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+A Stochastic ADMM Algorithm for Large-Scale Ptychography with Weighted
+Difference of Anisotropic and Isotropic Total Variation∗
+Kevin Bui† and Zichao (Wendy) Di ‡
+Abstract. Ptychography is an imaging technique that has various scientific applications, ranging from biology to
+optics. The method scans the object of interest in a series of overlapping positions, thereby generating
+a set of multiple Fourier magnitude measurements that are potentially corrupted by noise. From
+these measurements, an image of the object can be reconstructed depending on how the related
+inverse problem is formulated and solved. In this paper, we propose a class of variational models
+that incorporate the weighted anisotropic–isotropic total variation (AITV), an effective regularizer
+for image recovery. This class of models is applicable to measurements corrupted by either Gaussian
+or Poisson noise. In order to have the models applicable for large number of ptychographic scans,
+we design an efficient stochastic alternating direction method of multipliers algorithm and establish
+its convergence.
+Numerical experiments demonstrate that from a large set of highly corrupted
+Fourier measurements, the proposed stochastic algorithm with AITV regularization can reconstruct
+complex-valued images with satisfactory quality, especially for the phase components.
+AMS subject classifications. 65F22, 65K10, 68U10, 90C06, 90C15, 90C26
+Key words. phase retrieval, total variation, ptychography, ADMM, Poisson/Gaussian noise, nonconvex opti-
+mization, stochastic optimization
+1. Introduction. Ptychography is a popular imaging technique that combines both co-
+herent diffractive imaging and scanning transmission microscopy. It has been used in various
+industrial and scientific applications, including biology [34, 45, 58], crystallography [14], and
+optics [44, 48]. To perform a ptychographic experiment (see Figure 1), a coherent beam is
+scanned across the object of interest, where each scan may have overlapping positions with
+another. The scanning procedure provides a set of phaseless measurements that can be used
+to reconstruct an image of the object of interest.
+We describe the 2D ptychography in the discrete setting. Let z ∈ Cn2 be the object of
+interest with n × n pixels and ω ∈ Cm2 be the localized 2D probe with m × m pixels, where
+m < n. Both the object z and the probe ω are expressed as vectors in lexiographical order.
+We denote the set of N masks by {Sj}N
+j=1, where each Sj ∈ Rm2×n2 is a binary matrix that
+represents a (m×m)-size window over the image z. The set of phaseless measurements {dj}N
+j=1
+is obtained by dj = |F(Pjz)|2 = |F(ω◦Sjz)|2, where Pj := ω◦Sj is the jth probe, F ∈ Cm2×m2
+is the 2D discrete Fourier operator, the operation ◦ is elementwise multiplication, and the
+operation | · | is the elementwise absolute value of a vector. We aim to solve the following
+ptychographic phase retrieval problems. When the probe is unknown, the blind ptychographic
+∗Submitted to the editors DATE.
+Funding: This material is based upon work supported by the U.S. Department of Energy, Office of Science,
+under contract number DE-AC02-06CH11357.
+†Department of Mathematics, University of California at Irvine, Irvine, CA 92697 USA (kevinb3@uci.edu).
+‡Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL (wendydi@anl.gov).
+1
+arXiv:2301.02386v1  [math.NA]  6 Jan 2023
+
+2
+KEVIN BUI AND ZICHAO (WENDY) DI
+Probe
+Beam
+Object
+Measurements
+Figure 1: Schematic of a ptychography experiment.
+phase retrieval problem is expressed by
+BP-PR:
+To find ω ∈ Cm2 and z ∈ Cn2 such that |F(ω ◦ Sjz)|2 = dj, j = 1, . . . , N.
+(1.1)
+When the probe ω is known, (1.1) reduces to the non-blind case where we only find z ∈ Cn2.
+Multiple algorithms have been developed to solve the non-blind and blind phase retrieval
+problems. One of the most popular methods is the ptychographical iterative engine (PIE) [42],
+where later refinements led to ePIE [33] and rPIE [32]. The PIE methods are based on gradient
+descent applied to each measurement sequentially. Other gradient-based methods for phase
+retrieval include Wirtinger flow [6] and its variants [13, 56, 57], which use careful initialization
+by a spectral method and adaptive step sizes. PIE is also one class of projection algorithms
+for phase retrieval.
+Other projection-based algorithms are hybrid projection-reflection [1],
+Douglas-Rachford splitting [39, 46], and the relaxed averaged alternating reflections [31]. The
+phase retrieval problem can be formulated as a semidefinite optimization problem. For exam-
+ple, PhaseLift [7] solves the phase retrieval problem as a trace (nuclear) norm minimization
+problem. A nonconvex variant called PhaseLiftOff subtracts the trace norm by the Frobenius
+norm in the objective function [55]. PhaseCut proposes a different semidefinite formulation
+of the phase retrieval problem by explicitly separating the amplitude and phase variables and
+optimize only the values of the phase variables [47]. The phase retrieval problem can alter-
+natively be written as a saddle point problem [51], solved by alternating direction method
+of multipliers (ADMM) [4]. A globally convergent ADMM algorithm has recently been de-
+veloped to solve the BP-PR problem [8]. Another globally convergent algorithm is proximal
+alternating linearized minimization (PALM) [2], which has also been adapted to solve the
+BP-PR problem [12, 21]. For a detailed survey of numerical algorithms for phase retrieval,
+please refer to [17].
+For large-scale ptychography, when a huge number of scans are collected, many of the
+aforementioned algorithms may be inapplicable or may need to be adapted because of the
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+3
+demanding memory footprint and computational cost. Various parallel algorithms have been
+developed. For example, an asynchronous parallel version of ePIE has been implemented on
+GPUs, where each partition of a measurement set is asynchronously processed to obtain a sub-
+image and the sub-images are later fused together to form the entire image [35]. A parallel
+version of relaxed averaged alternating reflections has recently been developed for GPU impl-
+mentation [15]. Unfortunately, some of these parallel algorithms require a GPU, which many
+computers do not have. However, there are efficient algorithms for large-scale ptychography
+without the need for a GPU, although having one could speed up the processing time. A
+multigrid optimization framework has been proposed to accelerate large-scale gradient-based
+methods for phase retrieval [52]. An overlapping domain decomposition method combined
+with ADMM leads to a highly parallel algorithm with good load balance [9]. To the best
+of our knowledge, there does not yet exist a stochastic optimization algorithm for large-scale
+ptychography that iteratively processes a batch of measurements. Such an algorithm would be
+useful for practitioners who do not have access to multiple cores to perform parallel computing.
+To improve the image reconstruction quality in phase retrieval, total variation (TV) [43]
+has been incorporated for the cases when the measurements are corrupted with Gaussian noise
+[11] or with Poisson noise [10]. Both cases consider the isotropic TV approximation:
+∥∇z∥2,1 =
+n2
+�
+i=1
+�
+|(∇xz)i|2 + |(∇yz)i|2,
+(1.2)
+where ∇x and ∇y are the horizontal and vertical difference operators, respectively, and (∇xz)i
+and (∇yz)i are the ith entries of the vectors ∇xz and ∇yz, respectively. However, it has been
+known that isotropic TV tends to blur oblique edges.
+An alternative approximation that
+preserves sharper edges is the anisotropic TV [16]:
+∥∇z∥1 =
+n2
+�
+i=1
+(|(∇xz)i| + |(∇yz)i|) .
+(1.3)
+Overall, TV is meant to approximate the ℓ0 “norm” of the image gradient, i.e., ∥∇z∥0, be-
+cause TV is based on the ℓ1 norm, a convex relaxation of ℓ0. A nonconvex alternative to ℓ1 is
+ℓ1 − αℓ2, 0 < α ≤ 1, which performs well in recovering sparse solutions in various compressed
+sensing problems [27, 28, 29, 54]. The superior performance of ℓ1 − αℓ2 in sparse recovery has
+motivated the development of the weighted difference of anisotropic and isotropic total varia-
+tion (AITV) [30], which applies ℓ1 −αℓ2 on each gradient vector of an image. Mathematically,
+AITV is formulated by
+∥∇z∥1 − α∥∇z∥2,1 =
+n2
+�
+i=1
+�
+|(∇xz)i| + |(∇yz)i| − α
+�
+|(∇xz)i|2 + |(∇yz)i|2
+�
+.
+(1.4)
+AITV has demonstrated better performance than TV in image denoising, image deconvolution,
+image segmentation, and MRI reconstruction [5, 30, 38], especially in preserving sharper edges.
+In this work, we consider the large-scale ptychography problem where the measurements
+are corrupted by either Gaussian or Poisson noise.
+To improve the image reconstruction
+
+4
+KEVIN BUI AND ZICHAO (WENDY) DI
+quality, the AITV regularization is incorporated. The overall problem is formulated as a gen-
+eral variational problem, where we develop an ADMM algorithm to solve it. The ADMM
+algorithm has subproblems that can be approximately solved by stochastic gradient descent
+(SGD) [40]. Although SGD is a generic and popular algorithm for unconstrained optimization
+problems whose objective functions have a finite-sum structure, it may not be directly appli-
+cable to the subproblem being solved in the case of ptychography. Hence, we show how to
+appropriately apply SGD in order to develop our specialized stochastic ADMM algorithm. To
+further modify the algorithm, we incorporate adaptive step size based on the PIE algorithms
+[32, 33, 42]. Instead of using all measurements per iteration, this stochastic ADMM algorithm
+can iteratively process a batch of measurements to accurately perform image reconstruction.
+The paper is organized as follows. In Section 2, we review notations and definitions that
+will be used throughout the paper.
+Next in Section 3 we describe the AITV-regularized
+variational models to solve the image ptychography problem. Within this section, we design
+the stochastic ADMM algorithms to solve these models. Convergence analysis of the stochatsic
+ADMM algorithm follows in Section 4. In Section 5, we illustrate the performance of our
+proposed stochastic ADMM algorithms and compare them with other competing algorithms.
+Lastly, in Section 6, we conclude the paper with summary and future works.
+2. Preliminaries. In this section, we describe basic notations used throughout the paper.
+Let z ∈ Cn2. The ith entry of z is denoted by (z)i. The vector 1 is a vector whose entries
+are all ones. The vector 0 is also defined similarly. The real transpose and conjugate transpose
+of z are denoted by z⊤ and z∗, respectively. The same superscript notations are used for the
+real transpose and conjugate transpose of matrices. The sign of a complex value z′ ∈ C is
+given by
+sgn(z′) =
+�
+�
+�
+z′
+|z′|
+if z′ ̸= 0,
+c ∈ {c′ ∈ C : |c′| = 1}
+if z′ = 0.
+The sign of a vector z ∈ Cn2 is denoted by sgn(z) and is defined elementwise by sgn(z)i =
+sgn(zi), i = 1, . . . , n2. The standard basis vectors of Cn2 are denoted by {ei}n2
+i=1, where ei is
+a vector whose ith component is 1 while all other components are zeros. The diagonal matrix
+of a vector z ∈ Cn2 is denoted by Dz = Diag(z) = z1⊤ ◦ In2×n2.
+For p = (px, py) ∈ Cn2 × Cn2, its ith entry is pi =
+�(px)i
+(py)i
+�
+∈ C2. We define the following
+norms on Cn2 × Cn2:
+∥p∥1 =
+n2
+�
+i=1
+|(px)i| + |(py)i|,
+∥p∥2 =
+�
+�
+�
+�
+n2
+�
+i=1
+|(px)i|2 + |(py)i|2,
+∥p∥2,1 =
+n2
+�
+i=1
+�
+|(px)i|2 + |(py)i|2 =
+n2
+�
+i=1
+∥pi∥2.
+The discrete gradient operator ∇ : Cn2 → Cn2 × Cn2 when specifically applied to the image
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+5
+z is given by ∇z = (∇xz, ∇yz), where ∇x and ∇y are the forward horizontal and vertical
+difference operators.
+We also define the proximal operator of a function f : Cn2 → R ∪ {+∞} as
+proxf(·)(z′) = arg min
+z
+f(z) + 1
+2∥z − z′∥2
+2, ∀z′ ∈ Cn2.
+3. Proposed Model. Throughout the paper, we assume that among the mask set {Sj}N
+j=1,
+there exists j′ such that ∥Sj′ei∥1 = 1 for each i = 1, . . . , n2. This assumption ensures that
+each pixel of an image z ∈ Cn2 is sampled at least once.
+Suppose that the measurements {dj}N
+j=1 are corrupted by independent and identically
+distributed (iid) noise, i.e., dj
+iid
+∼ Noise(|F(ω ◦ Sjz)|2) for j = 1, . . . , N. We assume that the
+noise is either Gaussian or Poisson, both of which are common in phase retrieval. Given an
+unknown probe ω, the blind variational model [8] is
+min
+ω∈Cm2,z∈Cn2
+N
+�
+j=1
+B(|F(ω ◦ Sjz)|2, dj),
+(3.1)
+where
+B(g, f) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+2∥√g −
+�
+f∥2
+2,
+amplitude Gaussian metric (AGM) [51],
+1
+2⟨g − f ◦ log(g), 1⟩,
+intensity Poisson metric (IPM) [10].
+(3.2)
+Note that √· is elementwise square root. When the probe ω is known, (3.1) simplifies to the
+non-blind case as a special case where we only need to find z ∈ Cn2. Hence, throughout the
+rest of this section, we will focus on the blind case. To improve image recovery, we propose a
+class of AITV-regularized variants of (3.1).
+3.1. AITV model. For image ptychography, we propose the following AITV-regularized
+model:
+min
+ω∈Cm2,z∈Cn2
+N
+�
+j=1
+B(|F(ω ◦ Sjz)|2, dj) + λ (∥∇z∥1 − α∥∇z∥2,1) , λ > 0, α ∈ [0, 1].
+(3.3)
+To develop an ADMM algorithm of (3.3), we introduce auxiliary variables u = (u1, . . . , uN) ∈
+Cm2×N and v = (vx, vy) ∈ Cn2 ×Cn2 so that we obtain an equivalent constrained optimization
+problem
+min
+u,v,z
+N
+�
+j=1
+B(|uj|2, dj) + λ (∥v∥1 − α∥v∥2,1) s.t.
+uj = F(ω ◦ Sjz), j = 1, . . . , N, and v = ∇z.
+(3.4)
+
+6
+KEVIN BUI AND ZICHAO (WENDY) DI
+The augmented Lagrangian of (3.4) is
+L(u, ω, v, z, Λ, y) =
+N
+�
+j=1
+�
+B(|uj|2, dj) + R (⟨Λj, uj − F(ω ◦ Sjz)⟩) + β1
+2 ∥uj − F(ω ◦ Sjz)∥2
+2
+�
++ λ (∥v∥1 − α∥v∥2,1) + R (⟨y, v − ∇z⟩) + β2
+2 ∥v − ∇z∥2
+2,
+(3.5)
+where R(·) denotes the real component of a complex number; ⟨·, ·⟩ denotes the complex inner
+product between two vectors; Λ = (Λ1, . . . , ΛN) ∈ Cm2×N and y = (yx, yy) ∈ Cn2×n2 are
+Lagrange multipliers; and β1, β2 > 0 are penalty parameters. The ADMM algorithm iterates
+as follows:
+ut+1 ∈ arg min
+u
+L(u, ωt, vt, zt, Λt, yt),
+(3.6a)
+ωt+1 ∈ arg min
+ω
+L(ut+1, ω, vt, zt, Λt, yt),
+(3.6b)
+vt+1 ∈ arg min
+v
+L(ut+1, ωt+1, v, zt, Λt, yt),
+(3.6c)
+zt+1 ∈ arg min
+z
+L(ut+1, ωt+1, vt+1, z, Λt, yt),
+(3.6d)
+Λt+1
+j
+= Λt
+j + β1
+�
+ut+1
+j
+− F(ωt+1 ◦ Sjzt+1)
+�
+,
+j = 1, . . . , N,
+(3.6e)
+yt+1 = yt + β2
+�
+vt+1 − ∇zt+1�
+.
+(3.6f)
+Next we explain how to solve each subproblem.
+3.1.1. u-subproblem. In (3.6a), we solve uj independently of each other. For each j =
+1, . . . , N, we have
+ut+1
+j
+∈ arg min
+uj
+B(|uj|2, dj) + R
+�
+⟨Λt
+j, uj − F(P t
+j zt)⟩
+�
++ β1
+2 ∥uj − F(P t
+j zt)∥2
+2
+= arg min
+uj
+1
+β1
+B(|uj|2, dj) + 1
+2
+����uj − F(P t
+j zt) + 1
+β1
+Λt
+j
+����
+2
+2
+= prox 1
+β1 B(|·|2,dj)
+�
+F(P t
+j zt) − 1
+β1
+Λt
+j
+�
+,
+(3.7)
+where P t
+j = ωt ◦ Sj. The proximal operator for each fidelity term in (3.2) has a closed-form
+solution provided in [10, 11], so we have
+ut+1
+j
+=
+�
+�
+�
+�
+�
+�
+�
+√
+dj+β1
+���F(P t
+j zt)− 1
+β1 Λt
+j
+���
+1+β1
+◦ sgn
+�
+F(P t
+j zt) − 1
+β1 Λt
+j
+�
+,
+AGM,
+β1|F(P t
+j zt)− 1
+β1 Λt
+j|+
+��
+β1|F(P t
+j zt)− 1
+β1 Λt
+j|
+�2
++4(1+β1)dj
+2(1+β1)
+◦ sgn
+�
+F(P t
+j zt) − 1
+β1 Λt
+j
+�
+,
+IPM.
+(3.8)
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+7
+3.1.2. ω-subproblem. The ω-subproblem (3.6b) can be rewritten as
+ωt+1 ∈ arg min
+ω
+N
+�
+j=1
+�
+�β1
+2
+�����F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ω ◦ Sjzt
+�����
+2
+2
+�
+� ,
+(3.9)
+which shows that updating ω requires access to all N probes. Hence, we develop an alternative
+update scheme that uses only b ≤ N probes. Instead of solving (3.6b) exactly, we linearize it
+as done in [26, 37] to obtain
+ωt+1 ∈ arg min
+ω
+⟨∇ωL(ut+1, ωt, zt, Λt, yt), ω − ωt⟩ +
+1
+2δtω
+∥ω − ωt∥2
+2
+(3.10)
+for some constant δt
+ω > 0 at iteration t. Then (3.10) is equivalent to performing gradient
+descent with step size δt
+ω:
+ωt+1 = ωt − δt
+ω∇ωL(ut+1, ωt, vt, zt, Λt, yt).
+(3.11)
+Next we approximate ∇Lω by its stochastic estimator ˜∇ωL. thereby updating ωt+1 by SGD
+with step size δt
+ω > 0:
+ωt+1 = ωt − δt
+ω ˜∇ωL(ut+1, ωt, vt, zt, Λt, yt).
+(3.12)
+We derive some candidates for ˜∇ωL. Let Gt
+j(ω) = β1
+2
+���F−1 �
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ω ◦ Sjzt���
+2
+2, which
+means that ∇Gt
+j(ω) = −β1(Sjzt)∗ ◦
+�
+F−1 �
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ω ◦ Sjzt�
+. (3.11) can be rewritten as
+a gradient descent step with Nδt
+ω:
+ωt+1 = ωt − Nδt
+ω
+�
+� 1
+N
+N
+�
+j=1
+∇Gt
+j(ωt)
+�
+� .
+(3.13)
+The SGD estimator [3, 40] of 1
+N
+�N
+j=1 ∇Gt
+j(ωt) is 1
+b
+�
+j∈nt ∇Gt
+j(ωt), where nt ⊂ {1, . . . , N} is
+a sub-batch of N masks such that |nt| = b. Then the SGD variant (after scaling δt
+ω) of (3.11)
+is
+ωt+1 = ωt − δt
+ω
+�
+�1
+b
+�
+j∈nt
+∇Gt
+j(ωt)
+�
+� .
+(3.14)
+This implies that one candidate stochastic estimator for ∇ωL is
+˜∇SGD
+ω
+L(ut+1, ωt, vt, zt, Λt, yt) = 1
+b
+�
+j∈nt
+∇Gt
+j(ωt)
+= −β1
+b
+�
+j∈nt
+(Sjzt)∗ ◦
+�
+F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ωt ◦ Sjzt
+�
+.
+(3.15)
+
+8
+KEVIN BUI AND ZICHAO (WENDY) DI
+We can further modify (3.15) by incorporating spatially varying step sizes inspired by the PIE
+algorithms [32, 33, 42]. Let
+Φt
+j =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+∥Sjzt∥2∞
+,
+ePIE [33],
+∥Sjzt∥11
+∥Sjzt∥∞ (|Sjzt|2 + γω∥Sjzt∥2∞1),
+PIE [42],
+1
+(1 − γω)|Sjzt|2 + γω∥Sjzt∥2∞1,
+rPIE [32],
+(3.16)
+where γω ∈ [0, 1] for PIE and rPIE and division is elementwise. Incorporating Φt
+j into (3.15),
+we have another class of stochastic estimators
+˜∇PIE
+ω
+L(ut+1, ωt, vt, zt, Λt, yt) = −β1
+b
+�
+j∈nt
+Φt
+j ◦ (Sjzt)∗ ◦
+�
+F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− ωt ◦ Sjzt
+�
+.
+(3.17)
+3.1.3. v-subproblem. Expanding (3.6c) gives
+vt+1 ∈ arg min
+v
+λ
+β2
+(∥v∥1 − α∥v∥2,1) + 1
+2
+����v − ∇zt + yt
+β2
+����
+2
+2
+= arg min
+v
+n2
+�
+i=1
+λ
+β2
+(∥vi∥1 − α∥vi∥2) + 1
+2
+����vi − (∇zt)i + (yt)i
+β2
+����
+2
+2
+,
+(3.18)
+which means that the solution vt+1 can be solved elementwise. As a result, the subproblem
+simplifies to
+(vt+1)i = prox λ
+β2 (∥·∥1−α∥·∥2)
+�
+(∇zt)i − (yt)i
+β2
+�
+.
+(3.19)
+A closed-form solution for the proximal operator of ℓ1 − αℓ2 is provided in [27] but only for
+real-valued vectors. We generalize it to the complex case in Lemma 3.1, whose proof is delayed
+to Appendix A.
+Lemma 3.1. Given x′ ∈ Cn, λ > 0. and α ≥ 0, we have the following cases:
+1. When ∥x′∥∞ > λ, we have
+x∗ = (∥ξ∥2 + αλ)
+ξ
+∥ξ∥2
+, where ξ = sgn(x′) ◦ max(|x′| − λ, 0).
+2. When (1 − α)λ < ∥x′∥∞ ≤ λ, we have x∗ as a 1-sparse vector such that one chooses
+an index i ∈ arg maxj(|(x′)j|) and have
+(x∗)j =
+�
+(|(x′)j| + (α − 1)λ) sgn((x′)j)
+if j = i,
+0
+if j ̸= i.
+3. When ∥x′∥∞ ≤ (1 − α)λ, we have x∗ = 0.
+Then x∗ is an optimal solution to
+proxλ(∥·∥1−α∥·∥2)(x′) = arg min
+x
+∥x∥1 − α∥x∥2 + 1
+2λ∥x − x′∥2
+2.
+(3.20)
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+9
+3.1.4. z-subproblem. (3.6d) can be rewritten as
+zt+1 ∈ arg min
+z
+N
+�
+j=1
+�
+�β1
+2
+�����ut+1
+j
+− F(P t+1
+j
+z) +
+Λt
+j
+β1
+�����
+2
+2
+���
+� + β2
+2
+����vt+1 − ∇z + yt
+β2
+����
+2
+2
+,
+(3.21)
+which implies that zt+1 must satisfy the first-order optimality condition
+�
+�β1
+N
+�
+j=1
+(P t+1
+j
+)∗P t+1
+j
+− β2∆
+�
+� zt+1 =
+N
+�
+j=1
+β1(P t+1
+j
+)∗F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
++ β2∇⊤
+�
+vt+1 + yt
+β2
+�
+,
+(3.22)
+where the Laplacian ∆ = −∇⊤∇. Since the coefficient matrix of zt+1 is invertible, solving
+(3.22) can be performed exactly, but it could be computationally expensive if the matrix
+system is extremely large because of the image size of z. Since the coefficient matrix tends
+to be sparse, conjugate gradient [22] can be used to solve (3.22) like in [10, 11], but it needs
+access to all N probes and requires at most n2 iterations to attain an exact solution, assuming
+exact arithmetic. Moreover, it is sensitive to roundoff error [19].
+Alternatively, like in Section 3.1.2 we linearize (3.21) to obtain the gradient descent step
+with step size δt
+z > 0:
+zt+1 = zt − δt
+z∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt).
+(3.23)
+We approximate ∇zL by its stochastic estimator ˜∇zL that only has access to b ≤ N probes.
+Replacing ∇zL with ˜∇zL in (3.23) gives
+zt+1 = zt − δt
+z ˜∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt).
+(3.24)
+To design candidates for ˜∇zL, we will use the following lemma:
+Lemma 3.2. Let S ∈ Rm2×n2. If ei ∈ ker(S) for some index i, then for any x ∈ Cm2, we
+have (S⊤x)i = 0.
+Proof. We have (S⊤x)i = ⟨S⊤x, ei⟩ = ⟨x, Sei⟩ = ⟨x, 0⟩ = 0.
+For brevity, we denote the vectors
+At
+j = −β1
+�
+(P t+1
+j
+)∗F−1
+�
+ut+1
+j
++
+Λt
+j
+β1
+�
+− (P t+1
+j
+)∗P t+1
+j
+zt
+�
+,
+Bt = −β2
+�
+∇⊤
+�
+vt+1 + yt
+β2
+�
++ ∆zt
+�
+.
+(3.25)
+At each element i = 1, . . . , n2, (3.23) becomes
+(zt+1)i = (zt)i − δt
+z(∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt))i = (zt)i − δt
+z
+�
+�
+N
+�
+j=1
+(At
+j)i + (Bt)i
+�
+� .
+(3.26)
+
+10
+KEVIN BUI AND ZICHAO (WENDY) DI
+By Lemma 3.2, since (P t+1
+j
+)∗ = (ωt+1 ◦ Sj)∗ = S⊤
+j D(ωt+1)∗, we have (At
+j)i = 0 if ei ∈ ker(Sj),
+which means that element i is not scanned by the mask matrix Sj. For each i = 1, . . . , n2,
+we define Ni = {j : ei ̸∈ ker(Sj)} to be the set of indices corresponding to the mask matrices
+that scan element i. As a result, (3.26) reduces to and can be rewritten as
+(zt+1)i = (zt)i − δt
+z
+�
+� �
+j∈Ni
+(At
+j)i + (Bt)i
+�
+� = (zt)i − |Ni|δt
+z
+�
+� 1
+|Ni|
+�
+j∈Ni
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+��
+� .
+(3.27)
+Comparing (3.26) and (3.27), we observe that
+1
+|Ni|
+�
+j∈Ni
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+�
+∝ (∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt))i.
+Thus, a candidate for the stochastic estimator ˜∇zL is the SGD estimator ˜∇SGD
+z
+L given by
+�
+˜∇SGD
+z
+L(ut+1, ωt+1, vt+1, zt, Λt, yt)
+�
+i =
+1
+|nt
+i|
+�
+j∈nt
+i
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+�
+,
+(3.28)
+where nt
+i ⊂ Ni is a mini-batch sampled from Ni at iteration t [3, 40].
+We can further modify (3.28) by incorporating spatially varying step sizes inspired by the
+PIE algorithms [32, 33, 42]. We define
+Ψi,j =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+∥ωt+1∥2∞
+ePIE [33],
+∥P t+1
+j
+ei∥1
+∥ωt+1∥∞
+�
+∥P t+1
+j
+ei∥2
+1 + γz∥ωt+1∥2∞
+�
+PIE [42],
+1
+(1 − γz)∥P t+1
+j
+ei∥2
+1 + γz∥ωt+1∥2∞
+rPIE [32],
+(3.29)
+with γz ∈ [0, 1] for PIE and rPIE. A class of PIE candidates for the stochastic estimator is
+�
+˜∇PIE
+z
+L(ut+1, ωt+1, vt+1, zt, Λt, yt)
+�
+i =
+1
+|nt
+i|
+�
+j∈nt
+i
+Ψi,j
+�
+(At
+j)i +
+1
+|Ni|(Bt)i
+�
+.
+(3.30)
+The overall stochastic ADMM algorithm that solves (3.3) is provided by Algorithm 3.1.
+Notice that the non-blind problem is just a special case with the probe ω fixed.
+4. Convergence Analysis. We discuss the convergence of Algorithm 3.1. Although global
+convergence for ADMM can be established using Kurdyka-�Lojasiewicz assumptions [49], the
+result does not apply for our models because our models contain the gradient operator, which
+does not satisfy the necessary surjectivity assumption. Hence, we will prove up to subse-
+quential convergence. The convergence analysis is based on the analyses done in [10, 11, 51],
+where under certain assumptions, they showed that the iterate subsequences of the ADMM
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+11
+Algorithm 3.1 Stochastic ADMM to solve (3.3)
+Input:
+set of masks {Sj}N
+j=1; model parameters λ > 0, α ∈ [0, 1]; penalty parameters β1, β2 > 0; sequence of step sizes
+{(δt
+ω, δt
+z)}∞
+t=1; batch size b ≤ N; PIE factors γz, γω ∈ [0, 1].
+1: Initialize ω0, z0, {u0
+j}N
+j=1 = {Λ0
+j}N
+j=1, y0 = ∇z0.
+2: for t = 0 to T − 1 do
+3:
+Uniformly sample without replacement the subset nt ⊂ {1, . . . , N} of batch size b.
+4:
+Compute nt
+i from nt, i.e., nt
+i =
+�
+j∈nt
+∥Sjei∥1.
+5:
+Update ut+1
+j
+according to (3.8) for each j ∈ nt.
+6:
+if ω is unknown then
+7:
+Update ωt+1 = ωt − δt
+ω ˜∇ωL(ut+1, ωt, vt, zt, Λt, yt). See (3.15) and (3.17) for a candidate ˜∇ωL.
+8:
+else
+9:
+ωt+1 = ωt.
+10:
+end if
+11:
+Compute
+(vt+1)i = prox λ
+β2 (∥·∥1−α∥·∥2)
+�
+(∇zt)i − (yt)i
+β2
+�
+for all i = 1, . . . , n2; see Lemma 3.1.
+12:
+Update (zt+1)i = (zt)i − δt
+z
+�
+˜∇zL(ut+1, ωt+1, vt+1, zt, Λt, yt)
+�
+i for all i such that nt
+i ̸= 0. See (3.28) and (3.30)
+for a candidate ˜∇zL.
+13:
+Compute
+Λt+1
+j
+= Λt
+j + β1
+�
+ut+1
+j
+− F(ωt+1 ◦ Sjzt+1)
+�
+, ∀j ∈ nt,
+yt+1 = yt + β2
+�
+vt+1 − ∇zt+1�
+.
+14: end for
+Output: ω∗ = ωT , z∗ = zT
+algorithms converge to Karush-Kuhn-Tucker (KKT) points. To simplify notation, let Z =
+(u, ω, v, z) and Ω = (Λ, y). We also write L(ω), for example, to represent the Lagrangian with
+respect to ω with all other variables fixed at their most recent values. A KKT point (Z⋆, Ω⋆)
+of the Lagrangian (3.5) satisfies the KKT conditions given by
+0 ∈
+�
+�
+�
+�
+�
+∂|u⋆
+j|(|u⋆
+j| −
+�
+dj) + Λ⋆
+j,
+if AGM,
+∂|u⋆
+j|
+�
+|u⋆
+j| −
+�
+dj
+|u⋆
+j|
+�
++ Λ⋆
+j,
+if IPM,
+for j = 1, . . . , N,
+(4.1a)
+−y⋆
+λ ∈ ∂(∥v⋆∥1 − α∥v⋆∥2,1),
+(4.1b)
+u⋆
+j = F(ω⋆ ◦ Sjz⋆)
+for j = 1, . . . , N,
+(4.1c)
+v⋆ = ∇z⋆,
+(4.1d)
+∇ωL(Z⋆, Ω⋆) = 0,
+(4.1e)
+∇zL(Z⋆, Ω⋆) = 0.
+(4.1f)
+
+12
+KEVIN BUI AND ZICHAO (WENDY) DI
+Because we implement SGD to solve for the probe ω and the image z in the ADMM algorithm,
+we replace (4.1e) and (4.1f) with the following conditions, respectively:
+E
+�
+∥∇ωL(Z⋆, Ω⋆)∥2
+2
+�
+= 0
+(4.1e′)
+E
+�
+∥∇zL(Z⋆, Ω⋆)∥2
+2
+�
+= 0.
+(4.1f′)
+We say a point (Z⋆, Ω⋆) is a stochastic KKT point if it satisfies (4.1a)-(4.1d) and (4.1e′)-(4.1f′).
+Where Et denotes the expectation conditioned on the first t iterations of the stochastic
+ADMM algorithm, we impose the following assumption adapted from [3, Assumption 4.3]
+relating to the stochastic gradient estimators ˜∇ωL and ˜∇zL.
+Assumption 4.1. Let {(Zt, Ωt)}∞
+t=1 be a sequence of iterates generated by Algorithm 3.1.
+Suppose that at each iteration t, the stochastic gradient estimators
+˜∇ωL(ωt) := ˜∇ωL(ut+1, ωt, vt, zt, Λt, yt) and ˜∇zL(zt) := ˜∇ωL(ut+1, ωt+1, vt+1, zt, Λt, yt) sat-
+isfy the following:
+(a) There exist constants KU ≥ KL > 0 such that
+R
+�
+Et
+�
+⟨∇ωL(ωt), ˜∇ωL(ωt)⟩
+��
+≥ KLEt
+�
+∥∇ωL(ωt)∥2
+2
+�
+(4.3)
+���Et[ ˜∇ωL(ωt)]
+���
+2
+2 ≤ KUEt
+�
+∥∇ωL(ωt)∥2
+2
+�
+(4.4)
+R
+�
+Et
+�
+⟨∇zL(zt)), ˜∇zL(zt)⟩
+��
+≥ KLEt
+�
+∥∇zL(zt)∥2
+2
+�
+(4.5)
+���Et[ ˜∇zL(zt)]
+���
+2
+2 ≤ KUEt
+�
+∥∇zL(zt)∥2
+2
+�
+.
+(4.6)
+(b) There exists constant M, MV ≥ 0 such that
+Et
+�
+∥ ˜∇ωL(ωt)∥2
+2
+�
+−
+���Et[ ˜∇ωL(ωt)]
+���
+2
+2 ≤ M + MV Et
+���∇ωL(ωt)
+��2
+2
+�
+(4.7)
+Et
+�
+∥ ˜∇zL(zt)∥2
+2
+�
+−
+���Et[ ˜∇zL(zt)]
+���
+2
+2 ≤ M + MV Et
+���∇zL(zt)
+��2
+2
+�
+.
+(4.8)
+To prove the convergence of Algorithm 3.1, we require the following preliminary results.
+Lemma 4.2 provides useful inequalities while Proposition 4.3 bounds the iterates {(Zt, Ωt)}∞
+t=1
+and establishes some bounded property of the gradients {(∇ωL(ωt), ∇zL(zt))}∞
+t=1.
+Lemma 4.2. Let {(Zt, Ωt)}∞
+t=1 be a sequence of iterates generated by Algorithm 3.1 that
+satisfies Assumption 4.1. Suppose that {(ωt, zt)}∞
+t=1 is bounded. For each iteration t, we have
+Et[L(ωt+1)] − Et[L(ωt)] ≤ −
+�
+KL − δt
+ωLω(MV + KU)
+2
+�
+δt
+ωEt
+���∇ωL(ωt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
+(4.9)
+Et[L(zt+1)] − Et[L(zt)] ≤ −
+�
+KL − δt
+zLz(MV + KU)
+2
+�
+δt
+zEt
+���∇zL(zt)
+��2
+2
+�
++ (δt
+z)2LzM
+2
+(4.10)
+for some constants Lω, Lz > 0.
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+13
+Proposition 4.3. Let {(Zt, Ωt)}∞
+t=1 be a sequence of iterates generated by Algorithm 3.1 that
+satisfies Assumption 4.1. Suppose {(ωt, zt)}∞
+t=1 is bounded, �∞
+t=1 ∥Ωt+1 − Ωt∥2
+2 < ∞, and
+∞
+�
+t=1
+δt
+ω = ∞,
+∞
+�
+t=1
+(δt
+ω)2 < ∞,
+∞
+�
+t=1
+δt
+z = ∞,
+∞
+�
+t=1
+(δt
+z)2 < ∞.
+(4.11)
+Then {(Zt, Ωt)}∞
+t=1 is bounded and
+∞
+�
+t=1
+E
+�
+δt
+ω∥∇ωL(ωt)∥2
+2 + δt
+z∥∇zL(zt)∥2
+2
+�
+< ∞.
+(4.12)
+The convergence of Algorithm 3.1 is finally established below (see proofs in Appendix B).
+Theorem 4.4. Let {(Zt, Ωt)}∞
+t=1 be generated by Algorithm 3.1. Under the same assump-
+tion as Proposition 4.3, there exists a subsequence of {(Zt, Ωt)}∞
+t=1 whose accumulation point
+(Z⋆, Ω⋆) is a stochastic KKT point almost surely (a.s.) of (3.5).
+We note that the requirement �∞
+t=1 ∥Ωt+1 − Ωt∥2
+2 < ∞ is rather strong, but similar
+assumption was made in other nonconvex ADMM algorithms [24, 25, 53] that do not satisfy
+the necessary assumptions for global convergence [49].
+5. Numerical Results. In this section, we evaluate the performance of Algorithm 3.1 on
+two complex images presented in Figure 2. The probe size used for both images is 256 × 256,
+and the probe is scanned across an image from left to right and top to bottom, giving us
+N = 100 measurements. The measurements {dj}N
+j=1 are either corrupted by Gaussian noise
+or Poisson noise. More specifically, when the measurements are corrupted by Gaussian noise,
+we have
+dj = (|F(Pjz)| + ϵ)2,
+(5.1)
+where ϵ is an i.i.d. Gaussian random vector. When the measurements are corrupted by Poisson
+noise, we have
+dj = Poisson(|F(Pjzζ)|),
+(5.2)
+where zζ = ζz for some constant ζ > 0. Note that Poisson noise is stronger when ζ is smaller.
+For numerical evaluation, we compute the SSIMs [50] of the magnitudes and phases be-
+tween the reconstructed image z∗∗ and the ground-truth image zg, where z∗∗
+i
+= ζ∗z∗
+i+t∗ is
+adjusted for scaling by ζ∗ and translation by t∗ and (ζ∗, t∗) = arg min
+ζ∈C,t∈Z
+n2
+�
+i=1
+|ζz∗
+i+t − zg
+i |2. We
+compare the proposed stochastic ADMM algorithms with its deterministic, full-batch coun-
+terparts (i.e., (3.22) and (3.9) are solved exactly) and its isotropic TV counterparts based on
+
+14
+KEVIN BUI AND ZICHAO (WENDY) DI
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+(a)
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+(b)
+5
+10
+15
+20
+25
+30
+(c)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+(d)
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+(e)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+(f)
+Figure 2: Two complex sample images and their probes examined in the experiments. Left
+column: sample magnitude; middle column: sample phase with inserted proportionally sized
+probe magnitude; left column: the magnitude differences between the ground-truth probe and
+the initial probe ω0.
+Table 1: Parameter settings for each method. Note that b refers to the batch size.
+Total
+Epochs
+β1 = β2
+δt
+z
+Ψi,j
+δt
+ω
+Φi,j
+AGM
+600
+0.25
+�
+�
+�
+�
+�
+2
+√
+b
+1 ≤ t ≤ 300
+1
+5
+√
+b
+300 < t ≤ 450
+1
+50
+√
+b
+450 < t ≤ 600
+rPIE
+(γz = 0.1)
+�
+�
+�
+�
+�
+√
+b × 10−3
+1 ≤ t ≤ 300
+√
+b × 10−4
+300 < t ≤ 450
+√
+b × 10−5
+450 ≤ t ≤ 600
+rPIE
+(γω = 0.025)
+IPM
+300
+0.25
+�
+�
+�
+�
+�
+15
+√
+b
+1 ≤ t ≤ 150
+3
+2
+√
+b
+150 < t ≤ 225
+3
+20
+√
+b
+225 < t ≤ 300
+ePIE
+�
+�
+�
+�
+�
+2
+√
+b × 10−3
+1 ≤ t ≤ 300
+2
+√
+b × 10−4
+300 < t ≤ 450
+2
+√
+b × 10−5
+450 ≤ t ≤ 600
+ePIE
+[10, 11]. The results are also compared with Douglas-Rachford splitting [46], rPIE [33], and
+PHeBIE [21]. We follow the implementation of Douglas-Rachford from [8].
+We initialize z0 =
+1
+√
+2(1+i1) when using AGM for Gaussian-corrupted measurements and
+z0 =
+ζ
+√
+2(1 + i1) when using IPM for Poisson-corrupted measurements. When performing the
+blind experiments using Algorithm 3.1, ω0 is initialized as the perturbation of the ground-
+truth probe. The magnitude differences between the initial probe and the ground-truth probe
+are shown in Figure 2. The selected parameters, except for λ, are summarized in Table 1.
+The initial step sizes for δt
+z and δt
+ω are determined empirically, and motivated by (4.11), we
+decrease them by a factor of 10 at the 1/2 and 3/4 of the total epochs. Decreasing the step
+size in this way is a popular technique, especially in the deep learning community [18, 20].
+Inspired from [18], the step sizes are multiplied by a factor of
+√
+b so that they are scaled
+accordingly to the batch size b. For AITV regularization, we examine α ∈ {0.2, 0.4, 0.6, 0.8}
+
+-
+-
+-
+-
+-
+-
+-STOCHASTIC ADMM FOR PTYCHOGRAPHY
+15
+Table 2: SSIM results of the algorithms applied to the Gaussian corrupted measurements
+with SNR = 40. The stochastic algorithms (e.g., AITV and isoTV, b ∈ {5, 10, 20, 50}) are ran
+three times to obtain the average SSIM values. Bold indicates best value; underline indicates
+second best value.
+Non-blind
+Blind
+Chip
+Cameraman/Baboon
+Chip
+Cameraman/Baboon
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+DR
+0.8130
+0.8089
+0.8701
+0.5191
+0.8008
+0.7642
+0.8009
+0.3207
+rPIE
+0.8886
+0.9073
+0.8930
+0.6055
+0.9070
+0.9120
+0.8890
+0.6145
+PHeBIE
+0.8004
+0.8019
+0.8725
+0.5718
+0.8612
+0.8438
+0.8846
+0.5756
+isoTV (b = 5)
+0.9501
+0.9027
+0.9393
+0.7578
+0.9426
+0.8919
+0.9324
+0.7547
+isoTV (b = 10)
+0.9498
+0.9004
+0.9387
+0.7475
+0.9429
+0.8891
+0.9326
+0.7477
+isoTV (b = 20)
+0.9514
+0.8981
+0.9385
+0.7302
+0.9447
+0.8850
+0.9298
+0.7289
+isoTV (b = 50)
+0.9355
+0.9193
+0.9294
+0.7050
+0.9322
+0.9047
+0.9153
+0.7025
+isoTV (full batch)
+0.9578
+0.9145
+0.9769
+0.7338
+0.9527
+0.8698
+0.9589
+0.5774
+AITV (b = 5)
+0.9585
+0.9556
+0.9438
+0.7720
+0.9490
+0.9477
+0.9373
+0.7775
+AITV (b = 10)
+0.9620
+0.9579
+0.9515
+0.7747
+0.9534
+0.9481
+0.9450
+0.7772
+AITV (b = 20)
+0.9629
+0.9583
+0.9538
+0.7707
+0.9547
+0.9470
+0.9468
+0.7690
+AITV (b = 50)
+0.9585
+0.9550
+0.9490
+0.7358
+0.9514
+0.9432
+0.9391
+0.7342
+AITV (full batch)
+0.9674
+0.9513
+0.9814
+0.7463
+0.9676
+0.9296
+0.9725
+0.5956
+and determine that α = 0.8 yields the best results across all of our numerical examples. The
+batch sizes we examine are b ∈ {5, 10, 20, 50} for Gaussian noise and b ∈ {5, 10, 20, 25} for
+Poisson noise. For each parameter setting and image, we run three trials to obtain the mean
+SSIM values.
+The code for the experiments is available at https://github.com/kbui1993/Stochastic
+ADMM Ptycho.
+5.1. Gaussian noise. The SNR of the noisy measurements [10] is given by
+SNR
+�
+{
+�
+dj}N
+j=1, {|F(Pjz)|}N
+j=1
+�
+= −10 log10
+�
+�
+�
+�
+�
+�
+�
+N
+�
+j=1
+∥
+�
+dj − |F(Pjz)|∥2
+2
+N
+�
+j=1
+∥F(Pjz)∥2
+2
+�
+�
+�
+�
+�
+�
+�
+,
+so determined by the SNR value, the noise level ϵ in (5.1) can be calculated by
+ϵ =
+�
+�
+�
+�
+�
+�
+10−SNR/10
+N
+�
+j=1
+∥F(Pjz)∥2
+2
+Nm2
+.
+For both the non-blind and blind case, we examine the case when SNR = 40 for the
+noisy measurements. We set λ = 10.0. The numerical results are recorded in Table 2. For
+
+16
+KEVIN BUI AND ZICHAO (WENDY) DI
+(a) isoTV (b = 50)
+(b) AITV (b = 20)
+(c) AITV (full)
+(d) DR
+(e) rPIE
+(f) PHeBIE
+(g) isoTV (b = 50)
+(h) AITV (b = 20)
+(i) AITV (full)
+(j) DR
+(k) rPIE
+(l) PHeBIE
+(m) isoTV (b = 5)
+(n) AITV (b = 10)
+(o) AITV (full)
+(p) DR
+(q) rPIE
+(r) PHeBIE
+(s) isoTV (b = 5)
+(t) AITV (b = 10)
+(u) AITV (full)
+(v) DR
+(w) rPIE
+(x) PHeBIE
+Figure 3: Reconstructions of the non-blind case for the Gaussian noise. Top two rows: recon-
+structions of Figures 2a-2b; bottom two rows: reconstructions of Figs. 2d-2e.
+both the non-blind and blind cases, DR, rPIE, and PHeBIE yield magnitude images with
+the worst SSIM values, and AITV outperforms its corresponding isotropic TV counterpart by
+having better SSIM values for both the magnitude and phase images. The stochastic AITV,
+particularly b = 10 or b = 20, has slightly lower magnitude SSIM values by at most 0.04 than
+the best results obtained from the deterministic, full-batch AITV. In fact, stochastic AITV
+attains the second best magnitude SSIM values in three out of the four cases considered.
+On the other hand, stochastic AITV with either b = 10 or b = 20 has the best phase SSIM
+values, outperforming its deterministic version by up to 0.19.
+For the blind case of the
+cameraman/baboon image, the deterministic AITV and isotropic TV have worse phase SSIM
+values than their stochastic counterparts. In general, the stochastic algorithm does best in
+recovering phase images with superior SSIM values while recovering the magnitude images
+with comparable SSIM values as the deterministic algorithm.
+The reconstructed images for the non-blind experiments are presented in Figure 3. DR,
+rPIE, PHeBIE, and the stochastic algorithms have artifacts in all four corners of the magnitude
+images because the corners are scanned significantly less than in the middle of the image.
+
+-STOCHASTIC ADMM FOR PTYCHOGRAPHY
+17
+(a) isoTV (b = 5)
+(b) AITV (b = 20)
+(c) AITV (full)
+(d) DR
+(e) rPIE
+(f) PHeBIE
+(g) isoTV (b = 5)
+(h) AITV (b = 20)
+(i) AITV (full)
+(j) DR
+(k) rPIE
+(l) PHeBIE
+(m) isoTV (b = 10) (n) AITV (b = 10)
+(o) AITV (full)
+(p) DR
+(q) rPIE
+(r) PHeBIE
+(s) isoTV (b = 10)
+(t) AITV (b = 10)
+(u) AITV (full)
+(v) DR
+(w) rPIE
+(x) PHeBIE
+Figure 4: Reconstructions of the blind case for Gaussian noise.
