𝑧
+𝜕Φ
+𝜕𝜌′ = 0,
+(4.2)
+𝜕Φ
+𝜕𝜑 − d
+d𝑧
+𝜕Φ
+𝜕𝜑′ = 0,
+(4.3)
+where 𝜌′ =
+d𝜌
+d𝑧 and 𝜑′ =
+d𝜑
+d𝑧 . This system of equations allows one to find the values of 𝜌 and 𝜑
+corresponding to stationary state. Following the work [27], PLF use a constant-amplitude approximation,
+43704-6
+
+Temperature dependence of dielectric permittivity in incommensurately modulated phase of ammonium fluoroberyllate
+according to which the amplitude of the order parameter does not depend on the coordinates [𝜌(𝑧) = 𝜌0]:
+𝜌2
+0 = 𝛼′
+𝛽1
+(𝑇i − 𝑇).
+(4.4)
+This enables the authors [5] to come to something like a time-independent sine-Gordon equation,
+d2𝜑
+d𝑧2 = 𝛽2
+4𝛿 𝜌2
+0 sin 4𝜑.
+(4.5)
+The explicit solution of this equation is found for the temperature behavior of the phase 𝜑 (see also [28–
+30]:
+2𝜑 = am(2𝑞𝑧, 𝜖),
+(4.6)
+with 𝑞2 = 2𝛽2𝜌2
+0/(𝛿𝜖2), 𝜖2 = 2𝛽2𝜌4
+0/(𝐶 + 𝛽2𝜌4
+0) and 𝐶 being an integration constant. Here, am(2𝑞𝑧, 𝜖)
+represents the Jacobian elliptic function with the modulus 𝜖 (0 ⩽ 𝜖 ⩽ 1). Such an approach would imply
+a full-scale consideration of anisotropy 𝛽2.
+LS treat the same problem in another manner. They completely neglect the anisotropy (𝛽2 = 0) in the
+initial stage so that the equation (4.5) is simplified to d2𝜑/d𝑧2 = 0. This yields a standard formula for the
+plane-wave region of incommensurate phase [31]:
+𝜑 = 𝑘0𝑧,
+(4.7)
+with the wave vector 𝑘0 = |𝜎|/𝛿. On the other hand, in fact LS also start from the constant-amplitude
+approximation (4.4) under the condition 𝛽2 = 0. As a result, their approach seems to be notably simpler
+than that of PLF. However, after that LS take into account higher-order corrections to formula (4.7)
+given by a power series in 𝛽2 (more exactly, in the parameter Δ =
+𝛼′ 𝛿 |𝛽2 |
+𝜎2𝛽1 (𝑇i − 𝑇) =
+|𝛽2 |
+𝛽1 𝑡 ≪ 1), the
+lowest-order of which is proportional to Δ2 [3]. This corresponds to what can be termed as a ‘weak
+anisotropy approximation’, which would eventually affect the final solution for the amplitude, too.
+Since the dielectric permittivity 𝜀 is defined as 𝜀 = d𝑃/d𝐸, we obtain
+𝜀 = 1
+𝜅 − 2𝑎𝜌
+𝜅
+� 𝜕𝜌
+𝜕𝐸 cos 2𝜑 − 𝜕𝜑
+𝜕𝐸 𝜌 sin 𝜑
+�
+.
+(4.8)
+It is obvious that, unlike the PLF model, the phase in (4.7) does not depend on thermal changes in the
+approximation Δ = 0. Taking derivatives in (4.8), we arrive at a temperature-independent expression 𝜀(𝑇)
+which coincides with that obtained for the commensurate phase [3]:
+𝜀com(𝑇) = 1
+𝜅 +
+2𝑎2
+𝜅2(𝛽′
+1 − |𝛽′
+2|) ,
+(4.9)
+with 𝛽′
+1 and 𝛽′
+2 being renormalized coefficients (𝛽′
+1 = 𝛽1 − 2𝑎2/𝜅 and 𝛽′
+2 = 𝛽2 − 2𝑎2/𝜅). Having left a
+zero approximation, one can obtain a more complex expression in some vicinity of the incommensurate
+PT (at Δ ≪ 1) (cf. also with the earlier, less correct formula in the work [32]):
+𝜀LS(𝑇) = 1
+𝜅 +
+𝑎2
+𝜅2𝛽′
+1
+𝑡 6 + 𝑡
+4 − 𝑡 .