+100
+101
+102
+103
+epoch
+105
+106
+107
+108
+Figure 5: Amplitude Gaussian metric plotted across 600 epochs for the blind algorithms on
+the complex image given by Figure 2d-2e.
+
+-18
+KEVIN BUI AND ZICHAO (WENDY) DI
+20
+30
+40
+50
+60
+70
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Phase SSIM
+20
+30
+40
+50
+60
+70
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Magnitude SSIM
+Figure 6: Magnitude and phase SSIMs over different Gaussian noise level for the complex
+image given by Figure 2d-2e for the blind case.
+However, the deterministic AITV has no artifacts because (3.22) is solved exactly for the
+image solution z. As a result, it has higher magnitude SSIM values than their stochastic
+counterparts. Nevertheless, the stochastic algorithms yield better phase images with less noise
+artifacts than any other algorithms. For example, the phase images of Figure 2e reconstructed
+from stochastic isoTV and AITV have the least amount of cameraman remnants.
+Figure 4 shows the results of the blind algorithms. The phase images reconstructed by
+the deterministic AITV, DR, ePIE and PHeBIE are significantly worse than the stochastic
+algorithms.
+For example in Figure 2b, the contrasts of the reconstructed images by the
+deterministic AITV, DR, and PHeBIE are inconsistent as they become darker from left to
+right while the contrasts are more consistent with the stochastic algorithms. For Figure 2e,
+the stochastic algorithms perform the best in recovering the phase image while deterministic
+AITV is unable to recover the left half of the image and DR, rPIE, and PHeBIE have strong
+remnants of the cameraman present. Like in the non-blind case, stochastic AITV reconstructs
+the phase image the best.
+In Figure 5, we examine the convergence of the blind algorithms applied to the Camera-
+man/Baboon image by recording their AGM values for each epoch. We omit the convergence
+curves for isoTV since their curves are similar to their AITV counterparts. Overall, the curves
+for our proposed stochastic algorithms are decreasing, validating the numerical convergence
+of Algorithm 3.1 with AGM fidelity. However, their curves are slightly above the determinis-
+tic ADMM algorithm and rPIE. The reason why rPIE outperforms the AITV algorithms is
+because it seeks to only minimize AGM while the AITV algorithms minimize a larger objec-
+tive function given by (3.5). Overall, after several hundred epochs, our proposed stochastic
+algorithms can give comparable AGM values as the deterministic AITV and rPIE algorithms.
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+19
+Table 3: SSIM results of the algorithms applied to the Poisson corrupted measurements with
+η = 0.01. The stochastic algorithms (e.g., AITV and isoTV, b ∈ {5, 10, 20, 25}) are ran three
+times to obtain the average SSIM values. Bold indicates best value; underline indicates second
+best value.
+Non-blind
+Blind
+Chip
+Cameraman/Baboon
+Chip
+Cameraman/Baboon
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+mag.
+SSIM
+phase
+SSIM
+DR
+0.8523
+0.8455
+0.8704
+0.5043
+0.8431
+0.7387
+0.7630
+0.2529
+PHeBIE
+0.9404
+0.9398
+0.9271
+0.6791
+0.9280
+0.9082
+0.8678
+0.5470
+isoTV (b = 5)
+0.9460
+0.9151
+0.9301
+0.7193
+0.9394
+0.9001
+0.9238
+0.7105
+isoTV (b = 10)
+0.9365
+0.9212
+0.9250
+0.7085
+0.9342
+0.9064
+0.9188
+0.6979
+isoTV (b = 20)
+0.9335
+0.9397
+0.9224
+0.7008
+0.9355
+0.9248
+0.9153
+0.6905
+isoTV (b = 25)
+0.9353
+0.9465
+0.9220
+0.6992
+0.9376
+0.9315
+0.9150
+0.6907
+isoTV (full batch)
+0.9767
+0.9590
+0.9773
+0.7093
+0.9655
+0.9192
+0.9588
+0.4920
+AITV (b = 5)
+0.9526
+0.9680
+0.9319
+0.7477
+0.9409
+0.9530
+0.9242
+0.7418
+AITV (b = 10)
+0.9590
+0.9685
+0.9375
+0.7366
+0.9472
+0.9533
+0.9321
+0.7318
+AITV (b = 20)
+0.9598
+0.9682
+0.9383
+0.7171
+0.9494
+0.9526
+0.9322
+0.7114
+AITV (b = 25)
+0.9585
+0.9676
+0.9373
+0.7112
+0.9492
+0.9525
+0.9307
+0.7055
+AITV (full batch)
+0.9803
+0.9644
+0.9782
+0.7084
+0.9741
+0.9354
+0.9671
+0.4975
+Lastly, we analyze the robustness of the blind algorithms applied to Figures 2d-2e cor-
+rupted by different levels of Gaussian noise, from SNR 25 to 65. The fidelity parameter λ
+varies for different noise level of the image: λ = 100 for SNR = 25; λ = 50 for SNR = 30,
+35; λ = 10 for SNR =40, 45; λ = 5 for SNR = 50, 55, 60; and λ = 3 for SNR = 65. The
+SSIMs for the magnitude and phase images across different SNRs are plotted in Figure 6.
+For SNR ≥ 40, the deterministic algorithms have the best magnitude SSIMs than the other
+algorithms while their stochastic counterparts have slightly lower SSIMs. When SNR < 40,
+the stochastic algorithms perform the best. In fact, stochastic AITV has magnitude SSIM at
+least 0.90 across different noise levels. For the phase image, the stochastic algorithms have
+the highest SSIMs up to SNR = 55. For SNR ≥ 60, the rPIE algorithm has the best phase
+SSIM while stochastic AITV has the second best. Overall, stochastic AITV is the most stable
+across different levels of Gaussian noise.
+5.2. Poisson noise. For both the non-blind and blind case, we examine the measurements
+corrupted with Poisson noise with η = 0.01 according to (5.2). We set λ = 0.15. The numeri-
+cal results are recorded in Table 3. Note that rPIE results are excluded because the algorithm
+is tailored towards measurements corrupted with Gaussian noise [32]. Across all cases, de-
+terministic AITV attains the highest magnitude SSIM values and stochastic AITV attains
+the highest phase SSIM values while DR performs the worst in reconstructing images from
+Poisson-corrupted measurements. We observe general improvement in SSIM values for both
+magnitude and phase images by using AITV over isoTV. Although the stochastic algorithms
+have lower SSIM values than their deterministic counterparts for the magnitude images, the
+difference is at most 0.047 for AITV and at most 0.056 for isoTV. Moreover, the SSIM val-
+ues of the magnitude images from the stochastic algorithms are at least 0.91. Similar to the
+
+20
+KEVIN BUI AND ZICHAO (WENDY) DI
+Gaussian noise case, stochastic AITV reconstructs the phase image well while recovering the
+magnitude image with satisfactory quality.
+We examine the robustness of the blind algorithms on Figures 2d-2e with different level of
+Poisson noise. The noise levels we examine are η ∈ {0.005k}9
+k=1. We set the fidelity parameter
+to be λ = 15 × η. The SSIMs for the magnitude and phase images across different Poisson
+noise levels are plotted in Figure 7. We observe that the deterministic algorithms yield the
+best magnitude SSIMs while the stochastic algorithms yield the best phase SSIMs. DR yields
+the worst results for both magnitude and phase components. Although stochastic AITV yields
+the third best SSIMs for the magnitude image, its SSIMs are at least 0.90. Moreover, it has
+the best phase SSIMs, significantly more than its deterministic counterpart by at about 0.20.
+In summary, stochastic AITV is a robust method across different levels of Poisson noise.
+6. Conclusion. In this work, we propose AITV-regularized variational models for image
+ptychography, where the measurements are corrupted by either Gaussian or Poisson noise.
+To adapt the algorithm for large number of measurements, we design a stochastic ADMM
+algorithm that incorporates adaptive step sizes based on the PIE algorithms. Overall, us-
+ing both AITV regularization and stochastic ADMM, we are able to reconstruct an image
+of satisfactory quality from heavily corrupted measurements as demonstrated in our numer-
+ical experiments. In fact, the phase component of the image is best recovered through our
+proposed algorithms. Lastly, we prove theoretical convergence for the proposed stochastic
+ADMM algorithm under certain conditions and demonstrate numerical convergence in our
+experiments.
+Future directions include the design of a globally convergent algorithm for the AITV-
+regularized ptychography model, and incorporation of variance-reduced stochastic gradient
+estimators, such as SVRG [23] and SARAH [36], to accelerate convergence and improve re-
+construction quality.
+Appendix A. Proof of Lemma 3.1.
+Proof. If x′ = 0, then it is trivial, so for the rest of the proof, we assume that x′ ̸= 0.
+Suppose x∗ is the optimal solution to (3.20). We show that sgn(x∗) = sgn(x′). If x∗ = 0,
+then we can choose ci ∈ {c′ ∈ C : |c′| = 1} such that sgn(x∗)i = ci = sgn(x′)i for each
+i = 1, . . . , n2, giving the desired result. Suppose that x∗ ̸= 0. Because ∥ · ∥1 − α∥ · ∥2,1 is
+rotation invariant, we only need to examine and expand the quadratic term in (3.20). We see
+that
+∥x∗ − x′∥2
+2 =
+n2
+�
+i=1
+|(x∗)i − (x′)i|2 =
+n2
+�
+i=1
+�
+|(x∗)i|2 + |(x′)i|2 − 2|(x∗)i||(x′)i| cos θi
+�
+,
+where θi is the angle between the components (x∗)i and (x′)i. This term is minimized when
+θi = 0 for all i. This means that sgn(x∗)i = sgn(x′)i for all i, or otherwise x∗ would not be an
+optimal solution to (3.20). Hence, sgn(x∗) = sgn(x′).
+After establishing that sgn(x∗) = sgn(x′), we simplify (3.20) to an optimization problem
+with respect to |x| given by
+|x∗| =
+arg min
+ρ∈Rn, (ρ)i≥0
+∥ρ∥1 − α∥ρ∥2 + 1
+2λ∥ρ − |x′|∥2
+2.
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+21
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Phase SSIM
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+Magnitude SSIM
+Figure 7: Magnitude and phase SSIMs over different Poisson noise level for the complex image
+given by Figure 2d-2e for the blind case.
+Therefore, by applying [27, Lemma 1] to the optimization problem followed by multiplying
+the solution |x∗| by sgn(x∗), we obtain the desired results.
+Appendix B. Proofs of Section 4.
+Before proving our main results, we present prelimi-
+nary tools necessary for the convergence analysis.
+Definition B.1 ([41]). Let h : Rn2 → (−∞, +∞] be a proper and lower semicontinuouous
+function and dom h := {x ∈ Rn2 : h(x) < ∞}.
+(a) The Fr´echet subdifferential of h at the point x ∈ dom h is the set
+ˆ∂h(x) =
+�
+v ∈ Rn2 : lim inf
+y̸=x,y→x
+h(y) − h(x) − ⟨v, y − x⟩
+∥y − x∥
+≥ 0
+�
+.
+(b) The limiting subdifferential of h at the point x ∈ dom h is the set
+∂h(x) =
+�
+v ∈ Rn2 : ∃{(xt, yt)}∞
+t=1 s.t. xt → x, h(xt) → h(x), ˆ∂h(xt) ∋ yt → y
+�
+.
+We note that the limiting subdifferential is closed [41]:
+(xt, yt) → (x, y), h(xt) → h(x), yt ∈ ∂h(xt) =⇒ y ∈ ∂h(x).
+After establishing the definitions of subdifferentials in the real case, we extend them to the
+complex case. If z = z1 + z2i, where z1, z2 ∈ Rn2, then the limiting subdifferential of a proper
+and lower semicontinuous function f : Cn2 → (−∞, +∞] is defined by
+∂f(z) := ∂z1f(z) + ∂z2f(z)i.
+(B.1)
+If the function f is continuously differentiable at the point z, then with slight abuse of notation,
+we denote its gradient by ∇f(z), and ∂f(z) = {∇f(z)} [41].
+
+22
+KEVIN BUI AND ZICHAO (WENDY) DI
+B.1. Proof of Lemma 4.2.
+Proof. If {(ωt, zt)}∞
+t=1 is bounded, then there exists a constant C such that ∥ωt∥∞, ∥zt∥∞ ≤
+C for all t ∈ N. We establish that L(ω) has a Lipschitz continuous gradient with respect to
+ω. For any ω1, ω2 ∈ Cm2 at iteration t, we have
+∥∇ωL(ω2) − ∇ωL(ω1)∥2 ≤
+������
+N
+�
+j=1
+β1(Sjzt)∗ ◦ (ω2 − ω1) ◦ (Sjzt)
+������
+2
+≤
+�
+�
+N
+�
+j=1
+β1
+��(Sjzt)∗ ◦ (Sjzt)
+��
+∞
+�
+� ∥ω2 − ω1∥2 ≤ β1NC2∥ω2 − ω1∥2.
+Hence, we observe that L(ω) has a Lipschitz continuous gradient with Lipschitz constant
+Lω := β1NC2. By the descent property [8, Definition 1], at iteration t we have
+L(ωt+1) − L(ωt) ≤ R(⟨∇ωL(ωt), ωt+1 − ωt⟩) + Lω
+2 ∥ωt+1 − ωt∥2
+2
+= −δt
+ωR(⟨∇ωL(ωt), ˜∇ωL(ωt)⟩) + Lω(δt
+ω)2
+2
+∥ ˜∇ωL(ωt)∥2
+2,
+where the last equality is due to (3.12). Taking the expectation with respect to the first t
+iterations, we obtain
+Et[L(ωt+1)] − Et[L(ωt)] = −δt
+ωR
+�
+Et
+�
+⟨∇ωL(ωt), ˜∇ωL(ωt)⟩
+��
++ Lω(δt
+ω)2
+2
+Et
+�
+∥ ˜∇ωL(ωt)∥2
+2
+�
+≤ −
+�
+KL − δt
+ωLω(MV + KU)
+2
+�
+δt
+ωEt
+���∇ωL(ωt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
+,
+where the last inequality is due to combining (4.3), (4.4). and (4.7). Similarly, we can estimate
+(4.10) because we can compute that L(z) has a Lipschitz continuous gradient with Lipschitz
+constant Lz := β1NC2 + β2∥∆∥ and follow the same steps as above.
+B.2. Proof of Proposition 4.3.
+Proof. Because �∞
+t=1 ∥Ωt+1 −Ωt∥2
+2 < ∞, we have lim
+t→∞ Λt+1
+j
+−Λt
+j = 0 for each j = 1, . . . , N
+and lim
+t→∞ yt+1 − yt = 0, which implies from (3.6e)-(3.6f) that
+lim
+t→∞ ut
+j − F(ωt ◦ Sjzt) = 0
+∀j = 1, . . . , N,
+(B.2)
+lim
+t→∞ vt − ∇zt = 0.
+(B.3)
+It follows that {(ut, vt)}∞
+t=1 is bounded since {(ωt, zt)}∞
+t=1 is bounded. By (3.8), when B(·, ·)
+is AGM, we have
+∥ut+1
+j
+∥2 =
+������
+�
+dj + β1
+���F(ωt ◦ Sjzt) − 1
+β1 Λt
+j
+���
+1 + β1
+������
+2
+≥
+β1
+1 + β1
+� 1
+β1
+��Λt
+j
+��
+2 − ∥F(ωt ◦ Sjzt)∥2
+�
+,
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+23
+or equivalently,
+(1 + β1)∥ut+1
+j
+∥2 + β1∥F(ωt ◦ Sjzt)∥2 ≥ ∥Λt
+j∥2.
+(B.4)
+Similarly, when B(·, ·) is IPM, we have the same inequality as (B.4). As a result, {Λt}∞
+t=1 is
+bounded. Finally, we show that {yt}∞
+t=1 is bounded. By Lemma 3.1, we have two cases. When
+����(∇zt)i − (yt)i
+β2
+����
+∞
+≤ λ
+β2
+,
+we have
+λ
+β2
+≥
+����(∇zt)i − (yt)i
+β2
+����
+∞
+≥ 1
+β2
+��(yt)i
+��
+∞ −
+��(∇zt)i
+��
+∞ ,
+or λ + β2∥(∇zt)i∥∞ ≥ ∥(yt)i∥∞. Otherwise, we have
+∥(vt+1)i∥∞ ≥
+����
+����(∇zt)i − (yt)i
+β2
+���� − λ
+β2
+����
+∞
+≥
+����(∇zt)i − (yt)i
+β2
+����
+∞
+− λ
+β2
+≥ 1
+β2
+��(yt)i
+��
+∞ −
+��(∇zt)i
+��
+∞ − λ
+β2
+,
+or β2
+�
+∥(vt+1)i∥∞ +
+��(∇zt)i
+��
+∞
+�
++ λ ≥
+��(yt)i
+��
+∞. Altogether, {yt}∞
+t=1 is bounded since
+{(vt, zt)}∞
+t=1 is bounded. Therefore, we establish that {(Zt, Ωt)}∞
+t=1 is bounded.
+We see that
+L(Z, Ω) =
+N
+�
+j=1
+�
+B(|uj|2, dj) + β1
+2
+����uj − F(ω ◦ Sjz) + Λj
+β1
+����
+2
+2
+−
+1
+2β1
+∥Λj∥2
+2
+�
++ λ(∥v∥1 − α∥v∥2,1) + β2
+2
+����v − ∇z + y
+β2
+����
+2
+2
+−
+1
+2β2
+∥y∥2
+2
+≥
+N
+�
+j=1
+�
+B(|uj|2, dj) −
+1
+2β1
+∥Λj∥2
+2
+�
+−
+1
+2β2
+∥y∥2
+2.
+Because B(·, ·) is bounded below according to (3.2) and {Ωt}∞
+t=1 is bounded, {L(Zt, Ωt)}∞
+t=1
+is bounded below by some constant Linf. By (3.6a) and (3.6c), we have L(ut+1) ≤ L(ut) and
+L(vt+1) ≤ L(vt), respectively, so taking expectation with respect to the first t iterations, we
+obtain
+Et[L(ωt)] = Et[L(ut+1)] ≤ Et[L(ut)],
+(B.5)
+Et[L(zt)] = Et[L(vt+1)] ≤ Et[L(vt)] = Et[L(ωt+1)].
+(B.6)
+In addition, we have
+L(Λt+1) − L(Λt) =
+N
+�
+j=1
+R(⟨Λt+1
+j
+− Λt
+j, ut+1
+j
+− F(ωt+1 ◦ Sjzt+1)⟩)
+
+24
+KEVIN BUI AND ZICHAO (WENDY) DI
+= 1
+β1
+N
+�
+j=1
+���Λt+1
+j
+− Λt
+j
+���
+2
+2 = 1
+β1
+∥Λt+1 − Λt∥2
+2,
+where the second to last equality is due to (3.6e). Taking expectation with respect to the first
+t iterations gives
+Et[L(Λt+1)] − Et[L(Λt)] = 1
+β1
+Et
+���Λt+1 − Λt��2
+2
+�
+.
+(B.7)
+Similarly, we obtain
+Et[L(yt+1)] − Et[L(yt)] = 1
+β2
+Et[∥yt+1 − yt∥2
+2].
+(B.8)
+Summing up (4.9)-(4.10), (B.5)-(B.8) and taking total expectation, we have
+E[L(Zt+1, Ωt+1)] − E[L(Zt, Ωt)] ≤ 1
+β1
+Et
+���Λt+1 − Λt��2
+2
+�
++ 1
+β2
+E[∥yt+1 − yt∥2
+2]
+−
+�
+KL − δt
+ωLω(MV + KU)
+2
+�
+δt
+ωE
+���∇ωL(ωt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
+−
+�
+KL − δt
+zLz(MV + KU)
+2
+�
+δt
+zE
+���∇zL(zt)
+��2
+2
+�
++ (δt
+z)2LzM
+2
+.
+(B.9)
+By (4.11), lim
+t→∞ δt
+ω = 0 and lim
+t→∞ δt
+z = 0, which means that we can assume without generality
+that δt
+ωLω, δt
+zLz <
+KL
+MV +KU for all t ∈ N. Hence, summing up t = 1, . . . , T, we obtain
+Linf − E[L(Z0, Ω0)] ≤ E[L(ZT+1, ΩT+1)] − E[L(Z0, Ω0)]
+≤
+T
+�
+t=1
+�
+C∥Ωt+1 − Ωt∥2
+2 − KLδt
+ω
+2
+E
+���∇ωL(ωt)
+��2
+2
+�
+− KLδt
+z
+2
+E
+���∇zL(zt)
+��2
+2
+�
++ (δt
+ω)2LωM
+2
++ (δt
+z)2LzM
+2
+�
+,
+where C = max{ 1
+β1 , 1
+β2 }. Rearranging the inequality and letting T → ∞ give us
+0 ≤KL
+2
+∞
+�
+t=1
+�
+δt
+ωE
+���∇ωL(ωt)
+��2
+2
+�
++ δt
+zE
+���∇zL(zt)
+��2
+2
+��
+≤
+E[L(Z0, Ω0)] − Linf +
+∞
+�
+t=1
+�
+C∥Ωt+1 − Ωt∥2
+2 + (δt
+ω)2LωM
+2
++ (δt
+z)2LzM
+2
+�
+.
+By the assumption, the right-hand side is bounded, so therefore it implies (4.12).
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+25
+B.3. Proof of Theorem 4.4.
+Proof. If {(wt, zt)}∞
+t=1 is bounded, then ∥wt∥2, ∥zt∥2 ≤ C for all t ∈ N for some constant
+C > 0. At a given iteration t, we have ∇ωL(ωt+1) = 0 by the first-order optimality condition
+of (3.6b) and ∇ωL(ω) to be Lipschitz. It follows that
+∥∇ωL(ωt)∥2 = ∥∇ωL(ωt+1) − ∇ωL(ωt)∥2 ≤ Lω∥ωt+1 − ωt∥2 ≤ 2CLω.
+Similarly, we have ∥∇zL(zt)∥2 ≤ 2CLz. As a result, by squaring the inequalities and tak-
+ing their expectation,
+��
+E
+���∇ωL(ωt)
+��2
+2
+�
+, E
+���∇zL(zt)
+��2
+2
+���∞
+t=1 is bounded. The step size
+condition (4.11) implies that lim
+t→∞ δt
+ω = 0 and lim
+t→∞ δt
+z = 0. Since the SGD steps for ω and z
+are
+ωt+1 = ωt − δt
+ω ˜∇L(ωt), zt+1 = zt − δt
+z ˜∇L(zt),
+it follows that by taking expectation with respect to the first t iterations and using Assumption
+4.1,
+Et[
+��ωt+1 − ωt��2
+2] = (δt
+ω)2Et[∥ ˜∇ωL(ωt)∥2
+2] ≤ (δt
+ω)2 �
+M + (MV + KU)Et
+���∇ωL(ωt)
+��2
+2
+��
+,
+Et[
+��zt+1 − zt��2
+2] = (δt
+z)2Et[∥ ˜∇zL(zt)∥2
+2] ≤ (δt
+z)2 �
+M + (MV + KU)Et
+���∇zL(zt)
+��2
+2
+��
+.
+Because
+��
+E
+���∇ωL(ωt)
+��2
+2
+�
+, E
+���∇zL(zt)
+��2
+2
+���∞
+t=1 is bounded, we apply total expectation and
+take the limit to obtain
+lim
+t→∞ E[
+��ωt+1 − ωt��2
+2] ≤ lim
+t→∞(δt
+ω)2 �
+M + (MV + KU)E
+���∇ωL(ωt)
+��2
+2
+��
+= 0,
+(B.10)
+lim
+t→∞ E[
+��zt+1 − zt��2
+2] ≤ lim
+t→∞(δt
+z)2 �
+M + (MV + KU)E
+���∇zL(zt)
+��2
+2
+��
+= 0.
+(B.11)
+Earlier, in the proof of Proposition 4.3, we obtain
+lim
+t→∞ ut
+j − F(ωt ◦ Sjzt) = 0,
+∀j = 1, . . . , N,
+(B.12)
+lim
+t→∞ vt − ∇zt = 0.
+(B.13)
+By Proposition 4.3, we have {(Zt, Ωt)}∞
+t=1 to be bounded and (4.12) to be true.
+Because {Zt}∞
+t=1 is bounded, there exists a convergent subsequence {(Ztk, Ωtk)}∞
+k=1 and
+a point (Z⋆, Ω⋆) such that lim
+k→∞(Ztk, Ωtk) = (Z⋆, Ω⋆). In addition, (4.12) and (B.10)-(B.13)
+hold for the subsequence and its further subsequences. (B.10)-(B.11) imply that there is a
+further subsequence {(Ztkℓ, Ωtkℓ)}∞
+ℓ=1 such that
+lim
+ℓ→∞ ωtkℓ+1 − ωtkℓ = 0 a.s.,
+(B.14)
+lim
+ℓ→∞ ztkℓ+1 − ztkℓ = 0 a.s.
+(B.15)
+
+26
+KEVIN BUI AND ZICHAO (WENDY) DI
+From (4.12), we obtain lim
+ℓ→∞ E
+�
+δ
+tkℓ
+ω ∥∇ωL(ωtkℓ)∥2
+2 + δ
+tkℓ
+z ∥∇zL(ztkℓ)∥2
+2
+�
+= 0, so we obtain
+lim inf
+ℓ→∞ E[∥∇ωL(ωtkℓ)∥2
+2] = 0
+(B.16)
+lim inf
+ℓ→∞ E[∥∇zL(ztkℓ)∥2
+2] = 0.
+(B.17)
+Now we show that (Z⋆, Ω⋆) is a stochastic KKT point a.s. From (B.12)-(B.13), we have
+u⋆
+j = lim
+ℓ→∞ u
+tkℓ
+j
+= lim
+ℓ→∞ F(ωtkℓ ◦ Sjztkℓ) = F(ω⋆ ◦ Sjz⋆),
+(B.18)
+v⋆ = lim
+ℓ→∞ vtkℓ = lim
+ℓ→∞ ∇ztkℓ = ∇z⋆.
+(B.19)
+By (B.12) and (B.14)-(B.15), we have
+lim
+ℓ→∞ utkℓ+1 = lim
+ℓ→∞ F
+�
+ωtkℓ+1 ◦ Sjztkℓ+1�
+= F
+�
+lim
+ℓ→∞ ωtkℓ+1 ◦ Sj
+�
+lim
+ℓ→∞ ztkℓ+1
+��
+= F
+�
+lim
+ℓ→∞ ωtkℓ ◦ Sj
+�
+lim
+ℓ→∞ ztkℓ
+��
+= lim
+ℓ→∞ F
+�
+ωtkℓ ◦ Sjztkℓ�
+= lim
+ℓ→∞ utkℓ a.s.
+As a result, we have
+lim
+ℓ→∞ utkℓ+1 − utkℓ = 0 a.s.
+(B.20)
+Similarly, by (B.13) and (B.15), we have
+lim
+ℓ→∞ vtkℓ+1 − vtkℓ = 0 a.s.
+(B.21)
+For the sake of brevity, we will omit “a.s.” henceforth.
+Next we prove (4.1a). Suppose that B(·, ·) is AGM. At iteration tkℓ, the first-order opti-
+mality condition of (3.6a) is
+0 ∈ ∂
+���u
+tkℓ+1
+j
+���
+�
+|u
+tkℓ+1
+j
+| −
+�
+dj
+�
++ Λ
+tkℓ
+j
++ β1
+�
+u
+tkℓ+1
+j
+− F(ωtkℓ ◦ Sjzkℓ)
+�
+,
+(B.22)
+where
+�
+∂
+���u
+tkℓ+1
+j
+���
+�
+i =
+�
+�
+�
+�
+�
+�
+�
+(u
+tkℓ+1
+j
+)i
+|(u
+tkℓ+1
+j
+)i|
+,
+if (u
+tkℓ+1
+j
+)i ̸= 0
+{u ∈ C : |u| ≤ 1}
+if (u
+tkℓ+1
+j
+)i = 0.
+If (u⋆
+j)i ̸= 0 for some i ∈ {1, . . . , n2}, then there exists a neighborhood Br((u⋆
+j)i) = {u ∈ C :
+|u − (u⋆
+j)i| ≤ r} such that all points in Br((u⋆
+j)i) are nonzero. By (B.20) and the fact that
+lim
+ℓ→∞ utkℓ = u⋆, we have (u
+tkℓ+1
+j
+)i ̸= 0 for all tkℓ sufficiently large. As a result, (B.22) becomes
+0 =
+(u
+tkℓ+1
+j
+)i
+|(u
+tkℓ+1
+j
+)i|
+�
+|(u
+tkℓ+1
+j
+)i| − (
+�
+dj)i
+�
++ (Λ
+tkℓ
+j )i + β1
+�
+(u
+tkℓ+1
+j
+)i − (F(ωtkℓ ◦ Sjzkℓ))i
+�
+,
+(B.23)
+
+STOCHASTIC ADMM FOR PTYCHOGRAPHY
+27
+By (B.12) and (B.20), we take the limit to obtain
+0 =
+(u⋆
+j)i
+|(u⋆
+j)i|
+�
+|(u⋆
+j)i| − (
+�
+dj)i
+�
++ (Λ⋆
+j)i.
+(B.24)
+On the other hand, when (u⋆
+j)i = 0 for some index i, we have (u
+tkℓ
+j )i ∈ Br((u⋆
+j)i) \ {(u⋆
+j)i} to
+be nonzero. Taking the absolute value of (B.23) gives us
+���(Λ
+tkℓ
+j )i + β1
+�
+(u
+tkℓ+1
+j
+)i − (F(ωtkℓ ◦ Sjzkℓ))i
+���� =
+���|(u
+tkℓ+1
+j
+)i| − (
+�
+dj)i
+��� .
+Again, by (B.12) and (B.20), taking the limit yields
+��(Λ⋆
+j)i
+�� =
+���(
+�
+dj)i
+��� ,
+which implies that there exists u′ ∈ {u ∈ C : |u| ≤ 1} such that −u′(
+�
+dj)i + (Λ⋆
+j)i = 0.
+This result and (B.24) give us (4.1a) for the AGM case.
+As for the IPM case, because
+lim
+x→0+ x − d log x = +∞, it follows that (u⋆
+j)i ̸= 0 for all i, so we only need to worry about the
+nonzero case. Hence, verifying (4.1a) for IPM requires computation similar to the nonzero
+case for AGM, so it is omitted for brevity.
+At iteration tkℓ, the first-order optimality condition of (3.6c) is
+−ytkℓ
+λ
+− β2
+λ
+�
+vtkℓ − ∇ztkℓ�
+∈ ∂
+�
+∥vtkℓ+1∥1 − α∥vtkℓ+1∥2,1
+�
+.
+(B.25)
+By (B.13) and (B.21), we have
+lim
+ℓ→∞ −ytkℓ
+λ
+− β2
+λ
+�
+vtkℓ − ∇ztkℓ�
+= −y⋆
+λ ,
+lim
+ℓ→∞ vtkℓ+1 = lim
+ℓ→∞ vtkℓ = v⋆.
+By continuity, lim
+ℓ→∞ ∥vtkℓ+1∥1 − α∥vtkℓ+1∥2,1 = ∥v⋆∥1 − α∥v⋆∥2,1. Altogether, by closedness of
+limiting subdifferential, we establish (4.1b).
+Lastly, we prove (4.1e′)-(4.1f′). At iteration tkℓ, we have
+∇ωL(ωtkℓ) = −
+N
+�
+j=1
+�
+�
+�β1(Sjztkℓ)∗ ◦
+�
+�F−1
+�
+�u
+tkℓ+1
+j
++
+Λ
+tkℓ
+j
+β1
+�
+� − ωtkℓ ◦ Sjztkℓ
+�
+�
+�
+�
+� .
+Taking the limit, we see that
+∇ωL(Z⋆, Ω⋆) = −
+N
+�
+j=1
+�
+β1(Sjz⋆)∗ ◦
+�
+F−1
+�
+u⋆
+j +
+Λ⋆
+j
+β1
+�
+− ω⋆ ◦ Sjz⋆
+��
+= lim
+ℓ→∞ ∇ωL(ωtkℓ).
+Applying Fatou’s Lemma to (B.16), we have E
+�
+∥∇ωL(Z⋆, Ω⋆)∥2
+2
+�
+≤ lim inf
+ℓ→∞ E[∥∇ωL(ωtkℓ)∥2
+2] =
+0. Proving (4.1f′) is similar to proving (4.1e′), so we omit the proof. Altogether, (Z⋆, Ω⋆) is a
+stochastic KKT point a.s.
+
+28
+KEVIN BUI AND ZICHAO (WENDY) DI
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+

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+Under consideration for publication in J. Plasma Phys.
+1
+Energetic bounds on gyrokinetic instabilities.
+Part III. Generalized free energy.
+G. G. Plunk1†, and P. Helander1
+1Max-Planck-Institut für Plasmaphysik, 17491 Greifswald, Germany
+(Received xx; revised xx; accepted xx)
+Free energy, widely used as a measure of turbulence intensity in weakly collisional
+plasmas, has been recently found to be a suitable basis to describe both linear and
+nonlinear growth in a wide class gyrokinetic systems. The simplicity afforded by this
+approach is accompanied by some drawbacks, notably the lack of any explicit treatment
+of wave-particle effects, which makes the theory unable to describe things like stability
+thresholds or dependence on the geometry of the background magnetic field. As a step
+toward overcoming these limitations, we propose an extension of the theory based on
+a generalization of free energy. With this it is demonstrated that resonance effects
+are recovered, and the bounds on growth are significantly reduced. The simplicity
+and efficient computation of the associated “optimal” growth rates makes the theory
+potentially applicable to stellarator optimization.
+1. Introduction
+This is the third paper in a series (Helander and Plunk 2022; Plunk and Helander 2022),
+in which we develop a linear and nonlinear stability theory based on gyrokinetic energy
+balance. The last two papers used Helmholtz free energy, and introduced the concept of
+optimal mode growth for fully electromagnetic gyrokinetic. The present paper proposes
+a generalized energetic measure of fluctuations, allowing the inclusion of additional
+instability mechanisms. We do this first for a simple case, namely the electrostatic limit
+(low plasma β) with only one kinetic species (ions), with the electrons being treated
+adiabatically. These simplifications limit the application to ion-temperature-gradient
+(ITG) driven turbulence, though the central result of the paper is capable of treating
+completely general details of the magnetic geometry.
+Free energy is a useful concept for understanding nonlinear and linear aspects of
+plasma turbulence. At the level of linear instabilities it is common to speak of a source
+of free energy that drives modes. Indeed, without a source of free energy, provided
+by background plasma gradients (density, temperature, flows), there can be no linear
+instabilities (nor can there be subcritical turbulence Landreman et al. (2015); Plunk and
+Helander (2022)). However, there is usually another ingredient that arises in the detailed
+analysis of normal linear instabilities, namely the wave-particle resonance. In gyrokinetic
+theory, this involves parallel motion (along the magnetic field) and magnetic drift, and
+the resonance is physically linked to the work that the electrostatic field performs on
+gyrocenter motion. However, the terms needed to capture this do not contribute to free
+energy balance, and the influence of resonance therefore cannot be accounted for by the
+optimal modes that we introduced in our previous works.
+In this work we propose a new measure of gyrokinetic fluctuations, a generalization
+of the concept of free energy, that incorporates the resonance mechanism, and, via the
+† Email address for correspondence: gplunk@ipp.mpg.de
+arXiv:2301.00988v1  [physics.plasm-ph]  3 Jan 2023
+
+2
+G. G. Plunk and P. Helander
+magnetic drift, the full details of the background magnetic geometry. We demonstrate
+the existence of a class of quadratic measures closely related to Helmholtz free energy
+that behave as positive-definite norms for fluctuations in the distribution function. The
+corresponding energy balance equation is then used to derive a theory of optimal modes
+that most efficiently extract this energy from its source. The growth rate of these optimal
+modes provides a rigorous upper bound on the growth rate of linear instabilities, and this
+bound is shown to be lower than that obtained previously from Helmholtz free energy.
+By studying some simple limits, we show that we recover some expected behavior of both
+the slab and toroidal branches of the ITG mode.
+2. Definitions and gyrokinetic energy balance
+The ion gyrokinetic equation in the electrostatic limit is written
+∂gk
+∂t + v∥
+∂gk
+∂l + i˜ωdgk + 1
+B2
+�
+k′
+B · (k × k′)δφk′gk−k′ = eiF0
+Ti
+� ∂
+∂t + iωT
+∗
+�
+δφk,
+(2.1)
+where g is the gyro-center dependent part of the perturbed ion distribution function, i.e.
+fi = (1 − eiδφ(r)/Ti) Fi0 + g(R, Ei, µi, t). Its phase space variables are the energy Ea =
+mav2/2 + eaΦ(ψ) and the magnetic moment µa = mav2
+⊥/(2B), and the perpendicular
+wavenumber is k = k⊥ = kψ∇ψ + kα∇α with kψ and kα independent of the arc length
+l along the magnetic field, and ψ and α defined via B = Bb = ∇ψ × ∇α. We neglect
+collisions here†, and used the simplified notation gk = gi,k, and ω∗ = ω∗i, etc because the
+adiabatic approximation ge,k = 0 is assumed throughout‡. We will also assume kρi ∼ 1,
+implying kρe ≪ 1. The gyrokinetic free energy balance equation obtained in this limit
+reads
+d
+dt
+�
+k
+H = 2
+�
+k
+D,
+(2.2)
+where the drive term D is
+D(k, t) = Im ei
+��
+gkωT
+∗ δφ
+∗
+kd3v
+�
+,
+(2.3)
+and the free energy, expressed in terms of the gyrocenter distribution function
+H(k, t) =
+�
+Ti
+� |gk|2
+Fi0
+d3v −
+�
+a
+nae2
+a
+Ta
+|δφk|2
+�
+,
+(2.4)
+where the space average is defined as (see also Helander and Plunk (2022) for general-
+izations)
+⟨· · ·⟩ = lim
+L→∞
+� L
+−L
+(· · · )dl
+B
+� � L
+−L
+dl
+B .
+(2.5)
+The diamagnetic frequencies are
+† We do not retain collisions, since we will not be able to fix the sign of its contribution in
+our later analysis.
+‡ Here we do not include the customary correction for the zonal component (Dorland and
+Hammett 1993), but it does not affect the subsequent analysis, as the growth of this component
+is always zero because it has no free energy source (D = 0 for kα = 0).
+
+Energetic bounds. Part III
+3
+ω∗a = kαTa
+ea
+d ln na
+dψ
+,
+ωT
+∗a = ω∗a
+�
+1 + ηa
+�mav2
+2Ta
+− 3
+2
+��
+,
+and the magnetic drift frequency is
+˜ωd = k · vd,
+where the magnetic drift velocity is vd = ˆb × ((v2
+⊥/2)∇ ln B + v2
+∥κ)/Ωi, κ = ˆb · ∇ˆb, and
+Ωa = eaB/ma is gyrofrequency. Assuming ∇ ln B ≈ κ (low plasma β), we can separate
+the drift frequency into velocity-dependent and space-dependent factors following Plunk
+et al. (2014)†:
+˜ωd = ωd(l)
+�
+v2
+⊥
+2v2
+th
++
+v2
+∥
+v2
+th
+�
+.
+(2.6)
+The gyro-averaged electrostatic potential is denoted
+δφk = J0
+�k⊥v⊥
+Ωi
+�
+δφk,
+and the quasi-neutrality condition is
+�
+a
+nae2
+a
+Ta
+δφk = ei
+�
+gkJ0d3v, ,
+(2.7)
+where Jn = Jn(k⊥v⊥/Ωi). Following our previous convention, we define the free energy
+as twice that which appears in some other publications. Henceforth, we suppress the
+k-subscripts.
+2.1. Electrostatic energy and positive-definiteness of free energy
+It is useful to decompose the free energy into a part associated with a perturbed
+distribution function and a part associated with fluctuations in the electrostatic field,
+i.e.
+H = G + E,
+(2.8)
+where
+G = −TiSi =
+�
+Ti
+� |δF|2
+Fi0
+d3v
+�
+(2.9)
+E =
+�
+(τ + 1 − Γ0) nie2
+i
+Ti
+|δφ|2
+�
+.
+(2.10)
+Recall the conventional definitions Γn(b) = exp(−b)In(b) and b = k2
+⊥ρ2
+i = k2
+⊥Ti/(miΩ2
+i ),
+and τ = (eTi)/(eiTe). Note that δF = g−(eiδφ/Ti)F0 is the gyro-averaged perturbed dis-
+† Actually, there is spatial dependence in both v⊥ and v∥, since these are not the proper
+gyrokinetic phase-space variables, but a separation like this is useful to make contact with
+known limits from gyrokinetic theory of the ITG mode.
+
+4
+G. G. Plunk and P. Helander
+tribution function, and these two contributions to H can be identified as the gyrokinetic
+perturbed entropy and the gyrokinetic field energy.
+Although the general electromagnetic free energy admits a similar form as Eqn. 2.8
+(see for instance Helander and Plunk (2022)), we note that the electrostatic limit is
+distinguished by the fact that the field contribution E is itself a nonlinear invariant of
+the gyrokinetic system (Schekochihin et al. 2009), and its conservation may be viewed
+as an additional constraint on the nonlinear dynamics, with consequences e.g. for the
+cascade and production of large-scale E × B flows (PLUNK et al. 2010).
+For what follows, we need the electrostatic energy balance equation. This is obtained
+by multiplying the ion gyrokinetic equation by eiδφ
+∗ integrate over velocity, average over
+the parallel coordinate l, and sum over perpendicular wavenumber k, yielding (Helander
+et al. 2013)
+d
+dt
+�
+k
+E = 2
+�
+k
+K,
+(2.11)
+where the drive term K is
+K = −Re ei
+��
+δφ
+∗ �
+v∥
+∂
+∂l + iωd
+�
+gd3v
+�
+.
+(2.12)
+This is composed to two contributions, one coming from the parallel streaming term,
+and the other coming from the magnetic drift term. The first contribution has a simple
+physical interpretation, as the rate of energy exchanged between particles and the parallel
+electric field (i.e. the volume average of the parallel current multiplied by the parallel
+electric field), while the second term describes an analogous process in the perpendicular
+direction associated with the drift motion of gyrocenters.
+Eqn. 2.8 is a physically transparent form that makes it clear that the free energy H is
+a positive-definite norm for the distribution function g†, i.e.
+H ⩾ 0, and H = 0 iff g = 0
+(2.13)
+over all of phase space, ℓ and v. To see this, note that the quantities G and E are both
+positive, i.e. G ⩾ 0, obviously, and E ⩾ 0 because Γ0 ⩽ 1. Therefore if H = 0 then both
+E = 0 and G = 0. The first implies δφ = 0 everywhere, while the second implies δF = 0
+over all of phase space; δφ = 0 and δF = 0 obviously implies g = 0.
+We note that positive-definiteness is a desirable property of an energetic measure
+that can be useful for setting bounds on the growth rate of fluctuations; if a non-
+zero fluctuation (g ̸= 0) has zero measure M then the rate of growth d ln M/dt can
+be unbounded.
+Although we mainly consider a plasma with a single kinetic ion species and adiabatic
+electrons, the concepts and the formalism carry over to the more general case of a plasma
+with an arbitrary number of kinetic species, as shown in Appendix A. An important
+limitation, however, is that magnetic fluctuations and collisions are neglected.
+2.2. Generalized Free Energy
+The positive definiteness of H suggests a family of related quadratic energetic measures
+that are also positive definite. In particular it is clear that something of the form
+† By extension, using Eqn. 2.7, H can be shown to also be a positive-definite norm for the total
+deviation of the distribution function δf = g − (eiφ/Ti)F0 from the zeroth-order Maxwellian.
+
+Energetic bounds. Part III
+5
+˜H = H − ∆E,
+(2.14)
+will be positive-definite, by the same arguments of the previous section, for particular
+values of the parameter ∆. For instance the choice ∆ < 1 allows trivial generalization of
+the arguments, but we will see that the value can be extended beyond this.
+To find a range of permissible values of ∆, we will consider a diagonalization of ˜H,
+meaning that we will define a distribution function ˜g, which allows the energy to be
+expressed using to the Euclidean norm,
+˜H = ||˜g||2 = (˜g, ˜g),
+(2.15)
+where we have introduced the inner product
+(˜g1, ˜g2) =
+�
+Ti
+� ˜g∗
+1˜g2
+F0
+d3v
+�
+.
+(2.16)
+To find the relationship between ˜g and g, we introduce the Ansatz ˜g = g −νJ0F0eiδφ/Ti,
+substitute this into Eqn. 2.15, using also Eqn. 2.10, and solve for the free parameter ν,
+yielding
+ν = 1
+Γ0
+�
+1 + τ −
+�
+(1 + τ − Γ0)(1 + τ − ∆Γ0)
+�
+,
+(2.17)
+where we have taken the negative root for convenience. Observe that in order for ν to be
+real, we must have
+∆ ⩽ (1 + τ)/Γ0.
+(2.18)
+The parameter ∆ can of course be negative, in which case its magnitude is unbounded.
+Noting that Γ0 generally depends on k, we may also assume the more restrictive ∆ ⩽
+(1 + τ) to ensure that ˜H remains a nonlinear invariant.
+We pause to note that the choice ∆ = 0 yields a novel form of the conventional
+(Helmholtz) free energy, immediately suggesting what can be considered as the phase-
+space density of free energy, namely the quantity Ti|˜g|2/F0, for which there has not yet
+been an expression available.†
+It is useful now to write quasi-neutrality in terms of ˜g,
+ei
+Ti
+δφ = α
+ni
+�
+˜gJ0d3v,
+(2.19)
+where
+α =
+1
+�
+(1 + τ − Γ0)(1 + τ − ∆Γ0)
+.
+(2.20)
+Finally, we can show that ˜H is positive-definite. First, positivity follows from Eqn. 2.15,
+and it is obvious from Eqn. 2.7 that if g = 0 then δφ = 0 so that E and H both vanish,
+implying ˜H = 0. On the other hand, if we assume that ˜H = 0, then Eqn. 2.15 implies that
+˜g = 0, and Eqn. 2.19 implies that δφ = 0, from which we conclude g = 0. In summary,
+˜H ⩾ 0 and ˜H = 0 iff g = 0.
+† The idea for a phase-space density of free energy (i.e. a quantity that can be directly
+integrated over phase space to yield the total free energy) was suggested by Teaca.
+
+6
+G. G. Plunk and P. Helander
+3. Modes of optimal growth
+A key point in introducing the generalization of free energy ˜H is that this quantity
+introduces wave-particle effects (parallel resonance and drift resonance) that enter the
+electrostatic energy balance equation, Eqn. 2.11. Note that the case ∆ = 0 (i.e. the
+“conventional” Helmholtz free energy) is included as a limit ∆ = 0 and so the most
+stringent bound on growth obtained from the generalized free energy will be at least as
+good as the known bound obtained from the Helmholtz free energy.
+Note that, as long as the parameter ∆ is independent of k, the quantity ˜H is conserved
+by the nonlinearity, i.e. under summation over k. This is because it is a linear combination
+of two nonlinear invariants. One simply combines Eqns. 2.2 and 2.11 to obtain
+d
+dt
+�
+k
+˜H = 2
+�
+k
+(D − ∆K),
+for ∆ independent of k,
+(3.1)
+i.e. the change of this measure is due to the drive terms of electrostatic and free energy,
+and is otherwise conserved by the turbulent interactions. It is potentially useful to also
+consider ∆ that does depend on k, for the purpose of obtaining bounds on linear growth,
+but the nonlinear implications will be less clear in that case.
+In direct analogy to how modes of optimal free energy growth were defined, we
+introduce a rate Λ
+Λ = (D − ∆K)/ ˜H
+(3.2)
+to be optimized over the space of ion distribution functions g. We note the bound on
+conventional gyrokinetic instability growth rates,
+γL ⩽ max
+g
+Λ.
+(3.3)
+Having already found a diagonal form of the generalized free energy, Eqn. 2.15, we
+need not use a variational approach to find the states of extremal Λ. We simply identify
+the Hermitian linear operators associated with the input of free energy and electrostatic
+energy, i.e.
+D = (˜g, D˜g),
+(3.4)
+K = (˜g, K˜g)
+(3.5)
+Using Eqn. 2.19 and Eqns. 2.3 and 2.12, and some straightforward algebra (see Appendix
+B), we obtain
+D˜g = iα
+2ni
+J0F0ηω∗
+�
+d3v′J′
+0˜g′
+�� v
+vth
+�2
+−
+� v′
+vth
+�2�
+,
+(3.6)
+where primes denote evaluation at v′ and vth =
+�
+2Ti/mi. For convenience, the operator
+K can be split into its parallel and perpendicular components as K = K∥ + Kd, for which
+we obtain
+Kd˜g = iα
+2ni
+ωd(ℓ)F0J0
+�
+d3v′J′
+0˜g′
+�
+�
+�
+v⊥
+√
+2vth
+�2
++
+� v∥
+vth
+�2
+−
+�
+v′
+⊥
+√
+2vth
+�2
+−
+�
+v′
+∥
+vth
+�2�
+� ,
+(3.7)
+
+Energetic bounds. Part III
+7
+and
+K∥˜g = α
+2ni
+F0
+�
+J0
+�
+−B ∂
+∂l
+� 1
+B
+�
+d3v′v′
+∥J′
+0˜g′
+�
++
+�
+d3v′v′
+∥
+∂J′
+0
+∂l ˜g′
+�
++v∥
+∂
+∂l
+�
+J0
+�
+d3v′J′
+0˜g′
+��
+.
+(3.8)
+In deriving Eqn. 3.8, it is important to note that the parallel derivative is taken at fixed
+magnetic moment and particle energy, and that the velocity-space volume element d3v
+is proportional to B/v∥ in these variables. More details are given in Appendix A. The
+kinetic eigenvalue problem can be stated now as
+Λ˜g = (D − ∆K) ˜g,
+(3.9)
+where solutions (Λ, g(l, v)) realize modes of optimal growth of ˜H. The analysis of this
+eigenproblem is greatly simplified by adopting a moment form.
+3.1. Moment form of eigenproblem
+As found in the preceding papers, there are natural moments that appear in the
+energy input terms that can be identified to reduce the dimensionality of the problem
+substantially. Upon inspecting the energy balance equations one finds the following key
+dimensionless integrals:
+κ1 =
+�
+d3vJ0˜g/ni,
+(3.10)
+κ2 =
+�
+d3v
+� v2
+v2
+th
+�
+J0˜g/ni,
+(3.11)
+κ3 =
+�
+d3v
+�
+v2
+⊥
+2v2
+th
++
+v2
+∥
+v2
+th
+�
+J0˜g/ni,
+(3.12)
+κ4 =
+�
+d3v
+� v∥
+vth
+�
+J0˜g/ni,
+(3.13)
+κ5 =
+�
+d3v
+� v∥
+vth
+� ∂J0
+∂l ˜g/ni,
+(3.14)
+where κ1 is a density-like moment, κ2 and κ3 are pressure-like, κ4 is parallel ion flow,
+while κ5 is more abstract.
+It is easy to recognize these integrals on the right hand side of Eqns. 3.6, 3.7 and
+3.8, and straightforward to rewrite those equations in moment form. The dimensional
+reduction is achieved by taking moments of the these equations to obtain a coupled set
+of five fluid equations. These, which are given in Appendix C, can be combined, leading,
+after lengthy algebra, to a relatively simple second order ordinary differential equation,
+the main result of this paper:
+�4Λ2
+α2 + (∆ωdG3 − ηω∗G1)2 − G0
+�
+(ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2
+dG5
+��
+κ1
+= ∆2v2
+thG0B
+�
+− ∂
+∂l
+�G0,2
+B
+∂κ1
+∂l
+�
++ G′′
+0,2
+B κ1 − ∂
+∂l
+�G′
+0,2
+B
+�
+κ1
+�
+,
+(3.15)
+
+8
+G. G. Plunk and P. Helander
+The functions Gm,n, G′
+m,n, and G′′
+m,n, which depend on arc length via b(l) and B(l), are
+defined in terms of integrals involving Bessel functions, and are evaluated in Appendix
+D. The other b-dependent factors (G0-G5) can be expressed in terms of Gm,n, and are
+evaluated in terms of more elementary Bessel functions in Appendix D.1.
+In Eqn. 3.15 we see the eigenvalue Λ entering quadratically, reflecting the fact that
+there will be two real roots, one positive and one negative, owing to Hermiticity and
+time-reversal symmetriy of the full eigenproblem, Eqn. 3.9. Note that the terms arising
+from the parallel drive of electrostatic energy are placed on the right hand side. In the
+following section, we will consider some simple limits of this equation, and leave its more
+general solution for a future publication.
+4. Simple limits
+In this section we will consider some simple limits applied to Eqn. 3.15, and draw some
+comparison to linear theory of the main instability targeted by limit of this paper, the
+ion temperature gradient (ITG) mode (see for instance Plunk et al. (2014)). To start,
+we note that taking ∆ = 0, so that ˜H becomes the conventional Helmholtz free energy,
+yields
+Λ2 =
+(ηω∗)2
+4(1 + τ − Γ0)(1 + τ)
+�
+G0G2 − G2
+1
+�
+,
+(4.1)
+which matches Eqn. 6.20 of Helander and Plunk (2022).
+In considering other simplifications, we first should note that the adiabatic electron ap-
+proximation already neglects a trapped particle population, which is not really consistent
+unless we take the magnetic field strength to be independent of arc length
+∂B
+∂l = 0.
+(4.2)
+We have avoided making this approximation explicitly, since the present paper lays the
+foundation for extensions, in which it will be useful to include variation in B. Making
+the approximation now leads to minor simplifications of Eqn. 3.15, where all the explicit
+factors of B drop out of the right-hand side. A more significant simplification is achieved
+by assuming unsheared and uniform magnetic geometry, in particular
+∂b
+∂l = 0,
+(4.3)
+∂ωd
+∂l
+= 0,
+(4.4)
+In this limit, all of the coefficients of Eqn. 3.15 are constants, and a simple dispersion
+relation is the obtained by taking ∂κ1/∂l = ik∥κ1. We find
+4Λ2
+α2 + (∆ωdG3 − ηω∗G1)2 − G0
+�
+(ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2
+dG5
+�
+= ∆2k2
+∥v2
+thG2
+0/2.
+(4.5)
+were we have used G0,2 = G0/2. As noted in Section 2.2, the quantity ∆ is a free
+parameter, over which we can optimize Λ to improve the bounds on the growth rate of
+fluctuations.
+
+Energetic bounds. Part III
+9
+4.1. Slab ITG mode
+Setting ωd = 0 leaves only the slab branch of the ITG mode, driven by the temperature
+gradient, and involving ion parallel resonance. Eqn. 4.5 reduces to
+4Λ2
+(ηω∗)2α2 = G0G2 − G2
+1 + ∆2κ−2
+∥ G2
+0/2,
+(4.6)
+where κ∥ = ηω∗/(k∥vth). Because G0G2−G2
+1 ⩾ 0, the two contributions on the right hand
+side are both positive but the solution for which Λ is minimal is actually not obtained
+for ∆ = 0, due to the implicit dependence of α on ∆ given by Eqn. 2.20.
+To obtain the value of ∆ which yields an optimal bound, we can look for extrema of
+Λ2/(ηω∗)2, i.e.
+d
+d∆
+�
+G0G2 − G2
+1 + ∆2κ−2
+∥ G2
+0/2
+(1 + τ − Γ0)(1 + τ − ∆Γ0)
+�
+= 0.
+(4.7)
+This results in a quadratic equation for ∆ that is still rather complicated so we will
+consider the limit b → 0; see Appendix D.2 for the relevant limits of Gm,n, etc. Applying
+the limit to Eqn. 4.6 yields
+Λ2
+(ηω∗)2 =
+3 + ∆2/κ2
+∥
+8τ(1 + τ − ∆)
+(4.8)
+This solution diverges as ∆ approaches 1+τ; recall that this is the upper limit allowed
+by Eqn. 2.18. It also grows in an unbounded fashion as ∆ → −∞. There is an optimal
+value giving minimal |Λ|, obtained by solving Eqn. 4.7 in this limit. This solution, denoted
+as ∆min, is
+∆min = 1 + τ −
+�
+(1 + τ)2 + 3κ2
+∥
+(4.9)
+where the negative root has been selected to be consistent with Eqn. 2.18. Substituting
+this solution into Eqn. 4.8 gives
+Λ2
+min =
+(ηω∗)2
+4¯κ2
+∥τ(1 + τ)
+��
+1 + 3¯κ2
+∥ − 1
+�
+,
+(4.10)
+where we define ¯κ∥ = κ∥/(1 + τ). This reaches its maximum value in the limit ¯κ∥ → 0,
+and is a decreasing function of |¯κ∥|, i.e.
+Λ2
+min =
+�
+3
+8τ(τ+1)(ηω∗)2,
+for ¯κ∥ → 0,
+√
+3
+4τ |ηω∗k∥vth|,
+for |¯κ∥| ≫ 1,
+(4.11)
+Physically, the first result implies that when drive (ηω∗) is much smaller than the parallel
+transit frequency (k∥vth), the best bound is equal to that obtained by free energy (∆ = 0).
+In this case, the bound is consistent from expectations of the growth rate of a resonant
+slab ITG mode, i.e. γL ∼ ηω∗.
+In the opposite limit (¯κ∥ ≫ 1), however, when the drive large, i.e. in the so-called
+non-resonant or “fluid” limit, we obtain a much lower bound, essentially the geometric
+mean of the drive and the parallel transit frequency k∥vth. We note that this bound is
+not as low as what is obtained from the non-resonant solution of the dispersion relation
+(without density gradient), i.e. γL ∼ ηω1/3
+∗
+(k∥vth)2/3 (Plunk et al. 2014), but nevertheless
+captures the expected weakening (relative to the resonant result) qualitatively.
+
+10
+G. G. Plunk and P. Helander
+It is interesting to observe that this latter limit corresponds to ∆min → −∞, making
+˜H in some sense dominated by the electrostatic component.
+4.2. Toroidal ITG mode
+Now taking k∥vth to be small, we can neglect the right-hand side of Eqn. 4.5, leaving
+4Λ2
+α2 = G0
+�
+(ηω∗)2G2 − 2∆ωdηω∗G4 + ∆2ω2
+dG5
+�
+− (∆ωdG3 − ηω∗G1)2 .
+(4.12)
+To derive the optimal choice of ∆, we again take the b → 0 limit and obtain from
+Eqn. 4.12
+Λ2
+(ηω∗)2 = 3∆2 − 8∆κd + 6κ2
+d
+16κ2
+dτ(τ + 1 − ∆) ,
+(4.13)
+where we define κd = ηω∗/ωd. Note the similar qualitative behavior with ∆ as Eqn. 4.6,
+namely its divergence at the ∆ → 1 + τ, and unbounded growth as ∆ → −∞. The key
+difference here arises in the linear term in the drive parameter κ; this is expected from
+the theory of the toroidal ITG mode since the sign of the drift frequency (associated with
+so-called ‘good’ and ’bad’ magnetic curvature) is important for the resonance.
+We now find the value of ∆ that minimizes Λ:
+∆min = (1 + τ)
+�
+1 −
+�
+2¯κ2
+d − 8¯κd/3 + 1
+�
+,
+(4.14)
+where we define the parameter ¯κd = κd/(1 + τ). Substituting this into Eqn. 4.13 yields
+Λ2
+min = (ηω∗)2
+�
+8¯κd (ζ (¯κd) − 1) + 3 (ζ (¯κd) − 1)2 + 6¯κ2
+d
+16τ(τ + 1)¯κ2
+dζ (¯κd)
+�
+,
+(4.15)
+with ζ =
+�
+2¯κ2
+d − 8¯κd/3 + 1. This expression for Λmin is naturally separated into a
+factor that depends only on the instability parameter ¯κd, from which we can derive
+the asymptotic behavior. To show the behavior of this factor we plot the quantity
+τ(1 + τ)Λ2
+min/(ηω∗)2 in Fig. 1. The overall behavior of Λmin is captured by the following
+limits
+Λ2
+min =
+�
+�
+�
+�
+�
+�
+�
+|ηω∗ωd|
+�
+3
+√
+2+4σ
+8τ
+�
+,
+for |¯κd| ≫ 1,
+(ηω∗)2
+24τ(1+τ),
+for ¯κd → 0,
+3(ηω∗)2
+8τ(1+τ),
+for ¯κd → 4/3
+(4.16)
+where we denote σ = ±1 as the sign of κd. At large drive (|¯κd| ≫ 1; |ηω∗| ≫ |ωd|(1 + τ))
+we recover the expected non-resonant (“fluid”) behavior of the toroidal ITG mode with no
+density gradient, namely γL ∼ √ηω∗ωd. Note that this growth rate is much smaller than
+the bound found by merely considering the Helmholtz free energy (Helander and Plunk
+2022). Although we do not see the complete stabilization (Λ = 0) at negative values of
+¯κd (opposite sign of ωd and ηω∗) expected from theory, there is a strong asymmetry,
+with |Λ| having its larger values at positive ¯κd and being comparatively much smaller for
+negative ¯κd.
+The value ¯κd = 4/3 (i.e. ηω∗ = 4(1 + τ)ωd/3) achieves the maximal value of Λmin
+at fixed ηω∗, and therefore is evocative of the resonance condition for the toroidal ITG
+modes ηω∗ ∼ ωd (Biglari et al. 1989). This value of ¯κd is obtained by solving for ∆min = 0,
+
+Energetic bounds. Part III
+11
+-1
+1
+2
+4/3
+3
+4
+0.1
+0.2
+0.3
+Figure 1. Bound of the growth rate of the toroidal ITG mode, obtained from the optimal
+growth of generalized free energy, plotted versus the instability parameter
+¯κd = ηω∗/[ωd(1 + τ)].
+explaining why it produces the worst bound, i.e. that given by optimal growth of Helmoltz
+free energy.
+It is noteworthy that for the limit ¯κd → 0 (ωd ≫ ηω∗/(1 + τ)) our method yields a
+value of |Λ| that is a factor of 1/3 reduced as compared to the resonant case, again at
+least qualitatively reproducing the expected stabilization of the toroidal ITG mode in
+this limit.
+5. Conclusion
+We have demonstrated that the use of a generalized form of free energy ˜H introduces
+some of the physics of wave-particle resonance that is missing in the theory of optimal
+mode growth of Helmholtz free energy (Helander and Plunk 2021, 2022; Plunk and
+Helander 2022). The growth rates of optimal modes of generalized free energy provide
+a rigorous upper bound on the growth of conventional gyrokinetic instabilities (“normal
+modes”), which is always below the Helmholtz bound, as it must be, given that the
+Helmholtz free energy is a special case of the generalized measure. Moreover, optimal
+modes of generalized free energy depend on the magnetic-field geometry to a greater
+extent than those associated with Helmholtz free energy. The difference in growth rates
+can be very large. For instance, in the important case of a strongly driven toroidal ITG
+mode, the Helmholtz bound is larger by a factor of order ηω∗/ωd ≫ 1.
+A single ordinary differential equation has been derived for optimal modes, allowing
+general magnetic geometry. We found solutions of this equation in some simple limits to
+demonstrate that it indeed recovers, at least qualitatively, some of the physical effects
+expected from the theory of linear ITG modes, including sensitivity to the ratio of the
+frequencies associated with drive and resonance, and transition of the instability when
+this ratio is near one. Density gradient dependence of ITG mode is absent from both
+electrostatic and free energy input terms, assuming adiabatic electrons, so its effect is
+not accounted for by the theory presented here.
+The results of this work have possible implications for “turbulence optimization”,
+i.e. the endeavor to shape the equilibrium magnetic geometry for low turbulence in
+
+12
+G. G. Plunk and P. Helander
+stellarators. The general result, Eqn. 3.15, allows in principle for the inclusion of the
+complete geometric information contained in gyrokinetics, that is needed to run gyroki-
+netic simulations. However, the solution of this equation should be far simpler and more
+efficient, due to the reduction of velocity space to a single moment. The results found for
+the toroidal branch of the ITG hint at a possible optimization strategy. Consider fixed
+plasma conditions, i.e. a given temperature gradient (ηω∗) and temperature ratio (τ): At
+high drive (ηω∗/(1+τ) > 4ωd/3), minimization of the optimal growth rate |Λ| is achieved
+by minimization of the magnetic drift ωd (i.e. magnetic curvature), corresponding to
+minimization of the strongly-driven (non-resonant) toroidal ITG mode. On the other
+hand, at low drive (ηω∗/(1 + τ) < 4ωd/3) the increase of ωd is favored, corresponding to
+a weakening of the marginally unstable ITG mode, i.e. an increase of the threshold of
+instability. The latter case corresponds to “critical gradient” optimization, an idea which
+has recently been developed (Roberg-Clark et al. 2022a,b).
+More general solutions of the optimal mode equation, and the application to optimiza-
+tion will be pursued in future works. Other special limits can also be explored including,
+adiabatic ion limits, appropriate for studying trapped electron and universal instabilities.
+Electromagnetic generalizations are also possible: although it is not clear how to construct
+an electromagnetic form of generalized free energy ˜H that is a nonlinear invariant, it is
+certainly possible to consider related measures that focus on linear bounds.
+Funding. This work has been carried out within the framework of the EUROfusion
+Consortium, funded by the European Union via the Euratom Research and Training
+Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions
+expressed are however those of the author(s) only and do not necessarily reflect those of
+the European Union or the European Commission. Neither the European Union nor the
+European Commission can be held responsible for them. This work was partly supported
+by a grant from the Simons Foundation (560651, PH).
+Declaration of Interests. The authors report no conflict of interest.
+Author ORCID. G. G. Plunk, https://orcid.org/0000-0002-4012-4038; P. Helander,
+https://orcid.org/0000-0002-0460-590X.
+Appendix A. Several kinetic species
+For a plasma with an arbitrary number of particle species, we multiply each gyrokinetic
+equation
+∂ga,k
+∂t
++ v∥
+∂ga,k
+∂l
++ iωdaga,k + 1
+B2
+�
+k′
+B · (k × k′)δφk′ga,k−k′
+(A 1)
+= eaFa0
+Ta
+� ∂
+∂t + iωT
+∗a
+�
+δφk ,
+(A 2)
+by eaδφ
+∗
+k, integrate over velocity space, take the real part and the average ⟨· · · ⟩ over the
+flux tube, and sum over all species a and wave vectors k. In other words, we apply the
+operator
+Re
+�
+a,k
+ea
+��
+δφ
+∗
+k (· · · ) d3v
+�
+.
+(A 3)
+Since the expression
+Re (k × k′)δφ
+∗
+k′δφkga,k−k′ = 1
+2(k × k′)
+�
+δφ
+∗
+k′δφkga,k−k′ + δφk′δφ
+∗
+kg∗
+a,k−k′
+�
+(A 4)
+
+Energetic bounds. Part III
+13
+= 1
+2(k × k′)
+�
+δφ−k′δφkga,k−k′ + δφk′δφ−kga,−k+k′�
+(A 5)
+changes sign under an exchange of k and k′, the nonlinear terms cancel upon summation
+over k and k′, and we obtain
+Re
+�
+a,k
+ea
+��
+δφ
+∗
+k
+�∂ga,k
+∂t
++ v∥
+∂ga,k
+∂l
++ iωdaga,k − eaFa0
+Ta
+∂δφk
+∂t
+�
+d3v
+�
+= 0.
+(A 6)
+The quasineutrality equation (2.7) can be used to write the first term as
+Re
+�
+a,k
+ea
+�
+δφ
+∗
+k
+∂ga,k
+∂t d3v = d
+dt
+�
+k
+nae2
+a
+2Ta
+|δφk|2 .
+(A 7)
+We thus arrive at the electrostatic energy balance equation
+d
+dt
+�
+k
+nae2
+a
+2Ta
+�
+[1 − Γ0(bak)] |δφk|2�
+(A 8)
+= −Re
+�
+a,k
+�
+ea
+�
+δφ
+∗
+k
+�
+v∥
+∂ga,k
+∂l
++ iωdaga,k
+�
+d3v
+�
+,
+(A 9)
+which is the generalization of Eqn. 2.11 to several species. The right-hand side can be
+interpreted as minus the work done by the electric field on the various particle species.
+The first term in this expression contains
+�
+J0av∥
+∂ga,k
+∂l
+d3v =
+�
+σ
+2πσB
+m2a
+� ∞
+0
+dEa
+� Ea/B
+0
+J0a
+∂ga,k
+∂l
+dµa
+(A 10)
+= B ∂
+∂l
+� 1
+B
+�
+J0av∥ga,kd3v
+�
+−
+� ∂J0a
+∂l v∥ga,kd3v,
+(A 11)
+which we used in Eqn. 3.8, with σ = v∥/|v∥|, Ea = mav2/2 and µa = mav2
+⊥/(2B).
+Appendix B. Derivation of operators D and K
+We first write forms of D and K explicit in ˜g, noting that the contribution to D
+proportional to the density gradient is zero by use of quasi-neutrality with the adiabatic
+electron approximation (the factor ηω∗ ∝ dTi/dψ appears in what follows, but never
+ω∗ ∝ dni/dψ alone). For the same reason, the terms proportional to ν, involved in the
+transformation from g to ˜g, do not contribute, so expressing energy input in terms of ˜g
+merely has the consequence of introducing the overall factor α. The expressions are
+D = −Ti
+iα
+ni
+��
+˜g(v)˜g∗(v′) ηω∗
+� v2
+v2
+th
+�
+J0J′
+0d3vd3v′
+�
++ c.c.,
+(B 1)
+K∥ = −Ti
+α
+2ni
+��
+˜g∗(v′)
+�
+v∥
+∂˜g(v)
+∂l
+�
+J0J′
+0d3vd3v′
+�
++ c.c.,
+(B 2)
+Kd = −Ti
+iα
+2ni
+��
+˜g∗(v′)ωd˜g(v)J0J′
+0d3vd3v′
+�
++ c.c.,
+(B 3)
+
+14
+G. G. Plunk and P. Helander
+where we separate K = K∥ + Kd and ‘c.c.’ denotes complex conjutage. These can be
+re-expressed in terms of linear operators by writing them in the form
+D = (˜g, D˜g) =
+�
+Ti
+�
+d3v ˜g∗
+F0
+D˜g
+�
+,
+(B 4)
+K∥ = (˜g, K∥˜g) =
+�
+Ti
+�
+d3v ˜g∗
+F0
+K∥˜g
+�
+,
+(B 5)
+Kd = (˜g, Kd˜g) =
+�
+Ti
+�
+d3v ˜g∗
+F0
+Kd˜g
+�
+.
+(B 6)
+Identifying D and Kd is simply a matter of exchanging labels of dummy variables of
+integration (v for v′, etc.). Manipulating the expression for K∥ to reveal K∥ is more
+involved. We also need to integrate by parts in l and will need to use the fact that ∂/∂l
+is performed at fixed phase space variables Ei and µ = miv2
+⊥/(2B). The velocity space
+volume element contains an important factor of 1/v∥, which generally depends on l and
+does not itself commute with ∂/∂l:
+d3v = 2πv⊥dv⊥v∥ =
+�
+σ
+2πBdEidµi
+m2
+i |v∥|
+(B 7)
+where σ denotes the sign of v∥.
+Appendix C. Moment form of eigenproblem
+The three terms of Eqn. 3.9 can be rewritten in terms of the moments of ˜g (Eqn. 3.10):
+D˜g = iα
+2 ηω∗J0F0
+� v2
+v2
+th
+κ1 − κ2
+�
+,
+(C 1)
+K∥˜g = α
+2 F0
+�
+vthJ0
+�
+−B ∂
+∂l
+�κ4
+B
+�
++ κ5
+�
++ v∥
+∂
+∂l (J0κ1)
+�
+,
+(C 2)
+Kd˜g = iα
+2 ωdJ0F0
+��
+v2
+⊥
+2v2
+th
++
+v2
+∥
+v2
+th
+�
+κ1 − κ3
+�
+,
+(C 3)
+(C 4)
+Then, taking moments of Eqn. 3.9 yields the following five equations
+2Λ
+α κ1 = iηω∗ (G1κ1 − G0κ2) − i∆ωd (G3κ1 − G0κ3) − ∆G0vth
+�
+κ5 − B ∂
+∂l
+�κ4
+B
+��
+,(C 5)
+2Λ
+α κ2 = iηω∗ (G2κ1 − G1κ2) − i∆ωd (G2κ1 − G1κ3) − ∆G1vth
+�
+κ5 − B ∂
+∂l
+�κ4
+B
+��
+,(C 6)
+2Λ
+α κ3 = iηω∗ (G4κ1 − G3κ2) − i∆ωd (G5κ1 − G3κ3) − ∆G3vth
+�
+κ5 − B ∂
+∂l
+�κ4
+B
+��
+,(C 7)
+2Λ
+α κ4 = −∆vth
+�
+G′
+0,2κ1 + G0,2
+∂κ1
+∂l
+�
+,(C 8)
+2Λ
+α κ5 = −∆vth
+�
+G′′
+0,2κ1 + G′
+0,2
+∂κ1
+∂l
+�
+.
+(C 9)
+See the next section where the integrals Gm,n, etc., are defined and evaluated. Note that
+the final two equations can be immediately used to eliminate κ4 and κ5, leaving a system
+
+Energetic bounds. Part III
+15
+of three equations. The second and third equations are used together to find forms for κ2
+and κ3 in terms of κ1, and these forms are substituted into the first equation to obtain
+the final form, in terms of κ1 only, given by Eqn. 3.15.
+Appendix D. Bessel-type integrals
+The following definitions, mostly copied from Plunk and Helander (2022), are needed to
+perform the various integrals that appear in the moment equations for our eiegenproblem.
+First we need a general form of Weber’s integral,
+In(p, a1, a2) =
+� ∞
+0
+exp(−pt2)Jn(a1t)Jn(a2t)tdt
+= 1
+2p exp
+�−a2
+1 − a2
+2
+4p
+�
+In
+�a1a2
+2p
+�
+(D 1)
+where In is the modified Bessel function of order n. The integrals we need to evaluate
+can be conveniently found in terms of In. We define
+G⊥m(b) = 2
+� ∞
+0
+xm+1
+⊥
+exp(−x2
+⊥)J2
+0(
+√
+2bx2
+⊥)dx⊥,
+(D 2a)
+G(1)
+⊥m(b) = 2
+� ∞
+0
+xm+2
+⊥
+exp(−x2
+⊥)J0(
+√
+2bx2
+⊥)J1(
+√
+2bx2
+⊥)dx⊥,
+(D 2b)
+G(2)
+⊥m(b) = 2
+� ∞
+0
+xm+3
+⊥
+exp(−x2
+⊥)J2
+1(
+√
+2bx2
+⊥)dx⊥,
+(D 2c)
+where m is assumed to be even. Now we note that these integrals can be evaluated in
+terms of Weber’s integral:
+G⊥m(b) = 2
+��
+− d
+dp
+�m/2
+I0(p,
+√
+2b,
+√
+2b)
+�
+p=1
+,
+(D 3a)
+G(1)
+⊥m(b) = 2
+��
+− d
+dp
+�m/2 �
+− d
+dλ
+�
+I0(p, λ,
+√
+2b)
+�
+p=1,λ=
+√
+2b
+,
+(D 3b)
+G(2)
+⊥m(b) = 2
+��
+− d
+dp
+�m/2 �
+− d
+dλ1
+� �
+− d
+dλ2
+�
+I0(p, λ1, λ2)
+�
+p=1,λ1=λ2=
+√
+2b
+. (D 3c)
+The above relations allows us to evaluate the functions
+Gm,n(b) = G⊥m(b)G∥n,
+(D 4a)
+G(1)
+m,n(b) = G(1)
+⊥m(b)G∥n,
+(D 4b)
+G(2)
+m,n(b) = G(2)
+⊥m(b)G∥n.
+(D 4c)
+where
+G∥n =
+1
+√π
+� ∞
+−∞
+exp(−x2
+∥)xn
+∥dx∥ = 1 + (−1)n
+2√π
+ΓE
+�1 + n
+2
+�
+,
+(D 5)
+
+16
+G. G. Plunk and P. Helander
+and ΓE is the Euler gamma function. Finally, we can evaluate the integrals G′
+m,n, and
+G′
+m,n. We define
+G′
+⊥m(b) = 2
+� ∞
+0
+xm+1
+⊥
+exp(−x2
+⊥)J0
+∂J0
+∂l dx⊥,
+(D 6)
+G′′
+⊥m(b) = 2
+� ∞
+0
+xm+1
+⊥
+exp(−x2
+⊥)
+�∂J0
+∂l
+�2
+dx⊥,
+(D 7)
+Relating x⊥ to the proper gyrokinetic phase space variable µ, that is x2
+⊥ = µB/Ti allows
+the derivatives to be evaluated
+∂J0
+∂l = −
+�
+2/B ∂
+∂l(B
+√
+b)x2
+⊥J1
+(D 8)
+so that we can write
+G′
+⊥m(b) = −
+�
+2
+B
+∂
+∂l
+�
+B
+√
+b
+�
+G(1)
+⊥m(b),
+(D 9)
+G′′
+⊥m(b) = 2
+B
+� ∂
+∂l
+�
+B
+√
+b
+��2
+G(2)
+⊥m(b).
+(D 10)
+These expressions allow us to evaluate
+G′
+m,n = G′
+⊥mG∥n,
+(D 11)
+G′′
+m,n = G′′
+⊥mG∥n.
+(D 12)
+D.1. Explicit expressions for some Bessel integrals
+The b(l)-dependent factors in Eqn. 3.15 can be written as
+G0 = G0,0,
+(D 13)
+G1 = G2,0 + G0,2,
+(D 14)
+G2 = G4,0 + 2G2,2 + G0,4,
+(D 15)
+G3 = 1
+2G2,0 + G0,2,
+(D 16)
+G4 = 1
+2G4,0 + 3
+2G2,2 + G0,4,
+(D 17)
+G5 = 1
+4G4,0 + G2,2 + G0,4,
+(D 18)
+which can be evaluated using the identities of the previous section in terms of the familiar
+Γn(b) of gyrokinetic theory (suppressing its argument for succinctness):
+
+Energetic bounds. Part III
+17
+G0 = Γ0,
+(D 19)
+G1 =
+�3
+2 − b
+�
+Γ0 + bΓ1,
+(D 20)
+G2 = 1
+4
+��
+6b2 − 20b + 15
+�
+Γ0 + 2b ((10 − 4b)Γ1 + bΓ2)
+�
+,
+(D 21)
+G3 = 1
+2 (bΓ1 − (b − 2)Γ0) ,
+(D 22)
+G4 = 1
+4
+��
+3b2 − 11b + 10
+�
+Γ0 + b ((11 − 4b)Γ1 + bΓ2)
+�
+,
+(D 23)
+G5 = 1
+8
+��
+3b2 − 12b + 14
+�
+Γ0 + b (bΓ2 − 4(b − 3)Γ1)
+�
+.
+(D 24)
+where we recall
+Γn(b) = exp(−b)In(b)
+(D 25)
+For completeness, we evaluate the few remaining factors that enter Eqn. 3.15.
+G0,2 = Γ0
+2 ,
+(D 26)
+G(1)
+0,2 =
+√
+b (Γ0 − Γ1)
+2
+√
+2
+,
+(D 27)
+G(2)
+0,2 = 1
+8 (3bΓ0 + (2 − 4b)Γ1 + bΓ2) ,
+(D 28)
+and using Eqn. D 11
+G′
+0,2 = −
+�
+2
+B
+∂
+∂l
+�
+B
+√
+b
+�
+G(1)
+0,2,
+(D 29)
+G′′
+0,2 = 2
+B
+� ∂
+∂l
+�
+B
+√
+b
+��2
+G(2)
+0,2.
+(D 30)
+D.2. Limit b → 0
+In the limit b → 0 we obtain
+G0 = 1,
+(D 31)
+G1 = 3
+2,
+(D 32)
+G2 = 15
+4 ,
+(D 33)
+G3 = 1,
+(D 34)
+G4 = 5
+2,
+(D 35)
+G5 = 7
+4.
+(D 36)
+and
+
+18
+G. G. Plunk and P. Helander
+G0,2 = 1
+2,
+(D 37)
+G(1)
+0,2 = 0,
+(D 38)
+G(2)
+0,2 = 0,
+(D 39)
+REFERENCES
+P. Helander and G.G. Plunk.
+Energetic bounds on gyrokinetic instabilities. Part 1.
+Fundamentals. Journal of Plasma Physics, 88(2):905880207, 2022. .
+G.G. Plunk and P. Helander. Energetic bounds on gyrokinetic instabilities. Part 2. Modes of
+optimal growth. Journal of Plasma Physics, 88(3):905880313, 2022. .
+Matt Landreman, Gabriel G. Plunk, and William Dorland. Generalized universal instability:
+transient linear amplification and subcritical turbulence. Journal of Plasma Physics, 81
+(5):905810501, 2015. .
+W. Dorland and G. W. Hammett. Gyrofluid turbulence models with kinetic effects. Physics
+of Fluids B: Plasma Physics, 5(3):812–835, 1993. . URL https://doi.org/10.1063/1.
+860934.
+G. G. Plunk, P. Helander, P. Xanthopoulos, and J. W. Connor. Collisionless microinstabilities
+in stellarators. III. the ion-temperature-gradient mode. Physics of Plasmas, 21(3):032112,
+2014. . URL https://doi.org/10.1063/1.4868412.
+A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammett, G. G. Howes, E. Quataert, and
+T. Tatsuno. Astrophysical gyrokinetics: Kinetic and fluid turbulent cascades in magnetized
+weakly collisional plasmas. The Astrophysical Journal Supplement Series, 182(1):310, may
+2009. . URL https://dx.doi.org/10.1088/0067-0049/182/1/310.
+G. G. PLUNK, S. C. COWLEY, A. A. SCHEKOCHIHIN, and T. TATSUNO. Two-dimensional
+gyrokinetic turbulence. Journal of Fluid Mechanics, 664:407–435, 2010. .
+P. Helander, J. H. E. Proll, and G. G. Plunk. Collisionless microinstabilities in stellarators. i.
+analytical theory of trapped-particle modes. Physics of Plasmas, 20(12):122505, 2013. .
+URL https://doi.org/10.1063/1.4846818.
+Bogdan Teaca. private communication.
+H. Biglari, P. H. Diamond, and M. N. Rosenbluth. Toroidal ion-pressure-gradient-driven drift
+instabilities and transport revisited. Physics of Fluids B: Plasma Physics, 1(1):109–118,
+1989. . URL http://link.aip.org/link/?PFB/1/109/1.
+P. Helander and G. G. Plunk. Upper bounds on gyrokinetic instabilities in magnetized plasmas.
+Phys. Rev. Lett., 127:155001, Oct 2021. . URL https://link.aps.org/doi/10.1103/
+PhysRevLett.127.155001.
+G. T. Roberg-Clark, G. G. Plunk, and P. Xanthopoulos. Coarse-grained gyrokinetics for the
+critical ion temperature gradient in stellarators. Phys. Rev. Research, 4:L032028, Aug
+2022a. . URL https://link.aps.org/doi/10.1103/PhysRevResearch.4.L032028.
+G. T. Roberg-Clark, P. Xanthopoulos, and G. G. Plunk. Reduction of electrostatic turbulence
+in a quasi-helically symmetric stellarator via critical gradient optimization, 2022b. URL
+https://arxiv.org/abs/2210.16030.
+

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+arXiv:2301.12147v1  [math.AP]  28 Jan 2023
+LOGISTIC ELLIPTIC EQUATION WITH A NONLINEAR BOUNDARY
+CONDITION ARISING FROM COASTAL FISHERY HARVESTING II
+KENICHIRO UMEZU
+Abstract. We study the positive solutions of the logistic elliptic equation with a nonlinear
+Neumann boundary condition that models coastal fishery harvesting ([18]). An essential role is
+played by the smallest eigenvalue of the Dirichlet eigenvalue problem, with respect to which a
+noncritical case is studied in [32]. In this paper, we extend our analysis to the critical case and
+further study the noncritical case for a more precise description of the positive solution set. Our
+approach relies on the energy method, sub- and supersolutions, and implicit function analysis.
+1. Introduction
+This paper is devoted to the study of the positive solutions for the following logistic elliptic
+equation with a nonlinear boundary condition arising from coastal fishery harvesting ([18]):
+
+
+
+
+
+−∆u = u − up
+in Ω,
+u ≥ 0
+in Ω,
+∂u
+∂ν = −λuq
+on ∂Ω.
+(1.1)
+Here, Ω ⊂ RN, N ≥ 1, is a bounded domain with smooth boundary ∂Ω, ∆ = �N
+i=1
+∂2
+∂x2
+i is the
+usual Laplacian in RN, 0 < q < 1 < p, λ ≥ 0 is a parameter, and ν is the unit outer normal to
+∂Ω. Unless stated otherwise, throughout this paper we assume the subcritical condition
+p < N + 2
+N − 2
+for N > 2.
+(1.2)
+In the case of p = 2, the unknown function u ≥ 0 ecologically represents the biomass of fish that
+inhabit a lake Ω, obeying the logistic law ([8]), and the nonlinear boundary condition means
+fishery harvesting with the harvesting effort λ on the lake coast ∂Ω, obeying the Cobb–Douglas
+production function ([18, Subsection 2.1]).
+A nonnegative function u ∈ H1(Ω) is called a nonnegative (weak) solution of (1.1) if u satisfies
+�
+Ω
+�
+∇u∇ϕ − uϕ + upϕ
+�
++ λ
+�
+∂Ω
+uqϕ = 0,
+ϕ ∈ H1(Ω)
+(1.3)
+(we may regard (λ, u) as a nonnegative solution of (1.1)). It is seen that problem (1.1) has
+a solution (λ, 0) for every λ > 0, called a trivial solution.
+The sets {(λ, 0) : λ ≥ 0} and
+{(λ, 0) : λ > 0} are said to be the trivial lines. We know ([30]) that a nonnegative solution u of
+(1.1) belongs to the space W 1,r(Ω) for r > N (consequently, Cθ(Ω) for θ ∈ (0, 1)). Moreover, a
+nontrivial nonnegative solution u of (1.1) satisfies that u ∈ C2+θ(Ω) for θ ∈ (0, 1), and u > 0
+in Ω ([17], [27]), which is called a positive solution. Indeed, if u > 0 in Ω, then u ∈ C2+θ(Ω)
+by the bootstrap argument using elliptic regularity, and u satisfies (1.1) pointwisely in Ω in the
+classical sense. However, we do not know if u > 0 on the entirety of ∂Ω for a positive solution u
+2020 Mathematics Subject Classification. 35J65, 35B32, 35J25, 92D40.
+Key words and phrases. logistic elliptic equation, concave–convex nonlinearity, positive solution, uniqueness,
+stability, sub- and supersolutions, energy method, boundary harvesting.
+1
+
+of (1.1). As a matter of fact, Hopf’s boundary point lemma ([27]) does not work because of the
+lack of the one-sided Lipschitz condition [26, (4.1.19)] for mapping 0 ≤ u �→ (−uq) for u close to
+0.
+For a positive solution (λ, u) of (1.1) satisfying u > 0 in Ω, we call γ1 = γ1(λ, u) ∈ R the
+smallest eigenvalue of the linearized eigenvalue problem at (λ, u)
+�
+−∆ϕ = ϕ − pup−1ϕ + γϕ
+in Ω,
+∂ϕ
+∂ν = −λquq−1ϕ + γϕ
+on ∂Ω.
+(1.4)
+It is well known that γ1 is simple with a positive eigenfunction ϕ1 ∈ C2+θ(Ω) satisfying ϕ1 > 0
+in Ω. Indeed, γ1 is characterized by the variational formula
+γ1 = inf
+��
+Ω
+�
+|∇ϕ|2 − ϕ2 + pup−1ϕ2
+�
++ λ
+�
+∂Ω
+quq−1ϕ2 : ϕ ∈ H1(Ω),
+�
+Ω
+ϕ2 +
+�
+∂Ω
+ϕ2 = 1
+�
+.
+A positive solution u > 0 in Ω of (1.1) is said to be asymptotically stable, weakly stable, and
+unstable if γ1 > 0, γ1 ≥ 0, and γ1 < 0, respectively.
+Problem (1.1) possesses a sublinear nonlinearity at infinity and also a concave–convex nature.
+Thus, the global uniqueness of a positive solution of (1.1) for every λ > 0 would not be so easy
+to deduce. For nonlinear elliptic problems with a concave–convex nature, we refer to [4, 31, 1,
+5, 11, 12, 19]. The sublinear nonlinearity (−uq) that appears in (1.1) induces the absorption
+effect on ∂Ω. Sublinear boundary conditions of the uq type were explored in [16, 14, 15, 28, 29].
+The case of an incoming flux on ∂Ω was studied in [16, 15]. The mixed case of absorption and
+an incoming flux on ∂Ω was studied in [14]. The absorption case was also studied in [28, 29],
+where a similar type of logistic elliptic equation with an indefinite weight has been analyzed for
+the existence and multiplicity of nontrivial nonnegative solutions.
+An important role is played by the smallest eigenvalue βΩ > 0 of the Dirichlet eigenvalue
+problem
+�
+−∆φ = βφ
+in Ω,
+φ = 0
+on ∂Ω.
+It is well known that βΩ is simple with a positive eigenfunction φΩ ∈ H1
+0(Ω) (implying φΩ ∈
+C2+θ(Ω) by elliptic regularity). Indeed, φΩ > 0 in Ω, and
+c1 ≤ −∂φΩ
+∂ν ≤ c2
+on ∂Ω
+(1.5)
+for some 0 < c1 < c2. Moreover, βΩ is characterized by the variational formula
+βΩ = inf
+��
+Ω
+|∇φ|2 : φ ∈ H1
+0(Ω),
+�
+Ω
+φ2 = 1
+�
+.
+If βΩ < 1, then uD ∈ H1
+0(Ω) ∩ C2+θ(Ω) denotes the unique positive solution of the Dirichlet
+logistic problem ([7])
+�
+−∆u = u − up
+in Ω,
+u = 0
+on ∂Ω.
+(1.6)
+The existence, nonexistence, and multiplicity of positive solutions for (1.1) in the case where
+βΩ ̸= 1 were studied in the author’s previous work [32, Theorems 1.1, 1.2, 1.4, 1.5], which
+Theorem 0 summarizes.
+Theorem 0.
+(I) A positive solution u of (1.1) satisfies that u < 1 in Ω and u > 0 on Γ ⊂ ∂Ω with the
+condition |Γ| > 0.
+2
+
+(II) There exists λ∗ > 0 such that problem (1.1) has a positive solution curve
+C0 = {(λ, uλ) : 0 ≤ λ ≤ λ∗},
+(1.7)
+emanating from (λ, u) = (0, 1), that satisfies the following three conditions:
+• λ �→ uλ ∈ C2+θ(Ω) is C∞,
+• uλ > 0 in Ω,
+• uλ is asymptotically stable.
+Moreover, the positive solutions of (1.1) near (λ, u) = (0, 1) in R × C2+θ(Ω) form C0.
+Let λ be the positive value defined as
+λ = sup{λ > 0 : (1.1) has a positive solution for λ}.
+(1.8)
+Then, the following assertions hold.
+(III) Assume that βΩ < 1. Then, we have the following (as in Figure 1).
+(i) λ = ∞, and more precisely, problem (1.1) possesses a positive solution u for every
+λ > 0 such that u > 0 in Ω.
+(ii) (λ, uλ) ∈ C0 is a unique positive solution of (1.1) for 0 < λ ≤ λ∗ (by making λ∗ in
+(1.7) smaller if necessary).
+(iii) un → uD in H1(Ω) for a positive solution (λn, un) of (1.1) with λn → ∞.
+(iv) The positive solution set {(λ, u)} does not meet the trivial line {(λ, 0) : λ ≥ 0} in
+the topology of H1(Ω) (nor C(Ω)).
+(IV) Assume that βΩ > 1. Then, we have the following (as in Figure 2).
+(i) λ < ∞.
+(ii) There exists a bounded subcontinuum (closed connected subset) �C0 = {(λ, u)} of
+nonnegative solutions of (1.1) in [0, ∞) × C(Ω) joining (λ, u) = (0, 1) and (0, 0)
+such that �C0 \ {(0, 0)} includes C0 and consists of positive solutions of (1.1). Par-
+ticularly, problem (1.1) has at least two positive solutions for λ > 0 small.
+(iii) The positive solution set {(λ, u)} does not meet the trivial line {(λ, 0) : λ > 0} in
+the topology of H1(Ω) (nor C(Ω)).
+(iv) γ1(λn, un) < 0 for a positive solution (λn, un) of (1.1) such that (λn, un) → (0, 0)
+in R × H1(Ω), provided that un > 0 in Ω, i.e., un is unstable.
+Remark 0.
+(i) Assertions (I) and (II) hold for every case of βΩ > 0.
+(ii) Assertions (II) and (III-i) hold for any p > 1.
+(iii) Problem (1.1) with λ = 0 has exactly two nonnegative solutions (λ, u) = (0, 0), (0, 1).
+Thus, Theorem 0(I) is used to show easily that in every case of βΩ > 0, the positive
+solution set {(λ, u)} of (1.1) meets at most (0, 0) and (0, 1) on {(0, u) : u ≥ 0}, i.e., if
+(λn, un) is a positive solution of (1.1) such that (λn, un) → (0, u) in H1(Ω) (equivalently
+C(Ω) by elliptic regularity), then either u = 0 or 1.
+In this paper, we extend our consideration to the case where βΩ = 1 and further study the
+positive solution set in the case where βΩ < 1. Our first main result concerns the case where
+βΩ < 1. On the basis of Theorem 0(III), we present the uniqueness and stability of a positive
+solution of (1.1) for λ > 0 large and also the strong positivity of the positive solutions for every
+λ > 0.
+Theorem 1.1. Assume that βΩ < 1. Then, the following assertions hold (see Figure 1):
+3
+
+(i) There exists λ∗ ≥ λ∗ such that the positive solution of (1.1) ensured by Theorem 0(III-i)
+is unique for every λ > λ∗ (say uλ); more precisely, the positive solutions of (1.1) for
+λ > λ∗ form a C∞ curve C∞ = {(λ, uλ) : λ∗ < λ} (i.e., λ �→ uλ ∈ C2+θ(Ω) is C∞),
+which satisfies the following conditions:
+(a) uλ is asymptotically stable,
+(b) uλ −→ uD in H1(Ω) as λ → ∞,
+(c) uλ is decreasing, i.e., uλ1 > uλ2 in Ω if λ1 < λ2. Furthermore, if 0 < λ1 < λ2 with
+the condition that λ1 ≤ λ∗ < λ2, then u > uλ2 in Ω for a positive solution u of
+(1.1) for λ = λ1.
+(ii) u > 0 in Ω for a positive solution u of (1.1) for every λ > 0 (strong positivity).
+Figure 1. Possible positive solution sets in the case where βΩ < 1.
+Remark 1.2.
+(i) Assertions (i-a) and (i-b) hold for any p > 1 (see Remark 3.3).
+(ii) For (λ, uλ) ∈ C0 with 0 ≤ λ ≤ λ∗ in (1.7), we present similar results as those in assertions
+(i-c). Indeed, λ �→ uλ is decreasing for 0 < λ ≤ λ∗; if 0 < λ1 < λ2 with the condition
+that λ1 ≤ λ∗ < λ2, then uλ1 > u in Ω for a positive solution u of (1.1) with λ = λ2.
+(iii) It is an open question to get the global uniqueness for a positive solution of (1.1) for
+all λ > 0, i.e., λ∗ = λ∗. In this case, [0, ∞) ∋ λ �→ uλ is C∞ and decreasing.
+(iv) For uniqueness and stability analysis of positive solutions for large parameters in non-
+linear elliptic problems, we refer to [9, 34, 33, 24, 10, 25, 13, 20, 21, 22].