+(4.10)
+Formula (4.10) coincides with (2.3) with the notation 𝜀b = 1/𝜅 and 𝐴 = 𝑎2/𝛽′
+1. Finally, formulae (4.4),
+(4.6) and (4.8) obtained in frames of the PLF model result in
+𝜀PLF(𝑇) = 1
+𝜅 +
+𝑎2
+𝜅2𝛽′
+1
+�
+𝐸(𝜏)
+(1 − 𝜏2)𝐾(𝜏) − 1
+�
+,
+(4.11)
+where the substitutions 𝜀b = 1/𝜅 and 𝑐 = 𝜅𝛽′
+1/𝑎2 lead to formula (2.4).
+43704-7
+
+B. I. Horon, O. S. Kushnir, P. A. Shchepanskyi, V. Yo. Stadnyk
+Now we are in a position to compare the physical backgrounds of the LS and PLF models for the
+dielectric properties of AFB. At the first glance, the PLF result (4.6) underlying the formula (4.11) looks
+stronger compared to formula (4.7) obtained by LS, which is limited to a narrow vicinity of paraelectric–
+incommensurate PT. However, as pointed out in the work [30], any phenomenological model like those
+suggested by LS and PLF [3, 5] is anyway applicable only near the PT point𝑇i, where the spatial anisotropy
+is small enough. In some sense, the approximations of constant amplitude and weak anisotropy have close
+applicability regions. Then, the decision of PLF to maintain the exact solution for the phase 𝜑 and, at the
+same time, restrict themselves to the limit 𝜌(𝑧) = 𝜌0 can prove to be partly inconsistent, as if someone
+would exceed the accuracy of a given approximation. Probably, this is the main reason why the PLF
+formula is less accurate in describing the 𝜀(𝑇) function for the AFB crystals.
+On the other hand, the fact that the LS model has turned out to work fairly well in the overall range of
+incommensurate phase can be explained, at least partly, owing to the following circumstance: in terms of a
+variable (𝑇i −𝑇C)/𝑇i characterizing the temperature width of this phase, the latter is very narrow (∼ 0.03).
+Eventually, this factor also degrades a potential advantage of the PLF model associated with consideration
+of the spatial anisotropy, which would have played a more significant part in a wider temperature region.
+In this respect, we suppose that an LS-like model could hardly succeed when describing the dielectric
+properties of Rb2ZnCl4 where the incommensurate phase is very wide ((𝑇i−𝑇C)/𝑇i ∼ 0.36) and, moreover,
+the experimental data in the vicinity of 𝑇i are scarce [26].
+5. Potential influence of fluctuations and structural defects
+It is well known that the incommensurate phases in A2BX4 crystals are highly sensitive to any
+structural imperfections, e.g., due to pinning of the phase of the order parameter [33]. This implies that
+the dielectric permittivity can manifest some dependence on crystal samples or experimental conditions
+(heating or cooling run, temperature change rate, etc.). This poses a question of the potential influence
+of these phenomena on our data and conclusions. The next question is associated with the effect of the
+order-parameter fluctuations on the dielectric data.
+As stressed above, both the LS and PLF models represent mean-filed approaches. The temperature
+region 𝛿𝑇 = 𝑇 −𝑇C (or 𝛿𝜏′ in terms of a redefined reduced temperature, 𝛿𝜏′ = 𝛿𝑇/𝑇C) around the phase-
+transition point where the fluctuations and the critical phenomena begin to dominate and the Landau
+theory can no longer be employed is given by a so-called Ginzburg parameter 𝐺: 𝜏′ ≪ 𝐺 or, at least,
+𝜏′ < 𝐺 — see, e.g., [34, 35]). The corresponding results derived by us for AFB with a highly sensitive
+optical-birefringence technique (see [36]) will be reported elsewhere. Here, we only state that they yield
+in 𝐺 ≈ 0.0026. Then, we have the conditions 𝛿𝑇 ≪ 0.5 K or 𝛿𝑇 < 0.5 K. Inspection of the data in
+figure 1 (or, better, in figure 2b) testifies that some eight data points correspond to the region 0.5 K above
+𝑇C and only three data points correspond to the region 0.1 K. In other words, our study does not directly
+address the scaling region and includes, at the most, a region where the mean-filed theory can be applied
+with small fluctuation corrections. Then, the order-parameter fluctuations can hardly affect our results.