+Our second main result is the counterpart of Theorem 0(III-i) and (III-iii) for the case where
+βΩ = 1 and pq > 1.
+Theorem 1.3. Assume that βΩ = 1 and pq > 1. Then, problem (1.1) possesses a positive
+solution uλ for every λ > 0 such that uλ > 0 in Ω, which satisfies that
+uλ −→ 0
+and
+uλ
+∥uλ∥ −→ φΩ
+in H1(Ω)
+as λ → ∞.
+(1.9)
+Remark 1.4.
+(i) The existence assertion holds for any p > 1; thus, so does assertion (1.9) (see Remark
+3.3).
+(ii) Similarly as in Theorem 0(III-iii), assertion (1.9) is valid if we assume a positive solution
+(λ, uλ) (which may take zero value somewhere on ∂Ω) of (1.1) with λ → ∞.
+Our third main result is the counterpart of Theorem 0(III-iv), (IV-i), and (IV-iii) for the case
+where βΩ = 1.
+4
+
+u
+u
+A
+1
+1
+Co
+Co
+....
+入
+入
+*Y
+\*
+0
+入*
+\*
+0Theorem 1.5. Assume that βΩ = 1. Then, the following three assertions hold.
+(i) If pq ≤ 1, then λ < ∞ where λ > 0 is defined by (1.8).
+(ii) If pq ̸= 1, then the positive solution set {(λ, u)} of (1.1) does not meet the trivial line
+{(λ, 0) : λ > 0} in the topology of H1(Ω) (nor C(Ω)).
+(iii) If pq ≥ 1, then it does not meet {(0, 0)} in the topology of H1(Ω) (nor C(Ω)).
+Theorem 1.5 provides a guess (Remark 1.6) for the global extension of the C∞ positive solution
+curve C0 given by (1.7) in the case where βΩ = 1 and pq ≤ 1.
+Remark 1.6. Assume that βΩ = 1 and pq ≤ 1.
+Let �C0 = {(λ, u)} ⊂ [0, ∞) × C(Ω) be
+the component (maximal, closed, and connected subset) of nonnegative solutions of (1.1) that
+includes C0. From Theorems 0(I) and Theorem 1.5(i), �C0 \ {(0, 1)} ⊂ {(λ, u) ∈ [0, ∞) × C(Ω) :
+λ ≤ λ, u < 1 in Ω}. If we suppose that Γ0 :=
+�
+�C0 \ {(0, 1)}
+�
+∩{(λ, 0), (0, u) : λ ≥ 0, u ≥ 0} ̸= ∅,
+then Theorem 1.5(ii),(iii) show that Γ0 = {(0, 0)} and Γ0 ⊂ {(λ, 0) : λ ≥ Λ0} for some Λ0 > 0
+when pq < 1 and pq = 1, respectively. The existence of �C0 is still an open question. Suggested
+positive solution sets are illustrated in Figures 2 and 3.
+Figure 2. Suggested positive solution set in the case where βΩ = 1 and pq < 1.
+Figure 3. Suggested positive solution set in the case where βΩ = 1 and pq = 1,
+and λc ∈ Γ0.
+We conclude the Introduction by mentioning the stability of the trivial solution u = 0. A
+linearized stability analysis does not work for u = 0 because u �→ uq is not differentiable at
+u = 0.
+Instead, by the construction of suitable sub- and supersolutions of (1.1), we try to
+employ the Lyapunov stability criterion [26, Chapter 5] on the basis of the monotone iteration
+method, which is developed in Section 5.
+Notation: ∥ · ∥ denotes the usual norm of H1(Ω).
+un ⇀ u∞ means that un weakly
+converges to u∞ in H1(Ω). H1
+0(Ω) = {u ∈ H1(Ω) : u = 0 on ∂Ω}.
+�
+Ω fdx for f ∈ L1(Ω)
+5
+
+u
+1
+入
+入cu
+1and
+�
+∂Ω gdσ for g ∈ L1(∂Ω) are simply written as
+�
+Ω f and
+�
+∂Ω g, respectively. | · |
+represents both the Lebesgue measure in Ω and the surface measure on ∂Ω.
+The remainder of this paper is organized as follows. Sections 2 and 3 are devoted to the
+preparation for the proofs of Theorems 1.1, 1.3 and 1.5. In Section 2, we develop the energy
+method for the energy functional associated with (1.1).
+In Section 3, we use the sub- and
+supersolution method to prove existence and positivity results for positive solutions of (1.1).
+We give proofs for Theorems 1.1 and 1.3 in Section 3. In Section 4, we prove Theorem 1.5.
+Section 5 is devoted to a stability analysis of the trivial solution u = 0, which is based on
+Lemma 3.1 and Theorem 5.1.
+2. Energy method
+Let
+E(u) =
+�
+Ω
+(|∇u|2 − u2),
+u ∈ H1(Ω);
+then, the next lemma is used several times in the following arguments.
+Lemma 2.1. Let {un} ⊂ H1(Ω) satisfy E(un) ≤ 0, un ⇀ u∞, and un → u∞ in L2(Ω). Then,
+u∞ ̸= 0 if ∥un∥ ≥ C for some C > 0.
+Proof. By the weak lower semicontinuity, E(u∞) ≤ limn E(un) ≤ limn E(un) ≤ 0. If u∞ = 0,
+then ∥un∥ → 0, as desired.
+□
+We start by proving the following two propositions, which provide the asymptotic profile of
+a positive solution of (1.1) as λ → ∞. It is understood that uD = 0 if βΩ = 1.
+Proposition 2.2. Assume that βΩ ≤ 1. Let (λn, un) be a positive solution of (1.1) with λn → ∞.
+Then, un → uD in H1(Ω).
+Proof. We first assume that βΩ < 1. Because un < 1 in Ω, we substitute u = ϕ = un into (1.3)
+to deduce that
+�
+Ω
+|∇un|2 =
+�
+Ω
+�
+u2
+n − up+1
+n
+�
+− λn
+�
+∂Ω
+uq+1
+n
+(2.1)
+≤
+�
+Ω
+u2
+n ≤ |Ω|;
+thus, ∥un∥ is bounded. Immediately, up to a subsequence, un ⇀ u∞ ≥ 0, un → u∞ in L2(Ω)
+and L2(∂Ω), and un → u∞ a.e. in Ω for some u∞ ∈ H1(Ω). We then infer that
+�
+∂Ω
+uq+1
+n
+= 1
+λn
+�
+−
+�
+Ω
+|∇un|2 +
+�
+Ω
+�
+u2
+n − up+1
+n
+��
+≤ 1
+λn
+�
+Ω
+u2
+n −→ 0,
+(2.2)
+which implies that
+�
+∂Ω uq+1
+∞
+= 0; thus, u∞ ∈ H1
+0(Ω). From (1.3) with (λ, u) = (λn, un), it follows
+that
+�
+Ω
+�
+∇un∇ϕ − unϕ + up
+nϕ
+�
+= 0,
+ϕ ∈ H1
+0(Ω).
+Taking the limit, u∞ is a nonnegative solution of (1.6), where we have used the Lebesgue
+dominated convergence theorem to deduce that
+�
+Ω up
+nϕ →
+�
+Ω up
+∞ϕ.
+Then, we verify that u∞ ̸= 0. Since E(un) ≤ 0, the weak lower semicontinuity means that
+E(u∞) ≤ lim
+n→∞
+E(un) ≤ lim
+n→∞ E(un) ≤ 0.
+6
+
+If u∞ = 0, then it follows that ∥un∥ → 0. Here, we may assume that un → 0 a.e. in Ω. Say that
+wn =
+un
+∥un∥; then, up to a subsequence, wn ⇀ w∞ ≥ 0, wn → w∞ in L2(Ω) and L2(∂Ω), and
+wn → w∞ a.e. in Ω. Since ∥wn∥ = 1, we deduce that w∞ ̸= 0 using Lemma 2.1. However, we
+observe from (2.2) that
+�
+∂Ω
+wq+1
+n
+≤ 1
+λn
+�
+Ω
+w2
+n∥un∥1−q −→ 0.
+This implies that w∞ ∈ H1
+0(Ω). From (1.3) with (λ, u) = (λn, un), we see that
+�
+Ω
+�
+∇wn∇ϕ − wnϕ + wnϕ up−1
+n
+�
+= 0,
+ϕ ∈ H1
+0(Ω).
+Taking the limit,
+�
+Ω(∇w∞ϕ − w∞ϕ) = 0, where we have used the Lebesgue dominated conver-
+gence theorem to obtain that
+�
+Ω
+��wnϕ up−1
+n
+�� ≤
+��
+Ω
+w2
+n
+� 1
+2 ��
+Ω
+ϕ2u2(p−1)
+n
+� 1
+2
+≤ C
+��
+Ω
+ϕ2u2(p−1)
+n
+� 1
+2
+−→ 0.
+This implies that w∞ is a nontrivial nonnegative solution of the problem
+�
+−∆w = w
+in Ω,
+w = 0
+on ∂Ω.
+Thus, we deduce that βΩ = 1, which contradicts the assumption. The assertion that u∞ ≥ 0
+and u∞ ̸= 0 means that u∞ is the unique positive solution uD of (1.6) by the strong maximum
+principle.
+It remains to show that un → u∞ in H1(Ω).
+Observing that E(u∞) +
+�
+Ω up+1
+∞
+= 0 and
+E(un) ≤ −
+�
+Ω up+1
+n
+, we deduce that
+E(u∞) ≤ lim
+n→∞
+E(un) ≤ lim
+n→∞ E(un) ≤ − lim
+n→∞
+�
+Ω
+up+1
+n
+= −
+�
+Ω
+up+1
+∞
+= E(u∞),
+where we have used the Lebesgue dominated convergence theorem again. Thus, E(un) → E(u∞),
+i.e., ∥un∥ → ∥u∞∥. Since un ⇀ u∞, the desired assertion follows.
+Next, we assume that βΩ = 1. Then, u∞ is a nonnegative solution of (1.6), and indeed u∞ = 0
+because βΩ = 1 ([7]). Thus, ∥un∥ → 0, as desired.
+□
+In the case where βΩ = 1, we observe that E(u) ≥ 0 for u ∈ H1
+0(Ω). Indeed, we note that
+�
+u ∈ H1(Ω) : E(u) = 0
+�
+= ⟨φΩ⟩ := {sφΩ : s ∈ R}.
+We then investigate the asymptotic profile of a positive solution (λn, un) of (1.1) with ∥un∥ → 0.
+Proposition 2.3. Assume that βΩ = 1. Let (λn, un) be a positive solution of (1.1) such that
+λn ≥ λ for some λ > 0 and ∥un∥ → 0. Then, we obtain that
+un
+∥un∥ → φΩ in H1(Ω).
+Proof. Say that wn =
+un
+∥un∥ and, up to a subsequence, wn ⇀ w∞ ≥ 0, and wn → w∞ in L2(Ω)
+and L2(∂Ω) for some w∞ ∈ H1(Ω). From (2.1) it follows that
+λ
+�
+∂Ω
+uq+1
+n
+≤
+�
+Ω
+u2
+n.
+We use the condition ∥un∥ → 0 to infer that
+�
+∂Ω
+wq+1
+n
+≤ ∥un∥1−q
+λ
+�
+Ω
+w2
+n −→ 0;
+thus,
+�
+∂Ω wq+1
+∞
+= 0, i.e., w∞ ∈ H1
+0(Ω).
+7
+
+Since βΩ = 1 and E(wn) ≤ 0, the weak lower semicontinuity means that
+0 ≤ E(w∞) ≤ lim
+n→∞
+E(wn) ≤ lim
+n→∞ E(wn) ≤ 0,
+implying that E(wn) → E(w∞) = 0, i.e., ∥wn∥ → ∥w∞∥. Since wn ⇀ w∞, we deduce that
+wn → w∞ in H1(Ω) and w∞ = φΩ with ∥φΩ∥ = 1. Finally, because φΩ is unique, the desired
+conclusion follows.
+□
+Remark 2.4. If we construct a positive solution (λn, un) of (1.1) without using (1.2), then
+Propositions 2.2 and 2.3 remain valid for all p > 1.
+For further analysis of the asymptotic behavior of a positive solution (λn, un) of (1.1) with
+the condition that λn ≥ λ and ∥un∥ → 0, the orthogonal decomposition H1(Ω) = ⟨φΩ⟩⊕V using
+⟨φΩ⟩ is useful, where V denotes the orthogonal complement of ⟨φΩ⟩ that is given explicitly as
+V =
+�
+v ∈ H1(Ω) :
+�
+Ω
+�
+∇v∇φΩ + vφΩ
+�
+= 0
+�
+.
+Note that ⟨φΩ⟩ and V are both closed subspaces of H1(Ω) and ∥u∥ is equivalent to |s| + ∥v∥ for
+u = sφΩ + v ∈ H1(Ω) = ⟨φΩ⟩ ⊕ V .
+Using the orthogonal decomposition,
+un = snφΩ + vn ∈ ⟨φΩ⟩ ⊕ V
+(2.3)
+for a positive solution (λn, un) of (1.1) such that λn ≥ λ for some λ > 0 and ∥un∥ → 0 (under
+the assumption of Proposition 2.3). Since
+un
+∥un∥ → φΩ in H1(Ω), it follows that
+sn
+∥un∥ → 1,
+(2.4)
+∥vn∥
+∥un∥ → 0,
+(2.5)
+∥vn∥
+sn
+→ 0.
+(2.6)
+Because of (2.4), we may assume that sn > 0. Note that vn ≥ 0 on ∂Ω because φΩ = 0 on ∂Ω.
+We then deduce the following result, which plays a crucial role in the proof of Theorem 1.5.
+Lemma 2.5. Assume that βΩ = 1. Let {vn} be as introduced by (2.3). Then, there exists c > 0
+such that
+E(vn) + c
+�
+∂Ω
+vq+1
+n
+≤ 0
+for sufficiently large n,
+(2.7)
+provided that one of the following conditions is satisfied.
+(a) pq < 1,
+(b) pq = 1 and λn → ∞,
+(c) pq > 1 and λn is bounded above.
+Proof. Substituting un = snφΩ + vn into (2.1), we deduce that
+2sn
+��
+Ω
+∇φΩ∇vn − φΩvn
+�
++ E(vn) +
+�
+Ω
+(snφΩ + vn)p+1 + λn
+�
+∂Ω
+vq+1
+n
+= 0.
+(2.8)
+Using the divergence theorem,
+�
+Ω
+φΩvn =
+�
+Ω
+−∆φΩvn =
+�
+Ω
+∇φΩ∇vn +
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn;
+(2.9)
+8
+
+thus, (2.8) implies that
+−2sn
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn + E(vn) +
+�
+Ω
+(snφΩ + vn)p+1 + λn
+�
+∂Ω
+vq+1
+n
+= 0.
+It follows that
+E(vn) + λn
+2
+�
+∂Ω
+vq+1
+n
++ In ≤ 0
+(2.10)
+with
+In = λn
+2
+�
+∂Ω
+vq+1
+n
+− 2sn
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn.
+(2.11)
+Once we verify that
+In ≥ 0
+for sufficiently large n,
+(2.12)
+we obtain (2.7) and complete the proof. To verify (2.12), we use the test function ϕ = φΩ to
+deduce that
+�
+Ω
+�
+∇un∇φΩ − unφΩ + up
+nφΩ
+�
+= 0.
+(2.13)
+Substituting un = snφΩ + vn into (2.13) and combining (2.9) with (2.13) provide
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+� vn
+sp
+n
+=
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ.
+(2.14)
+We then consider either case (a) or (b). From (2.6), we deduce that
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ −→
+�
+Ω
+φp+1
+Ω
+> 0.
+Taking into account (1.5), we may derive from (2.14) that
+csp
+n ≤
+�
+∂Ω
+vn
+for some c > 0. By H¨older’s inequality, it follows that
+csp
+n ≤
+�
+∂Ω
+vn ≤ |∂Ω|
+q
+q+1
+��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+.
+(2.15)
+Combining (2.11) with (2.15) and using H¨older’s inequality, there exist c, ˜c > 0 such that
+In ≥ λn
+2
+�
+∂Ω
+vq+1
+n
+− c sn
+��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+=
+�
+λn
+2
+��
+∂Ω
+vq+1
+n
+�
+q
+q+1
+− c sn
+� ��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+≥ {˜c λn spq
+n − c sn}
+��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+= spq
+n
+�
+˜cλn − c s1−pq
+n
+� ��
+∂Ω
+vq+1
+n
+�
+1
+q+1
+.
+Since sn → 0 from (2.4), assertion (2.12) follows.
+9
+
+We next consider case (c) and verify (2.12). By combining (2.11) with (2.14), it follows from
+(2.11) that
+In = sp+1
+n
+�λn
+2
+�
+∂Ω
+vq+1
+n
+sp+1
+n
+− 2
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+.
+(2.16)
+Furthermore, we use the test function ϕ = 1 in (1.3) to infer that
+−
+�
+Ω
+un +
+�
+Ω
+up
+n + λn
+�
+∂Ω
+uq
+n = 0.
+Substituting un = snφΩ + vn,
+−
+�
+Ω
+�
+φΩ + vn
+sn
+�
++ sp−1
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
++
+�
+∂Ω
+λnvq
+n
+sn
+= 0,
+which implies that
+�
+∂Ω
+λnvq
+n
+sn
+−→
+�
+Ω
+φΩ > 0,
+where we have used condition (2.6). Thus, we may deduce that
+c sn
+λn
+≤
+�
+∂Ω
+vq
+n
+for some c > 0. Using H¨older’s inequality, we deduce that
+c
+� sn
+λn
+� q+1
+q
+≤
+�
+∂Ω
+vq+1
+n
+(2.17)
+for some c > 0. Combining (2.16) with (2.17),
+In ≥ sp+1
+n
+�
+c s
+1
+q −p
+n
+λ
+− 1
+q
+n
+− 2
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+for some c > 0. We observe that
+s
+1
+q −p
+n
+→ ∞,
+λ
+− 1
+q
+n
+≥ c
+by some constant c > 0,
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ →
+�
+Ω
+φp+1
+Ω
+> 0.
+Thus, assertion (2.12) follows.
+□
+We conclude this section with the establishment of the uniqueness and stability results for a
+positive solution of (1.1) in the case where βΩ < 1.
+Proposition 2.6. Assume that βΩ < 1. Then, there exists Λ > 0 such that if λ > Λ, then the
+following two assertions hold:
+(i) Problem (1.1) has at most one positive solution.
+(ii) A positive solution u of (1.1) satisfying u > 0 in Ω is asymptotically stable.
+Proof. We recall that the unique positive solution uD of (1.6) is asymptotically stable, i.e.,
+γ1,D = inf
+��
+Ω
+�
+|∇ϕ|2 − ϕ2 + pup−1
+D
+ϕ2�
+: ϕ ∈ H1
+0(Ω),
+�
+Ω
+ϕ2 = 1
+�
+> 0.
+(2.18)
+10
+
+(i) Assume to the contrary that problem (1.1) has two distinct positive solutions (λn, un) and
+(λn, vn) with λn → ∞. Note that un, vn < 1 in Ω. We may assume that un, vn → uD in H1(Ω)
+and a.e. in Ω. The difference wn = un − vn (may change sign) allows that
+�
+Ω
+�
+∇wn∇ϕ − wnϕ
+�
++
+�
+Ω
+�
+(vn + wn)p − vp
+n
+�
+ϕ + λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+wnϕ = 0
+(2.19)
+for ϕ ∈ H1(Ω), where Γn = {x ∈ ∂Ω : wn(x) ̸= 0}. Note that ∥wn∥ → 0, and wn → 0 a.e. in Ω.
+Substituting ϕ = wn into (2.19), the mean value theorem shows the existence of θn ∈ (0, 1)
+such that
+E(wn) +
+�
+Ω
+p(vn + θnwn)p−1w2
+n + λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+w2
+n = 0;
+thus, ψn =
+wn
+∥wn∥ implies that
+E(ψn) +
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n + λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+ψ2
+n = 0.
+(2.20)
+From ∥ψn∥ = 1, we infer that up to a subsequence, ψn ⇀ ψ∞, ψn → ψ∞ in L2(Ω) and L2(∂Ω)
+for some ψ∞ ∈ H1(Ω). Then, we claim that ψ∞ ∈ H1
+0(Ω) and ψ∞ ̸= 0. From (2.20), we deduce
+that
+λn
+�
+Γn
+(vn + wn)q − vq
+n
+wn
+ψ2
+n ≤
+�
+Ω
+ψ2
+n ≤ 1.
+We observe that
+(vn + wn)q − vq
+n
+wn
+≥ q
+on Γn
+because un, vn < 1 in Ω. It follows that λnq
+�
+∂Ω ψ2
+n ≤ 1. Passing to the limit, we deduce that
+ψ∞ ∈ H1
+0(Ω). Indeed, ψ∞ ̸= 0 by Lemma 2.1.
+Then, we assert in (2.20) that
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n −→
+�
+Ω
+pup−1
+D
+ψ2
+∞.
+Indeed, we use
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n =
+�
+Ω
+p(vn + θnwn)p−1ψ2
+∞ +
+�
+Ω
+p(vn + θnwn)p−1(ψ2
+n − ψ2
+∞).
+Since vn → uD and wn → 0 a.e. in Ω and un, vn < 1 in Ω, the Lebesgue dominated convergence
+theorem shows that
+�
+Ω
+p(vn + θnwn)p−1ψ2
+∞ −→
+�
+Ω
+pup−1
+D
+ψ2
+∞.
+Using the fact
+�
+Ω |ψ2
+n − ψ2
+∞| → 0 yields
+�
+Ω
+p(vn + θnwn)p−1(ψ2
+n − ψ2
+∞) −→ 0,
+as desired.
+Then, the weak lower semicontinuity allows us to deduce from (2.20) that
+�
+Ω
+�
+|∇ψ∞|2 − ψ2
+∞ + pup−1
+D
+ψ2
+∞
+�
+≤ lim
+n→∞
+�
+E(ψn) +
+�
+Ω
+p(vn + θnwn)p−1ψ2
+n
+�
+≤ 0,
+which contradicts ψ∞ ∈ H1
+0(Ω) and ψ∞ ̸= 0 in view of (2.18).
+11
+
+(ii) On the basis of (1.4), we claim that γ1 > 0 for sufficiently large λ > 0.
+Assume by
+contradiction that a positive solution (λn, un) of (1.1) with the condition that λn → ∞ and
+un > 0 in Ω satisfies γn := γ1,n(λn, un) ≤ 0. This means that
+�
+Ω
+�
+|∇ϕn|2 − ϕ2
+n + pup−1
+n
+ϕ2
+n
+�
++ λnq
+�
+∂Ω
+uq−1
+n
+ϕ2
+n = γn ≤ 0,
+(2.21)
+where ϕn := ϕ1,n, normalized as
+�
+Ω
+ϕ2
+n +
+�
+∂Ω
+ϕ2
+n = 1.
+(2.22)
+Because
+�
+Ω |∇ϕn|2 ≤
+�
+Ω ϕ2
+n ≤ 1 from (2.21) and (2.22), ∥ϕn∥ is bounded, which implies that up
+to a subsequence, ϕn ⇀ ϕ∞, ϕn → ϕ∞ in L2(Ω) and L2(∂Ω), and ϕn → ϕ∞ a.e. in Ω for some
+ϕ∞ ∈ H1(Ω). Since uq−1
+n
+≥ 1 from Theorem 0(I), assertion (2.21) gives us
+λnq
+�
+∂Ω
+ϕ2
+n ≤
+�
+Ω
+ϕ2
+n ≤ 1.
+Passing to the limit, ϕ∞ = 0 on ∂Ω; thus, ϕ∞ ∈ H1
+0(Ω). From (2.22),
+�
+Ω ϕ2
+∞ = 1 is also deduced.
+By the weak lower semicontinuity, we derive from (2.21) that
+�
+Ω
+�
+|∇ϕ∞|2 − ϕ2
+∞ + pup−1
+D
+ϕ2
+∞
+�
+≤ lim
+n→∞
+�
+Ω
+�
+|∇ϕn|2 − ϕ2
+n + pup−1
+n
+ϕ2
+n
+�
+≤ 0
+(2.23)
+Indeed, on the basis of the facts that un → uD in H1(Ω) and un < 1 in Ω (see Theorem 0(I)
+and Proposition 2.2), the Lebesgue dominated convergence theorem shows that
+�
+Ω
+up−1
+n
+ϕ2
+n =
+�
+Ω
+up−1
+n
+ϕ2
+∞ +
+�
+Ω
+up−1
+n
+(ϕ2
+n − ϕ2
+∞) −→
+�
+Ω
+up−1
+D
+ϕ2
+∞,
+where we have used that un → uD a.e. in Ω. Assertion (2.23) contradicts ϕ∞ ∈ H1
+0(Ω) and
+�
+Ω ϕ2
+∞ = 1 in view of (2.18).
+□
+Remark 2.7. If we construct a positive solution u of (1.1) such that u > 0 in Ω without using
+(1.2), then assertion (ii) of Proposition 2.6 remains valid for all p > 1.
+3. Sub- and supersolutions
+Consider the case where βΩ < 1 or where βΩ = 1 and pq > 1. Then, we first construct small
+positive subsolutions of (1.1) and use them to establish an a priori lower bound for positive
+solutions (λ, u) of (1.1) satisfying u > 0 in Ω. For a fixed τ > 0 and with a parameter ε > 0, we
+set
+φε(x) = ε(φΩ(x) + ετ),
+x ∈ Ω,
+which implies that φε ∈ C2+θ(Ω) and φε > 0 in Ω.
+We then use φε to formulate the following a priori lower bound for positive solutions u > 0
+in Ω of (1.1).
+Lemma 3.1. Assume that βΩ < 1 or that βΩ = 1 and pq > 1. Let τ > 1−q
+q
+when βΩ < 1, and
+let 1−q
+q
+< τ < p − 1 when βΩ = 1 and pq > 1. Then, for Λ > 0 there exists ε = ε(τ, Λ) > 0 such
+that φε is a subsolution of (1.1), provided that λ ∈ [0, Λ] and ε ∈ (0, ε]. Furthermore, u ≥ φε
+in Ω for a positive solution u > 0 in Ω of (1.1) with λ ∈ [0, Λ]. Here, ε does not depend on
+λ ∈ [0, Λ].
+12
+
+Proof. We only consider the case where βΩ = 1 and pq > 1. The case where βΩ < 1 is proved
+similarly. First, we verify the former assertion. We take 0 < ε ≤ 1 and then use the condition
+p − τ − 1 > 0 to deduce that
+−∆φε − φε + φp
+ε ≤ ε1+τ
+�
+−1 + εp−τ−1
+�
+1 + max
+Ω
+φΩ
+�p�
+≤ 0
+in Ω
+if ε > 0 is small. For Λ > 0 we use (1.5) and the condition τ > 1−q
+q
+to deduce that
+∂φε
+∂ν + λφq
+ε ≤ −c1ε + λε(1+τ)q ≤ ε(−c1 + Λεq+τq−1) ≤ 0
+on ∂Ω
+if 0 < ε ≤ (c1/Λ)1/(q+τq−1). The desired assertion follows.
+Next, we argue by contradiction to verify the latter assertion. Assume by contradiction that
+u ̸≥ φε in Ω for some positive solution u > 0 in Ω with λ ∈ [0, Λ]. Because ε �→ φε is increasing
+and φε → 0 uniformly in Ω, we can take ε1 ∈ (0, ε) such that
+�
+u ≥ φε1
+in Ω,
+u(x1) = φε1(x1)
+for some x1 ∈ Ω.
+(3.1)
+Take c > 0 such that u, φε1 ≥ c in Ω; then, choose K > 0 sufficiently large so that fK(t) =
+Kt+t−tp is increasing for t ∈
+�
+0, maxΩ u
+�
+and M > 0 sufficiently large so that M −Λqcq−1 > 0.
+We use the subsolution φε1 (not a positive solution of (1.1)) to deduce that
+(−∆ + K)(u − φε1) ≥ fK(u) − fK(φε1) ≥ 0 (and ̸≡ 0)
+in Ω,
+(3.2)
+and for x ∈ ∂Ω satisfying u > φε1,
+� ∂
+∂ν + M
+�
+(u − φε1) ≥ −λuq + λφq
+ε1 + M(u − φε1)
+=
+�
+M − λuq − φq
+ε1
+u − φε1
+�
+(u − φε1)
+≥ (M − Λqcq−1)(u − φε1) > 0.
+(3.3)
+Thus, the strong maximum principle and boundary point lemma are applicable to infer that
+u − φε1 > 0 in Ω, which contradicts (3.1).
+□
+On the basis of Lemma 3.1, we construct minimal and maximal positive solutions of (1.1) as
+follows.
+Proposition 3.2. Assume that βΩ < 1 or that βΩ = 1 and pq > 1. Then, problem (1.1) has
+a minimal positive solution uλ ∈ C2+θ(Ω) and a maximal positive solution uλ ∈ C2+θ(Ω) for
+each λ > 0 such that 0 < uλ ≤ uλ in Ω, meaning that any positive solution u of (1.1) with the
+condition that u > 0 in Ω satisfies uλ ≤ u ≤ uλ in Ω. Moreover, both uλ and uλ are weakly
+stable, i.e., γ1(λ, uλ), γ1(λ, uλ) ≥ 0.
+Proof. It is clear that (λ, 1) is a supersolution of (1.1). Choose ε0 > 0 such that φε0 ≤ 1 in Ω;
+then, Lemma 3.1 states that (λ, φε0) is a subsolution of (1.1). By Theorem 0(I) and Lemma 3.1,
+a positive solution u > 0 in Ω of (1.1) implies that φε0 ≤ u ≤ 1 in Ω. Thus, this proposition is
+a direct consequence of applying [3, (2.1)Theorem] and [2, Proposition 7.8].
+□
+Remark 3.3. In view of the construction, Lemma 3.1 and Proposition 3.2 remain valid for
+any p > 1; therefore, Propositions 2.2, 2.3, 2.6(ii) hold for any p > 1 with the positive solution
+(λn, un) of (1.1) with λn → ∞ that is constructed by Proposition 3.2 (see Remarks 2.4 and 2.7).
+13
+
+In the case where βΩ < 1, Propositions 2.6 and 3.2 ensure the existence of a unique positive
+solution u(λ) of (1.1) for λ > Λ, which is asymptotically stable. Using the implicit function
+theorem provides us with the following result.
+Corollary 3.4. Assume that βΩ < 1. Then, {(λ, u(λ)) : λ > Λ} is a C∞ curve, i.e., λ �→
+u(λ) ∈ C2+θ(Ω) is C∞. Moreover, it is decreasing, i.e., u(λ1) > u(λ2) in Ω for λ2 > λ1 > Λ.
+Proof. We verify the first assertion.
+Let (λ0, u(λ0)) be the unique positive solution of (1.1)
+for λ0 > Λ. Since γ1(λ0, u(λ0)) > 0, the implicit function theorem applies at (λ0, u0); then,
+we deduce, thanks to the uniqueness, that {(λ, u(λ)) : λ1 < λ ≤ λ2} is a C∞ curve for λ1 <
+λ0 < λ2 such that u(λ) is asymptotically stable. The implicit function theorem applies again at
+(λ2, u(λ2)); then, the curve is continued until λ = λ3 > λ2. Repeating the same procedure, the
+curve is continued to λ = ∞ thanks to the a priori upper and lower bounds (Theorem 0(I) and
+Lemma 3.1), as desired.
+We next verify the second assertion. If λ1 < λ2, then u(λ1) is a supersolution of (1.1) for
+λ = λ2. By Lemma 3.1, it is possible to construct a subsolution φε of (1.1) for λ = λ2 such
+that 0 < φε ≤ u(λ1) in Ω. The sub- and supersolution method applies, and problem (1.1) has
+a positive solution u for λ = λ2 such that φε ≤ u ≤ u(λ1) in Ω, where u ̸≡ u(λ1). Thanks
+to Proposition 2.6(i), we obtain u = u(λ2). The desired assertion follows by using the strong
+maximum principle and boundary point lemma (as developed in (3.2) and (3.3)).
+□
+We conclude this section by employing the weak sub- and supersolution method [23] to show
+global strong positivity for a positive solution of (1.1) in the case where βΩ < 1.
+Proposition 3.5. Assume that βΩ < 1. Then, a positive solution (λ, u) of (1.1) satisfies that
+u > 0 in Ω.
+Proof. Assume by contradiction that problem (1.1) possesses a positive solution (λ0, u0) for
+λ0 > 0 such that u0 = 0 somewhere on ∂Ω. Let λ = λ1 > max(λ0, Λ), for which problem (1.1)
+has at most one positive solution by Proposition 2.6(i). Then, u0 is a weak supersolution of
+(1.1) for λ = λ1. Indeed,
+0 =
+�
+Ω
+�
+∇u0∇ϕ − u0ϕ + up
+0ϕ
+�
++ λ0
+�
+∂Ω
+uq
+0ϕ
+≤
+�
+Ω
+�
+∇u0∇ϕ − u0ϕ + up
+0ϕ
+�
++ λ1
+�
+∂Ω
+uq
+0ϕ,
+ϕ ∈ H1(Ω) and ϕ ≥ 0.
+We next construct a weak subsolution of (1.1) for λ = λ1 that is smaller than or equal to
+u0. Note that u0 ∈ Cθ(Ω) and u0 > 0 in Ω. From βΩ < 1, it follows by the continuity and
+monotonicity of βΩ with respect to Ω that we can choose a subdomain Ω1 ⋐ Ω with smooth
+boundary ∂Ω1 such that βΩ1 < 1. Then, we deduce that
+u0 ≥ c
+in Ω1
+(3.4)
+for some c > 0. We also deduce that if ε > 0 is sufficiently small, then
+−∆(εφΩ1) ≤ εφΩ1 − (εφΩ1)p
+in Ω1,
+where φΩ1 is a positive eigenfunction associated with βΩ1. Consequently, the divergence theorem
+is applied to
+�
+Ω −∆(εφΩ1)ϕ for ϕ ∈ H1(Ω) and ϕ ≥ 0 to deduce that
+�
+Ω1
+�
+∇(εφΩ1)∇ϕ − (εφΩ1)ϕ + (εφΩ1)pϕ
+�
+≤ 0.
+(3.5)
+14
+
+Define
+Φε =
+�
+εφΩ1
+in Ω1,
+0
+in Ω \ Ω1,
+and Φε ∈ H1(Ω) ∩ C(Ω). By virtue of (3.5), the linking technique [6, Lemma I.1] yields
+�
+Ω
+�
+∇Φε∇ϕ − Φεϕ + Φp
+ε ϕ
+�
++ λ1
+�
+∂Ω
+Φq
+ε ϕ =
+�
+Ω1
+�
+∇Φε∇ϕ − Φεϕ + Φp
+ε ϕ
+�
+≤ 0.
+Thanks to (3.4), we can take ε > 0 such that Φε ≤ u0 in Ω, as desired.
+The weak sub- and supersolution method [23, Subsection 2.2] is now applicable to deduce
+the existence of a positive solution (λ1, u1) of (1.1) such that Φε ≤ u1 ≤ u0 in Ω. Particularly,
+u1 = 0 somewhere on ∂Ω because so is u0. However, this contradicts Proposition 3.2 in view of
+the uniqueness.
+□
+Proof of Theorem 1.1. The uniqueness assertion follows from Proposition 2.6(i). Assertions (i-
+a) and (i-b) follow from Propositions 2.6(ii) and 2.2, respectively. Assertion (i-c) is verified by
+Corollary 3.4 and an analogous argument as in the proof of Corollary 3.4. Assertion (ii) follows
+from Proposition 3.5.
+□
+Proof of Theorem 1.3. The existence part follows from Proposition 3.2. Assertion (1.9) follows
+from Propositions 2.2 and 2.3.
+□
+4. Proof of Theorem 1.5
+This section is devoted to the proof of Theorem 1.5.
+(i) We prove assertion (i). Assume by contradiction that problem (1.1) has a positive solution
+(λn, un) with λn → ∞. Then, Proposition 2.2 shows that ∥un∥ → 0; thus, Proposition 2.3 shows
+that
+un
+∥un∥ → φΩ in H1(Ω). We apply Lemma 2.5(a) and (b); then, for un = snφΩ+vn ∈ ⟨φΩ⟩⊕V
+as in (2.3), we have (2.7) with (2.4)–(2.6).
+Observe from (2.5) that ∥vn∥ → 0. Say that ψn =
+vn
+∥vn∥; then, up to a subsequence, ψn ⇀
+ψ∞ ≥ 0, and ψn → ψ∞ in L2(Ω) and L2(∂Ω) for some ψ∞ ∈ H1(Ω). From (2.7), it follows that
+c
+�
+∂Ω
+ψq+1
+n
+≤ −E(ψn)∥vn∥1−q −→ 0,
+so that
+�
+∂Ω ψq+1
+∞
+= 0, i.e., ψ∞ ∈ H1
+0(Ω). Lastly, we use the condition E(ψn) ≤ 0 derived from
+(2.7) to follow the argument in the last paragraph of the proof of Proposition 2.3; then, we arrive
+at the contradiction ψ∞ = φΩ ∈ ⟨φΩ⟩ ∩ V = {0}.
+(ii) We verify assertion (ii). We remark that the convergences (λn, un) → (λ∞, 0) with λ∞ ≥ 0
+in R × H1(Ω) and R × C(Ω) are equivalent for a positive solution (λn, un) of (1.1) with λn > 0.
+This is verified by the bootstrap argument [32, Lemma 3.3]. In fact, the proof of assertion (ii)
+is similar to that for assertion (i). Assume by contradiction that problem (1.1) has a positive
+solution (λn, un) with the condition that λn → λ∞ > 0 and ∥un∥ → 0. Lemma 2.5(a) and (c)
+apply; then, we arrive at a contradiction.
+(iii) To verify assertion (iii), we prove the following three auxiliary lemmas. Say that Un =
+λ
+−
+1
+1−q
+n
+un.
+Lemma 4.1. There exists C > 0 such that ∥Un∥ ≤ C for a positive solution (λn, un) of (1.1)
+with λn > 0 satisfying that (λn, un) → (0, 0) in R × H1(Ω).
+15
+
+Proof. Assume by contradiction that ∥Un∥ → ∞. Say that wn =
+Un
+∥Un∥; then, up to a subse-
+quence, wn ⇀ w∞ ≥ 0, and wn → w∞ in L2(Ω) and L2(∂Ω) for some w∞ ∈ H1(Ω). Since
+E(wn) ≤ 0, Lemma 2.1 provides w∞ ̸= 0.
+Recall that (λ, U) = (λn, Un) satisfies
+�
+Ω
+�
+∇U∇ϕ − Uϕ + λ
+p−1
+1−q U pϕ
+�
++
+�
+∂Ω
+U qϕ = 0,
+ϕ ∈ H1(Ω).
+(4.1)
+Using the test function ϕ = 1 in (4.1), we deduce that
+�
+Ω
+Un = λ
+p−1
+1−q
+n
+�
+Ω
+U p
+n +
+�
+∂Ω
+U q
+n =
+�
+Ω
+up−1
+n
+Un +
+�
+∂Ω
+U q
+n,
+implying
+�
+Ω
+wn =
+�
+Ω
+up−1
+n
+wn +
+�
+∂Ω
+wq
+n∥Un∥q−1.
+(4.2)
+We may assume that un → 0 a.e. in Ω, and since un < 1 in Ω, we deduce that
+�
+Ω
+up−1
+n
+wn =
+�
+Ω
+up−1
+n
+w∞ +
+�
+Ω
+up−1
+n
+(wn − w∞) −→ 0,
+by applying the Lebesgue dominated convergence theorem and using the condition wn → w∞
+in L2(Ω).
+Then, passing to the limit in (4.2) yields
+�
+Ω w∞ = 0, i.e., w∞ = 0, which is a
+contradiction.
+□
+Lemma 4.2. Assume that βΩ = 1. Then, there is no positive solution U of (4.1) for λ = 0.
+Proof. If it exists, then from (4.1) with λ = 0 and ϕ = 1, it follows that U > 0 on Γ ⊂ ∂Ω with
+|Γ| > 0, implying
+�
+∂Ω
+∂φΩ
+∂ν U < 0. We use the test function ϕ = φΩ to deduce that
+�
+Ω
+�
+∇U∇φΩ − UφΩ
+�
+= 0.
+However, the divergence theorem leads us to the contradiction
+�
+Ω
+φΩU =
+�
+Ω
+−∆φΩU =
+�
+Ω
+∇φΩ∇U −
+�
+∂Ω
+∂φΩ
+∂ν U >
+�
+Ω
+∇φΩ∇U.
+□
+Lemma 4.3. Assume that βΩ = 1 and pq ≥ 1. Then, there exists C > 0 such that ∥Un∥ ≥ C
+for a positive solution (λn, Un) of (4.1) with λn → 0+.
+Proof. Assume by contradiction that (λn, Un) → (0, 0) in R × H1(Ω) for a positive solution
+(λn, Un) of (4.1). Say that wn =
+Un
+∥Un∥; then, up to a subsequence, wn ⇀ w∞ ≥ 0, wn → w∞ in
+Lp+1(Ω) and L2(∂Ω) for some w∞ ∈ H1(Ω). From (4.1) with (λ, U) = (λn, Un) and ϕ = Un, it
+follows that
+�
+Ω
+�
+|∇Un|2 − U 2
+n + λ
+p−1
+1−q
+n
+U p+1
+n
+�
++
+�
+∂Ω
+U q+1
+n
+= 0.
+(4.3)
+We then deduce that
+�
+∂Ω wq+1
+n
+≤
+�
+Ω w2
+n∥Un∥1−q → 0; thus,
+�
+∂Ω wq+1
+∞
+= 0, i.e., w∞ ∈ H1
+0(Ω). We
+also deduce from (4.3) that E(wn) =
+�
+Ω(|∇wn|2 − w2
+n) ≤ 0. Thus, we derive that wn → φΩ in
+H1(Ω) using a similar argument as in the last paragraph of the proof of Proposition 2.3.
+For a contradiction, we use the same strategy developed in the proof of assertion (i). To this
+end, we consider the orthogonal decomposition Un = snφΩ + vn ∈ ⟨φΩ⟩ ⊕ V as in (2.3); then,
+16
+
+we obtain (2.4) to (2.6) with un replaced by Un. As in the proof of Lemma 2.5, we deduce the
+following counterpart of (2.10) and (2.11) for (4.3):
+E(vn) + 1
+2
+�
+∂Ω
+vq+1
+n
++ Jn ≤ 0,
+with
+Jn = 1
+2
+�
+∂Ω
+vq+1
+n
+− 2sn
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn.
+(4.4)
+In the same spirit of Lemma 2.5 ((2.12)), we establish
+E(vn) + 1
+2
+�
+∂Ω
+vq+1
+n
+≤ 0
+for sufficiently large n,
+(4.5)
+by verifying that
+Jn ≥ 0
+for sufficiently large n.
+(4.6)
+Analogously to (2.14), we obtain
+�
+∂Ω
+�
+−∂φΩ
+∂ν
+�
+vn = λ
+p−1
+1−q
+n
+sp
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ.
+Using this assertion, we deduce from (4.4) that
+Jn = sp+1
+�1
+2
+�
+∂Ω
+vq+1
+n
+sp+1 − 2λ
+p−1
+1−q
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+.
+(4.7)
+Furthermore, we use the test function ϕ = 1 in (4.1) to obtain
+−
+�
+Ω
+Un + λ
+p−1
+1−q
+n
+�
+Ω
+U p
+n +
+�
+∂Ω
+U q
+n = 0.
+Substituting Un = snφΩ + vn,
+−
+�
+Ω
+�
+φΩ + vn
+sn
+�
++ λ
+p−1
+1−q
+n
+sp−1
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
++
+�
+∂Ω
+vq
+n
+sn
+= 0,
+from which we use (2.6) with Un to infer that
+�
+∂Ω
+vq
+n
+sn
+−→
+�
+Ω
+φΩ > 0.
+Then, we may deduce that
+csn ≤
+�
+∂Ω
+vq
+n
+for some c > 0. By H¨older’s inequality, we deduce that
+cs
+q+1
+q
+≤
+�
+∂Ω
+vq+1
+for some c > 0. We use this inequality to derive from (4.7) that
+Jn ≥ sp+1
+�
+cs
+1
+q −p
+n
+− 2λ
+p−1
+1−q
+n
+�
+Ω
+�
+φΩ + vn
+sn
+�p
+φΩ
+�
+for some c > 0; thus, (4.6) follows. Assertion (4.5) has been now established.
+We end the proof of this lemma. Observe from (2.5) with Un that ∥vn∥ → 0. Then, we
+develop the same argument as in the second paragraph of the proof of assertion (i) to arrive at
+the same contradiction.
+□
+17
+
+Employing the above lemmas, we then verify assertion (iii). Assume by contradiction that
+(λn, un) → (0, 0) in R × H1(Ω) for a positive solution (λn, un) of (1.1) with λn > 0. Then,
+(λn, Un) with Un = λ
+−
+1
+1−q
+n
+un admits a positive solution of (4.1). Since Un is bounded in H1(Ω)
+by Lemma 4.1, we deduce that up to a subsequence, Un ⇀ U∞ ≥ 0, and Un → U∞ in Lp+1(Ω)
+and L2(∂Ω) for some U∞ ∈ H1(Ω). Thanks to Lemma 4.3, we apply Lemma 2.1 to obtain
+U∞ ̸= 0.
+Furthermore, substituting (λ, U) = (λn, Un) into (4.1) and then taking the limit, we deduce
+that
+�
+Ω
+�
+∇U∞∇ϕ − U∞ϕ
+�
++
+�
+∂Ω
+U q
+∞ϕ = 0.
+This implies that U∞ is a nonnegative solution of (4.1) for λ = 0. Finally, Lemma 4.2 provides
+U∞ = 0, which is a contradiction.
+The proof of Theorem 1.5 is complete.
+□
+Remark 4.4. Assertions (ii) and (iii) of Theorem 1.5 are also derived from Lemma 3.1 when
+βΩ = 1 and pq > 1.
+5. Stability analysis of the trivial solution
+In the last section, we consider the stability of the trivial solution u = 0. It is worthwhile
+to mention that a linearized stability analysis does not work for u = 0 because u �→ uq is not
+differentiable at u = 0.
+The corresponding initial-boundary value problem is formulated as
+follows:
+
+
+
+
+
+∂u
+∂t (t, x) = ∆u + u − up
+in (0, ∞) × Ω,
+∂u
+∂ν = −λuq
+on (0, ∞) × ∂Ω,
+u(0, x) = u0(x) ≥ 0
+in Ω.
+(5.1)
+We use the method of monotone iterations to determine the Lyapunov stability of the trivial
+solution u = 0 (see [26, Definition 5.1.1]).
+When βΩ < 1 or when βΩ = 1 and pq > 1, we observe from Lemma 3.1 that u = 0 is unstable
+in the following sense: for u0 ∈ C2(Ω) sufficiently small such that u0 > 0 in Ω, the positive
+solution u(t, x) of (5.1) corresponding to the initial value u0 moves away from 0 as t → ∞.
+When βΩ > 1, for ε, δ, τ > 0, we set
+ψδ,ε,τ(x) = δ(φΩ(x) + ε)τ,
+x ∈ Ω.
+Let Ωρ := {x ∈ Ω : dist(x, ∂Ω) < ρ} for ρ > 0 be a tubular neighborhood of ∂Ω. Then, by (1.5),
+for ρ0 > 0 small, we can choose a constant c3 = c3(ρ0) > 0 such that |∇φΩ|2 ≥ c3 in Ωρ for
+0 < ρ ≤ ρ0. If 0 < ρ ≤ ρ0, then there exists c4 = c4(ρ) > 0 such that φΩ ≥ c4 in Eρ := Ω \ Ωρ.
+The following result would then provide useful information about the stability of the trivial
+solution u = 0.
+Theorem 5.1. Assume that βΩ > 1. Then, for
+1
+βΩ < τ < 1 and ε > 0 small, there exists δ1 > 0
+such that ψδ,ε,τ is a supersolution of (1.1) whenever 0 < δ ≤ δ1.
+Proof. We write ψδ,ε,τ simply as φδ,ε. By direct computations, we obtain
+∇ψδ,ε = δτ(φΩ + ε)τ−1∇φΩ,
+(5.2)
+∆ψδ,ε = δτ(τ − 1)(φΩ + ε)τ−2|∇φΩ|2 + δτ(φΩ + ε)τ−1∆φΩ.
+(5.3)
+18
+
+We see from (5.3) that for x ∈ Ωρ,
+∆ψδ,ε + ψδ,ε − ψp
+δ,ε ≤ δτ(τ − 1)(φΩ + ε)τ−2|∇φΩ|2 + δ(φΩ + ε)τ
+= δ(φΩ + ε)τ−2
+
+
+−τ(1 − τ)c3 +
+�
+ε + max
+Ωρ
+φΩ
+�2
+
+ .
+We then find 0 < ρ1 ≤ ρ0 and ε1 > 0 such that
+�
+ε + max
+Ωρ1
+φΩ
+�2
+≤ τ(1 − τ)c3
+for 0 < ε ≤ ε1,
+and then,
+−∆ψδ,ε + ψδ,ε − ψp
+δ,ε ≤ 0
+in Ωρ1.
+Let us fix c4 = c4(ρ1), and let 0 < ε ≤ ε1. We also see from (5.3) that for x ∈ Eρ1,
+∆ψδ,ε + ψδ,ε − ψp
+δ,ε ≤ δτ(φΩ + ε)τ−1(−βΩ)φΩ + δ(φΩ + ε)τ
+≤ δ(φΩ + ε)τ−1 {(1 − τβΩ)c4 + ε} .
+We then determine 0 < ε2 ≤ ε1 such that (1 − τβΩ)c4 + ε2 ≤ 0, and then,
+−∆ψδ,ε2 + ψδ,ε2 − ψp
+δ,ε2 ≤ 0
+in Eρ1.
+Finally, using (1.5), we see from (5.2) that
+∂ψδ,ε2
+∂ν
++ λψq
+δ,ε2 ≥ δq(−δ1−qτετ−1
+2
+c2 + λετq
+2 ) ≥ 0
+on ∂Ω,
+if 0 < δ ≤ δ1 for some δ1 > 0.
+In summary, ψδ,ε2, 0 < δ ≤ δ1, is as desired.
+□
+From Theorem 5.1, it might be claimed that u = 0 is asymptotically stable for the case where
+βΩ > 1, meaning that for u0 in the order interval [0, ψδ1,ε2,τ], the positive solution u(t, x) of (5.1)
+associated with u0 tends to 0 as t → ∞. If this occurs, then Theorem 0(II) means that problem
+(5.1) is bistable with two nonnegative stable equilibria for 0 < λ ≤ λ∗ (one is uλ, and the other
+is u = 0), which presents ecologically a conditional persistence strategy for the harvesting effort
+λ. However, the difficulty arises from the fact that the monotone iteration scheme does not
+work for (5.1) in the order interval [0, ψδ1,ε2,τ] because u �→ (−uq) does not satisfy the one-sided
+Lipschitz condition [26, (4.1.19)] for u close to 0. Rigorous verification of the claim is an open
+question.
+References
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+autonomous problems with concave–convex nonlinearities, Nonlinear Anal. 93 (2013), 226–235.
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+Soc. Edinburgh Sect. A 136 (2006), 779–784.
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+Math. J. 55 (2013), 399–409.
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+Glasg. Math. J. 58 (2016), 461–469.
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+lems, Topol. Methods Nonlinear Anal. 28 (2006), 87–103.
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+Differential Equations 134 (1997), 183–215.
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+Englewood Cliffs, N.J. 1967.
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+of the positive solutions set of a logistic equation, J. Differential Equations 257 (2014), 3935–3977.
+[29] H. Ramos Quoirin, K. Umezu, Bifurcation for a logistic elliptic equation with nonlinear boundary
+conditions: a limiting case, J. Math. Anal. Appl. 428 (2015), 1265–1285.
+[30] J. D. Rossi, Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem, Sta-
+tionary partial differential equations. Vol. II, 311–406, Handb. Differ. Equ., Elsevier/North-Holland,
+Amsterdam, 2005.
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+Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), 161–163.
+20
+
+[32] K. Umezu, Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery
+harvesting, Nonlinear Anal. Real World Appl. 70 (2023), Paper No. 103788, 29 pp.
+[33] H. Wiebers, S-shaped bifurcation curves of nonlinear elliptic boundary value problems, Math. Ann.
+270 (1985), 555–570.
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+rameter, Math. Ann. 270 (1985), 401–402.
+Department of Mathematics, Faculty of Education, Ibaraki University, Mito 310-8512, Japan
+Email address: kenichiro.umezu.math@vc.ibaraki.ac.jp
+21
+

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+Sample efficient graph classification using binary Gaussian boson sampling
+Amanuel Anteneh∗
+Department of Computer Science, University of Virginia, Charlottesville, Virginia 22903, USA†
+Olivier Pfister‡
+Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA
+(Dated: January 4, 2023)
+We present a variation of a quantum algorithm for the machine learning task of classification with
+graph-structured data. The algorithm implements a feature extraction strategy that is based on
+Gaussian boson sampling (GBS) a near term model of quantum computing. However, unlike the
+currently proposed algorithms for this problem, our GBS setup only requires binary (light/no light)
+detectors, as opposed to photon number resolving detectors. These detectors are technologically
+simpler and can operate at room temperature, making our algorithm less complex and less costly
+to implement on the physical hardware. We also investigate the connection between graph theory
+and the matrix function called the Torontonian which characterizes the probabilities of binary GBS
+detection events.
+I.
+INTRODUCTION
+Graphs are one of the most versatile data structures
+used in computing, and developing machine learning
+methods for working with graph-structured data has
+been a growing sub-field of machine learning research.
+Graph classification, in particular, has useful applica-
+tions in fields such as bioinformatics, network science
+and computer vision as many of the objects studied in
+these fields can easily be represented as graphs. How-
+ever, using graph-structured data with machine learning
+models is not a straightforward task.
+This is because
+one of the most common ways of representing a graph
+for computational applications, i.e., as an adjacency ma-
+trix, cannot be easily used as an input to machine learn-
+ing classifiers which primarily take vector-valued data as
+their inputs. Therefore, a common way of working with
+graph-structured data is by defining a feature map φ that
+maps a graph G to a vector in a Hilbert space called a
+feature space. From there a function κ, called a kernel,
+is defined that measures the similarity of two graphs in
+the feature space.
+An example of a feature map from
+R2 → R3 is shown in Fig. 1.
+Kernel methods refer to machine learning algorithms
+that learn by comparing pairs of data points using this
+similarity measure. In our context we have a set of graphs
+G and we call a kernel κ a graph kernel if it is a function
+of the form κ : G × G → R [1, 2]. The most common
+example of a kernel function is the feature space’s inner
+product κ(x, x′) = ⟨φ(x), φ(x′)⟩. The goal of such meth-
+ods is to construct mappings to feature vectors whose en-
+tries (the features) relate to relevant information about
+the graphs. Using a Gaussian boson sampling (GBS) de-
+vice to construct graph kernels was an idea first proposed
+∗ asa2rc@virginia.edu
+† Current address:
+Booz Allen Hamilton, Arlington, Virginia
+22202, USA
+‡ olivier.pfister@gmail.com
+FIG. 1: In the original input space R2 the data points,
+which belong either to the class ‘red’ or ‘blue’, are not
+separable by a linear function (the decision boundary)
+but after mapping the points to feature vectors in a
+higher dimensional space R3 a linear function is able to
+separate the two classes. This linear decision boundary
+can be calculated by supervised machine learning
+models such as a support vector machine. In our case
+the input space is the set of all undirected graphs which
+we denote as G.
+by Schuld et al. in Ref. 3.
+Boson sampling was first proposed by Aaronson and
+Arkhipov [4] as a task—generating the photon-counting
+outcomes of the “quantum Galton board” constituted by
+an M ×M optical interferometer fed with single photons
+into some of its input ports—that is strongly believed
+to be intractable to classical computers. The reason for
+this intractability is that calculating the probability dis-
+tribution for generating random outcomes using Monte
+Carlo simulations requires calculating the permanent of
+an M × M matrix. Calculating the permanent of a gen-
+eral matrix is known to be #P-complete [5] which is a
+class of problems comparable to the class of NP-complete
+problems in their difficulty. Gaussian boson sampling [6]
+is a variant of boson sampling in which the single-photon
+inputs are replaced with single-mode squeezed states, as
+produced, for example, by two-photon-emitting optical
+arXiv:2301.01232v1  [quant-ph]  3 Jan 2023
+
+Linear Decision
+Boundary
+Input Space
+Feature Space
+d3
+d22
+parametric amplifiers [7]. The GBS probability distribu-
+tion is governed by the Hafnian of an M × M matrix.
+Calculating the Hafnian of a general square matrix can
+be reduced to the task of calculating permanents there-
+fore calculating the Hafnian is also #P-complete. In both
+cases, a quantum machine implementing boson sampling
+or GBS can easily sample from these hard-to-calculate
+probability distributions, just because they are “wired-
+in,” and this constitutes the “quantum advantage” that
+was recently demonstrated in optical experiments [8, 9].
+Note also that the initial “quantum supremacy” result
+obtained by Google on a superconducting qubit array [10]
+was a quantum (circuit) sampling result as well.
+Beyond these necessary initial steps of demonstrating
+that quantum hardware can indeed reach regions inacces-
+sible to classical hardware, a subsequent question is that
+of the utility of a sampling task. Whereas the usefulness
+of sampling in and of itself is far from established, we
+know that the histograms produced by statistically sig-
+nificant sampling constitute empirical probability distri-
+butions that tend toward the true, classically intractable
+probability distributions for sample numbers linear in
+the number of possible outcomes [11]. The problem is
+that this very number of possible outcomes grows expo-
+nentially with M in a M-qubit quantum circuit in gen-
+eral [12], and exponentially or super-exponentially with
+M in an M-optical-mode boson or Gaussian boson sam-
+pler, which dispels any notion of quantum advantage for
+calculating the corresponding quantum probability dis-
+tributions.
+One direction that has been explored out of this co-
+nundrum is the binning of GBS measurements results
+into outcome classes whose cardinality scales favorably
+(e.g. polynomially) with the problem size (the GBS mode
+number). The immediate downside of such an approach
+is loss of information it entails, which impacts usefulness.
+However, graph classification using feature vectors and
+coarse-graining might provide advantageous GBS appli-
+cations. This was first pointed out by Schuld et al. [3].
+In this paper, we show that a technologically simpler
+version of GBS, which we term binary GBS, can achieve
+comparable or better performance. The paper is struc-
+tured as follows. In Sec.II we give broad reminders about
+GBS and graph theory (with details in Appendix A) and
+the current GBS graph kernel from Ref. 3.
+We then
+present our graph kernel in Sec.III along with results
+from numerical experiments and analyses of its complex-
+ity, features and advantages.
+II.
+REMINDERS ABOUT GAUSSIAN BOSON
+SAMPLING (GBS) AND GRAPH THEORY
+A.
+Gaussian Boson Sampling
+As mentioned above, an M-mode GBS devise com-
+prises M single-mode-squeezing (SMS) inputs, an M ×M
+optical interferometer, and M photon-number-resolving
+FIG. 2: Example of a 3-mode Gaussian boson sampler.
+Mode i ∈ {1, 2, 3} starts in the vacuum state |0⟩, is then
+squeezed by ˆS(ri) and passes through the network of
+two beamsplitters (the interferometer) before the
+number of photons in each mode is measured by the
+detectors Di∈{1,2,3}.
+(PNR) detectors, see Fig.2 for an example. The latter
+have come of age in superconducting devices such as tran-
+sition edge sensors [13] and superconducting nanowire
+single-photon detectors [14].
+Both the former and the
+latter have recently been used to make PNR measure-
+ments of as many as 100 photons [15, 16].
+An M-mode Gaussian boson sampler prepares a Gaus-
+sian (Wigner function) quantum state by the M squeez-
+ers and the interferometer.
+The squeezers output
+squeezed light into the interferometer and the photons
+are then passed through the interferometer after which
+the M detectors detect what modes the photons end up
+in resulting in a detection event.
+We denote a detec-
+tion event as n = (n1, ..., nM), where ni is the photon
+count in the ith mode and the total number of photons
+is n = �M
+i=1 ni.
+We know consider binary detectors, such as single-
+photon avalanche photodiodes, which are single-photon
+sensitive but aren’t PNR and give the same signal how-
+ever many photons were absorbed. In this case, we have
+ni ∈ {0, 1} where ni = 0 indicated zero photons were
+detected in that mode and ni = 1 indicates that at least
+one photon was detected. When using binary detectors
+we no longer know the total photon number n so we use
+N to denote the number of detectors that detect photons
+leading to �M
+i=1 ni = N ≤ M.
+An M-mode Gaussian state is fully described by a co-
+variance matrix Σ ∈ R2M×2M and a displacement vector
+d ∈ R2M [17].
+B.
+Graph theory
+In this paper we define a graph G = (V, E) as a
+set of vertices V
+= {v1, v2, ...} and a set of edges
+E = {(v1, v1), (v1, v2), ...(vi, vj), ...} that connect vertices
+if the edge value is not zero. A graph can be unweighted,
+with all nonzero edge weights equal to 1, or weighted, for
+example with real edge weights in GBS. For undirected
+graphs, which is what we will exclusively work with in
+
+<ol
+S(r1)
+D
+B(p2, T2)
+<ol
+S(r2)
+D.
+B(p1, T1)
+<ol
+S(r3)
+D3
+this paper, (vi, vj) = (vj, vi), ∀i, j. The size of a graph is
+equal to the cardinality |V | of its vertex set. The degree
+of a vertex v is the number of edges that are connected to
+it. The maximum degree of a graph is the largest degree
+of a vertex in its vertex set.
+Graphs can be represented in a number of ways such as
+a diagram, Fig.3a, or a more computationally useful way
+as an adjacency matrix, Fig.3b. The adjacency matrix
+of an undirected graph G with |V | vertices is a |V | × |V |
+symmetric matrix A with entries aij where aij is the
+weight of the edge connecting vertices i and j.
+1
+2
+3
+4
+3
+6
+4
+9
+(a) Undirected weighted
+4-vertex graph with 4 edges
+�
+���
+0 4 3 0
+4 0 6 9
+3 6 0 0
+0 9 0 0
+�
+���
+(b) Adjacency matrix of graph
+FIG. 3: 4-vertex graph and its corresponding adjacency
+matrix
+C.
+Sample complexity of GBS
+1.
+The problem with using GBS beyond sampling
+The sample complexity of a machine learning algo-
+rithm refers to the number of samples or amount of data
+required to learn some target function. In the case of
+GBS applications it refers to the number of samples we
+need to generate from the GBS device to learn or approx-
+imate a probability distribution over some set of the pho-
+ton detection events. This complexity type is extremely
+important to examine for any applications of GBS as it
+could potential render certain applications of GBS in-
+tractable for larger problem sizes.
+For example it was shown that the GBS device uti-
+lizing PNR detectors can encode the graph isomorphism
+problem [18]. This is done by encoding two graphs into
+two GBS devices and sampling each S times. The S sam-
+ples could then be used, in principle, to reconstruct the
+probability distribution over all possible detection events
+n for a given M and n. However, this cannot be done effi-
+ciently enough to provide a quantum advantage. Indeed,
+we know from Refs. 11, 19 that reconstructing a proba-
+bility distribution D over a discrete finite set Ω of cardi-
+nality |Ω| from an empirical distribution ˆD constructed
+from samples from D we require
+S =
+�2(ln(2)|Ω| + ln( 1
+δ ))
+ϵ2
+�
+(1)
+samples to guarantee that
+p(||D − ˆD||1 ≥ ϵ) ≤ δ
+(2)
+where ||D − ˆD||1 denotes the L1 distance between D and
+ˆD. In other words we require
+O
+�|Ω| + ln( 1
+δ )
+ϵ2
+�
+(3)
+samples to ensure with probability at most δ that the
+sum of the absolute values of the errors on the empirical
+probability distribution is ϵ or greater.
+This means the number of samples we need to approx-
+imate a probability distribution scales linearly with the
+number of elements in it’s sample space i.e. the num-
+ber of outcomes. In the case where D is the probability
+distribution over the set of all possible PNR detection
+events the number of such events for a given number of
+modes M and maximum number of photons n is
+|Ω| =
+�n + M − 1
+M − 1
+�
+= (n + M − 1)!
+n!(M − 1)!
+(4)
+which in number theory is also known as the formula for
+the number of weak compositions of an integer n into M
+parts. As shown in appendices B and C under the as-
+sumption that the number of mode scales quadratically
+with the number of photons, M ∈ O(n2), this quantity
+grows super exponentially with M and in general scales
+as O((n + M − 1)M−1) meaning that as the size of the
+graphs increase, and therefore as the number of modes
+of the GBS device increase, we require an exponential
+number of samples to ensure the algorithm can give us
+the correct result within a certain probability. Therefore
+while the algorithm may in principle be able to decide
+graph isomorphism, it is sample inefficient to an expo-
+nential degree making it intractable to implement even
+with a fault tolerant quantum computer.
+2.
+Coarse graining of sample distributions
+However a method was suggested in [18] to coarse-grain
+the probability distribution by combining outcomes into
+groups called orbits. Coarse-graining in this sense means
+to construct a new probability distribution over the set
+of these groups, the cardinality of which is less than the
+original set of all possible detection events. An orbit On
+consists of a detection event n and all of its permutations.
+For example the orbit that contains the detection event
+n = (1, 2, 2) also contains the detection events (2, 1, 2)
+and (2, 2, 1). The number of orbits for a 4-mode GBS
+device is equal to the number of ways one can write n1 +
+n2 + n3 + n4 = n, where the order of the summands does
+not matter. This is called the number of partitions of
+the integer n into M parts and from the number theory
+literature [20] it is known to behave asymptotically as
+|Ω| ≈
+eπ
+�
+2(n−M)
+3
+4
+√
+3(n − M), M ≤ n ≤ 2M.
+(5)
+
+4
+If we assume the number of photons grows linearly with
+the number of modes, n ∈ Θ(M) → n = 2M, we have
+the following asymptotic bound on the number of orbits
+1
+4
+√
+3M eπ√
+2M
+3 ∈ O(eπ√
+2M
+3
+M
+).
+(6)
+This means the number of orbits, which is now the num-
+ber of outcomes |Ω| from Eq. 1, grows like M −1eπ√
+2M/3
+meaning we would have a sample complexity of
+O
+�
+M −1eπ√
+2M/3 + ln( 1
+δ )
+ϵ2
+�
+(7)
+which is subexponential but still intractable for large M.
+3.
+Sample complexity of previously proposed GBS graph
+kernels
+The first GBS based graph kernel proposed in [3]
+maps a graph G to feature vectors in a feature space
+φ : G → f = (f1, f2, ..., fD) ∈ RD. Where fi = p(Oi
+n)
+is the probability of detecting a detection event from the
+orbit Oi
+n. This kernel was shown to perform well against
+three of the four classical kernels we use as benchmarks
+in this paper. However a shortcoming of this method is
+that the sample complexity is O(M −1eπ√
+2M/3).
+The second GBS kernel is of the form φ : G →
+f = (f1, f2, ..., fD) ∈ RD, with fi = p(Mi
+n,∆s) where
+p(Mi
+n,∆s) is the probability of detecting a detection event
+that belongs to the “meta-orbit” Mi
+n,∆s. A meta-orbit
+Mn,∆s is uniquely defined by a total photon number n
+and ∆s which is defined as
+∆s = {n :
+�
+i
+ni = n ∧ ∀i : ni ≤ s }.
+(8)
+Therefore a meta-orbit consists of all detection events
+where total photon number is equal to n, where no detec-
+tor counts more than s photons. It is claimed that this
+strategy partitions the set of all PNR detection events
+into a polynomial number of subsets in n [21].
+III.
+THE ALGORITHM
+A.
+GBS with binary detectors and it relation to
+graph theory
+While the relationship between GBS with PNR de-
+tectors and graph theory has been thoroughly explored,
+there has been little exploration of how GBS with bi-
+nary detectors fits into the picture. In this section we
+shed some light on the relationship between the two.
+As stated before when using binary detectors the detec-
+tion outcomes are of the form nbin = (n1, ..., nM) where
+ni ∈ {0, 1} ∀i and ni = 1 indicates the ith detector de-
+tected one or more photons. The probability of detect-
+ing a detection outcome with binary detectors is charac-
+terized by a matrix function called the Torontonian, to
+which the same arguments for classical intractability as
+for the Hafnian can be extended [22]. The probability of
+a given binary detection event nbin is given by
+p(nbin) = Tor(Onbin)
+�
+det(Q)
+= Tor(X ˜Anbin)
+�
+det(Q)
+(9)
+where Q is defined in Eq. (A2), O = I − Q−1, and Tor()
+is the Torontonian of a 2N × 2N matrix A defined as
+Tor(A) =
+�
+Z∈P ([N])
+(−1)|Z|
+1
+�
+det(I − AZ)
+.
+(10)
+Where P([N]) is the power set, the set of all possible
+subsets, of the set [N] = {1, 2, ..., N}. The probability of
+a PNR detection event n can be written in terms of the
+matrix O as
+p(n) =
+1
+�
+det(Q)
+Haf( ˜An)
+n!
+=
+1
+�
+det(Q)
+Haf(XOn)
+n!
+(11)
+The probability of a binary GBS detection event is
+simply the sum of all probabilities of the corresponding
+PNR detection events.
+A useful example to illustrate
+this is a 4-mode Gaussian boson sampler programmed
+according to some adjacency matrix A of a graph G. Sup-
+pose we use binary detectors and measure the detection
+event nbin = (1, 0, 1, 0).
+The corresponding detection
+events when using PNR detectors would be of the form
+n = (n1, 0, n3, 0) where n1, n3 > 0. We will define N to
+be the set of all possible 4-mode PNR detection events
+with 0’s in the 2nd and 4th index, i.e. only the 2nd and
+4th detectors detect no photons. From this we have
+p((1, 0, 1, 0)) = Tor(X ˜A(1,0,1,0))
+�
+det(Q)
+=
+�
+n∈N
+p(n) =
+�
+n∈N
+Haf2(An)
+n!
+�
+det(Q)
+.
+(12)
+This means the Torontonian of X ˜A is proportional to an
+infinite sum of Hafnians as there are an infinite number
+of integer lists of the form (n1, 0, n3, 0) where n1, n3 > 0.
+In a real GBS experiment, however, the energy is finite
+and therefore the measured probabilities of these events
+would be equal to a finite version of this sum where all
+detection events with total photon number greater than
+some cutoff photon number vanish from the series.
+In terms of graph theory this means the probability of
+detecting nbin = (1, 0, 1, 0) is proportional to the sum of
+the squared Hafnians of all possible subgraphs of G of
+unbounded size with their 2nd and 4th vertices removed.
+But again in practice the maximum size of the subgraphs
+
+5
+will always be bounded by some maximum photon num-
+ber for a real GBS experiment. More generally we have
+p(nbin) = Tor(Onbin)
+�
+det(Q)
+= Tor(X ˜Anbin)
+�
+det(Q)
+=
+�
+n∈N
+Haf2(An)
+n!
+�
+det(Q)
+(13)
+where N is the set of all PNR events that correspond to
+the binary detection event nbin [23].
+B.
+Constructing the feature vectors
+Once the GBS device is programmed we generate S
+samples from the device. For our algorithm we use bi-
+nary detectors so each sample is a list of length M with
+entries either 0 or 1. Once we have these samples we use
+them to construct the feature vector of which we have
+two definitions based on two coarse-graining strategies.
+The first is based on what we call the µ coarse-graining
+strategy where we group together detection events that
+contain exactly i detector ‘clicks’ or ones.
+For exam-
+ple the detection events (1, 0, 0) and (0, 0, 1) would be
+grouped together since they both contain exactly 1 detec-
+tor click. These groups can also be thought of as ‘binary
+orbits’ since they contain a detection event and all its per-
+mutations. This strategy partitions the set of all binary
+detection events into a linear number of disjoint subsets
+in N. Using this strategy we can define the feature map
+as φ : G → f = (f0, f1, ..., fN) ∈ RN. Where N is the
+maximum number of detector clicks and fi =
+Si
+S with
+Si being the number of samples which contain exactly i
+ones. Equivalently this is the probability of detecting an
+event where exactly i detectors detect a photon.
+The second feature map is based on what we call the
+ν coarse-graining strategy. For a 5 mode boson sampler
+utilizing binary detectors with maximum click number
+5 there are |Ω| = 32 possible detection outcomes. This
+coarse-graining strategy groups together detection events
+whose first 5 modes are one of these 32 outcomes. For
+example the detection event nbin = (0, 1, 0, 0, 1, 0, 1) be-
+longs in the group associated with the detection event
+(0, 1, 0, 0, 1) since they are equal if one is only concerned
+with the first 5 modes.
+This strategy partitions the
+set of all detection events of 5 or more modes into a
+constant number of subsets, i.e. 32. The feature map
+based on this strategy is defined as φ : G → f =
+(f[0,0,0,0,0], f[1,0,0,0,0], ..., fn) ∈ R32. Where fn is the prob-
+ability of detecting an event where the first 5 modes cor-
+respond to one of the 32 possible detection outcomes. For
+example f[1,0,0,0,0] is the probability that the first detec-
+tor detects photons and the following 4 detectors detect
+vacuum.
+Once we construct the feature vector for each graph in
+the data set we input them to a machine learning classi-
+fier such as a support vector machine.
+C.
+Numerical experiments
+We used The Walrus python library to classically sam-
+ple from the GBS output distribution when running our
+experiments and the GraKel python library to fetch the
+data sets and simulate the classical graph kernels [24, 25].
+Classically sampling from a GBS output distribution is
+very time intensive even when using binary detectors so
+we choose to follow the choice made in [3] and discard
+graphs with greater than 25 and less than 6 vertices for
+each data set. Before sampling from the GBS device we
+have four parameters we can set: the maximum number
+of detector clicks allowed N, the average photon number
+¯n, the displacement on each mode of the GBS device d
+and lastly the number of samples generated by the GBS
+device S. We set N = 6, ¯n = 5 and d = 0 for our re-
+sults reported here leading to probability distribution of
+32 outcomes using the ν coarse-graining strategy and 7
+outcomes using the µ coarse-graining strategy. Using Eq.
+1 with δ = 0.01 and ϵ = 0.06 we require about S = 15000
+samples for the ν feature vectors and about S = 6000
+samples for the µ feature vectors.
+For the machine learning classifier we use a support
+vector machine with an RBF kernel κrbf. We obtain the
+accuracies in Table II by running a double 10-fold cross-
+validation 10 times. The inner fold performs a grid search
+through the discrete set of values [10−4, 10−3, ..., 102, 103]
+on the C hyper-parameter of the SVM which controls
+the penalty on misclassifications. We tested our graph
+kernel on the same data sets used in [3]. We also ignored
+vertex labels, vertex attributes and edge attributes and
+converted all adjacency matrices to be unweighted.
+Four classical graph kernels were used as a bench-
+mark for our algorithms classification accuracy.
+The
+subgraph matching kernel (SM) with time complexity
+O(kM k+1) where M is the number of vertices and k the
+size of the subgraphs being considered [26], the graphlet
+sampling kernel (GS) with worst case time complexity
+O(M k) which can be optimized to O(Mdk−1) for graphs
+of bounded degree with the restriction that k ∈ {3, 4, 5},
+where k is the graphlet size and d is the maximum de-
+gree of the graph [27], the random walk kernel (RW) with
+time complexity O(M 3) [28] and the shortest path kernel
+(SP) with time complexity O(M 4) [29]. For the graphlet
+sampling kernel we set maximum graphlet size to k = 5
+and draw 5174 samples, for the random walk kernel we
+use fast computation and a geometric kernel type with
+the decay factor set to λ = 10−3, for the subgraph match-
+ing kernel we set maximum subgraph size to k = 5 and
+for the shortest path kernel we used the Floyd–Warshall
+algorithm to calculate shortest paths. The accuracies of
+all four classical kernels, the original GBS graph kernels
+from [3] with n = 6 and our kernel are shown in Table II.
+
+6
+TABLE I: Graph data set statistics after prepossessing. A more detailed description of these data sets can be found
+in appendix B of [3].
+Data set
+# of graphs # of classes avg. # of vertices avg. # of edges
+AIDS
+1723
+2
+11.11
+11.29
+BZR MD
+257
+2
+20.10
+197.69
+COX2 MD
+118
+2
+23.90
+274.40
+ENZYMES
+204
+6
+18.56
+36.30
+ER MD
+357
+2
+19.27
+185.15
+FINGERPRINT
+1080
+3
+10.58
+9.10
+IMDB-BINARY
+806
+2
+15.98
+63.32
+MUTAG
+179
+2
+17.48
+19.23
+NCI1
+1853
+2
+19.77
+21.27
+PROTEINS
+515
+2
+15.77
+29.37
+PTC FM
+284
+2
+13.64
+13.99
+TABLE II: Average test accuracies of the support vector machine with different data sets and graph kernels. The
+error reported is the standard deviation between 10 repeats of double cross validation. GS, RW, SM and SP refer to
+the graphlet sampling, random walk, subgraph matching and shortest path kernels respectively. GBSbin
+ν
+and GBSbin
+µ
+denotes our GBS kernel with binary detectors that use the ν and µ coarse-graining strategies to construct the
+feature vectors respectively. GBSbin+
+ν
+denotes that the feature associated with detecting vacuum [0, 0, 0, 0, 0] in the
+first 5 modes was dropped from all feature vectors. GBSPNR and GBSPNR+ refer to the original GBS kernels with
+PNR detectors that use orbit and meta-orbit probabilities as features respectively with a displacement of d on each
+mode. *Runtime > 7 days
+Data set
+GBSbin+
+ν
+GBSbin
+ν
+GBSbin
+µ
+GS
+RW
+SM
+SP
+AIDS
+98.47(0.10)
+98.74(0.20)
+99.53(0.05)
+99.30(0.07)
+53.11(11.90)
+77.85(2.44)
+99.34(0.09)
+BZR MD
+60.14(1.28)
+61.73(0.89)
+58.79(1.17)
+51.42(3.51)
+64.54(0.36)
+time out*
+50.82(1.76)
+COX2 MD
+51.62(2.76)
+50.18(2.96)
+51.30(3.86)
+49.01(3.18)
+48.98(4.78)
+time out*
+48.11(4.30)
+ENZYMES
+48.10(1.18)
+41.75(2.35)
+19.83(1.43)
+34.59(2.54)
+19.50(2.29)
+37.38(1.60)
+22.15(1.88)
+ER MD
+67.74(0.94)
+69.19(0.33)
+68.84(0.50)
+48.88(4.53)
+70.32(0.02)
+time out*
+45.23(4.35)
+FINGERPRINT
+64.45(0.78)
+65.53(0.86)
+63.56(0.67)
+65.25(1.30)
+33.63(3.57)
+46.89(0.56)
+46.22(1.02)
+IMDB-BINARY
+60.69(0.84)
+61.35(0.98)
+67.34(0.38)
+68.49(0.63)
+67.78(0.38)
+time out*
+65.50(0.27)
+MUTAG
+84.63(0.91)
+85.94(0.98)
+81.37(0.90)
+80.80(0.91)
+83.22(0.04)
+83.24(1.27)
+82.74(1.65)
+NCI1
+63.45(0.57)
+56.99(1.69)
+59.09(1.02)
+50.34(3.22)
+50.96(3.58)
+time out*
+53.40(2.25)
+PROTEINS
+65.95(1.03)
+63.38(0.73)
+63.11(0.55)
+65.75(0.94)
+56.91(1.39)
+62.93(0.83)
+63.63(0.41)
+PTC FM
+52.63(3.95)
+57.47(2.72)
+59.17(1.58)
+60.74(1.48)
+50.95(3.68)
+56.36(2.66)
+55.38(4.04)
+Data set
+GBSPNR (d = 0)
+GBSPNR (d = 0.25)
+GBSPNR+ (d = 0)
+GBSPNR+ (d = 0.25)
+AIDS
+99.60(0.05)
+99.62(0.03)
+99.58(0.06)
+99.61(0.05)
+BZR MD
+62.73(0.71)
+62.13(1.44)
+62.01(1.43)
+63.16(2.11)
+COX2 MD
+44.98(1.80)
+50.11(0.97)
+57.84(4.04)
+57.89(2.62)
+ENZYMES
+22.29(1.60)
+28.01(1.83)
+25.72(2.60)
+40.42(2.02)
+ER MD
+70.36(0.78)
+70.41(0.47)
+71.01(1.26)
+71.05(0.83)
+FINGERPRINT
+65.42(0.49)
+65.85(0.36)
+66.19(00.84)
+66.26(4.29)
+IMDB-BINARY
+64.09(0.34)
+68.71(0.59)
+68.14(0.71)
+67.60(0.75)
+MUTAG
+86.41(0.33)
+85.58(0.59)
+85.64(0.78)
+84.46(0.44)
+NCI1
+63.61(0.00)
+62.79(0.00)
+63.59(0.17)
+63.11(0.93)
+PROTEINS
+66.88(0.22)
+66.14(0.48)
+65.73(0.69)
+66.16(0.76)
+PTC FM
+53.84(0.96)
+52.45(1.78)
+59.14(1.72)
+56.25(2.04)
+D.
+Complexity analysis
+In this section we discuss, in addition to the time and
+space complexity, the sample complexity of our algo-
+rithm.
+1.
+Sample Complexity
+Since the ni’s for binary detection events can be either
+0 or 1 we can think of the detection outcomes as binary
+strings of length M with at most M ones. The number
+of binary strings of length M with exactly i ones is
+�M
+i
+�
+.
+So the number of possible binary detection events, the
+number of binary strings of length M with at most M
+
+7
+ones, is given by
+|Ω| =
+M
+�
+i=0
+�M
+i
+�
+.
+(14)
+We can show this function grows like 2M using the bino-
+mial expansion
+2M = (1 + 1)M =
+M
+�
+i=0
+�M
+i
+�
+1M−i1i =
+M
+�
+i=0
+�M
+i
+�
+.
+(15)
+Therefore we could not simply use the probability of the
+individual detection events as features without coarse-
+graining even when using binary detectors as we would
+still need a prohibitively large number of samples to ap-
+proximate their probabilities to within a constant error.
+This was the reason for introducing the ν and µ coarse-
+graining strategies.
+Since the number of outcomes of the µ distribution
+scales linearly with N which is ≤ M the sample com-
+plexity of approximating the µ coarse-grained probability
+distribution is
+O
+�M + ln( 1
+δ )
+ϵ2
+�
+(16)
+which reduces to O(M) for constant ϵ and δ. The sample
+complexity of approximating the ν coarse-grained prob-
+ability distribution is
+O
+�32 + ln( 1
+δ )
+ϵ2
+�
+(17)
+which reduces to O(1) for constant ϵ and δ.
+2.
+Space Complexity
+The size of the ν feature vectors is constant with re-
+spect to the graph size so the space required is O(1) and
+for the µ feature vectors the size grows linearly with N
+which is ≤ M so the space required is O(M).
+How-
+ever storing the adjacency matrix of the graphs requires
+O(M 2) space complexity.
+3.
+Time Complexity
+The time complexity is determined by the most com-
+putationally time intensive step of the algorithm which
+is encoding the adjacency matrix into the GBS device.
+This is the case because the encoding process requires
+taking the Takagi decomposition of the matrix A which
+for a M × M matrix has time complexity O(M 3) as it
+is a special case of the singular value decomposition [30].
+However there do exist quantum algorithms for comput-
+ing the singular value decomposition of a matrix with
+complexity that is polylogarithmic in the size of the ma-
+trix [31]. In particular the quantum singular value esti-
+mation algorithm for a m × n matrix presented in [32]
+has complexity O(polylog(mn)/ϵ) where ϵ is an additive
+error.
+E.
+Feature analysis & comparison to classical
+kernels
+Fig. 4 shows the results of performing a principal com-
+ponent analysis on the feature vectors generated using
+the ν coarse-graining strategy for various datasets. The
+analysis shows that the feature associated with vacuum
+[0, 0, 0, 0, 0] contributes by far the most in the support of
+the first principal component. The analysis also suggests
+that in some cases the first 10 or so features contribute
+the most to the support of all of the first four principal
+components but in other cases, such as with FINGER-
+PRINT, most features contribute more or less equally.
+Our graph kernel has a time complexity that is equiva-
+lent to the random walk kernel and better than the short-
+est path kernel by a factor of M while outperforming both
+on most data sets. Furthermore the time complexity of
+our kernel is not exponential in the size of the subgraphs
+we are probing like the subgraph matching kernel. The
+graphlet sampling kernel does have a more favorable com-
+plexity of O(Mdk−1) for graphs with maximum degree
+d. However it’s important to note that many real world
+graphs are what are called ‘scale-free networks’ and from
+the network science literature [33] the maximum degree
+of these graphs grows polynomially with the graph size.
+Therefore it is possible that the the maximum degree
+of these graphs grows linearly with the graph size i.g.
+d ∈ O(M) which would lead to a complexity of O(M k)
+for the graphlet sampling kernel. What is also interesting
+is that GBS kernels seems to provide more distinguish-
+ing power than some classical kernels for graphs with no
+vertex and edge labels like those used in our simulations.
+Take for example the ENZYMES dataset for which the
+binary GBS kernel achieves a classification accuracy of
+≈ 48% while the shortest path kernel reaches about 23%.
+If we instead choose to not ignore vertex labels we found
+the shortest path kernel gives a classification accuracy of
+about 50%. Since the GBS features are related to Haf-
+nians this suggests that features related to the number
+of perfect matchings of a graph could be more useful for
+distinguishing graphs of different classes when one has no
+information about the attributes of the graph nodes.
+IV.
+CONCLUSION
+We proposed a variation of an algorithm for the ma-
+chine learning task of classification with graph-structured
+data that uses a Gaussian boson sampler utilizing only
+binary detectors. We show that our algorithm out per-
+forms four classical graph kernels for the task of graph
+
+8
+FIG. 4: Results of the principal component analysis (PCA) on the ν feature vector entries for the ENZYMES,
+MUTAG, IMDB BINARY and FINGERPRINT datasets. The heatmaps show the weight/coefficient associated with
+each feature with regard to the first four principal components.
+classification on many data sets. This is most evident
+with regard to the ENZYMES data set where the ν fea-
+ture map outperforms all methods. The feature corre-
+sponding to detecting vacuum in the first 5 modes plays
+a particularity important role as shown by the princi-
+pal component analysis as it is related to the Hafnian of
+all possible subgraphs of G with their first 5 vertices re-
+moved. We also show that it is sample efficient, a major
+issue for applications of GBS, and has a time complexity
+that is comparable with the classical strategies.
+The fact that a GBS kernel using only binary detec-
+tors produces such accuracies suggests that technologi-
+cally more feasible—binary detectors such as SPADs do
+not operate at cryogenic temperatures such as supercon-
+ducting PNR ones—GBS devices could have useful appli-
+cations for machine learning with graph-structured data.
+We believe that GBS with PNR detectors should also be
+explored more for this application with particular atten-
+tion given to coarse-graining strategies that both reduce
+the sample complexity as well as provide features that
+capture useful information about the graphs.
+A number of questions remain open for investigation
+such as how vertex and edge labels can be encoded into
+the GBS device. Also as stated earlier it is known that
+the existence of a polynomial-time classical algorithm for
+exact sampling from the output probability distribution
+of a boson sampling or Gaussian boson sampling device
+would imply the collapse of the polynomial hierarchy to
+the third level and thus the existence of such an algorithm
+is believed to be very unlikely [34]. This result can also
+be extended to GBS with binary detectors [22]. However
+it is not known, although some work has been done in
+this area [21], if such arguments exist for algorithms that
+sample from coarse-grained versions of these probability
+distributions such as those defined in [3] or our work. It is
+vital to know if such arguments exist as they would imply
+these quantum kernels are also likely hard to simulate
+classically.
+ACKNOWLEDGMENTS
+We thank Maria Schuld, Kamil Br´adler, Scott Aaron-
+son, Ignacio Cirac, Miller Eaton, Nicol´as Quesada, An-
+drew Blance, Shreyas Murthy and Sefonias Maereg for
+useful advice and discussions. We thank Research Com-
+puting at the University of Virginia for providing access
+to, and support with, the Rivanna computing cluster.
+
+ENZYMES
+FINGERPRNT
+Feautures
+1
+1 
+01 - [0, 0, 0, 0, 0]
+02 - [1, 0, 0, 0, 0]
+5 -
+5 -
+1.00
+03 - [0. 1, 0, 0, 0
+04 - [0, 0, 1, 0, 0]
+9 -
+- 6
+05 - [0, 0, 0, 1, 0]
+tures
+eautures
+0.75
+06 - 0. 0. 0, 0, 1
+13
+ 13
+07 - [1, 1, 0, 0, 0]
+08 - [1, 0, 1, 0, 0]
+17
+17
+eau
+09 - [1, 0, 0, 1, 0]
+0.50
+10 - [1, 0, 0, 0, 1]
+21
+21
+11 - [0, 1, 1, 0, 0]
+25 -
+25 
+12 - [0, 1, 0, 1, 0
+- 0.25
+13 - [0, 1, 0, 0, 1]
+29 -
+29
+14 - [0, 0, 1, 1, 0]
+15 - [0, 0, 1, 0, 1]
+16 - [0, 0, 0, 1, 1]
+PC1
+PC2
+PC3
+PC4
+PC1
+PC2
+PC3
+PC4
+- 0.00
+MUTAG
+17 - [1, 1, 1, 0, 0]
+IMDBBINARY
+18 - [1, 1, 0, 1, 0]
+1 
+19 - [1, 1, 0, 0, 1]
+5
+-0.25
+5 -
+20 - [1, 0, 1, 1, 0]
+21 - [1, 0, 1, 0, 1]
+22 - [1, 0, 0, 1, 1
+23 - [0, 1, 1, 1, 0]
+-0.50
+es
+13
+13
+24 - [0, 1, 1, 0, 1]
+eautur
+25 - [0, 1, 0, 1, 1]
+17
+17
+26 - [0, 0, 1, 1, 1]
+-0.75
+27 - [1, 1, 1, 1, 0
+21
+E 21
+28 - [1, 1, 1, 0, 1]
+29 - [1, 1, 0, 1, 1
+25 -
+25
+30 - [1, 0, 1, 1, 1]
+-1.00
+31 - [0, 1, 1, 1, 1]
+29 -
+29 -
+32 - [1, 1, 1, 1, 1]
+PC1
+PC2
+PC3
+PC4
+PC1
+PC2
+PC3
+PC49
+This work was supported by NSF grant PHY-2112867.
+Appendix A: Reminders about standard GBS
+1.
+GBS with PNR detectors
+There has been substantial work done already on the
+connection between graph theory and Gaussain boson
+sampling with PNR detectors [18, 35, 36].
+Here we
+present the important concepts. Any undirected graph
+G with no self-loops and |V | = M vertices can be en-
+coded into a M-mode GBS setup consisting of a set of
+M squeezers followed by an interferometer of beamsplit-
+ters according to its adjacency matrix A. Once the graph
+is encoded into the GBS device the probability of detect-
+ing a specific detection event n = (n1, ..., nM) is equal
+to
+p(n) =
+1
+�
+det(Q)
+Haf( ˜An)
+n!
+=
+1
+�
+det(Q)
+Haf2(An)
+n!
+(A1)
+with
+Q = (I2M − X ˜A)−1,
+X =
+�
+0 I
+I 0
+�
+,
+(A2)
+n! = n1!×...×nM!, ˜A = (A⊕A) and Haf() denoting the
+Hafnian of a 2M × 2M matrix. The Hafnian is a matrix
+function defined mathematically as
+Haf(A) =
+�
+π∈SM
+�
+(u,v)∈π
+Au,v,
+(A3)
+where SM is the partition of the set {1, 2, ..., 2M} into
+unordered disjoint pairs.
+For example if M = 2 then
+SM = ({(1, 2), (3, 4)}, {(1, 4), (2, 3)}, {(1, 3), (2, 4)}).
+If
+A is the adjacency matrix of a unweighted graph then
+the Hafnian is equal to the number of perfect matchings
+of the vertices of the graph.
+A perfect matching is a
+partition of the vertex set of a graph into pairs such that
+each vertex is connected to exactly one edge from the
+edge set. All perfect matchings of a complete 4-vertex
+graph are shown in Fig.5.
+An is the n×n submatrix of A induced according to the
+photon detection event n. An is obtained by repeating
+the ith row and column according to the measurement
+pattern n. If ni = 0 then the ith row and column are
+deleted from A but if ni > 0 then the ith row and col-
+umn are repeated ni times. For example the probability
+of detecting the event where each mode has exactly one
+photon n = (1, 1, ..., 1) would be proportional to the Haf-
+nian of the original matrix A since An = A. What this
+means in terms of the graph is that vertex i and all its
+edges are either deleted if ni = 0 or duplicated ni times
+if ni > 0. Therefore the probability of a detection event
+n is proportional to the squared Hafnian of the subgraph
+Gn corresponding to the induced adjacency matrix An.
+1
+2
+3
+4
+(a) Complete graph of 4
+vertices
+1
+2
+3
+4
+1
+2
+3
+4
+1
+2
+3
+4
+(b) The three perfect matchings of
+the complete 4-vertex graph
+FIG. 5: The complete graph of 4 vertices and its
+corresponding perfect matching
+FIG. 6: Table of different photon detection events n
+and the corresponding subgraphs Gn they induce and
+the value of the squared Hafnians of those subgraphs.
+The probability of the detection event where each
+detector detects one photon corresponds to the Hafnian
+of the graph encoded into the GBS. We can see in the
+third graph from the top when a detector detects 2
+photons the corresponding vertex is duplicated.
+Examples of different detection events and their corre-
+sponding induced subgraphs are shown in Fig.6.
+These induced subgraphs are of even size since the
+number of photons detected is always even due to the
+fact that the inputs are squeezed states. However when
+displacement is applied to the modes of the GBS the
+probability of detecting an odd number of photons is in
+general not zero anymore and the probability of individ-
+ual detection events is characterized by the loop Hafnian
+lHaf() as opposed to the Hafnian [37, 38]. We don’t apply
+displacement for the experiments done in this paper.
+
+Gn
+Haf(An)
+n
+2
+4
+(1,1,1,1)
+3
+4
+(1, 0, 0, 1)
+1
+4
+2
+(1,1,1,2)
+0
+4
+3
+410
+2.
+Encoding a graph into a GBS device
+To map a graph to a feature vector we must first pro-
+gram the GBS device, by setting the squeezing parame-
+ters and beamsplitter angles of the device, according to
+the adjacency matrix A of the graph. Any adjacency ma-
+trix A ∈ RM×M of an undirected graph of M vertices can
+be mapped to a symmetric, positive definite 2M ×2M co-
+variance matrix Σ of a pure Gaussian state of M modes
+via the following procedure. First a doubled adjacency
+matrix ˜A is constructed,
+˜A = c
+�
+A 0
+0 A
+�
+= c(A ⊕ A),
+(A4)
+where c is a rescaling constant chosen such that 0 < c <
+1/λmax where λmax is the maximum singular value of
+A [3]. We use ˜A as, unlike A, it is guaranteed to map
+to a covariance matrix of a pure Gaussian state which
+is easier to prepare than a mixed one [35].
+This also
+has the advantage of allowing us to utilize the identity
+Haf(A ⊕ A) = Haf2(A) to relate ˜A to A.
+To map ˜A
+to a covariance matrix Σ we use the following matrix
+equations
+Σ = Q−I2M/2, with Q = (I2M −X ˜A)−1,
+X =
+�
+0 I
+I 0
+�
+.
+(A5)
+To program the GBS device to sample from the probabil-
+ity distribution corresponding to the covariance matrix Σ
+of the pure Gaussian state we need the unitary matrix
+U that characterizes the interferometer of the device as
+well as the squeezing parameters r1, ..., rM of each the
+M squeezers. We can obtain these values by taking the
+Takagi decomposition of A which is of the form
+A = Udiag(λ1, ..., λM)U T .