+The next point is concerned with ‘frozen-in fluctuations’, i.e., with structural defects (see [37]).
+Having no direct facilities for estimating a defect state of our sample, we rely upon indirect methods.
+Namely, it is known that the influence of structural defects can disguise itself as a fluctuation effect in
+a close vicinity of PT. Therefore, the defects usually widen the ‘fluctuation’ region, i.e., they contribute
+additively to the Ginzburg parameter (see [14, 36]). This enables us to perform a rough comparison of the
+structural perfection for different samples of a given crystal: the larger is the Ginzburg number obtained
+for a crystal sample, the higher is the concentration of its defects. The following fact is worthwhile in this
+respect. When comparing our results with the corresponding data for the other A2BX4 crystals [36], one
+observes that the Ginzburg parameter for our AFB crystal (𝐺 ∼ 0.003) is relatively small (although the
+same in the order of magnitude). This indirectly indicates that the structural imperfection typical of our
+crystal sample is not so high to dominate the temperature dependences of its physical properties.
+Moreover, the defects with heavy concentrations can even ‘smear’ the divergent-like anomalies
+detected at the PT points. However, we observe no such situation with our sample, thus confirming
+again that the effects studied by us are nor defect-driven. Another similar argument against a significant
+contribution of the defects into the dielectric behavior of our AFB crystal is as follows. One of the
+43704-8
+
+Temperature dependence of dielectric permittivity in incommensurately modulated phase of ammonium fluoroberyllate
+common consequences of strong influence of structural defects is a decrease in the dielectric peak 𝜀max.
+However, our parameter 𝜀max ≈ 55 is very close to the average value 𝜀avg
+max ≈ 57 found from [1–5, 12, 19]
+at comparable electric frequencies. This is another evidence that the structural defects should play only
+a secondary role in the dielectric behavior of our crystal sample.
+We would also like to emphasize that, in some other terms, our main conclusion is that the temperature
+anomaly of the dielectric permittivity in the incommensurate phase of improper ferroelectric AFB near
+the 𝑇C point is ‘slower’ than that predicted by the inverse power law (see, e.g., a gradual decrease in the
+slope — i.e., the power-law ‘exponent’ — with approaching 𝑇C, which is seen in the double logarithmic-
+scale plot in figure 2b), although this law is a common regularity known in the theory of PTs. A (very
+loose) analogy with the situation occurring in proper uniaxial ferroelectrics can be mentioned: therein, the
+leading temperature-dependent terms are also ‘slower’ than those given by the inverse power law, being
+described by logarithmic corrections. However, there is still no theory predicting such a ‘slow-down’ in
+the dielectric divergence near the PT point as a result of structural defects.
+Finally, an important question arises in view of a potential effect of structural defects on the dielectric
+properties of the AFB crystals: is the LS model universally better than the PLF model — or some
+experimental data could be found in the literature which prefer the latter model? Since we cannot rule
+out completely the sample dependence of the dielectric permittivity, it would be difficult to expect a
+straightforward answer. However, this situation seems to be quite unlikely because both of the LS and
+PLF models refer to defect-free crystals. Then, the application of these models to essentially imperfect
+crystal samples might have rather resulted in the failure of both models than in changing the balance of
+their efficiencies [13].
+6. Conclusions
+We have studied the dielectric properties of improper ferroelectric AFB crystals in their paraelectric,
+incommensurately modulated and commensurate ferroelectric phases. Similar to the previous experimen-
+tal studies, the dielectric permittivity of AFB is not affected by the incommensurate PT at 𝑇i but reveals
+a weak peak at the commensurate PT point 𝑇C. The experimental results for the incommensurate phase
+of AFB are compared with the data following from the four phenomenological theories: the Curie-Weiss
+and generalized Curie-Weiss laws and the LS and PLF models [3, 5]. It is ascertained that the PLF
+model provides the results very similar to those of the inverse power laws given by the Curie-Weiss
+and generalized Curie-Weiss formulae. According to the results of rigorous statistical tests, all of these
+models provide a much worse fit of the experimental data than the LS model. In addition, the latter model
+can be efficiently applied within the overall temperature range of the incommensurate phase.