+(A6)
+The squeezing parameters are determined by the sin-
+gular values λ1, ..., λM and c via the relationship ri =
+tanh−1(cλi).
+The singular values and c also uniquely
+determine the mean photon number ¯n of the device ac-
+cording to
+¯n =
+M
+�
+i=1
+(cλi)2
+1 − (cλi)2 =
+M
+�
+i=1
+sinh2(ri).
+(A7)
+The rescaling constant c can be used to adjust ¯n as multi-
+plying A by c scales it’s singular values without changing
+the structure of the graph other than scaling all edge
+weights by c. The matrix U can be decomposed to give
+the parameters of the beamsplitter gates of the interfer-
+ometer [39].
+The GBS device, if using PNR detectors, now samples
+from the probability distribution
+p(n) =
+1
+�
+det(Q)
+Haf( ˜An)
+n!
+=
+1
+�
+det(Q)
+Haf2(An)
+n!
+. (A8)
+Appendix B: Super Exponential Growth of GBS
+Detection Events for M ∈ O(n2)
+Lemma 1.
+(n+M−1)!
+n!(M−1)! ∈ ω(
+√
+M
+√
+M) for n = ⌊
+√
+M⌋
+Proof.
+(n + M − 1)!
+n!(M − 1)!
+n=⌊
+√
+M⌋
+−−−−−−→ (⌊
+√
+M⌋ + M − 1)!
+(⌊
+√
+M⌋)!(M − 1)!
+(⌊
+√
+M⌋ + M − 1)!
+(⌊
+√
+M⌋)!(M − 1)!
+= [�⌊
+√
+M⌋
+i=1
+(M − 1 + i)](M − 1)!
+[�⌊
+√
+M⌋
+i=1
+i](M − 1)!
+= [�⌊
+√
+M⌋
+i=1
+(M − 1 + i)]
+[�⌊
+√
+M⌋
+i=1
+i]
+=
+⌊
+√
+M⌋
+�
+i=1
+[M − 1
+i
++ 1]
+>
+⌊
+√
+M⌋
+�
+i=1
+[M − 1
+⌊
+√
+M⌋
++ 1]
+= (M − 1
+⌊
+√
+M⌋
++ 1)⌊
+√
+M⌋
+= (
+M
+⌊
+√
+M⌋
++ 1 −
+1
+⌊
+√
+M⌋
+)⌊
+√
+M⌋
+≥ ⌊
+√
+M⌋
+⌊
+√
+M⌋
+Therefore (n+M−1)!
+n!(M−1)! ∈ ω(
+√
+M
+√
+M) for n = ⌊
+√
+M⌋. We
+drop the floor function for simplicity as ⌊x⌋ ∈ Θ(x).
+Appendix C: Induction Proof for
+�n
+k
+�
+∈ Θ(nk)
+Lemma 2.
+�n
+k
+�
+∈ Θ(nk)
+Proof. Base Case: k = 2
+�n
+2
+�
+= n(n − 1)
+2!
+lim
+n→∞
+n(n−1)
+2!
+n2
+= 1
+2!
+0 < 1
+2! < ∞
+∴
+�n
+2
+�
+∈ Θ(n2)
+Assume result holds up to k = ℓ
+�n
+ℓ
+�
+= n(n − 1)(n − 2) · · · (n − ℓ − 1)
+ℓ!
+∈ Θ(nℓ)
+
+11
+Inductive Step: k = ℓ + 1
+� n
+ℓ + 1
+�
+= n(n − 1)(n − 2) · · · (n − (ℓ + 2))
+(ℓ + 1)!
+lim
+n→∞
+n(n−1)(n−2)···(n−ℓ−2)
+(ℓ+1)!
+nℓ+1
+= lim
+n→∞
+n(n−1)(n−2)···(n−ℓ−1)
+ℓ!
+nℓ
+(n−ℓ−2)
+ℓ+1
+n
+= lim
+n→∞
+n(n−1)(n−2)···(n−ℓ−1)
+ℓ!
+nℓ
+lim
+n→∞
+(n−ℓ−2)
+ℓ+1
+n
+≡ 1
+ℓ!
+1
+ℓ + 1
+=
+1
+(ℓ + 1)!
+0 <
+1
+(ℓ + 1)! < ∞
+∴
+� n
+ℓ + 1
+�
+∈ Θ(nℓ+1)
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+IRONFORGE: An Open, Secure, Fair, Decentralized
+Federated Learning
+Guangsheng Yu∗, Xu Wang†, Caijun Sun§, Qin Wang∗, Ping Yu‡,
+Wei Ni∗, Renping Liu†, Xiwei Xu∗
+∗CSIRO Data61, Australia
+†University of Technology Sydney, Australia
+‡Harbin University of Technology, China
+§Zhejiang Lab, China
+Abstract—Federated learning (FL) provides an effective ma-
+chine learning (ML) architecture to protect data privacy in a
+distributed manner. However, the inevitable network asynchrony,
+the over-dependence on a central coordinator, and the lack of an
+open and fair incentive mechanism collectively hinder its further
+development. We propose IRONFORGE, a new generation of FL
+framework, that features a Directed Acyclic Graph (DAG)-based
+data structure and eliminates the need for central coordinators
+to achieve fully decentralized operations. IRONFORGE runs in
+a public and open network, and launches a fair incentive
+mechanism by enabling state consistency in the DAG, so that the
+system fits in networks where training resources are unevenly
+distributed. In addition, dedicated defense strategies against
+prevalent FL attacks on incentive fairness and data privacy are
+presented to ensure the security of IRONFORGE. Experimental
+results based on a newly developed testbed FLSim highlight the
+superiority of IRONFORGE to the existing prevalent FL frame-
+works under various specifications in performance, fairness, and
+security. To the best of our knowledge, IRONFORGE is the first
+secure and fully decentralized FL framework that can be applied
+in open networks with realistic network and training settings.
+Index Terms—Federated Learning, DAG, Blockchain
+I. INTRODUCTION
+Federated learning (FL), officially introduced by Google
+in 2017 [1], has become the preference to aggregate data
+from distributed ends without breaching data privacy [1],
+[2]. By aggregating huge data with comprehensive extracted
+features in FL, critical issues such as model overfitting can be
+significantly addressed [3]. However, Πthe inevitable network
+asynchrony,  the over-dependence on a central coordinator,
+and Ž the lack of an open and fair incentive mechanism hinder
+the further development of FL in large and open scenarios [4].
+Traditional FL considers no or low delay throughout an ag-
+gregation process, namely, synchronous FL. However, network
+synchrony is unrealistic due to the inevitable capacity limit
+of computation, bandwidth, and storage, as well as the im-
+balanced capacities among the distributed participants. Thus,
+recent studies propose pseudo-asynchronous FL [5] and asyn-
+chronous FL [6]. The aggregation of pseudo-asynchronous
+FL allows a short interval for collecting the model caches in
+order to ensure that the number of models aggregated can be
+sufficiently large, while the central coordinator immediately
+updates the global model once receiving a new local model
+from any idle participants in asynchronous FL.
+Neither pseudo-asynchronous FL nor asynchronous FL can
+tolerate the single-point-of-failure (SPoF) of the central coor-
+dinator or even a malicious and corrupted coordinator (issue-
+). The over-dependence on the central coordinator could
+potentially degrade the system availability and the training
+flexibility in the sense that an FL network may be confined to
+specific training domains or tasks determined by the coordina-
+tor. Participants in many existing studies [7]–[9], once opting
+in an FL network, would have to obey the defined training
+target with no flexibility to go for different tasks at will.
+In addition to the weak training flexibility, the lack of an
+open and fair incentive mechanism results in participants who
+have fewer resources and a weaker capacity not willing to
+contribute their resources to the global aggregation. This issue
+deteriorates particularly in FL networks where resources are
+not evenly distributed, and potentially leads to the model over-
+fitting and weak generality against contingencies. Although
+the authors of [10] survey the incentive mechanisms in FL,
+all mentioned frameworks require a central coordinator, also
+leading to issue-.
+Existing studies propose to replace the central coordinator
+with a committee running a consensus process in a blockchain
+network to prevent the SPoF or a corrupted coordinator. Mean-
+while, by sharing the model collection during the consensus
+in the committee, pseudo-asynchronous FL can be achieved in
+a decentralized manner, i.e., BlockFL [7], [11]–[14]. Consid-
+ering only issue- being solved and issue-Œ being partially
+solved by BlockFL, the authors of [9] introduce a Directed
+Acyclic Graph (DAG)-based FL where both issue-Πand issue-
+ are solved using the concept of asynchronous FL [6] to
+fully decentralize the FL process. However, the paper [9] only
+considers an ideal network in which the training resources are
+evenly distributed. Moreover, the approach to enabling state
+consistency for a secure and fair incentive mechanism (issue-
+Ž) is missing in [9], which results in difficulty in adopting the
+mechanism in a public and open network.
+We propose IRONFORGE that is an open, secure, fair,
+and decentralized FL system. IRONFORGE solves the above
+mentioned pain points at one time. Openness: It features a
+DAG-based data structure in an open network. Decentral-
+ization: The need for a central coordinator is eliminated
+throughout the process by IRONFORGE, inheriting from the
+concept of asynchronous FL. As a result, the models are
+1
+arXiv:2301.04006v1  [cs.LG]  7 Jan 2023
+
+TABLE I: Qualitative comparisons between the proposed IRONFORGE and the existing FL frameworks
+FL Framework
+Data Structure
+Data Asynchrony
+Decentralization
+Openness
+Incentive
+Security
+Google FL [2]
+Isolated models
+Synchronous
+Centralized
+Private
+�
+�
+Asynchronous FL [6]
+Isolated models
+Asynchronous
+Centralized
+Private
+�
+�
+Block FL [15]
+Blockchain
+Synchronous
+Decentralized
+Private
+Reward
+�
+DAG FL [9]
+DAG
+Asynchronous
+Decentralized
+Public
+Reward
+Poisoning/Backdoor/Lazy
+IRONFORGE
+DAG
+Asynchronous
+Decentralized
+Public
+Reward, Penalty
+Poisoning/Backdoor/Stealing*/Collusion
+* The stealing attack considered in this paper includes the traditional lazy attack.
+The difference is that stealing attackers not only upload their previous models, but also fake the ownership of others’ previous models.
+� Lack of corresponding designs.
+maintained in a decentralized manner by all participants.
+Fairness: IRONFORGE considers a practical scenario, where
+resources are unevenly distributed among users. Each user,
+based on its resource amount, selects several existing models,
+verifies the correctness and evaluates the model accuracy over
+the local dataset, and conducts the aggregation. IRONFORGE
+also enables state consistency, by using which an open and
+fair incentive mechanism can be established to motivate more
+participants. Security: Moreover, dedicated defense strategies
+against malicious attacks on incentive fairness, and against
+dataset privacy breaching are presented to ensure the security
+of IRONFORGE. The key contributions are as follows.
+⊲ We propose a fully decentralized FL framework, namely,
+IRONFORGE, which features a DAG-based data structure.
+IRONFORGE addresses the network asynchrony typically
+undergone in an FL process, and improves the motivation
+of agents participating in the process in an open envi-
+ronment by enabling reliable token rewards with strong
+consistency and model prediction accuracy.
+⊲ We specifically design a new validation mechanism
+guarding against well-known FL attacks, including model
+poisoning attacks, backdoor attacks, lazy attacks, and
+model stealing attacks, among which the model of steal-
+ing attack has never been considered in any existing FL
+frameworks. By making use of noise-enabled Proof-of-
+Learning (PoL) to validate the gradient descent process,
+any malicious behaviors, such as faking the ownership
+or directly using the existing models, or embezzling the
+rewards for their conspirator by claiming a falsified source
+list, can be captured and given punishments.
+⊲ We build a flexible and efficient testbed, named FLSim,
+to simulate the workflow across all considered FL frame-
+works in this paper, including the proposed IRONFORGE.
+We conduct comprehensive experiments based on FLSim,
+comparing the system performance, security, and fairness
+between the existing FL frameworks and IRONFORGE.
+Insights are shed to provide guidelines on how to select
+strategies in IRONFORGE to meet different requirements.
+Extensive experiments corroborate that IRONFORGE outper-
+forms the prevalent FL frameworks with and without attacks
+leveraged, which highlights the holistic solution to the network
+asynchrony (issue-Œ) and the over-dependence on the central
+coordinators (issue-). Strictly and approximately monotonic
+increases of rewards are observed in experiments with increas-
+ing CPU cores, memory capacity, and bandwidth in different
+incentive settings. This indicates that fairness (issue-Ž) can be
+ensured in IRONFORGE under various definitions of fairness.
+The rest of the paper is organized as follows. Section I
+gives the introduction, followed by related works in Section II.
+Section III provides the system overview and Section IV
+details the design of IRONFORGE. Section V presents our
+implementation based on a new testbed with comprehensive
+experimental results. Section VI discusses system security and
+properties. Finally, Section VII concludes this work.
+II. RELATED WORK
+A conventional synchronous FL framework is constructed
+by a central coordinator and numbers of nodes, which main-
+tains the global model and perform FL iterations, respec-
+tively [2]. The coordinator periodically distributes the latest
+global model to the nodes, and then the nodes independently
+train the model with their local data and upload the trained
+local models to the coordinator [16]. After receiving updated
+models from nodes, the coordinator aggregates all the local
+models as a new global model. Such synchronous FL frame-
+work can hardly be adapted to large-scale and heterogeneous
+networks, where asynchrony is non-negligible.
+The issue of data asynchrony is tackled by the asyn-
+chronous FL enabling nodes to train the global model from
+central coordinators at any time, and the coordinators can
+update the global model immediately when any local model
+is collected. In [14], the authors introduced a cache layer
+between the coordinator and local nodes. Each node trains
+the global model with its local data and uploads its model to
+the cache. The coordinator periodically aggregates the local
+model in the cache and generates a new global model. Semi-
+asynchronous FL protocols address the problems in FL such
+as low round efficiency and poor convergence rate happened
+in asynchronous FL. The system [5] incorporates a client se-
+lection algorithm decoupling the coordinator and the selected
+clients for a reduction of average round time. The authors
+of [17] proposed an asynchronous federating-based detection
+approach for end devices. A pre-shared data training strategy
+for non-independent-and-identically-distributed (non-IID) data
+is developed to avoid convergence divergence under the non-
+IID patterns. After the collaborative model training procedure,
+each client further conducts an additional local training process
+to fit respective patterns.
+The aforementioned FL frameworks require central coor-
+dinators to schedule model training and aggregate models.
+The centralized architecture suffers inherent security risks,
+such as SPoF and malicious central coordinator, and limited
+2
+
+scalability with the bottleneck of the central coordinator. The
+most recent Distributed Ledger Technology (DLT) holds the
+potential to decentralize FL systems [18], [19]. Two key
+technologies in DLT are blockchain and DAG. In blockchain,
+a group of miners run the consensus protocol to generate hash-
+chained data blocks, which are assembled from transactional
+data, and synchronize the chained blocks. Blockchain assures
+strong consistency among blockchain nodes and enables smart
+contracts to be executed across the blockchain network in a
+consistent and trustworthy way. In DAG, transactions from
+decentralized DAG users are organized in a DAG structure
+where directed edges indicate the reference relationship be-
+tween the transactions. DAG can achieve high throughput with
+short latency compared with blockchain [20].
+DLT has been developed to remove the central coordinator
+and decentralize FL networks [9], [15], [21]. In BlockFL [15],
+[22], [23], decentralized blockchain miners conduct model
+verification and aggregation. To be specific, miners obtain
+trained local models from working nodes and other miners.
+After verification, miners aggregate local models for the
+updated global models and conduct Proof-of-Work (PoW) to
+create valid blocks containing the new global models. Then,
+the blocks are propagated to all miners to start the next FL
+iteration. The BlockFL relies on the resource-intensive PoW
+consensus protocol to slow down the system and keep miners
+synchronized. To reduce overhead and improve scalability,
+DAG technology [24] is introduced to FL networks [9], [21],
+where trained models are updated to a DAG topology by
+working nodes without any coordination. Working nodes can
+learn the latest local models in the DAG by exchanging data
+with other nodes. By themselves, working nodes select and
+verify aggregate local models and train the models using local
+datasets. Next, working nodes publish their trained models to
+the DAG with directed edges indicating the model reference.
+Existing works only consider homogeneous networks where
+the training resources are evenly distributed and thus lack
+open and fair incentive mechanisms. IRONFORGE proposed
+by this paper, on the other hand, improves the motivation of
+participants with rewards for training contributions and penal-
+ties for dishonest behaviors. IRONFORGE also tackles new
+vulnerabilities in open FL networks, including model stealing
+attacks where attackers steal models from others and claim
+rewards from the plagiarized model, and collusion attacks
+where attackers claim trained models are from conspirators.
+III. SYSTEM OVERVIEW
+In this section, we describe IRONFORGE from the aspects
+of its architecture, workflow, and system assumptions.
+A. System Overview
+We first introduce the roles that participate in the system
+and present our high-level design.
+Architecture. IRONFORGE is a decentralized FL system that
+features a DAG-based network structure to tackle the incon-
+sistency in the decentralized FL process, excessive reliance
+on central coordination, and ineffective motivation of con-
+tributing the learning resources at the same time. Specifically,
+Fig. 1. System model of IRONFORGE
+IRONFORGE builds a hybrid architecture (cf. Fig. 1) that
+involves two types of DAG, namely, Task-DAG and Global-
+DAG (details refer to Fig. 2 and Fig. 3, respectively). The
+training processes in both Task-DAG and Global-DAG are
+traceable owing to the DAG data structure. A DAG node
+published by a participant consists of a model update and the
+directed edges of the node indicate the aggregating relationship
+with existing models during the update, hence no central
+coordinator required to conduct the training processes.
+Global-DAG contains a variety of models adopted by all
+participants, which can be viewed as a “unique” and public
+model resource pool. No consistent testing dataset is given
+in Global-DAG. Each user comes to Global-DAG and hunts
+for models that uniquely meet its own local testing dataset.
+Without central coordinators, any user can fetch models from
+the pool for direct uses, release his task requests, or make con-
+tributions, such as training on Global-DAG or on uncompleted
+training tasks, or verifying the tasks.
+Each training task is managed by a Task-DAG, while IRON-
+FORGE can contain multiple Task-DAGs at the same time
+to handle a range of different training tasks (see the right-
+hand side in Fig. 1). Task-DAGs are task-specific and are
+released by users who aim at improving their local model
+prediction accuracy by virtue of the computational powers
+and resources of others. Within a task, the Task-DAG network
+contains multiple contributors who have the same training
+target provided by the publisher. The trained models for each
+task are broadcast and stored in the corresponding Task-DAG,
+and await the check and verification. The satisfied model of a
+task, observed by the publisher, is subsequently merged into
+Global-DAG, increasing the exposure to the public users. As a
+result, parallel learning on our hybrid DAG networks becomes
+possible, and the resultant models can be collected by Global-
+DAG for further involvement.
+Roles. In IRONFORGE, the users can take different roles:
+viewer, task publisher, verifier, and contributor. A user is a
+participant in the network. Each user can select one or multiple
+roles to perform specific functional activities (see the left-hand
+side in Fig. 1). Specifically, a viewer can directly fetch models
+from the public resource pool without further actions. The
+task publisher aims to propose new tasks and the proposed
+tasks are broadcast and await others’ contributions. In order
+3
+
+Global Network
+User
+Global DAG
+Contributor
+User
+Task DAG
+Global DAG
+Contributor
+Task DAG
+User
+Global DAG
+Contributor
+Task DAG
+Viewer
+Fetching 
+Evaluating
+Models
+Selected Models
+Satisfied Mode
+Task Publisher
+Aggregating
+Verification Committee
+Aggregated Model
+Local dataset
+Contributor
+Training
+Trained Model
+Selected Models
+Task Publisher
+sk
+Aggregated Model
+Verification Committee
+%
+Trained Model
+Task
+WorkNerifyTask
+Community
+……
+……
+t
+① Register and Release a task
+Publisher
+Workers
+② Announce
+③ Observe
+……
+b. Sync the DAG and validate DAG nodes
+c. Evaluate some models with local test dataset
+d. Pick up the best ones and aggregate them
+e. Start training with local training dataset
+f. Publish the model to the DAG
+④ Train and publish models
+Task-model node
+Header:
+Sender: …
+Timestamp: …
+Sources: […]
+Evaluations: […]
+Payload:
+Weights: ipfs_uri+hash
+④
+④
+④
+Local
+dataset
+Local
+dataset
+Local
+dataset
+Task-termination node
+Sender: Publisher
+Timestamp: …
+Winner: …
+Public testing dataset: Dtest
+Accuracy: …
+Balance update: …
+Task-genesis node
+Header:
+Sender: Publisher
+Timestamp: …
+Target: <e.g., 95%>
+Commitment of Dtest: …
+Prize: …
+Contest strategy: …
+Penalty strategy: …
+Verification committee:
+Payload:
+Weights: ipfs_uri+hash
+⑤ End a task by selecting the winner with the highest accuracy
+a. Register on the global FL-DAG
+V
+P
+Verify
+Fig. 2. Task-DAG. The figure illustrates an overview of starting a new DAG-based FL task, also known as Task-DAG. One
+with aims to improve his model accuracy to a certain target with the help of the community can release a task as the task
+publisher. Some amount of token is deposited as the prize which will be subsequently awarded to all eligible participants
+until the winning model is found and selected by the task publisher. The balance update of each participant is recorded in a
+task-termination node published by the task publisher, and can be subsequently settled by the Global-DAG network.
+to reap profits, a user can become a contributor to process a
+training process by selecting, aggregating and training models.
+He can either start the work on Global-DAG or enroll in others’
+published work from the uncompleted tasks. Also, a verifier in
+the system is to verify existing tasks in the resource pool. He
+can contribute or verify either one favorable task or multiple
+tasks in parallel for a higher profit. In short, the four roles
+cover all potential functional activities within IRONFORGE.
+B. Workflow Overview
+Then, we provide an overview of the workflow of IRON-
+FORGE. We focus on the procedures of task establishment and
+task processing by presenting the interactive steps of a user
+between the Task-DAG and Global-DAG.
+Step-1. The user registers a task in the Global-DAG network by
+depositing the committed prize. He obtains a task identifier
+and then broadcasts the task to the network. We assume
+that another user has accepted the proposed task prior and
+worked on the task as a task contributor.
+Step-2. The contributor enters the procedure of training mod-
+els. He first evaluates several existing models from the pool
+and selects a series of models for the shortlist.
+Step-3. Based on the selected models, the contributor aggre-
+gates all the short-listed models and integrates them with
+local datasets to train the model according to requirements.
+Step-4. Once completing the training, the contributor submits
+the trained model to the Task-DAG. Meanwhile, peer con-
+tributors may also work on the same task and generate com-
+petitively trained models. All these models are propagated
+within the Task-DAG network.
+Step-5. The publisher who obtains the trained model termi-
+nates the task by marking it with a termination tag. Once
+selected by users who are conducting the training process
+in Global-DAG, the trained model is deemed to be formally
+synchronized into Global-DAG.
+Notably, a user in the Global-DAG network can either
+contribute to other tasks proposed by peer users, or personally
+publish a task by himself. All the procedures follow similar
+steps, as described from Step-2 to Step-5.
+C. System Assumptions
+In this section, we list our assumptions on the network,
+security, and threat models of IRONFORGE.
+Resource assumption. We do not assume any resource dis-
+tribution in our work. The resource distribution in the entire
+network is random. This means different participants, with a
+high probability, hold different computing resources, including
+computing power, network bandwidth, memory space, storage
+capability, and training dataset quality. Addressing the system
+heterogeneity is one of the core contributions in this work, as
+we weaken the long-existing implicit assumption in previous
+work [9]: the even resource distribution. IRONFORGE enables
+any distribution of shares of any type of resources among the
+participants, making the system practical.
+User behavior assumption. We have two assumptions on user
+behaviors. First, the participants in the network are rational,
+meaning that they can select an arbitrary task, switch to
+others, or quit existing tasks for better profits. Second, different
+participants can focus on different training targets, including
+both task bundles (one task has dependency on another) and
+orthogonal tasks (one task is independent of the others). This
+enables the processing of multiple tasks in parallel, greatly
+improving the system’s overall scalability and performance.
+Security assumption. We assume that the honest nodes
+always conduct honest behaviors, where they obey all the
+policies during the model selection, model aggregation, model
+4
+
+Task-1
+(terminated)
+Task-2
+(terminated)
+Task-N
+(ongoing)
+G
+1
+t1
+t2
+tn
+2
+V
+P
+τ1
+3
+Local
+dataset
+Local
+dataset
+Local
+dataset
+……
+Community
+Publish
+Publish
+Publish
+Contribute
+Contribute
+VRF-
+consensus
+Settlement node
+Sender: <based on consensus>
+Timestamp: …
+PoL results: …
+Unverified PoL: …
+Balance: …
+Fig. 3. Global-DAG. This figure illustrates an overview of the Global-DAG network. With the absence of a centralized
+coordinator, each participant trains a model by selecting and aggregating as many models (including the outcomes of terminated
+tasks) published by others as possible (based on the local capability). Model targets are not unique and according to different
+needs, the network can be treated as a global resource pool containing a variety of models. Ones can either find a model which
+satisfies his local testing dataset from the pool, or make contributions to the pool and obtain token rewards by improving
+existing models. Token balances are periodically settled (endorsed by a verifiable random function (VRF)-driven consensus)
+by settlement nodes that employ a chain structure to achieve strong consistency.
+training, task verification, and other operations related to the
+defense strategies against adversaries. The adversaries have the
+ability to delay the model convergence and lower the model
+accuracy by leveraging popular FL attacks, including lazy
+attacks [9], poisoning attacks [25], and backdoor attacks [26].
+The adversaries also have the ability to breach the incentive
+fairness by leveraging model stealing attacks [27]–[30], and
+compromising the privacy of others’ training datasets. Adver-
+saries not only can upload their previous models (traditional
+lazy attacks), but also fake the ownership of others’ previous
+models or fake their own training process to embezzle rewards
+for their conspirators. These two faking types are defined as
+stealing attacks and collusion attacks, respectively, and both
+belong to the context of model stealing attacks in this paper.
+IV. DECENTRALIZED FEDERATED LEARNING
+IRONFORGE involves four novel mechanisms, i.e., the re-
+lease of Task-DAG networks, the decentralized model training,
+the defense strategy, and the incentive mechanism. IRON-
+FORGE features two types of network, public Global-DAG and
+task-specific Task-DAG. A training task can be outsourced to
+communities by releasing a Task-DAG network and following
+four steps including preparation, initialization, monitoring, and
+finalization. The decentralized training processes of Task-DAG
+and Global-DAG are specifically defined by the new decen-
+tralized model training mechanism, in which users aggregate
+existing models, train the aggregated models and publish the
+model updates to the network in a decentralized way. The
+training processes are guarded by a new defense strategy
+against model stealing attacks in the decentralized setting that
+has never been considered in existing studies. The crafted
+incentive mechanism assures the state consistency of networks
+and enables smart-contract-enhanced incentives including both
+rewards and penalties. Table II summarizes the notations.
+A. Managing Task-DAG
+Any user can outsource an FL training task by managing
+a DAG, as shown in Algo. 1. An FL training task can
+be described with a training target, i.e., the model to be
+trained, the targeted accuracy, and the testing dataset. To build
+incentives to motivate distributed workers and deter malicious
+workers, we design reward, penalty, and verification schemes
+for Task-DAG networks.
+1) Preparation: A training task Task𝑚 can be described
+with an initial model, i.e., 𝑊𝑚,𝑔, and an accuracy target 𝛼𝑚 of
+the model tested on the dataset 𝐷test
+𝑚
+from the task publisher
+𝑈𝑚,𝑝. To reduce the storage and bandwidth overhead, the
+model weights can be stored in external infrastructures, e.g.,
+the InterPlanetary File System (IPFS). The Uniform Resource
+Identifiers (URIs) and hash codes to the weights are embedded
+in DAG nodes for access and verification. The testing dataset
+𝐷test
+𝑚
+is committed in the genesis node of the Task-DAG by
+embedding the hash code, and is revealed by the end of the
+training for model verification. This commit-and-reveal design
+prevents direct access to 𝐷test
+𝑚 during the training process and
+ensures that the final selected model can be publicly verified.
+To run a Task-DAG with an incentive in a secure way, the
+task publisher 𝑈𝑚,𝑝 needs to design a contest strategy Φ𝑚 to
+allocate reward 𝜈𝑚 to contributors and a penalty strategy Υ𝑚
+to suppress the flooding of excessive trivial models and other
+malicious behaviors.
+Some examples of the plug-and-play contest strategy in-
+clude an egalitarian strategy where the prize is divided equally
+among contributors along the traversal of the final winner
+node, or an implementation of “to each according to his
+5
+
+TABLE II: Notation Definition
+Notation
+Definition
+𝑈𝑘
+The 𝑘-th user
+𝐵𝑘
+The balance of 𝑘-th user
+𝐷train
+𝑘
+The local training dataset of 𝑈𝑘
+𝐷test
+𝑘
+The local testing dataset of 𝑈𝑘
+𝛽
+The number of candidate weights
+𝜎
+The number of aggregated weights
+Task-DAG
+Task𝑚
+The 𝑚-th Task-DAG network
+𝛼𝑚
+The accuracy target of Task𝑚
+𝜈𝑚
+The committed prize of Task𝑚
+𝑈𝑚,𝑝
+The publisher of Task𝑚
+𝑁𝑚,𝑔
+The genesis node of Task𝑚
+𝑇𝑚,𝑔
+The creation timestamp of 𝑁𝑚,𝑔
+𝑊𝑚,𝑔
+The initial model weights of Task𝑚
+𝑁𝑚,𝑘,𝑖
+The 𝑖-th node published by the 𝑘-th user in Task𝑚
+M𝑚,𝑘,𝑖
+The source list of 𝑁𝑚,𝑘,𝑖
+𝐸𝑚,𝑘,𝑖
+The evaluation result for 𝑁𝑚,𝑘,𝑖 over 𝐷test
+𝑘
+E𝑚,𝑘,𝑖
+The evaluation result for M𝑚,𝑘,𝑖 over 𝐷test
+𝑘
+𝑊 ∗
+𝑚,𝑘,𝑖
+The model weights aggregated by M𝑚,𝑘,𝑖 before the local
+training
+𝑊𝑚,𝑘,𝑖
+The model weights after the local training
+𝜌𝑚,𝑘,𝑖
+The setting for training 𝑊 ∗
+𝑚,𝑘,𝑖 into 𝑊𝑚,𝑘,𝑖
+𝑇𝑚,𝑘,𝑖
+The creation timestamp of 𝑁𝑚,𝑘,𝑖
+𝑁𝑚,𝑒
+The task-termination node of Task𝑚
+𝑇𝑚,𝑒
+The creation timestamp of 𝑁𝑚,𝑒
+Ψ𝑚
+The prize allocation of Task𝑚
+Φ𝑚
+The contest strategy of Task𝑚
+Υ𝑚
+The penalty strategy for malicious attacks of Task𝑚
+Ω𝑚
+The verification committee of Task𝑚
+Global-DAG
+𝑁𝑔
+The genesis node of Global-DAG
+𝑁𝑘,𝑖
+The 𝑖-th node published by the 𝑘-th user in Global-DAG
+𝑇𝑘,𝑖
+The creation timestamp of 𝑁𝑘,𝑖
+𝑆ℎ
+The ℎ-th settlement node in the settlement sets S
+𝜆ℎ
+The subtree that is aggregated by 𝑆ℎ
+𝑉𝑘,𝑘′,𝑖
+A PoL-challenge raised by 𝑈𝑘′ for 𝑁𝑘,𝑖 where 𝑘 ≠ 𝑘′
+𝜋𝑘,𝑘′,𝑖
+The deposit to raise 𝑉𝑘,𝑘′,𝑖
+𝜖PoL
+The threshold of PoL-verification
+𝑃𝑘,𝑘′,𝑖
+The PoL-response replied by the publisher 𝑈𝑘 of 𝑁𝑘,𝑖 for
+𝑉𝑘,𝑘′,𝑖 raised by 𝑈𝑘′ where 𝑘 ≠ 𝑘′
+𝑅𝑘, ˆ𝑘,𝑖
+The PoL-result sent from 𝑈 ˆ𝑘 on 𝑃𝑘,𝑘′,𝑖 where 𝑘 ≠ ˆ𝑘
+𝜏ℎ
+The timeout for 𝑃𝑘,𝑘′,𝑖 to be published after 𝑉𝑘′,𝑘,𝑖 has
+been published and confirmed by 𝑆ℎ
+Θℎ
+The committee elected to conduct consensus for 𝑆ℎ
+contribution” whereby the prize is allocated based on the
+amount of contribution, or striking a balance in-between. Some
+examples of the penalty strategy include an implementation
+of S-Index
+H-Index (i.e., preventing excessive self-citations) [31] or the
+occupation ratio along the traversal of the final winner node.
+The publisher 𝑈𝑚,𝑝 also invokes the election to nominate a
+set of 𝑈𝑘 that constitute a task committee Ω𝑚 of Task𝑚 for
+evaluating trained models and conducting PoL-verification.
+The committee is elected via a Verifiable Random Function
+(VRF) [32] upon the balances 𝐵𝑘 of eligible 𝑈𝑘.
+2) Initialization: To initialize Task𝑚, the publisher 𝑈𝑚,𝑝
+firstly registers Task𝑚 to the task management smart contract
+Algorithm 1: Manage Task-DAG
+⊲ Initialize a training task
+1 𝑈𝑚,𝑝.Deposit(Task𝑚, 𝜈𝑚)
+2 Ω𝑚 ← 𝑈𝑚,𝑝.VRF(Task𝑚)
+⊲ Elect the nominated verification committee to Task𝑚
+3 𝑁𝑚,𝑝 ← {𝐻 (𝑊𝑚,𝑔), URI(𝑊𝑚,𝑔), 𝐻 (𝐷test
+𝑚 ), 𝛼𝑚,
+𝜈𝑚, Φ𝑚, Υ𝑚, Ω𝑚, 𝑇𝑚,𝑔 }
+4 𝑁𝑚,𝑝 ← 𝑈𝑚,𝑝.Sign(𝑁𝑚,𝑝)
+5 𝑈𝑚,𝑝.Announce(𝑁𝑚,𝑝)
+⊲ Broadcast to the network
+⊲ Observe the training
+6 while True do
+7
+𝑁𝑚,𝑘,𝑖 ← 𝑈𝑚,𝑝.Monitor(Task𝑚)
+8
+if 𝑁𝑚,𝑘,𝑖 breaches Υ𝑚 then
+9
+𝑈𝑚,𝑝.ApplyPenalty(Υ𝑚, 𝑁𝑚,𝑘,𝑖, 𝑈𝑘)
+10
+if 𝐸𝑚,𝑘,𝑖 > 𝛼𝑚 then
+11
+ˆ𝐸𝑚,𝑘,𝑖 ← 𝑈𝑚,𝑝.Evaluate(𝑊𝑚,𝑘,𝑖, 𝐷test
+𝑚 )
+12
+if ˆ𝐸𝑚,𝑘,𝑖 > 𝛼𝑚 then
+13
+ˆ𝑁𝑚,𝑘,𝑖 ← 𝑁𝑚,𝑘,𝑖
+14
+break
+⊲ Finalize the training task
+15 Ψ𝑚 ← 𝑈𝑚,𝑝.AllocatePrize(Φ𝑚, ˆ𝑁𝑚,𝑘,𝑖)
+16 𝑁𝑚,𝑒 ← { ˆ𝑁𝑚,𝑘,𝑖,
+ˆ𝑈𝑘, URI(𝐷test
+𝑚 ),
+ˆ𝐸𝑚,𝑘,𝑖, Ψ𝑚, 𝑇𝑚,𝑒 }
+𝑆𝐶𝑇 on the Global-DAG network by depositing the committed
+prize 𝜈𝑚. Next, 𝑈𝑚,𝑝 prepares a genesis node 𝑁𝑚,𝑔 including
+the commitment and the URI of the initial model to be trained,
+i.e., 𝐻(𝑊𝑚,𝑔) and URI(𝑊𝑚,𝑔), the model accuracy target 𝛼𝑚,
+the commitment of the public testing dataset 𝐻(𝐷test
+𝑚 ), the
+committed prize 𝜈𝑚, the contest strategy Φ𝑚, the penalty
+strategy Υ𝑚, the nominated verification committee Ω𝑚, and
+the creation timestamp 𝑇𝑚,𝑔1. Then, 𝑈𝑚,𝑝 can sign the genesis
+node 𝑁𝑚,𝑔 and announce the genesis node.
+3) Monitoring: Upon these operations, training starts in
+Task𝑚, and the publisher 𝑈𝑚,𝑝 observes the progress until the
+model becomes mature enough. Any 𝑈𝑘, who is interested in
+contributing the computational resources and competing for
+the prize 𝜈𝑚, continues training models and publishing node
+𝑁𝑚,𝑘,𝑖, i.e., the 𝑖-th node released by 𝑈𝑘 in Task𝑚, as a worker.
+If any nodes breaching the penalty strategy Υ𝑚 are found,
+𝑈𝑚,𝑝 issues fines and updates the balance of the publisher of
+the breaching nodes. Note that the balance change in regard
+to the penalty has yet to be finalized at this stage.
+4) Finalization: When the claim of reaching the targeted
+model accuracy 𝛼𝑚 is realized, 𝑈𝑚,𝑝 evaluates the model
+over the testing dataset 𝐷test
+𝑚 . Once the accuracy is surely
+met by the winner node
+ˆ𝑁𝑚,𝑘,𝑖, 𝑈𝑚,𝑝 executes the contest
+strategy Φ𝑚 and obtains the prize allocation Ψ𝑚. Next, 𝑈𝑚,𝑝
+terminates Task𝑚 by creating a task-termination node 𝑁𝑚,𝑒 that
+points to the winner node ˆ𝑁𝑚,𝑘,𝑖 and contains the winner’s
+address ˆ𝑈𝑘, the URI to the testing dataset URI(𝐷test
+𝑚 ), the
+achieved testing accuracy ˆ𝐸𝑚,𝑘,𝑖, the prize allocation Ψ𝑚, and
+the creation timestamp 𝑇𝑚,𝑒. The task publisher 𝑈𝑚,𝑝 then
+signs 𝑁𝑚,𝑒 and broadcasts 𝑁𝑚,𝑒 to the Global-DAG network.
+The balance change in regard to both the prize allocation Ψ𝑚
+and the penalty is subsequently finalized by settlement nodes
+once the termination node is revealed to the public and is
+1The trustworthiness of the timestamp is guaranteed by trusted timestamp-
+ing services.
+6
+
+Algorithm 2: Train Models
+1 for 𝑈𝑘 parallelly do
+⊲ Worker registration
+2
+𝑈𝑘.Register(Task𝑚)
+3
+while Task𝑚 is ongoing do
+⊲ Sync, verify and select nodes
+4
+while unsynchronized do
+5
+𝑁𝑚,𝑘′,𝑖′ ← 𝑈𝑘.SyncNodes(Task𝑚)
+6
+𝑈𝑘.VerifySignature(𝑁𝑚,𝑘′,𝑖′)
+7
+𝑈𝑘.VerifyRegistration(𝑈𝑘′)
+8
+𝑈𝑘.VerifyBalance(𝑈𝑘′)
+9
+if verification passes then
+10
+𝑈𝑘.Propagate(𝑁𝑚,𝑘′,𝑖′)
+11
+for 𝛽 number of 𝑁𝑚,𝑘′,𝑖′ ∈ Task𝑚 parallelly do
+12
+𝐸′
+𝑚,𝑘′,𝑖′ ← 𝑈𝑘.Evaluate(𝑊𝑚,𝑘′,𝑖′, 𝐷test
+𝑘 )
+13
+M𝑚,𝑘,𝑖, E𝑚,𝑘,𝑖 ← 𝑈𝑘.Select({𝑁𝑚,𝑘′,𝑖′, 𝐸′
+𝑚,𝑘′,𝑖′ },
+𝜎)
+⊲ Aggregate, train and contribute nodes
+14
+𝑊 ∗
+𝑚,𝑘,𝑖 ← 𝑈𝑘.Aggregate(M𝑚,𝑘,𝑖)
+15
+𝑊𝑚,𝑘,𝑖 ← 𝑈𝑘.T𝑚,𝑘 (𝑊 ∗
+𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖, 𝐷train
+𝑘
+)
+16
+𝐸𝑚,𝑘,𝑖 ← 𝑈𝑘.Evaluate(𝑊𝑚,𝑘,𝑖, 𝐷test
+𝑘 )
+17
+𝑁𝑚,𝑘,𝑖 ← {M𝑚,𝑘,𝑖, E𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖,
+𝐻 (𝑊𝑚,𝑘,𝑖), URI(𝑊𝑚,𝑘,𝑖), 𝐸𝑚,𝑘,𝑖, 𝑇𝑚,𝑘,𝑖}
+18
+𝑁𝑚,𝑘,𝑖 ← 𝑈𝑘.Sign(𝑁𝑚,𝑘,𝑖)
+19
+𝑈𝑘.Announce(𝑁𝑚,𝑘,𝑖)
+referred by any future model in Global-DAG; see details in
+Section IV-D.
+B. Decentralized Model Training
+As an open system, IRONFORGE allows the workers to
+contribute to the decentralized training in both a Task𝑚 and
+Global-DAG. The training process in a Task-DAG is given
+in Algo. 2, while the training process in Global-DAG shares
+the same algorithm except that there does not exist a public
+shared testing dataset 𝐷test
+𝑚 that decides the stopping point (i.e.,
+the training accuracy target 𝛼) in the Global-DAG network.
+Global-DAG acts as a public resource pool of diversified
+models, allowing for free hunting of models that uniquely
+meet customized training targets upon local testing datasets
+𝐷test
+𝑘
+for each 𝑘-th user 𝑈𝑘.
+Task-DAG Training. Any user 𝑈𝑘, who is interested in com-
+peting for the training rewards in a Task𝑚, needs to admit the
+contest and penalty strategies specified in 𝑁𝑚,𝑔 and registers to
+the task management smart contract 𝑆𝐶𝑇 on the Global-DAG
+network by depositing a certain amount of tokens. This can
+suppress Sybil attacks, distributed denial-of-service (DDoS)
+attacks, and other malicious behaviors under the regulation of
+the penalty strategy Υ𝑚.
+While Task𝑚 is running, 𝑈𝑘 can synchronize the view of
+Task𝑚 and obtain the latest nodes 𝑁𝑚,𝑘′,𝑖′ with (𝑘 ≠ 𝑘′ ∨
+𝑖 ≠ 𝑖′) ∧ (𝑇𝑚,𝑘′,𝑖′ < 𝑇𝑚,𝑘,𝑖). 𝑈𝑘 then verifies the signature
+of the nodes and confirms that the corresponding worker 𝑈𝑘′
+is registered and has enough balance from 𝑆𝐶𝑇 . Worker 𝑈𝑘
+drops the nodes that do not pass verification and propagates
+the success nodes to other workers.
+After verification, 𝑈𝑘 randomly evaluates several 𝑁𝑚,𝑘′,𝑖′
+over its local testing dataset 𝐷test
+𝑘
+for the testing accuracy
+𝐸 ′
+𝑚,𝑘′,𝑖′ until it collects 𝛽 candidated weights (Lines 11-13
+of Algo. 2). Next, 𝑈𝑘 picks up the top 𝜎 models constituting
+the source list M𝑚,𝑘,𝑖 which are then aggregated into the pre-
+trained model 𝑊∗
+𝑚,𝑘,𝑖 using a weighted aggregation function,
+as given by
+𝑊∗
+𝑚,𝑘,𝑖 =
+∑︁
+𝑊𝑝 in M𝑚,𝑘,𝑖
+𝐸𝑝 in E𝑚,𝑘,𝑖
+𝐸 𝑝
+�
+𝐸𝑞 ∈E𝑚,𝑘,𝑖 𝐸𝑞
+𝑊𝑝.
+(1)
+After that, 𝑈𝑘 trains the aggregated model 𝑊∗
+𝑚,𝑘,𝑖 over its local
+training dataset 𝐷train
+𝑘
+with training settings 𝜌𝑚,𝑘,𝑖 for a trained
+model 𝑊𝑚,𝑘,𝑖 , as given by
+𝑊𝑚,𝑘,𝑖 = T𝑚,𝑘 (𝑊∗
+𝑚,𝑘,𝑖, 𝜌𝑚,𝑘,𝑖, 𝐷train
+𝑘
+),
+(2)
+where T𝑚,𝑘 is the training function of 𝑈𝑘 for Task𝑚.
+Once the training is done, 𝑈𝑘 evaluates the model over
+its local testing dataset 𝐷test
+𝑘
+and obtains the testing accuracy
+𝐸𝑚,𝑘,𝑖. Next, 𝑈𝑘 prepares a model update node 𝑁𝑚,𝑘,𝑖 with the
+source list M𝑚,𝑘,𝑖, the corresponding accuracy list E𝑚,𝑘,𝑖, the
+hash code and URI to the trained model, i.e., 𝐻(𝑊𝑚,𝑘,𝑖) and
+URI(𝑊𝑚,𝑘,𝑖), the testing accuracy 𝐸𝑚,𝑘,𝑖, the training settings
+𝜌𝑚,𝑘,𝑖, and the creation timestamp 𝑇𝑚,𝑘,𝑖. Worker 𝑈𝑘 then
+signs 𝑁𝑚,𝑘,𝑖 and broadcasts the signed node. Note that the
+training settings 𝜌𝑚,𝑘,𝑖, such as the learning rate and batch
+size, are embedded in nodes as a record of the training process
+for any upcoming PoL processes.
+Global-DAG Training. Training in Global-DAG also goes
+through Algo. 2, except that there exists neither a public shared
+testing dataset 𝐷test
+𝑚 , nor a unique training target that decides
+the stopping point, hence an indefinitely growing DAG. The
+testing accuracy 𝐸 ′
+𝑚,𝑘′,𝑖′ is also removed in rewarding con-
+tributors who upload models to Global DAG. 𝑈𝑘 receives a
+reward whenever one of its models gets referred by any other
+subsequent models in the network, which becomes the default
+contest strategy for Global-DAG. Users conducting training
+in Global-DAG need to be responsible for their own training
+processes, including preparing their own goals and local
+testing datasets and hunting for appropriate models across the
+whole network. Note that, the task-termination nodes of each
+Task𝑚 are also included in Global-DAG, which enables tasks
+to be advertised to the wider public and to be further evolvable
+along with diversified models in Global-DAG.
+C. Proof-of-Learning: Defense against Model Stealing Attacks
+We design a privacy-preserving PoL scheme to prove the
+computing-extensive training work and suppress model steal-
+ing attacks [27]–[30]. The idea of the privacy-preserving PoL
+is based on the reproducibility of training and the PoL in [33]
+in which the training process from the same starting point
+over the same training dataset with the same training settings
+results in the same trained model or bounded differences
+among the trained models. In the proposed privacy-preserving
+PoL, provers can provide obfuscated training dataset and
+give an estimated bound of model differences. We propose
+a new dataset obfuscation process to protect the privacy of
+the training dataset. The proposed privacy-preserving PoL
+scheme in the Global-DAG network is given in Algo. 3. The
+privacy-preserving PoL scheme for Task-DAG networks can be
+7
+
+Algorithm 3: Proof-of-Learning
+⊲ Raise a PoL-challenge
+1 𝑉𝑘,𝑘′,𝑖 ← 𝑈𝑘′.Challenge(𝑁𝑘,𝑖 | 𝑘 ≠ 𝑘′ ∧ not been challenged)
+2 𝑉𝑘,𝑘′,𝑖 ← 𝑈𝑘′.Deposit(𝜋𝑘,𝑘′,𝑖)
+⊲ Broadcast to the network
+3 𝑉𝑘,𝑘′,𝑖 ∈ 𝜆ℎ is subsequently settled by 𝑆ℎ.
+⊲ 𝜏ℎ countdown starts
+⊲ Reply with a PoL-proof
+4
+�
+𝐷train
+𝑘
+←𝑈𝑘.Obfuscate(𝐷train
+𝑘
+)
+5 𝑃𝑘,𝑘′,𝑖 ← 𝑈𝑘.Prove(𝑉𝑘,𝑘′,𝑖, �
+𝐷train
+𝑘
+))
+⊲ Broadcast to the network
+6 𝑃𝑘,𝑘′,𝑖 ∈ 𝜆ℎ+𝑛 is subsequently settled by 𝑆ℎ+𝑛.
+⊲ Verify the PoL-proof
+7 if 𝑃𝑘,𝑘′,𝑖 presents before the timeout then
+8
+if 𝑇𝑘,𝑘′,𝑖 ≤ 𝑇ℎ + 𝜏ℎ then
+9
+for 𝑈 ˆ𝑘 ∈ Θℎ+𝑛+1 parallelly do
+10
+𝑊 replay
+𝑘,𝑖
+←𝑈 ˆ𝑘.LearningReplay(𝑃𝑘,𝑘′,𝑖.(𝑊 ∗
+𝑘,𝑖, �
+𝐷train
+𝑘
+,
+𝜌𝑘,𝑖))
+11
+if ∥ 𝑊 replay
+𝑘,𝑖
+− 𝑁𝑘,𝑖.𝑊𝑘,𝑖 ∥2< 𝜖PoL then
+12
+𝑅𝑘, ˆ𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖
+13
+R𝑘,𝑖 ←Consensus(𝑅𝑘, ˆ𝑘,𝑖 | 𝑈 ˆ𝑘 ∈ Θℎ+𝑛+1)
+14
+if R𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖 then
+15
+emit Challenge fails
+⊲ Learning proved
+16
+goto Finalization
+17 emit Challenge succeeds
+⊲ Learning invalidated
+⊲ Finalization
+18 if Challenge fails then
+19
+𝑈𝑘′ ←Refund(𝛽𝜋𝑘,𝑘′,𝑖 | 𝛽 ∈ (0, 1))
+20 else
+21
+𝑈𝑘′ ←Refund(𝜋𝑘,𝑘′,𝑖)
+22
+𝑈𝑘′ ←Penalty(𝑈𝑘, 𝜋𝑘,𝑘′,𝑖)
+23
+WithdrawReward(𝑁𝑘,𝑖)
+24 𝑆ℎ+𝑛+1 is generated via Algo. 4
+⊲ Notice
+The Consensus is conducted by the nominated
+verification committee Ω𝑚 when the PoL is
+done in a Task𝑚.
+conducted in the same way where the PoL-proof is verified by
+the nominated task committee Ω𝑚.
+1) Challenge: If a node 𝑁𝑘,𝑖 has not been challenged
+before, any worker 𝑁𝑘′ can raise a PoL challenge against the
+node as a challenger. 𝑈𝑘′ needs to deposit a certain amount
+of tokens 𝜋𝑘,𝑘′,𝑖 to the PoL smart contract 𝑆𝐶𝑃𝑜𝐿 for the
+challenging node 𝑉𝑘,𝑘′,𝑖. Then, 𝑈𝑘′ signs and broadcasts the
+challenging node 𝑉𝑘,𝑘′,𝑖 to the network and starts a countdown.
+2) Response: The publisher of 𝑁𝑘,𝑖, i.e., 𝑈𝑘, needs to reply
+to the challenge 𝑉𝑘,𝑘′,𝑖 as a prover within the PoL timeout 𝜏.
+𝑈𝑘 firstly obtains the obfuscated dataset �𝐷train
+𝑘
+by applying
+the noise 𝛿𝑘,𝑘′,𝑖 to its local training dataset 𝐷train
+𝑘
+. Next, 𝑈𝑘
+prepares a PoL proof node 𝑃𝑘,𝑘′,𝑖 with the hash code and
+URI to the obfuscated dataset, i.e., 𝐻( �𝐷train
+𝑘
+) and URI( �𝐷train
+𝑘
+).
+Then, 𝑈𝑘 signs 𝑃𝑘,𝑘′,𝑖 and broadcasts it to the network.
+3) Verification: At the PoL verification stage, the commit-
+tee Θℎ (see details in Section IV-D for the use of Θℎ) verifies
+𝑃𝑘,𝑘′,𝑖 in parallel if the timestamp of the prover node is within
+the timeout 𝜏ℎ. To be specific, a verifier 𝑈 ˆ𝑘 can fetch the
+obfuscated dataset �𝐷train
+𝑘
+with the URI given in 𝑃𝑘,𝑘′,𝑖 and
+confirm its integrity. Next, 𝑈 ˆ𝑘 conducts a training task with
+the starting model described in 𝑁𝑘,𝑖, the training settings 𝜌𝑘,𝑖
+embedded in 𝑁𝑘,𝑖, and the training dataset �𝐷train
+𝑘
+, i.e.,
+𝑊∗
+𝑘,𝑖 =
+∑︁
+𝑊𝑝 in M𝑘,𝑖
+𝐸𝑝 in E𝑘,𝑖
+��� 𝑝
+�
+𝐸𝑞 ∈E𝑘,𝑖 𝐸𝑞
+𝑊𝑝
+𝑊replay
+𝑘,𝑖
+= Tˆ𝑘 (𝑊∗
+𝑘,𝑖, 𝜌𝑘,𝑖, �𝐷train
+𝑘
+).
+(3)
+The verifier 𝑈 ˆ𝑘 then calculates the Frobenius-Norm (F-Norm)
+between trained 𝑊replay
+𝑘,𝑖
+and 𝑊𝑘,𝑖 in 𝑁𝑘,𝑖 as the PoL re-
+sult 𝑅𝑘, ˆ𝑘,𝑖 for 𝑃𝑘,𝑘′,𝑖 [34]. If 𝑅𝑘, ˆ𝑘,𝑖 is within the 𝜖PoL, the
+verification on 𝑈 ˆ𝑘 is success and 𝑅𝑘, ˆ𝑘,𝑖 roots for 𝑃𝑘,𝑘′,𝑖.
+All versifiers in the committee Θℎ run consensus algorithms,
+e.g., Practical Byzantine Fault Tolerance (PBFT) [35], on
+PoL results and get the final committee decision R𝑘,𝑖 as
+a proving node in the network. Notice that, to prevent the
+spoofing attacks against the PoL verification and improve
+the security and robustness [36], 𝜖PoL can be dynamically
+adjustable from the early stage to the later stage to circumvent
+the consistent F-norm-based model distance during a PoL
+spoofing attack. Also, any PoL prover 𝑃𝑘,𝑘′,𝑖, ∀𝑘, 𝑘′, 𝑖 could
+alternatively conduct Verifiable Computation (VC) by showing
+an additional VC-proof to guarantee the identity of 𝑊replay
+𝑘,𝑖
+.
+4) Clearing: At the finalization stage, the challenger 𝑈𝑘′
+can only get 𝛽𝜋𝑘,𝑘′,𝑖 from the challenge deposit, 𝛽 ∈ (0, 1),
+if the challenge fails (i.e., the learning is proved). Otherwise,
+𝑈𝑘′ can get a full refund of the challenge deposit and also
+receive a reward from the penalty on 𝑈𝑘. The reward to 𝑈𝑘
+from 𝑁𝑘,𝑖 is revoked as well if the learning cannot be proved.
+D. Achieving State Consistency: Incentive Basis
+IRONFORGE features settlement nodes to achieve state con-
+sistency for the Global-DAG network, enabling smart contracts
+in DAG and recording consistent account states. The process
+is shown in Algo. 4.
+Global-DAG periodically, with the time interval Δ𝑇 , elects
+the settlement committee Θ among which consensus is reached
+to generate settlement nodes. At the beginning of the ℎ-th
+interval, VRF is used to elect a committee securely Θℎ for
+the settlement node 𝑆ℎ and the committee leader 𝑈 ¯ℎ. The
+probability that any worker 𝑈𝑘 is selected for the committee
+depends on the balance of 𝑈𝑘, i.e., 𝐵𝑘; see (4) below,
+VRF-hash(prv𝑘, 𝑠𝑒𝑒𝑑) ∈ Λ𝑘,
+(4)
+where Λ𝑘 is the area portion occupied by 𝑈𝑘 in a hash ring,
+and Λ𝑘 ∝ 𝐵𝑘.
+The committee can then settle the status of Global-DAG.
+To be specific, the leader of the committee 𝑈 ¯𝑘 synchronizes
+the view of a particular preceding moment of Global-DAG
+via consensus with others, and identifies all the tip nodes
+in the ℎ-th interval, which do not have any successor in the
+current interval. From each of the tip nodes, 𝑈 ¯𝑘 traverses back
+according to every preceding node list M𝑘,𝑖 of 𝑁𝑘,𝑖 along the
+path. The search stops and the subtree 𝜆ℎ is obtained when all
+specific nodes are met, i.e., the first visible node which does
+belong to the previous subtree 𝜆ℎ−1 in each path or the genesis
+node 𝑁𝑔. Based on all nodes in 𝜆ℎ, 𝑈 ¯𝑘 updates the balances
+of involved workers according to the training contributions,
+PoL challenges, PoL proofs, and smart contract executions. In
+8
+
+Algorithm 4: Node Generation in the Settlement Set
+⊲ Generating a settlement node upon consensus
+1 while True do
+2
+if 𝑇
+mod Δ𝑇 = 0 then
+3
+Θℎ ← VRF(Balance(𝑈𝑘 | ∀𝑘))).
+⊲ Election
+4
+while Θℎ.Consensus(𝑈𝑘.view | ∀𝑘 ∧ (𝑈𝑘 ∈ Θℎ)) do
+5
+if consensus is reached then
+6
+break and obtain 𝑆ℎ
+7
+S.Append(𝑆ℎ)
+⊲ The latest balance can be found in S[latest]
+⊲ Consensus
+8 for 𝑈𝑘 in Θℎ parallelly do
+9
+Tips ← Prune({𝑁𝑘,𝑖 | 𝑇𝑘,𝑖 > 𝑇 })
+10
+while Traverse(start ←Tips) do
+11
+if (Path-𝑝 reaches 𝑁𝑔 OR 𝑁𝑘,𝑖 ∈ 𝜆ℎ−1) is True then
+12
+Stop Path-𝑝
+13
+if ALL paths have stopped then
+14
+break and obtain 𝜆ℎ
+15
+𝜆ℎ ← Prune(Tips)
+⊲ Obtain the subtree 𝜆ℎ for the creation of 𝑆ℎ
+16
+¯𝑆𝑘,ℎ ← Form(𝜆ℎ.balance, 𝜆ℎ.PoL,
+{𝑁𝑚,𝑒 | never been collected by S})
+⊲ Tips are excluded in balance calculation
+17
+if 𝑈𝑘 is the leader 𝑈 ¯𝑘 of Θℎ then
+18
+¯𝑆leader ← ¯𝑆𝑘,ℎ
+19
+Broadcast( ¯𝑆leader)
+20
+else
+21
+Verify( ¯𝑆leader, ¯𝑆𝑘,ℎ)
+22
+if verification passes then
+23
+emit Consensus is reached
+24
+else
+25
+Elect a new Θℎ ← Θ
+′
+ℎ and redo Consensus
+Global-DAG, each valid reference from 𝑁𝑘′,𝑖 awards the owner
+𝑈𝑘 (𝑘 ≠ 𝑘′) of the referred model 𝑁𝑘,𝑖 with a certain amount
+of tokens. Next, 𝑈 ¯𝑘 proposes a new settlement node ¯𝑆𝑘,ℎ
+covering the updated balances and PoL results. The settlement
+committee Θℎ verifies and votes ¯𝑆𝑘,ℎ. The committee Θℎ
+endorses ¯��𝑘,ℎ as 𝑆ℎ if the committee reaches consensus, or
+elects a new leader otherwise.
+V. IMPLEMENTATION AND EVALUATION
+In this section, we conduct comparisons between the pro-
+posed FL system, IRONFORGE, and other popular frameworks,
+including GoogleFL [2], AsyncFL [6], and BlockFL [15]. We
+experimentally assess IRONFORGE in terms of the model per-
+formance and expected amount of rewards that can be earned
+under a variety of different environment settings, including
+different aggregation strategies, different sizes of hardware and
+software resources, and different types and levels of malicious
+attacks such as lazy attacks [9], poisoning attacks [25], back-
+door attacks [26], and model stealing attacks [27].
+A. Experimental Configurations
+1) Hardware settings: The experiments are conducted on
+6 servers listed as follows.
+Type-A (#1-3):
+• CPU. 2 × Intel(R) Xeon(R) Gold 6230R CPU @
+2.10GHz, 2 × 52 cores
+• GPU. 1 × Quadro RTX 4000, 1 × 8GB
+• Memory. 528GB
+• Bandwidth. 1000Mb/s
+Type-B (#4-6):
+• CPU. 2 × Intel(R) Xeon(R) Gold 6138 CPU @ 2.00GHz,
+2 × 40 cores
+• GPU. 8 × NVIDIA PCIe A100, 8 × 40GB
+• Memory. 250GB
+• Bandwidth. 1000Mb/s
+2) Software settings: We carry out the experiments upon
+Ubuntu 18.04.6 LTS with Keras 2.7 in Python 3.7.13 and
+Docker 20.10.12. We use FastDFS as the distributed file
+system with 15TB storage space for the model weights.
+3) A new testbed - FLSim: To benchmark the considered FL
+frameworks, we build an FL testbed named FLSim as shown
+in Fig. 4. FLSim is docker-containerized upon our servers (#1–
+6). Choosing different FL frameworks is flexibly plug-and-play
+in FLSim via three generic interfaces, i.e., the event emitter,
+model channel, and capability configuration.
+Event emitter. FLSim is event-driven where all events are
+delivered through Redis which serves as a message queue.
+Each runner can receive events in the network in real-time
+by the subscription function of Redis, and can broadcast
+corresponding events according to their role.
+Fig. 4. Architecture of the new FL testbed FLSim
+Model channel. Indexing models are done via MySQL where
+properties such as the URIs of weights are included, while
+the actual model weights are stored in FastDFS. As a result,
+the query efficiency can be significantly improved with no
+need of retaining the large weights unless they are required
+for evaluation or aggregation.
+Capability configuration. The tasks are trained on runners
+deployed on docker clusters. Each docker container represents
+9
+
+GoogleFL
+BlockFL
+FL Framework
+AsyncFL
+IronForge
+Interface
+Event emitter
+Redis
+Model channel
+CapabilityCofig
+MySQL+FastDF
+Master
+Operational Layer
+Minier
+Worker
+Runner
+Virtualisation Layer
+Docker Engine
+Docker Orchestration
+Physical Layer
+Server 1
+Server nTABLE III: Hyper-parameter settings
+Notation
+Definition
+Value (unit)
+P
+idle probability
+0.1
+E
+global epoch
+2000
+𝑒
+default local epoch
+5
+𝑙
+learning rate
+0.002
+𝜂
+sampled weights
+30
+𝛽
+default number of candidate weights
+6
+𝜎
+default number of aggregated weights
+5
+B
+default batch size
+100
+V
+validation set size
+100
+a runner with different resource settings, such as CPU, mem-
+ory, and bandwidth. Specifications of the containers are craft-
+specified with strong scalability and flexibility by defining the
+capability configuration to simulate various scenarios, such
+as the resource imbalance considered in the experiments.
+Moreover, each runner is categorized into different roles based
+on which FL framework has been plugged in, e.g., “workers”
+in all considered frameworks, “masters” in GoogleFL and
+AsyncFL, and “miners” in BlockFL.
+4) Training settings: The tuned hyper-parameters of the
+experiments are summarized in Table III. We perform training
+over the MNIST with 60,000 data samples and a Convolutional
+Neural Network (CNN) model illustrated in Fig. 5.
+Totally 60,000 MNIST samples are randomly split into two
+parts, 48,000 samples are used as the training set and 12,000
+samples are used as the testing set. We create non-IID training
+shards for contributors from the training set to simulate a
+practical network condition. The training set is divided into
+two subsets, i.e., 24,000 for each. The samples in the first
+subset are sorted by labels, and are subsequently distributed
+into 120 shards, 200 for each shard. Thus, the sample labels
+in each shard are relatively concentrated. The other half of
+the samples are randomly selected and distributed into 120
+shards, i.e., 200 samples for each shard. Thus, the sample
+labels in each shard are relatively uniform. Finally, we repeat
+the sampling operation 120 times, and each time we take a
+shard from each of the two subsets for merging. We end up
+obtaining 120 new shards each of which contains 400 samples.
+5) Environment settings: To demonstrate the state-of-the-
+art of IRONFORGE, a comprehensive comparison between
+IRONFORGE and the other three FL frameworks are conducted
+in our experiments, i.e., the synchronous GoogleFL, AsyncFL,
+and BlockFL. We launch 120 runners as workers to train
+models with a probability of P. We also define two events
+that represent receiving two types of intermediate models:
+• GLOBAL MODEL UPLOADED EVENT
+(GMUE):
+the model acquired by aggregating the uploaded local
+models prior to training.
+• LOCAL MODEL UPLOADED EVENT (LMUE): the
+model trained by workers with their local datasets
+These events are broadcast to notify each runner associated
+with the next step to take. Note that the genesis model of a
+task is tagged as a global model in order to initiate any selected
+FL framework. This indicates that emitting GMUE is used to
+notify the network when the task is published.
+Fig. 5. The CNN model is a lightweight version of the model
+in [2]. It contains one convolution layer of which the filter
+size is 32 and the kernel size is 5 × 5, one 2 × 2 max-pooling
+layers, one fully connected layer with 256 units, and ReLu
+activation. The output is processed by a fully connected layer
+with 10 units and softmax activation.
+Synchronous GoogleFL. One additional runner is launched
+as the master to aggregate local model weights. For GoogleFL,
+workers are activated by GMUE and the master is activated
+by LMUE. The workers, when receiving a GMUE, train on
+top of a global model downloaded from the master with
+their own local datasets. The master receives an LMUE when
+the workers upload the trained models, and subsequently
+aggregates all collected local models after a timeout, followed
+by uploading the aggregated result as the global model. The
+above iterating process continues until the task reached the
+max iteration threshold E.
+AsyncFL. One additional runner is launched as the master to
+aggregate local models. AsyncFL shares the same procedure
+as GoogleFL, except that the master, when receiving an
+LMUE during its idle period, creates a new global model by
+aggregating the most recent global model and newly-collected
+local models with an identical weighting factor.
+BlockFL. Five additional runners are launched as miners for
+BlockFL. In each iteration, the miners behave the same way
+as the master does in GoogleFL or AsyncFL. An additional
+step is that the miners compute the nonce to finalize the block
+and compete for the rewards with a synchronized lock being
+used in Redis to ensure the mining order.
+IRONFORGE. No additional runners are launched for IRON-
+FORGE. Each runner acts as a worker and a master at the same
+time, i.e., it supports both aggregating and training operations,
+thus receiving both GMU and LMUE during each iteration.
+6) Security settings: We implement five types of con-
+tributors: normal contributors, poison contributors, backdoor
+contributors, and stealing and colluding contributors.
+Normal contributors act honestly and independently across
+all phases.
+10
+
+input:
+[(None, 28, 28, 1)]
+conv2d 2 input
+InputLayer
+output:
+[(None, 28, 28, 1)]
+input:
+(None,28,28,1)
+conv2d 2
+Conv2D
+output:
+(None,24,24,32)
+input:
+(None,24,24,32)
+max_pooling2d_2
+MaxPooling2D
+output:
+(None,12,12,32)
+input:
+(None, 12,12,32)
+flatten 2
+Flatten
+output:
+(None,4608)
+input:
+(None,4608)
+dense_4
+Dense
+output:
+(None, 256)
+input:
+(None, 256)
+dense 5
+Dense
+output:
+(None, 10)Poisoning contributors aim to undermine the integrity and
+availability of the global model by crafting local poisoning
+models [25]. In this paper, we simulate poison contributors by
+adopting the label-flipping strategy that fakes labels and then
+conducting training on the forged datasets.
+Backdoor contributors aim to fail the global model on
+targeted tasks, typically by adhering crafted triggers to training
+samples, conducting training on the amended samples, and
+then uploading the attack models [26]. In this paper, backdoor
+contributors layer 5 × 5 white patches to training samples and
+change the label of the manipulated samples to a fixed one.
+Stealing contributors aim to gain rewards by stealing model
+weights trained by others and uploading the plagiarized
+weights as their own work [9]. In this paper, a stealing
+contributor 𝑘′ selects and directly uploads one of the existing
+weights by simply changing the ownership from 𝑘 to 𝑘′.
+Colluding contributors aim to embezzle training rewards for
+their conspirators by performing honest training processes but
+claiming source list M from the conspirators. In this paper, a
+certain proportion of contributors tamper the source list M in
+their uploaded weights, with a certain probability.
+B. Results and Evaluations
+Several experiments are conducted from two perspectives,
+i.e., the performance comparison between IRONFORGE and
+others with and without attacks, and the fairness comparison
+between different resource levels in terms of rewards.
+1) Performance - with and without attacks: Fig. 6 shows
+that the proposed IRONFORGE outperforms AsyncFL and
+BlockFL with only slightly slower convergence than the base-
+line GoogleFL across 2,000 iterations with no attacks lever-
+aged. It is worth noting that the red curve increases sharply
+with as few oscillations as that of the baseline, particularly
+highlighting the stability of IRONFORGE.
+Fig. 6. Comparison between IRONFORGE and others in terms
+of accuracy with no attacks leveraged
+The performance comparison with different levels of attack
+behaviors being applied to IRONFORGE is shown in Fig. 7. It
+is realized that IRONFORGE is resistant to the stealing attack
+the most, followed by the resistance to the backdoor attacks
+and poisoning attacks. It is worth noting that the performance
+of a stealing ratio of 20% can be as good as that of others.
+This is because the native validation process in IRONFORGE
+can capture and eliminate the plagiarized models which are
+reused or whose ownerships are fake. The remaining 80% of
+models are still sufficient for contributors to aggregate and
+train by offering strong diversity of the non-IID data samples.
+On the other hand, an evident degradation of the accuracy,
+around 5%, is shown in both poisoning (cf. Fig. 7(b)) and
+backdoor (cf. Fig. 7(c)) contexts. Nevertheless, it can be found
+from Fig. 8(a) and 8(b) that IRONFORGE outperforms all
+the other FL frameworks with either 20% ratio of poisoning
+attackers or 20% ratio of backdoor attackers. This significantly
+highlights the superiority of IRONFORGE in terms of its strong
+resistance to malicious model updating. Fig. 8(c) highlights
+the resistance to stealing attacks and collusion attacks of
+IRONFORGE. Note that only BlockFL is considered in the
+comparison as GoogleFL and AsyncFL do not support incen-
+tives natively. It is found that, by using the native validation
+process in IRONFORGE, including the PoL verification, none
+of the dishonest contributors who leverage either the stealing
+attack or collusion attack can gain rewards. This prevents the
+malicious contributors from faking the ownership of or directly
+using the existing models, or embezzling the rewards for their
+conspirators by claiming a falsified source list.
+Fig. 9(a) shows the accuracy ranges of adjusting the candi-
+date size (𝛽) for different levels of aggregation sizes (𝜎), e.g.,
+the blue band representing the accuracy ranged from 1-of-2
+to 1-of-8 with 𝜎 = 1 and 𝛽 ∈ [2, 8]. The results reveal that
+increasing the aggregation size 𝜎 stabilizes the performance
+in the beginning stage, and allows for an increasingly higher
+convergence point. The effect of increasing the candidate size
+for a certain aggregation size becomes gradually weakened
+as the aggregation size increases, as you can see that the
+width of each band turns more and more narrow. The effect
+of increasing the aggregation size also becomes weakened, as
+you can see from Fig. 9(a) that the performance of 5-of-6
+is as good as that of 7-of-8. The same insight can also be
+shed by observing the performance comparison of adjusting
+the aggregation sizes (𝜎) for different levels of candidate size
+(𝛽) is shown in Fig. 9(b), e.g., the blue band representing the
+accuracy ranged from 1-of-3 to 2-of-3. The bottom line of the
+red band for 𝛽 = 9 performs as poorly as the 2-of-3 strategy
+does, while the top line of the red band performs the best
+among the kinds. It can thus be concluded that knowing that
+no attacks are leveraged, aggregating more weights is more
+beneficial for obtaining high accuracy than merely aggregating
+a low number of weights from more candidate weights.
+Fig. 9(c) shows the running time of different aggregation
+strategies all the way from downloading the weights to up-
+loading a new model. There is a stronger effect on the running
+time when adjusting the candidate size (𝛽) than adjusting the
+aggregation size (𝜎). An implication can be realized based on
+this observation that the bandwidth could be the bottleneck in
+IRONFORGE. We design dedicated experiments that learn this
+phenomenon, as explained in the following Section V-B2.
+2) Fairness - earn rewards: The fairness is investigated in
+terms of the difference in rewards between different levels
+of hardware specifications. The bottleneck of gaining more
+11
+
+0.95
+06'0
+08'0
+BlockFL
+0.75
+AsyncFL
+GoogleFL
+IRONFORGE
+0.70
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Iteration(a) Stealing attacks
+(b) Poisoning attacks
+(c) Backdoor attacks
+Fig. 7. Comparison between different levels of stealing attacks, poisoning attacks, and backdoor attacks applied to IRONFORGE
+in terms of accuracy
+(a) Poisoning attacks with a poisoning ratio of 20% (b) Backdoor attacks with a poisoning ratio of 20%
+(c) Stealing and collusion attacks
+Fig. 8. Comparison between IRONFORGE and others in terms of accuracy or rewards with a certain level of poisoning attacks,
+backdoor attacks, and stealing and colluding attacks
+rewards in IRONFORGE can also be realized. We select two
+different contest strategies, i.e., Immediate settlement and
+Winner traverse. The immediate-settlement is the strategy used
+in Global-DAG by default, rewarding every model every time
+it gets referred by others. The winner-traverse could be one of
+the main options used in a Task-DAG, rewarding every model
+that exists in the traversal path all the way from the winner
+node to the genesis node (excluded).
+According to Fig. 10, A monotonic increase of the rewards
+can be realized for the immediate-settlement with increasing
+CPU cores, memory capacity, and bandwidth, and for the
+winner-traverse only with increasing bandwidth (the winner-
+traverse appears to have similar characteristics to BlockFL
+which also fails to offer a pure monotonic increase of rewards
+in all specifications). This is because the winner-traverse strat-
+egy could include moderate models being aggregated during
+each iteration in the winner-traversal path while the winner
+could be highly random when the competition is intense
+and the convergence is near. Therefore, many models with
+high performance could be excluded by the unique winner
+traversal path at the end. On the contrary, the immediate-
+settlement allows every model not to be missed so long as
+a valid reference relationship is confirmed. Nevertheless, the
+result difference between these two strategies does not tell
+the superiority of fairness. Different requirements may lead
+to different principles of fairness. Rooting for an egalitarian
+strategy, or “to each according to his contribution”, or striking
+a balance in between is a flexible option that the proposed
+IRONFORGE offers back to users without a harsh setting.
+On the other hand, the monotonic increase in the band-
+width comparison for both strategies, as shown in Fig. 10(c),
+highlights the bandwidth being the most critical effect for
+earning rewards in IRONFORGE. That is to say, relatively
+poorer users being more active in uploading models to the
+network with higher frequency can help them be more likely
+to share the rewards, rather than spending much time on a
+strongly performant model.
+Fig. 11 learns the effect of data quality upon the rewards by
+adding a random perturbation to 50% of the training samples
+of half of the users with a mean of 0 and a standard deviation
+of 1. We define the affected nodes as “Poor” nodes, while
+“Excellent” nodes own normal data samples only. The result
+shows that the poor nodes that use the immediate-settlement
+strategy enjoy a narrower range of rewards compared to that
+of the winner-traverse strategy and BlockFL. This highlights
+that poor nodes earning rewards via the immediate-settlement
+12
+
+0.9
+0.8
+0.7
+Accuracy
+0.6
+0.5
+0.4
+BlockFL
+0.3
+AsyncFL
+0.2
+GoogleFL
+IRONFORGE
+0.1
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration0.8
+Accuracy
+0.6
+0.4
+BlockFL
+AsyncFL
+0.2
+GoogleFL
+IRONFORGE
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+IterationBlockFL
+30
+IRONFORGE
+25
+20
+Rewards
+15
+10
+5
+0
+Normal
+Dishonest
+ContributorType,<Normal>/<Stealing/Colluded>1.0
+0.8
+0.6
+0.4
+Stealing Ratio:0%
+StealingRatio:5%
+0.2
+StealingRatio:10%
+Stealing Ratio:20%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration1.0
+0.8
+Accuracy
+0.6
+0.4
+Poisoning Ratio:0%
+Poisoning Ratio: 5%
+0.2
+PoisoningRatio:10%
+PoisoningRatio:20%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration1.0
+0.8
+0.6
+0.4
+BackdoorRatio:0%
+BackdoorRatio:5%
+0.2
+BackdoorRatio:10%
+BackdoorRatio:20%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Iteration(a) Accuracy difference between aggregation sizes
+(b) Accuracy difference between candidate sizes
+(c) Time difference between aggregation strategies
+Fig. 9. Comparison between different combination of the aggregation strategies (𝜎-of-𝛽) applied to IRONFORGE in terms of
+accuracy and execution time
+(a) Reward difference between different CPU cores
+(b) Reward difference between memory capacity
+(c) Reward difference between bandwidths
+Fig. 10. Comparison between different levels of the CPU core, memory capacity, and bandwidth in terms of rewards
+Fig. 11. Comparison between IRONFORGE and others in terms
+of rewards under the influence of the data quality issue
+in IRONFORGE can be more stable and predictable than
+BlockFL and the winner-traverse, and can be less affected by
+unexpected data degradation or network noise in unreliable
+channels. Excellent nodes have more opportunities to earn
+higher rewards than poor nodes while the median value and
+the minima of rewards remain as high as that of poor nodes.
+This reflects the fairness between excellent and poor nodes,
+i.e., offering stable rewards to the poor while the excellent are
+given chances to make a great fortune.
+VI. DISCUSSION AND ANALYSIS
+This analysis focuses on the security of IRONFORGE. Every
+role in the system is involved in the attack model except that
+the timestamp in IRONFORGE is considered synchronous via
+external trustworthy servers.
+Adversaries target to break the state consistency in Global-
+DAG so that operations such as incentive and consensus fail to
+be executed. The adversaries also target to leverage the model
+stealing attack in order to:
+• forge the “amount of work” by simply stealing others’
+models with no more effort being put into the training;
+• collude with attackers by creating a model referring to
+models from colluded attackers.
+In addition, the adversaries can target on breaching the dataset
+privacy during the dataset sharing in a PoL process. At the
+same time, the adversaries can also unbalance the competition
+by abusing others’ datasets to enrich local resources.
+State consistency: The state consistency is guaranteed over a
+sufficiently long period Δ𝑇 as long as the seeds being used in
+each VRF process are secure and the lower bounds of faulty
+tolerance of consensus protocol (e.g., 33% for PBFT) are
+satisfied in the VRF-elected committees. We consider the time
+gap between two settlement nodes Δ𝑆ℎ,𝑆ℎ−1 is sufficiently large
+in IRONFORGE to expect that each user who gets registered
+13
+
+0.9
+0.970
+0.8
+0.965
+Accuracy
+0.960
+Aggregation Size(α)
+0.950
+0.7
+0.945
+1
+2
+0.940
+0.935
+3
+0.930
+4
+0.6
+1000
+1200
+1400
+1600
+1800
+2000
+5
+7
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Iteration0.96
+0.94
+Accurao
+0.92
+0.90
+Candidate Size (β)
+3
+0.88
+5
+7
+9
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+Iteration1-of-3
+2-of-3
+1-of-5
+2-of-5
+3-of-5
+(o-of-
+4-of-5
+Strategy(
+1-of-7
+3-of-7
+5-of-7
+1-of-9
+3-of-9
+5-of-9
+7-of-9
+0
+20
+40
+60
+80
+Running Time (s)600
+BlockFL
+IronForge(Immediatesettlement)
+500
+IronForge(Winnertraverse)
+400
+Rewards
+300
+200
+100
+0
+2
+6
+8
+CPUCores400
+BlockFL
+350
+IronForge(lmmediatesettlement)
+ronForge(Winnertraverse
+300
+250
+rds
+Rewar
+200
+150
+100
+50
+0
+4
+6
+8
+10
+Memory(GB)400
+BlockFL
+IronForge(immediatesettlement)
+350
+IronForge(Winnertraverse)
+300
+Rewards
+250
+200
+150
+100
+50
+0
+2
+4
+8
+12
+Bandwidth(MB/s)BlockFL
+25
+IronForge(lmmediate settlement)
+IronForge (Winner traverse
+20
+Rewards
+15
+10
+5
+0
+Excellent
+Poor
+Data Quality,Poor/Excellentfor committee election has an identical “view” of Global-DAG
+starting from 𝑇ℎ−1 to 𝑇ℎ−1 + Δ𝑇 . This prevents the consensus
+process in the committee from being trapped into an indefinite
+disagreement due to the network asynchrony. On the other
+hand, an unbiased and unpredictable random seed is crucial for
+a fair VRF process where (4) cannot be manipulated. This can
+be achieved by implementing existing randomness generators
+such as RANDAO [37] or RandHound-VRF [38].
+Model stealing attack: Offering the proposed incentive mech-
+anism in a decentralized FL attracts attackers to leverage
+model stealing attacks, by either stealing the model ownership
+or faking the training processes. Attackers can, with no effort
+on local training, steal others’ models and fake ownership with
+ease. Attackers can alternatively fake the source lists upon an
+honest local training process so that their accomplices, who are
+instead placed in the source list, can reap profits against other
+honest users. These two types of model stealing attacks are
+used for misleading those who wish to ensure the necessary
+training overhead and the efforts in the source list. They are
+particularly useful when attackers intend to steal the rewards
+and share them with their accomplices. IRONFORGE enables
+PoL-challenge where users can choose to challenge a model
+via idle resources, and the model owner requires to provide
+the valid PoL-proof in time for a public verification during
+the consensus process. By the committee replaying parts of the
+training from scratch and reaching the consensus, the “amount
+of work” and the source list M can be explicitly determined.
+Dataset privacy and model melting: This security metric is
+an implementation of our work [34]. Dataset obfuscation helps
+to preserve dataset privacy when datasets require to be publicly
+shared for PoL-challenge. Experimental results in [34] show
+that an obfuscated dataset satisfies PoL verification without
+sacrificing the privacy level while being able to decrease the
+model utility against the abuse of collecting provers’ obfus-
+cated datasets, namely, model melting. In addition, applying
+training over different data samples or using non-IID noise
+significantly can reduce the risks of privacy decline when a
+sufficient number of challenges against the same model owner
+are deliberately raised by attackers.
+VII. CONCLUSION
+This paper proposed IRONFORGE, a new generation of FL
+framework constructed by DAG-based structure, which for
+the first time eliminates the need for the central coordinator
+to solve the issues of network asynchrony and the excessive
+reliance on the central coordinator while at the same time
+enabling an open and fair incentive mechanism to encourage
+more participants, particularly in networks where training
+resources are unevenly distributed. Experimental results based
+on a newly developed testbed FLSim along with the security
+analysis highlight the superiority of IRONFORGE over the
+existing prevalent FL frameworks under various specifica-
+tions regarding performance, fairness, and security. To our
+knowledge, this is the first paper proposing a secure and
+fully decentralized FL framework that can be applied in open
+networks with realistic network and training settings.
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+

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+OPTIMIZATION PERSPECTIVES ON SHELLSORT
+Oscar Skean
+Department of Computer Science
+University of Kentucky
+oscar.skean@uky.edu
+Richard Ehrenborg
+Department of Mathematics
+University of Kentucky
+Jerzy W. Jaromczyk
+Department of Computer Science
+University of Kentucky
+ABSTRACT
+Shellsort is a sorting method that is attractive due to its simplicity, yet it takes effort to analyze its
+efficiency. The heart of the algorithm is the gap sequence chosen a priori and used during sorting.
+The selection of this gap sequence affects the efficiency of Shellsort, and thus drives both its theo-
+retical and experimental analysis. We contribute to Shellsort by identifying efficient gap sequences
+based on new parameterized functions. Specifically, a parameter grid-search identifies optimal pa-
+rameters for different input sizes for sorting by observing minimal overhead in three categories:
+number of comparisons, number of exchanges, and running time. We report that our method finds
+sequences that outperform state-of-the-art gap sequences concerning the number of comparisons for
+chosen small array sizes. Additionally, our function-based sequences outperform the running time
+of the Tokuda sequences for chosen large array sizes. However, no substantial improvements were
+observed when minimizing the number of exchanges.
+1
+Introduction
+The Shellsort algorithm is a sorting method that was among the first to be discovered. Published in 1959 [1], it saw
+early interest due to its low memory requirements and simple implementation. Despite this, its analysis is difficult and
+remains incomplete. The algorithm has found practical use today in memory-constrained environments, embedded
+systems, and the bzip2 compressor. Recently it has also found use in data-oblivious sorting [2] and in fully homomor-
+phic encryption [3].
+Shellsort is an in-place comparison sort and can be viewed as a generalization of insertion sort. For a data array
+A of size N, Shellsort operates using a predetermined gap sequence 1 = k1 < · · · < km < N. The algorithm
+performs m passes over A: starting with the largest km and ending with k1. During pass j, for a given gap km−j,
+insertion sort occurs for the km−j subarrays consisting of the data elements A(i), A(i + km−j), A(i + 2 · km−j), . . .
+for i = 0, . . . , km−j − 1. We say a k-inversion is a pair (i, i + k) such that the inequality A(i) > A(i + k) holds.
+After pass j, all km−j-inversions that were originally in A have been solved. We say an array with no k-inversions
+is k-sorted. Note that the final pass with k1 is equivalent to insertion sort and is necessary to guarantee sortedness.
+Therefore, the purpose of the gap sequence is to presort A as much as possible before the expensive final insertion sort
+pass.
+The main results of this paper are:
+1. New efficient Shellsort sequences derived from experimentally optimizing sequence-generating functions.
+For prescribed array sizes, these sequences outperform well-known efficient sequences (e.g. Tokuda, Ciura)
+with respect to the number of comparisons. These sequences also outperform the running time of the Tokuda
+sequence, making them the fastest function-based sequence on the tested array sizes.
+2. We demonstrate results of experimental analysis comparing our proposed approaches with well-known se-
+quences by measuring the number of comparisons, exchange operations, and running time needed to sort
+randomized permutations.
+Traditionally, improvements for Shellsort have come from finding gap sequences with theoretical properties. We
+discuss some particularly important sequences in Section 2. Then in Section 3, we introduce parameterized sequence-
+arXiv:2301.00316v1  [cs.DS]  1 Jan 2023
+
+generating functions that generate a Shellsort sequence. The parameters are then optimized in a grid-search finding
+the best possible sequence that can be produced from that function for a chosen array size. In Section 4 we discuss our
+experimental methodology to compare the performance of the optimized template sequences to the baseline sequences
+mentioned in Section 2.
+2
+Background
+The selection of a good gap sequence is critical to the performance of Shellsort. There has been a plethora of work
+focused on selecting good sequences [4, 5, 6]. Some of the earliest proposed sequences were based on powers of
+2 [7, 1]. Then Pratt showed that the sequence of 2p3q obtains a number of inversions that is Θ(Nlog2N) in the
+worst case [8]. We call this sequence Pratt-23 in Table 1. This sequence still has the best known asymptotic time
+complexity for any Shellsort sequence. However, it has a very large constant factor which spurred the development of
+new sequences.
+The proof technique used to show the time complexity of Pratt-23 was based on counting the inversions of a sequence
+that has already been 2-sorted and 3-sorted. A natural extension of this is to apply the Frobenius problem to place
+bounds on what has already been sorted in prior passes. A typical formulation of the Frobenius problem is as follows:
+Suppose that you have k coins of denominations u1, u2, . . . , uk. What is the largest value which cannot be made
+with a nonnegative linear combination of these coins? This largest value is known as the Frobenius number [9]. In
+the context of Shellsort, the coins can be equated to gap size and the Frobenius number can be equated to the largest
+remaining inversion after sorting with the gaps. Using the Frobenius problem, several sequences were proposed that
+had a lower constant factor than Pratt-23 despite having a worse time-complexity [10, 11].
+Following those sequences, the focus of gap sequence selection shifted from finding theoretically good sequences
+to finding experimentally good ones. For example, one property that was observed was that a geometric sequence
+with a growth of 2.25 often performed well in practice. This observation was the basis of the Tokuda sequence [6].
+See Table 1 for a functional form. To the best of our knowledge, this remains the most competitive function in the
+literature.
+The next improvement came from Ciura in 2002 [4]. The Ciura sequence disregarded the idea of a function-based
+sequence, and instead searched for the best set of gap elements themselves. Ciura found the best sequence for array
+sizes of 128 and 1000, as well as a sequence that was conjectured to perform better for much larger array sizes. We
+call these Ciura-128, Ciura-1000, and Ciura-Large in Table 1.
+The Ciura sequences also marked a transition in how Shellsort performance was measured. Previously, most works
+counted the number of exchanges used by the algorithm [8]. This partly because some proof techniques relied on
+counting the number of inversions. Ciura instead focused on optimizing the number of comparisons, which was found
+to be more directly related to the computation time of Shellsort. A comparison is defined as checking if a pair of array
+elements are inverted. An exchange is typically defined as the variable swap used to fix an inversion. Minimizing the
+number of comparisons is especially beneficial when comparisons are expensive to make, such as when sorting large
+satellite data. Similarly, minimizing exchanges is beneficial in memory-constrained systems. In this work, we make
+clear distinctions in Section 4 about what measurement we’re optimizing for. For a full treatment of the history of gap
+sequences, we point the reader to [5].
+3
+Parameterized Template Functions
+The approach of directly optimizing the gap sequence, as in [4], grows in computational cost very quickly as N
+increases. This growth is due the fact that as N increases, both the expected number of sequence elements and their
+possible range of values increases. To help alleviate this, the authors of [4] found a suitable sequence prefix and
+optimize the extension of it by a few values. However, at very large N even finding this prefix would be very costly.
+Here, we formulate the problem of finding the optimal sequence as optimizing the parameters of a pre-defined
+sequence-generating function. Guided by principles that we have seen in sequences that perform well, we define
+two functions as follows.
+kA(i) = ⌊(a⌊ i
+b ⌋ · c⌊ i
+d ⌋)f + e⌋
+(1)
+kB(i) = ⌊(a · b⌊ i
+c ⌋) + d⌋
+(2)
+We refer to (1) as Ours-A and (2) as Ours-B in Table 1. Both formats contain the floor function in exponents. We
+found this to allow the function to express a more ”chaotic” sequence, which helped improve performance. The unique
+characteristic of Ours-A is the parameter f, an exponent that helps regulate growth. The parameter a of Ours-B was
+2
+
+Sequence Name
+Function
+Optimized for N
+Parameters
+Initial Terms
+Ciura [4]
+-
+128
+-
+1 4 9 24 85 126
+-
+1000
+-
+1 4 10 23 57 156 409 995
+-
+Large
+-
+1 4 10 23 57 132 301 701 1750
+Tokuda [6]
+⌈ (9/4)k−1
+(9/4)−1 ⌉
+-
+-
+1 4 9 20 46 103 233 525 . . .
+Ours A
+⌊(a⌊ i
+b ⌋ · c⌊ i
+d ⌋)f + e⌋
+128 (Comp)
+Table 2
+1 4 9 24 85 150 . . .
+1000 (Comp)
+Table 2
+1 4 10 23 57 153 400 . . .
+1000 (Time)
+Table 2
+1 3 7 16 33 85 179 472 . . .
+Ours B
+⌊(a · b⌊ i
+c ⌋) + d⌋
+10000 (Comp)
+Table 2
+1 4 10 27 72 187 488 . . .
+Pratt-23 [8]
+Ordered 2p · 3q
+-
+-
+1 2 3 4 6 8 9 . . .
+Pratt-25
+Ordered 2p · 5q
+-
+-
+1 2 4 5 8 10 15 16 . . .
+Pratt-34
+Ordered 3p · 4q
+-
+1 3 4 9 12 16 24 . . .
+Table 1: Gap sequences that are compared during experiments
+Template
+a
+b
+c
+d
+e
+f
+Ours-A128-Comp
+2.6321
+1.6841
+2.1570
+0.7360
+3
+0.7630
+Ours-A1000-Comp
+3.5789
+2.6316
+3.8158
+2.1579
+3
+0.7632
+Ours-A1000-Time
+2.75
+2.75
+3.7142
+2.4286
+2
+0.7429
+Ours-B10000-Comp
+4.0816
+8.5714
+2.2449
+0
+-
+-
+Table 2: Optimized parameters for template functions
+designed to have a similar purpose, albeit via multiplication. Ours-B contains fewer parameters which allows for
+quicker optimization. For conciseness, we only optimize Ours-A for array sizes 128 and 1000, and Ours-B for 10000.
+Furthermore, we denote as Ours-A1000-Comp if optimizing Format A for number of comparisons on arrays of size
+1000. In the following section, we discuss the experimental procedure for optimizing (1) and (2), as well as for
+comparing them to other baseline sequences.
+4
+Experimental Procedure
+Because the function is highly non-convex, it is difficult to utilize efficient techniques such as gradient descent. In-
+stead, we employ a grid-search approach. One benefit of using a function grid-search, as opposed to direct sequence
+optimization, is that the size of the search space has no relation to N. It depends only on the granularity and bounds
+of the search. This is in constrast to the methodology used by Ciura, in which the number of tested sequences grows
+with N [4].
+For Ours-A, we define the grid for parameters a, b, c, d, f as 20 linearly spaced values between 0.5 and 5. We also
+allow e to be an integer value between 0 and 10, including both endpoints. Because Ours-B has fewer parameters,
+we can take a more fine-grained approach. For parameters a, b, c, we test 50 linearly spaced values from 0 to 10. For
+parameter d, we constrain it to be the same as e in Ours-A.
+The data array that we test on contains N distinct values 1 through N, and we shuffle it with the Fischer-Yates shuffle.
+For each set of parameters in the grid-space, we compute the mean cost over 1000 iterations. We then take the set of
+parameters producing the lowest mean cost as optimal. This cost can be defined as number of comparisons, number
+of exchanges, or time.
+Because Ciura sequences are optimized for specific array sizes with no means of extending them, it would be inap-
+propriate to directly compare them with any function-based sequence at large array sizes. One method of extending a
+Ciura sequence is by starting a geometric series on its last term with a ratio of 2.25. We adopt this method of extension
+when measuring the performance of Ciura sequences.
+3
+
+Sequence
+N=20
+N=128
+N=200
+µCO
+µEX
+µCO
+µEX
+µCO
+µEX
+Ours-A128-Comp
+76 ± 6
+38 ± 6
+998 ± 33
+531 ± 33
+1786 ± 46
+948 ± 48
+Ours-A1000-Comp
+76 ± 6
+39 ± 7
+1004 ± 32
+516 ± 31
+1787 ± 44
+919 ± 45
+Ours-A1000-Time
+79 ± 5
+39 ± 7
+1035 ± 26
+468 ± 27
+1832 ± 38
+846 ± 39
+Ours-B10000-Comp
+76 ± 7
+33 ± 5
+1096 ± 52
+535 ± 36
+1775 ± 49
+960 ± 49
+Ciura-128
+76 ± 6
+37 ± 6
+998 ± 32
+531 ± 33
+1800 ± 46
+970 ± 49
+Ciura-1000
+76 ± 7
+39 ± 7
+1006 ± 31
+519 ± 34
+1787 ± 45
+920 ± 44
+Ciura-Long
+76 ± 7
+39 ± 7
+1004 ± 32
+516 ± 32
+1794 ± 44
+907 ± 42
+Tokuda
+76 ± 6
+37 ± 6
+1020 ± 28
+490 ± 28
+1808 ± 42
+891 ± 43
+Pratt-25
+111 ± 4
+27 ± 4
+1732 ± 16
+345 ± 17
+3207 ± 21
+610 ± 24
+Pratt-23
+136 ± 3
+25 ± 4
+2209 ± 13
+333 ± 15
+4095 ± 19
+589 ± 21
+Pratt-34
+95 ± 4
+29 ± 4
+1424 ± 16
+374 ± 19
+2593 ± 25
+660 ± 26
+Table 3: Number of operations to sort small arrays averaged over 1000 random array permutations. µCO denotes the
+number of comparisons, µEX denotes the number of exchanges.