+The analysis of the experimental data shows that the temperature slopes of both the reciprocal
+permittivity with subtracted dielectric background and the permittivity plotted on the double logarithmic
+scale change continuously with temperature. However, any inverse power law would have implied a
+constant slope. This is a formal reason why the models (1), (2) and (4) fail in describing the experimental
+results. On the contrary, the temperature dependence of the permittivity in frames of the LS model (3) is
+governed by a combination of terms linear in temperature, including a divergent term in the denominator.
+Then, the peak at the PT point 𝑇C is ‘damped’ by the temperature-dependent terms in the numerator.
+This mathematical structure provides a necessary change in the slope and so appropriately describes the
+experimental data.
+In order to compare different phenomenological models more in detail, the main physical hypotheses
+underlying the LS and PLF approaches are elucidated. In particular, it is stressed that the LS model is
+based upon the approximation of weak spatial anisotropy of the order parameter and small corrections to
+the approximation of constant amplitude of the order parameter. These approximations are fully justified
+only within the plane-wave region of the incommensurate phase. On the other hand, the PLF model
+employs the constant-amplitude approximation and finds an exact solution for the phase of the order
+parameter, thus not relying on the assumption of weak anisotropy. However, the two approximations
+partly contradict each other, which may be the reason of a lower efficiency of the PLF model, compared
+to the LS model. Most likely, the LS model remains applicable within the entire incommensurate phase
+in AFB due to a very narrow temperature range of the latter. This fact also undermines a potential
+43704-9
+
+B. I. Horon, O. S. Kushnir, P. A. Shchepanskyi, V. Yo. Stadnyk
+advantage of the PLF approach, i.e., its applicability outside the plane-wave region, when it is applied to
+the incommensurate crystals like AFB.
+Possible contributions of the structural defects and the critical fluctuations into the 𝜀(𝑇) function of
+our AFB crystals are discussed. It is shown that the influence of the defects can hardly be decisive, while
+the fluctuations typical of a very close vicinity of the PT point are out of the scope of our study and so
+cannot affect its main conclusions.
+Acknowledgements
+This study has been supported by the Ministry of Education and Science of Ukraine (the Project
+#0120U102320).
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+
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+33. Cummins H. Z., Phys. Rep., 1990, 185, 211–409, doi:10.1016/0370-1573(90)90058-A.
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+37. Levanyuk A. P., Sigov A. S., Defects and Structural Phase Transitions, Gordon and Breach, New York, 1988.
+Температурна залежнiсть дiелектричної проникностi в
+несумiрно модульованiй фазi фторберилату амонiю
+Б. I. Горон1,2, О. С. Кушнiр2, П. А. Щепанський1, В. Й. Стадник1
+1 Кафедра загальної фiзики, Львiвський нацiональний унiверситет iменi Iвана Франка,
+вул. Драгоманова, 23, 79005 Львiв, Україна
+2 Кафедра оптоелектронiки та iнформацiйних технологiй, Львiвський нацiональний унiверситет
+iменi Iвана Франка, вул. ген. Тарнавського, 107, 79013 Львiв, Україна
+Дослiджено температурну залежнiсть дiелектричної проникностi вздовж полярної осi сегнетоелектрично-
+го кристала фторберилату амонiю (ФБА) в околi точок його фазових переходiв. Експериментальнi данi для
+несумiрно модульованої фази ФБА порiвняно з передбаченнями феноменологiчних моделей, вiдомих з
+лiтератури: закону Кюрi–Вейса (КВ), узагальненого закону Кюрi–Вейса (УКВ), а також моделей Леванюка i
+Саннiкова (ЛС) i Преловшека, Левстiка та Фiлiпiча (ПЛФ), запропонованих для невласних сегнетоелектри-
+кiв. Показано, що пiдхiд ЛС краще описує температурну поведiнку дiелектричної проникностi для кристала
+ФБА, нiж моделi КВ, УКВ i ПЛФ. З’ясовано основнi фiзичнi причини такої ситуацiї.
+Ключовi слова: фазовi переходи, несумiрнi фази, невласнi сегнетоелектрики, дiелектрична
+проникнiсть, фторберилат амонiю
+43704-11
+
+