+Sequence
+N=1000
+N=2000
+N=5000
+µCO
+µEX
+µCO
+µEX
+µCO
+µEX
+Ours-A128-Comp
+13250 ± 203
+7847 ± 199
+30530 ± 378
+18611 ± 384
+91122 ± 973
+57728 ± 904
+Ours-A1000-Comp
+12941 ± 167
+7004 ± 155
+29596 ± 293
+16234 ± 282
+86821 ± 768
+50349 ± 770
+Ours-A1000-Time
+13193 ± 144
+6461 ± 146
+30120 ± 263
+14913 ± 257
+87455 ± 548
+44305 ± 552
+Ours-B10000-Comp
+12980 ± 186
+7245 ± 177
+29643 ± 305
+17241 ± 325
+86514 ± 617
+57388 ± 817
+Ciura-128
+13300 ± 166
+7003 ± 168
+30359 ± 318
+15987 ± 310
+88193 ± 629
+46689 ± 627
+Ciura-1000
+12918 ± 161
+7002 ± 155
+29534 ± 282
+16138 ± 274
+86641 ± 757
+47852 ± 751
+Ciura-Long
+13035 ± 142
+6701 ± 149
+29567 ± 246
+15427 ± 261
+86232 ± 502
+45347 ± 496
+Tokuda
+13116 ± 143
+6556 ± 142
+29888 ± 241
+14952 ± 228
+86838 ± 454
+44116 ± 472
+Pratt-25
+26211 ± 68
+4318 ± 72
+62722 ± 122
+9755 ± 131
+194196 ± 263
+28195 ± 278
+Pratt-23
+34380 ± 64
+4253 ± 69
+82785 ± 106
+9669 ± 116
+259088 ± 242
+28354 ± 257
+Pratt-34
+20974 ± 89
+4671 ± 87
+50038 ± 153
+10543 ± 160
+154298 ± 372
+30448 ± 372
+Table 4: Number of operations to sort medium-sized arrays averaged over 1000 random array permutations
+4.1
+Filtering
+There are two techniques we employ to reduce our grid-search space.
+First, we notice that different sets of parameters could produce the same sequence. For example, for the template
+function Ours-A, the ordering of (a, b) and (c, d) do not matter. We precalculate all of the sequences produced in the
+grid and only experiment on the unique ones. For example, for N = 10000, this reduces the grid search space from
+over 1.5 million sets of parameters to about 1 million.
+Second, we use sequential analysis to act as a low-pass filter for screening out obviously poor sequences. This statis-
+tical approach was first applied to Shellsort in [4]. Given bounds for the mean and an upper bound for the variance,
+sequential analysis is able to tell in just a few repetitions whether or not a sample mean falls below the mean bounds
+with a certain confidence. Sequential analysis allows us to quickly accept good gap sequences that have a low mean
+number of comparisons. Any sequence that is accepted by the filter is then run for the full 1000 iterations to obtain a
+more accurate estimate of the mean. We adopt the same setup as in [4].
+The Ciura sequences were optimized with respect to the number of comparisons, and because they are well-known to
+be some of the most practical sequences, we optimize our template functions with respect to the number of comparisons
+as well.
+4.2
+Hardware
+The experiments were performed on a Ubuntu machine with an 8-core Intel Xeon W-3225. Experiments counting
+number of comparisons and exchanges were multithreaded. Experiments involving measurement of time were done
+4
+
+Sequence
+N=10000
+µCO
+µEX
+Ours-A128-Comp
+206356 ± 1796
+132351 ± 1797
+Ours-A1000-Comp
+196336 ± 1707
+119012 ± 1710
+Ours-A1000-Time
+194052 ± 879
+98952 ± 883
+Ours-B10000-Comp
+192029 ± 992
+209292 ± 1293
+Ciura-128
+195256 ± 1106
+105544 ± 1109
+Ciura-1000
+193778 ± 1895
+111338 ± 1897
+Ciura-Long
+191435 ± 892
+101680 ± 897
+Tokuda
+192574 ± 795
+98071 ± 796
+Pratt-25
+450131 ± 516
+62191 ± 526
+Pratt-23
+604502 ± 451
+66923 ± 725
+Pratt-34
+355382 ± 723
+63272 ± 462
+Table 5: Number of operations to sort an array size of 10000 averaged over 1000 random array permutations
+Sequence
+Running Time (ms)
+Ours-A128-Comp
+3.15 ± 0.08
+Ours-A1000-Comp
+3.02 ± 0.06
+Ours-A1000-Time
+3.01 ± 0.06
+Ours-B10000-Comp
+3.04 ± 0.07
+Ciura-128
+3.07 ± 0.06
+Ciura-1000
+3.01 ± 0.06
+Ciura-Long
+3.04 ± 0.07
+Tokuda
+3.06 ± 0.08
+Pratt-25
+5.00 ± 0.09
+Pratt-23
+6.35 ± 0.11
+Pratt-34
+4.17 ± 0.08
+Table 6: Time to sort an array size of 1000 averaged over 1000 random array permutations
+Figure 1: (Left) For varying array sizes, shows the difference in number of comparisons between baseline sequences
+and Ours-A128. A positive value means Ours-A128 uses fewer comparisons. (Right) Number of comparisons for
+varying array sizes larger than what Ours-A128 was optimized for.
+5
+
+Difference inMeanNumberof Comparisons
+35
+Ciura-128
+Ciura-Large
+30 
+Tokuda
+25
+differences
+20
+15
+10
+5
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Array SizeMeanNumberofComparisonsvsArraySize
+1800
+Ours-A128
+Ciura-128
+Ciura-Large
+1700
+Tokuda
+NumberofComparisons
+1600
+1500
+1400
+1300
+1200
+150
+160
+170
+180
+190
+200
+Array Sizesingle threaded, as any multithreaded applications could cause discrepancies in time measurement. All code was
+written in Python.
+4.3
+Results
+The best parameters that we found for Ours-A and Ours-B are in Table 2. Additionally, the first few terms of the
+sequences are shown in Table 1.
+For array size 200, we have found that Ours-B10000-Comp outperforms all other tested sequences in terms of number
+of comparisons, as shown in Table 3. Figure 1 is a graphical aid to this table. These graphs show that sequences
+generated by template functions can still perform well for array sizes larger, but not significantly so, than what they
+were optimized for. Furthermore, we found that both Ours-A128-Comp and Ours-A1000-Comp met the number of
+comparisons of, but did not surpass, the Ciura and Tokuda sequences. It’s interesting to note that several of the initial
+terms are equivalent between Ciura-128 and Ours-A128-Comp.
+We also test medium and large arrays, with results shown in Tables 4 and 5. For a graphical representation of Table 5,
+see Figure 1 in the Appendix. Our new sequences approach the performance of Ciura sequences without surpassing
+them. However, Ours-B10000 still surpasses the Tokuda sequence for all array sizes that we have tested here. Recall
+that the Tokuda sequence is currently the best known sequence to be generated from a function. Therefore, we have
+shown a new function-based sequence that outperforms other function-based sequences in terms of the number of
+comparisons. As mentioned previously, this is particularly useful if comparisons are a dominant operation such as
+when sorting large satellite data.
+On the other hand, our experiments with optimizing sequences to minimize running time are shown in Table 6. The
+relevant sequence, Ours-A1000-Time, takes a similar running time to the Ciura-1000 sequence. Both are faster than
+any other tested sequence. The sequence Ours-A1000-Time is particularly interesting because its first few elements (1
+3 7) are different than most other fast sequences (typically starting with 1 4 9). This difference may imply that while
+sequences beginning with (1 4 9) may be very good at minimizing number of comparisons, it does not guarantee that
+they have a good overall running time.
+5
+Conclusion
+Improvements for Shellsort traditionally come from finding gap sequences with better theoretical properties. Here
+we introduced an experimental framework to find improved gap sequences, following in the footsteps of [4]. Our
+generated gap sequences outperformed all well-known gap sequences in terms of number of comparisons on prescribed
+array sizes. Furthermore, the sequence Ours-A1000-Time is, to our knowledge, the function-based sequence with the
+quickest running time. However, it meets the performance of the Ciura sequence but does not surpass it. This may be
+improved with different sequence-generating functions or experimental setup, which we leave to future work. While
+the sequences presented here were optimized for chosen array sizes, the optimization may be repeated for any array
+size of interest.
+References
+[1] D. L. Shell, “A high-speed sorting procedure,” Communications of the ACM, vol. 2, no. 7, pp. 30–32, 1959.
+[2] M. T. Goodrich, “Randomized shellsort: A simple data-oblivious sorting algorithm,” Journal of the ACM (JACM), vol. 58,
+no. 6, pp. 1–26, 2011.
+[3] J.-W. Lee, Y.-S. Kim, and J.-S. No, “Analysis of modified shell sort for fully homomorphic encryption,” IEEE Access, vol. 9,
+pp. 126198–126215, 2021.
+[4] M. Ciura, “Best increments for the average case of shellsort,” in International Symposium on Fundamentals of Computation
+Theory, pp. 106–117, Springer, 2001.
+[5] R. Sedgewick, “Analysis of shellsort and related algorithms,” in European Symposium on Algorithms, pp. 1–11, Springer,
+1996.
+[6] N. Tokuda, “An improved shellsort,” in Proceedings of the IFIP 12th World Computer Congress on Algorithms, Software,
+Architecture-Information Processing’92, Volume 1-Volume I, pp. 449–457, 1992.
+[7] T. N. Hibbard, “An empirical study of minimal storage sorting,” Communications of the ACM, vol. 6, no. 5, pp. 206–213,
+1963.
+[8] V. R. Pratt, “Shellsort and sorting networks,” tech. rep., STANFORD UNIV CA DEPT OF COMPUTER SCIENCE, 1972.
+[9] E. S. Selmer, “On the linear diophantine problem of Frobenius,” 1977.
+6
+
+[10] J. Incerpi and R. Sedgewick, “Improved upper bounds on shellsort,” Journal of Computer and System Sciences, vol. 31, no. 2,
+pp. 210–224, 1985.
+[11] E. S. Selmer, “On shellsort and the Frobenius problem,” 1987.
+7
+

BtAyT4oBgHgl3EQfePgk/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf,len=507
+page_content='OPTIMIZATION PERSPECTIVES ON SHELLSORT  Oscar Skean  Department of Computer Science  University of Kentucky  oscar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='skean@uky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='edu  Richard Ehrenborg  Department of Mathematics  University of Kentucky  Jerzy W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Jaromczyk  Department of Computer Science  University of Kentucky  ABSTRACT  Shellsort is a sorting method that is attractive due to its simplicity, yet it takes effort to analyze its  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The heart of the algorithm is the gap sequence chosen a priori and used during sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The selection of this gap sequence affects the efficiency of Shellsort, and thus drives both its theo-  retical and experimental analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We contribute to Shellsort by identifying efficient gap sequences  based on new parameterized functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Specifically, a parameter grid-search identifies optimal pa-  rameters for different input sizes for sorting by observing minimal overhead in three categories:  number of comparisons, number of exchanges, and running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We report that our method finds  sequences that outperform state-of-the-art gap sequences concerning the number of comparisons for  chosen small array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Additionally, our function-based sequences outperform the running time  of the Tokuda sequences for chosen large array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, no substantial improvements were  observed when minimizing the number of exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  1  Introduction  The Shellsort algorithm is a sorting method that was among the first to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Published in 1959 [1], it saw  early interest due to its low memory requirements and simple implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Despite this, its analysis is difficult and  remains incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The algorithm has found practical use today in memory-constrained environments, embedded  systems, and the bzip2 compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Recently it has also found use in data-oblivious sorting [2] and in fully homomor-  phic encryption [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Shellsort is an in-place comparison sort and can be viewed as a generalization of insertion sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For a data array  A of size N, Shellsort operates using a predetermined gap sequence 1 = k1 < · · · < km < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The algorithm  performs m passes over A: starting with the largest km and ending with k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' During pass j, for a given gap km−j,  insertion sort occurs for the km−j subarrays consisting of the data elements A(i), A(i + km−j), A(i + 2 · km−j), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' , km−j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We say a k-inversion is a pair (i, i + k) such that the inequality A(i) > A(i + k) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  After pass j, all km−j-inversions that were originally in A have been solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We say an array with no k-inversions  is k-sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Note that the final pass with k1 is equivalent to insertion sort and is necessary to guarantee sortedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Therefore, the purpose of the gap sequence is to presort A as much as possible before the expensive final insertion sort  pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The main results of this paper are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' New efficient Shellsort sequences derived from experimentally optimizing sequence-generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For prescribed array sizes, these sequences outperform well-known efficient sequences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Tokuda, Ciura)  with respect to the number of comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' These sequences also outperform the running time of the Tokuda  sequence, making them the fastest function-based sequence on the tested array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We demonstrate results of experimental analysis comparing our proposed approaches with well-known se-  quences by measuring the number of comparisons, exchange operations, and running time needed to sort  randomized permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Traditionally, improvements for Shellsort have come from finding gap sequences with theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We  discuss some particularly important sequences in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Then in Section 3, we introduce parameterized sequence-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='00316v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='DS]  1 Jan 2023  generating functions that generate a Shellsort sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The parameters are then optimized in a grid-search finding  the best possible sequence that can be produced from that function for a chosen array size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In Section 4 we discuss our  experimental methodology to compare the performance of the optimized template sequences to the baseline sequences  mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  2  Background  The selection of a good gap sequence is critical to the performance of Shellsort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' There has been a plethora of work  focused on selecting good sequences [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Some of the earliest proposed sequences were based on powers of  2 [7, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Then Pratt showed that the sequence of 2p3q obtains a number of inversions that is Θ(Nlog2N) in the  worst case [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We call this sequence Pratt-23 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This sequence still has the best known asymptotic time  complexity for any Shellsort sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, it has a very large constant factor which spurred the development of  new sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The proof technique used to show the time complexity of Pratt-23 was based on counting the inversions of a sequence  that has already been 2-sorted and 3-sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A natural extension of this is to apply the Frobenius problem to place  bounds on what has already been sorted in prior passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A typical formulation of the Frobenius problem is as follows:  Suppose that you have k coins of denominations u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' , uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' What is the largest value which cannot be made  with a nonnegative linear combination of these coins?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This largest value is known as the Frobenius number [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In  the context of Shellsort, the coins can be equated to gap size and the Frobenius number can be equated to the largest  remaining inversion after sorting with the gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Using the Frobenius problem, several sequences were proposed that  had a lower constant factor than Pratt-23 despite having a worse time-complexity [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Following those sequences, the focus of gap sequence selection shifted from finding theoretically good sequences  to finding experimentally good ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For example, one property that was observed was that a geometric sequence  with a growth of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='25 often performed well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This observation was the basis of the Tokuda sequence [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  See Table 1 for a functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' To the best of our knowledge, this remains the most competitive function in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The next improvement came from Ciura in 2002 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The Ciura sequence disregarded the idea of a function-based  sequence, and instead searched for the best set of gap elements themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Ciura found the best sequence for array  sizes of 128 and 1000, as well as a sequence that was conjectured to perform better for much larger array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We  call these Ciura-128, Ciura-1000, and Ciura-Large in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The Ciura sequences also marked a transition in how Shellsort performance was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Previously, most works  counted the number of exchanges used by the algorithm [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This partly because some proof techniques relied on  counting the number of inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Ciura instead focused on optimizing the number of comparisons, which was found  to be more directly related to the computation time of Shellsort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A comparison is defined as checking if a pair of array  elements are inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' An exchange is typically defined as the variable swap used to fix an inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Minimizing the  number of comparisons is especially beneficial when comparisons are expensive to make, such as when sorting large  satellite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Similarly, minimizing exchanges is beneficial in memory-constrained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In this work, we make  clear distinctions in Section 4 about what measurement we’re optimizing for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For a full treatment of the history of gap  sequences, we point the reader to [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  3  Parameterized Template Functions  The approach of directly optimizing the gap sequence, as in [4], grows in computational cost very quickly as N  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This growth is due the fact that as N increases, both the expected number of sequence elements and their  possible range of values increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' To help alleviate this, the authors of [4] found a suitable sequence prefix and  optimize the extension of it by a few values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, at very large N even finding this prefix would be very costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Here, we formulate the problem of finding the optimal sequence as optimizing the parameters of a pre-defined  sequence-generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Guided by principles that we have seen in sequences that perform well, we define  two functions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  kA(i) = ⌊(a⌊ i  b ⌋ · c⌊ i  d ⌋)f + e⌋  (1)  kB(i) = ⌊(a · b⌊ i  c ⌋) + d⌋  (2)  We refer to (1) as Ours-A and (2) as Ours-B in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Both formats contain the floor function in exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We  found this to allow the function to express a more ”chaotic” sequence, which helped improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The unique  characteristic of Ours-A is the parameter f, an exponent that helps regulate growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The parameter a of Ours-B was  2  Sequence Name  Function  Optimized for N  Parameters  Initial Terms  Ciura [4] 128 1 4 9 24 85 126 1000 1 4 10 23 57 156 409 995 Large 1 4 10 23 57 132 301 701 1750  Tokuda [6]  ⌈ (9/4)k−1  (9/4)−1 ⌉ 1 4 9 20 46 103 233 525 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Ours A  ⌊(a⌊ i  b ⌋ · c⌊ i  d ⌋)f + e⌋  128 (Comp)  Table 2  1 4 9 24 85 150 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  1000 (Comp)  Table 2  1 4 10 23 57 153 400 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  1000 (Time)  Table 2  1 3 7 16 33 85 179 472 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Ours B  ⌊(a · b⌊ i  c ⌋) + d⌋  10000 (Comp)  Table 2  1 4 10 27 72 187 488 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Pratt-23 [8]  Ordered 2p · 3q 1 2 3 4 6 8 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Pratt-25  Ordered 2p · 5q 1 2 4 5 8 10 15 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Pratt-34  Ordered 3p · 4q 1 3 4 9 12 16 24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Table 1: Gap sequences that are compared during experiments  Template  a  b  c  d  e  f  Ours-A128-Comp  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6321  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6841  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1570  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7360  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7630  Ours-A1000-Comp  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5789  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6316  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='8158  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1579  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7632  Ours-A1000-Time  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7142  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4286  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7429  Ours-B10000-Comp  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='0816  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5714  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2449  0 Table 2: Optimized parameters for template functions  designed to have a similar purpose, albeit via multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Ours-B contains fewer parameters which allows for  quicker optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For conciseness, we only optimize Ours-A for array sizes 128 and 1000, and Ours-B for 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Furthermore, we denote as Ours-A1000-Comp if optimizing Format A for number of comparisons on arrays of size  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In the following section, we discuss the experimental procedure for optimizing (1) and (2), as well as for  comparing them to other baseline sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  4  Experimental Procedure  Because the function is highly non-convex, it is difficult to utilize efficient techniques such as gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' In-  stead, we employ a grid-search approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' One benefit of using a function grid-search, as opposed to direct sequence  optimization, is that the size of the search space has no relation to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' It depends only on the granularity and bounds  of the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This is in constrast to the methodology used by Ciura, in which the number of tested sequences grows  with N [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For Ours-A, we define the grid for parameters a, b, c, d, f as 20 linearly spaced values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We also  allow e to be an integer value between 0 and 10, including both endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Because Ours-B has fewer parameters,  we can take a more fine-grained approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For parameters a, b, c, we test 50 linearly spaced values from 0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For  parameter d, we constrain it to be the same as e in Ours-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The data array that we test on contains N distinct values 1 through N, and we shuffle it with the Fischer-Yates shuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For each set of parameters in the grid-space, we compute the mean cost over 1000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We then take the set of  parameters producing the lowest mean cost as optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This cost can be defined as number of comparisons, number  of exchanges, or time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Because Ciura sequences are optimized for specific array sizes with no means of extending them, it would be inap-  propriate to directly compare them with any function-based sequence at large array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' One method of extending a  Ciura sequence is by starting a geometric series on its last term with a ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We adopt this method of extension  when measuring the performance of Ciura sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='38 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='998 ± 33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='531 ± 33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1786 ± 46  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='948 ± 48  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1004 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='516 ± 31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1787 ± 44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='919 ± 45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='79 ± 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1035 ± 26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='468 ± 27  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1832 ± 38  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='846 ± 39  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-B10000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='33 ± 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1096 ± 52  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='535 ± 36  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1775 ± 49  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='960 ± 49  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='37 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='998 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='531 ± 33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1800 ± 46  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='970 ± 49  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1006 ± 31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='519 ± 34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1787 ± 45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='920 ± 44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-Long  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='39 ± 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1004 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='516 ± 32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1794 ± 44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='907 ± 42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Tokuda  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='76 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='37 ± 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1020 ± 28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='490 ± 28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1808 ± 42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='891 ± 43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='111 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='27 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1732 ± 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='345 ± 17  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3207 ± 21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='610 ± 24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='136 ± 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='25 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2209 ± 13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='333 ± 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4095 ± 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='589 ± 21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='95 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29 ± 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1424 ± 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='374 ± 19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2593 ± 25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='660 ± 26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Table 3: Number of operations to sort small arrays averaged over 1000 random array permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' µCO denotes the  number of comparisons, µEX denotes the number of exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=5000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13250 ± 203  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7847 ± 199  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30530 ± 378  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='18611 ± 384  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='91122 ± 973  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='57728 ± 904  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='12941 ± 167  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7004 ± 155  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29596 ± 293  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='16234 ± 282  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86821 ± 768  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='50349 ± 770  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13193 ± 144  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6461 ± 146  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30120 ± 263  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='14913 ± 257  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='87455 ± 548  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='44305 ± 552  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-B10000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='12980 ± 186  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7245 ± 177  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29643 ± 305  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='17241 ± 325  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86514 ± 617  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='57388 ± 817  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13300 ± 166  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7003 ± 168  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30359 ± 318  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='15987 ± 310  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='88193 ± 629  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='46689 ± 627  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='12918 ± 161  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='7002 ± 155  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29534 ± 282  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='16138 ± 274  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86641 ± 757  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='47852 ± 751  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-Long  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13035 ± 142  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6701 ± 149  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29567 ± 246  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='15427 ± 261  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86232 ± 502  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='45347 ± 496  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Tokuda  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='13116 ± 143  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='6556 ± 142  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='29888 ± 241  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='14952 ± 228  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='86838 ± 454  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='44116 ± 472  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='26211 ± 68  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4318 ± 72  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='62722 ± 122  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='9755 ± 131  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='194196 ± 263  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='28195 ± 278  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='34380 ± 64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4253 ± 69  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='82785 ± 106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='9669 ± 116  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='259088 ± 242  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='28354 ± 257  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='20974 ± 89  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4671 ± 87  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='50038 ± 153  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='10543 ± 160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='154298 ± 372  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='30448 ± 372  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Table 4: Number of operations to sort medium-sized arrays averaged over 1000 random array permutations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='1  Filtering  There are two techniques we employ to reduce our grid-search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  First, we notice that different sets of parameters could produce the same sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For example, for the template  function Ours-A, the ordering of (a, b) and (c, d) do not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We precalculate all of the sequences produced in the  grid and only experiment on the unique ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For example, for N = 10000, this reduces the grid search space from  over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='5 million sets of parameters to about 1 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  Second, we use sequential analysis to act as a low-pass filter for screening out obviously poor sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This statis-  tical approach was first applied to Shellsort in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Given bounds for the mean and an upper bound for the variance,  sequential analysis is able to tell in just a few repetitions whether or not a sample mean falls below the mean bounds  with a certain confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Sequential analysis allows us to quickly accept good gap sequences that have a low mean  number of comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Any sequence that is accepted by the filter is then run for the full 1000 iterations to obtain a  more accurate estimate of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' We adopt the same setup as in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  The Ciura sequences were optimized with respect to the number of comparisons, and because they are well-known to  be some of the most practical sequences, we optimize our template functions with respect to the number of comparisons  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='2  Hardware  The experiments were performed on a Ubuntu machine with an 8-core Intel Xeon W-3225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Experiments counting  number of comparisons and exchanges were multithreaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Experiments involving measurement of time were done  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='N=10000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µCO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='µEX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='206356 ± 1796  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='132351 ± 1797  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='196336 ± 1707  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='119012 ± 1710  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A1000-Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='194052 ± 879  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='98952 ± 883  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-B10000-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='192029 ± 992  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='209292 ± 1293  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-128  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='195256 ± 1106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='105544 ± 1109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='193778 ± 1895  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='111338 ± 1897  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ciura-Long  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='191435 ± 892  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='101680 ± 897  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Tokuda  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='192574 ± 795  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='98071 ± 796  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='450131 ± 516  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='62191 ± 526  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='604502 ± 451  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='66923 ± 725  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Pratt-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='355382 ± 723  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='63272 ± 462  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Table 5: Number of operations to sort an array size of 10000 averaged over 1000 random array permutations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Sequence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Running Time (ms)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='Ours-A128-Comp  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='08  Ours-A1000-Comp  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ours-A1000-Time  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ours-B10000-Comp  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='07  Ciura-128  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ciura-1000  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06  Ciura-Long  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='07  Tokuda  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='08  Pratt-25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='09  Pratt-23  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='11  Pratt-34  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='08  Table 6: Time to sort an array size of 1000 averaged over 1000 random array permutations  Figure 1: (Left) For varying array sizes, shows the difference in number of comparisons between baseline sequences  and Ours-A128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' A positive value means Ours-A128 uses fewer comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' (Right) Number of comparisons for  varying array sizes larger than what Ours-A128 was optimized for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  5  Difference inMeanNumberof Comparisons  35  Ciura-128  Ciura-Large  30  Tokuda  25  differences  20  15  10  5  0  25  50  75  100  125  150  175  200  Array SizeMeanNumberofComparisonsvsArraySize  1800  Ours-A128  Ciura-128  Ciura-Large  1700  Tokuda  NumberofComparisons  1600  1500  1400  1300  1200  150  160  170  180  190  200  Array Sizesingle threaded, as any multithreaded applications could cause discrepancies in time measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' All code was  written in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='3  Results  The best parameters that we found for Ours-A and Ours-B are in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Additionally, the first few terms of the  sequences are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  For array size 200, we have found that Ours-B10000-Comp outperforms all other tested sequences in terms of number  of comparisons, as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Figure 1 is a graphical aid to this table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' These graphs show that sequences  generated by template functions can still perform well for array sizes larger, but not significantly so, than what they  were optimized for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Furthermore, we found that both Ours-A128-Comp and Ours-A1000-Comp met the number of  comparisons of, but did not surpass, the Ciura and Tokuda sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' It’s interesting to note that several of the initial  terms are equivalent between Ciura-128 and Ours-A128-Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  We also test medium and large arrays, with results shown in Tables 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' For a graphical representation of Table 5,  see Figure 1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Our new sequences approach the performance of Ciura sequences without surpassing  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, Ours-B10000 still surpasses the Tokuda sequence for all array sizes that we have tested here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Recall  that the Tokuda sequence is currently the best known sequence to be generated from a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Therefore, we have  shown a new function-based sequence that outperforms other function-based sequences in terms of the number of  comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' As mentioned previously, this is particularly useful if comparisons are a dominant operation such as  when sorting large satellite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  On the other hand, our experiments with optimizing sequences to minimize running time are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The  relevant sequence, Ours-A1000-Time, takes a similar running time to the Ciura-1000 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Both are faster than  any other tested sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' The sequence Ours-A1000-Time is particularly interesting because its first few elements (1  3 7) are different than most other fast sequences (typically starting with 1 4 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This difference may imply that while  sequences beginning with (1 4 9) may be very good at minimizing number of comparisons, it does not guarantee that  they have a good overall running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  5  Conclusion  Improvements for Shellsort traditionally come from finding gap sequences with better theoretical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Here  we introduced an experimental framework to find improved gap sequences, following in the footsteps of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Our  generated gap sequences outperformed all well-known gap sequences in terms of number of comparisons on prescribed  array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Furthermore, the sequence Ours-A1000-Time is, to our knowledge, the function-based sequence with the  quickest running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' However, it meets the performance of the Ciura sequence but does not surpass it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' This may be  improved with different sequence-generating functions or experimental setup, which we leave to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' While  the sequences presented here were optimized for chosen array sizes, the optimization may be repeated for any array  size of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  References  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' Shell, “A high-speed sorting procedure,” Communications of the ACM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content=' 30–32, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtAyT4oBgHgl3EQfePgk/content/2301.00316v1.pdf'}
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+Bifurcation instructed design of multistate machines
+Teaya Yang,1 David Hathcock,1 Yuchao Chen,1 Paul
+McEuen,2 James P. Sethna,1 Itai Cohen,2 and Itay Griniasty1
+1Laboratory of Atomic and Solid State Physics,
+Cornell University, Ithaca, New York 14853-2501, USA
+2Laboratory of Atomic and Solid State Physics,
+Cornell University, Ithaca, New York 14853-2501,
+USA and Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, NY, USA
+(Dated: January 5, 2023)
+We propose a novel design paradigm for multi-state machines where transitions from one state
+to another are organized by bifurcations of multiple equilibria of the energy landscape describing
+the collective interactions of the machine components. This design paradigm is attractive since,
+near bifurcations, small variations in a few control parameters can result in large changes to the
+system’s state providing an emergent lever mechanism. Further, the topological configuration of
+transitions between states near such bifurcations ensures robust operation, making the machine less
+sensitive to fabrication errors and noise. To design such machines, we develop and implement a
+new efficient algorithm that searches for interactions between the machine components that give
+rise to energy landscapes with these bifurcation structures. We demonstrate a proof of concept for
+this approach by designing magneto elastic machines whose motions are primarily guided by their
+magnetic energy landscapes and show that by operating near bifurcations we can achieve multiple
+transition pathways between states. This proof of concept demonstration illustrates the power of
+this approach, which could be especially useful for soft robotics and at the microscale where typical
+macroscale designs are difficult to implement.
+Systems composed of a large number of interact-
+ing elements such as meta-materials, elastic mem-
+branes, and proteins can exhibit emergent behaviors
+that arise from the collaborative interaction of the
+system components. Designing functionality in such
+systems is a formidable task that requires searches
+in a high dimensional parameter space of the sys-
+tem components and their interactions. Developing
+organizing principles for effectively designing such
+systems remains an outstanding problem in the field
+[1–6]. Here, we propose that designing multi-state
+machines around bifurcations of multiple equilibria
+is a powerful paradigm that can be used to system-
+atically organize such searches.
+Bifurcations, where a single equilibrium configu-
+ration splits into multiple equilibria as a function of
+a control parameter is a canonical dynamical sys-
+tems structure that has been used to explain vari-
+ous natural phenomena ranging from phase transi-
+tions [7] to the operation of simple machines. Exam-
+ples of simple machines include Venus flytraps and
+hummingbird beaks that have been shown to open
+smoothly and then snap shut by operating about
+a cusp bifurcation where three equilibria converge
+[8]. Designing systems to operate near such bifur-
+cations provides several advantages. Since the split-
+ting of the equilibria has a power law dependence
+on the control parameters [4, 7], operating near bi-
+furcations automatically provides a lever mechanism
+by which small variations in the control parameters
+lead to large changes in the system state [12, 13].
+In the case of the Venus fly trap, slight changes in
+hydrostatic pressure can drive large motions of the
+trap. Similarly in hummingbirds, slight twisting of
+the jaw bones enables rapid closing of a wide open
+beak.
+Further, such bifurcations organize a topo-
+logically protected structure of saddle node mani-
+folds. As such, provided that the system trajectory
+encircles the cusp bifurcation where the saddle node
+manifolds meet, the system is guaranteed to exhibit
+a smooth change in state followed by a snap.
+In
+the Venus fly trap and hummingbird examples, this
+topological protection guarantees that the opening
+and snapping of the trap or beak is robust against
+variations in the applied hydrostatic or muscle forces
+driving the transitions in the system state.
+Here,
+we propose that moving beyond cusp bifurcations
+to design systems that operate near bifurcations of
+arbitrarily many equilibria preserves the lever ad-
+vantage and topological protection of cusp bifurca-
+tions.
+Such systems can be driven by only a few
+control parameters to undergo snapping transitions
+between multiple states making the design of ma-
+chines near such bifurcations a powerful paradigm
+for organizing complex functions. To develop and
+demonstrate this paradigm, we experimentally in-
+vestigate increasingly sophisticated magneto elastic
+machines whose function is organized by such bifur-
+cations.
+We start by constructing a simple magneto elas-
+tic machine consisting of a control panel that can
+be translated in the x − y plane and a second panel
+arXiv:2301.01507v1  [cond-mat.soft]  4 Jan 2023
+
+2
+FIG. 1.
+Magneto-Elastic
+machine
+capable
+of
+adopting multiple configurations due to operat-
+ing near a cusp bifurcation (a.) System: Panels P1
+and P2 are decorated with identical magnets. Panel P1 is
+actuated externally to translate in the x and y directions,
+in response Panel P2 rotates about a hinge, the dynam-
+ics are over-damped.
+(b.)
+Magnetic potential energy
+landscapes: We plot the potential for dx = −0.09 and
+dy ∈ [0.28, 0.17, 0.02], where dx and dy are deviations
+from the cusp’s position. Varying y we cross two saddle
+node bifurcations where the number of extrema of the
+magneto elastic landscape changes. (c.)
+Equilibrium
+manifold: The system’s equilibria θ(dx, dy) are plotted
+as a function of the deviation of the parameters, color
+signifies the value of θ. Brown curve marks saddle node
+bifurcations where the number of equilibria change, and
+the light red curve denotes the experimental trajectory.
+(d.) Experimentally observed snap-through transition:
+The system follows a parametric trajectory marked by a
+red curve and colored tube whose color denotes the pre-
+dicted state θ, around a cusp bifurcation. The colored
+disks represent the experimentally measured state θ. As
+expected a single snap through transition at a saddle
+node bifurcation (curves colored according to the bifur-
+cating state θ and converging at the cusp) is observed.
+that is free to rotate about a hinge connecting the
+two panels (Fig. 1a and experimental apparatus
+schematic Fig.S1). The state of the system, is given
+by the angle θ between the panels. By decorating
+the panels with magnets, we are able to design a
+magneto elastic landscape with different numbers of
+minima as a function of the parameters x and y
+(Fig. 1b).
+Transitions between these minima cor-
+respond to changes in the state of the system. To
+understand the various pathways for making such
+transitions we construct the manifold defined by the
+local equilibria as a function of the parameters x and
+y. For this particular arrangement of magnets, we
+calculate (see SI) that the resulting manifold has a
+domain with multiple solutions delineated by saddle
+node bifurcation curves (Brown). These curves in-
+tersect and terminate at a cusp bifurcation beyond
+which there is only a single equilibrium state. By
+translating the control panel in the x-y plane, the
+system can undergo either smooth or abrupt changes
+in θ. For example, starting the system at point (i)
+and moving through points (ii-v), the hinge angle
+increases smoothly. A further slight increase in the
+control parameter y, however, leads to an abrupt
+transition from a high to a low angle, correspond-
+ing to points (v) and (vi) respectively. These pre-
+dictions are born out by the experiments (Fig. 1d),
+which also show a smooth increase in θ for a path-
+way that encircles the cusp (i-v) and an abrupt tran-
+sition in θ when crossing a saddle node curve (v-vi).
+In this 2D representation the system makes a transi-
+tion when the color of the path (yellow) matches the
+color denoting the state associated with the saddle
+node curve (yellow). This magneto elastic mecha-
+nism is reminiscent of the cocking and snapping of
+a Venus flytrap or a humming bird’s beak.
+In addition to providing a mechanism for abrupt
+transitions, operating near a cusp bifurcation creates
+a lever mechanism where small variations in the con-
+trol parameters lead to large variations in the system
+state. This mechanism resolves the generic problem
+that creating large variations in the system state of-
+ten requires unfeasibly large variations in the control
+parameters. Lever mechanisms are generic near bi-
+furcations of equilibria since the magnitude of the
+transition in the system state is typically propor-
+tional to the square root of the parameter distance
+from the bifurcation.
+To characterize this lever mechanism in our exper-
+iment, we map the snapping transition curves asso-
+ciated with the saddle node bifurcations.
+Specifi-
+cally, for a given value of y (or x) we toggle x (y)
+so that the system snaps back and forth, and record
+the values of the control parameters x and y, and θ
+immediately after each transition (Fig. 2a).
+To test the scaling relations, we define the normal
+
+a)
+HingeAxis
+N
+s
+Parameter
+SystemState
+X-ControlParameter
+b)
+[nrun qe] A
+L
+2.2
+2.952.2
+2.95
+2.2
+2.95
+e[rad]-SystemState
+c)
+111
+2.9
+ [rad] - System State
+Snap-through
+S
+2.5
+d)
+LS
+Cusp Point
+0.03
+0.3
+0.15
+-0.06
+0
+dy [cm]-Control Parameter
+-0.15
+-0.093
+FIG. 2.
+Parametric levers The change in the state
+of the system after a snap through transition near a
+cusp bifurcation scales sub linearly with the normal
+form parameters. This sublinear scaling leads to large
+variation of the state in response to small variation
+of the system parameters.
+a) Measurements of snap
+through transitions near a cusp: The blue points mark
+the state of the magneto elastic system of Fig. 1 af-
+ter a snap through transition.
+The dashed curve is a
+fit of snap through transitions near a cusp bifurcation
+to the data, derived from the normal form potential
+˜V = δθ4 + a2δθ2 + a1δθ.
+The normal form parame-
+ters a1 and a2 are locally given by re-scaled rotations
+of dx and dy, which are the deviations of the parame-
+ters away from the cusp. b) Scaling laws near a cusp:
+The predicted scaling laws are demonstrated by project-
+ing the measurements and fit onto log-log plots. Near
+the cusp the system response to a1 acts as a giant lever,
+∂δθ/∂a1 ∼ 50.
+form parameters a1 and a2 as rotations of the dis-
+placement of the parameters x and y from the cusp.
+We then fit the predicted scaling form ∆θ ∝ √a2
+and a1 ∝ a3/2
+2
+near a cusp to determine the cusp’s
+position and the rotation of the normal form param-
+eters. The fitted model then predicts that ∆θ ∝ a1/3
+1
+(see SI). Because the scaling exponents for ∆θ are
+fractions of unity, small variations of the parameters
+along a1 and a2 lead to large variations of the sys-
+tem’s state.
+For example, in our experiments the
+range of actuation for panel 1’s position is approx-
+imately 1 cm and the range of angles accessible to
+panel 2 is 180◦ or π radians. Near the bifurcation
+a translation along a1 of 0.1% of its range (∼ 10µ
+m) leads to a snap that changes θ by ∼ 5% of it
+range (∼ 0.1 rad) providing a lever advantage of
+∼50 (Fig. 2b ).
+The complexity of the actions achieved by such
+magneto elastic mechanisms is dictated by the range
+and number of stable states that the system can ac-
+cess. This complexity can be achieved by designing
+the magneto elastic potentials such that the system
+operates near bifurcations between multiple states.
+For example, working near a hypothetical symmetric
+butterfly bifurcation associated with the potential
+V = θ6 + a4θ4 + a2θ2 + a1θ should enable smooth
+and abrupt transitions between three stable states
+in any order depending on the chosen trajectory for
+the control parameters. In Fig. 3a we show a cut
+through parameter space of the saddle node surfaces
+near this butterfly bifurcation. If the system starts
+in the S (Small) state and moves along the depicted
+trajectory (black arrows), it would first snap to the
+M (Medium) state when the system crosses the pur-
+ple saddle node bifurcation and then the L (Large)
+state when it crosses the green curve. For the return
+path, however, the system would transition from
+the L minimum directly to the S minimum when
+it crosses the yellow saddle node bifurcation curve.
+Moreover, by working near the bifurcation, the lever
+mechanism should allow for transitioning between
+these distinct states within an accessible range of
+experimental control parameters.
+SEARCH ALGORITHM FOR
+BIFURCATIONS OF MULTIPLE
+EQUILIBRIA
+To design parametric configurations correspond-
+ing to bifurcations of multiple equilibria we develop
+a search gradient continuation algorithm that takes
+advantage of their nested structure. Bifurcations as-
+sociated with k equilibria (minima plus maxima) are
+degenerate singularities where the first k derivatives
+of the potential vanish. Thus they can be found iter-
+atively by searching for singularities of the potential
+with increasing order, solving for one constraint at a
+time. We find that this method is especially efficient
+in finding experimentally realizable parametric con-
+figurations corresponding to bifurcations of multiple
+equilibria. Moreover, this method naturally extends
+to searching for bifurcations with desired properties
+by introducing further constraints, for example opti-
+mizing the robustness of the bifurcation’s associated
+states to external noise.
+For ease of illustration we describe how to use this
+approach to find the symmetrized butterfly bifurca-
+tion described above with parameters a1, a2, a4 and
+variable θ. For a random combination of parame-
+ters we find an equilibrium angle where dV/dθ = 0.
+Generically, this point is part of a smooth man-
+ifold over which this constraint holds.
+We then
+vary a1, a2, a4 and θ within this manifold to min-
+imize the next constraint |d2V/dθ2|.
+The trajec-
+tory follows the gradient of the second constraint as
+closely as possible while maintaining the first con-
+straint dV/dθ = 0 until we reach a point on a sad-
+
+a)
+b)
+0.09 h
+ [rad]
+System State
+0.05
+0.06
+0.04
+0
+0.05
+0.1
+0.18
+-0.05
+a2 [cm] - Normal Form Parameter
+] - System State
+0.09
+0
+- 0.02
+dx [cm]
+0.05.
+0.06
+ControlParameter
+73
+0.1
+80 [rad]
+0.04
+0.15
+1×10-4
+3×10-4
+0.001
+dy [cm] - Control Parameter
+a,[cm] - Normal Form Parameter4
+dle node surface, which is a manifold where both
+the first and second constraints hold.1 Minimizing
+the third derivative within the saddle node manifold
+maintains the first two constraints and allows for
+finding a cusp bifurcation associated with two sta-
+ble equilibria. Successive iterations allow for identi-
+fying bifurcations between an increasing number of
+equilibria and eventually the butterfly bifurcation.
+Our gradient continuation algorithm adapts stan-
+dard algorithms from the dynamical systems liter-
+ature [1, 2, 15] and retools them to locally follow
+the gradient of the unsatisfied constraint (see SI for
+further details). We depict the resulting search path
+in Fig. 3b, which highlights the fact that, indepen-
+dent of the number of parameters, the search algo-
+rithm follows a 1D trajectory, which is organized
+by the nested structure of the intermediate bifurca-
+tions. These properties enable the algorithm to find
+realizable bifurcations for systems with hundreds of
+parameters.
+THREE STATES AND THE BUTTERFLY
+BIFURCATION
+As a proof of concept for our approach we demon-
+strate the construction and operation of a magneto
+elastic machine with 3 stable states operating near
+a bifurcation of multiple equilibria. The first step in
+designing such a machine is to implement our gra-
+dient continuation algorithm to design a magneto
+elastic potential with a butterfly bifurcation between
+three stable states. To realize a system operating
+near such a bifurcation where only three control pa-
+rameters (x,y,z positions of panel 1) are actively var-
+ied, we allowed the algorithm to also determine the
+x,y positions of two of the nine magnets on panel
+1.2 With these seven parameters, the algorithm was
+able to identify multiple butterfly bifurcations that
+satisfied these criteria (See SI for details).
+Having found an appropriate butterfly bifurca-
+tion, we use standard dynamical systems continua-
+tion algorithms[1, 17] to compute and plot the saddle
+node surfaces in the control parameter space (x,y,z)
+near the bifurcation (Fig. 4). We find multiple dis-
+tinct surfaces where the color denotes the angle θ
+1 A local minimum of |∂2
+θV | with respect to variation of all
+parameters, that lies on the fixed point manifold will throw
+the algorithm off, but this is a co-dimension m point for a
+system with m parameters, and so highly unlikely.
+2 Typically, a butterfly bifurcation requires four control pa-
+rameters to navigate between all of the stable states. Here,
+we have identified a nonlinear mapping of the three active
+control parameters (x,y,z) onto the four dimensional space,
+which enables transitions between arbitrary minima.
+FIG. 3.
+Bifurcations of multiple equilibria.
+a)
+Work cycle near a butterfly: A system operating near a
+hypothetical symmetrized butterfly bifurcation can cycle
+between three states. The bifurcation is associated with
+a potential V = θ6+a4θ4+a2θ2+a1θ and three accessible
+states denoted by large (L), medium (M) and small (S).
+As the system follows the trajectory denoted by black ar-
+rows with colored background marking its state θ, it cy-
+cles between the three states snapping from S to M to L
+and back to S by changing a2 and a1 while a4 = 0.1. The
+snaps occur at saddle node bifurcations (colored curves)
+whose color signifies the state θ of the minima that is an-
+nihilated at each boundary. b) Gradient Continuation
+algorithm: The search algorithm finds bifurcations of
+multiple equilibria by following a one dimensional curve.
+Starting from a bifurcation of k equilibria the algorithm
+searches for a bifurcation of k+1 equilibria by following a
+curve in the augmented parameter space, tangent to the
+gradient of |V k+1| in the kth bifurcation manifold. We
+draw a search for a butterfly bifurcation in its symmet-
+ric normal form potential.
+The entire volume denotes
+the equilibrium manifold. Starting from a fixed point,
+the algorithm finds a saddle node bifurcation (along the
+white curve), Parameters are then varied on the saddle
+node surface (yellow), and cusp surface (thin lines) to re-
+spectively find a cusp bifurcation (along the gray curve)
+and a swallow tail bifurcation (along the black curve)
+near a butterfly bifurcation (black point).
+at which the saddle node bifurcation occurs3. In-
+structed by these surfaces, we design a cyclic path
+3 There is further local data in the potential at a saddle node
+
+D[a.u.]
+a1
+1.0
+L
+s
+0.5
+SL
+SM
+M
+0
+M
+SML
+12
+SL
+SL
+-0.5
+L
+-1.0
+Butterfly
+ Saddle node
+Cusp
+Equilibrium
+a1
+a4
+a25
+through the parameter space such that the system
+snaps between the large, medium, and small min-
+ima. The path color at each point denotes the sys-
+tem state, θ.
+As with the cusp and symmetrized
+butterfly bifurcations depictions in Figs. 1d and 3a,
+transitions occur at intersections of the path and
+saddle node surfaces where their colors match. We
+note that for the generic butterfly bifurcation, the
+surface structure can be quite complicated as shown
+by the two projections in Fig. 4a,b.
+In contrast
+to the symmetrized butterfly bifurcation structure
+(Fig. 2a), this complicated structure necessitate us-
+ing all three control parameters x, y, and z, to design
+a pathway that cycles between the three states. Im-
+portantly, despite the surface complexity the design
+is robust. Specifically, since the trajectory crosses
+surfaces, slight deviations in the control parameters
+should still lead to similar snaps, snap sequences,
+and ultimately the resulting complex actions of the
+entire magneto elastic machine.
+Using the design parameters determined by our
+search algorithm, we built a magneto elastic machine
+similar to that depicted in Fig. 1a, but with a dif-
+ferent magnetic dipole pattern and with two of the
+magnets in panel 1 displaced in the panel plane (See
+SI). By following the theoretically predicted path, we
+found three snap through transitions from small to
+large, large to medium, and medium to small (Fig. 4c
+and Movie S1). Two of the transitions occurred at
+the predicted locations, while the large to medium
+transition was displaced by 0.4 cm from its predicted
+location. In addition, we found excellent fidelity be-
+tween the predicted and measured angles θ for the
+equilibrium states. Using the same magneto elastic
+machine, we also designed and demonstrated cycli-
+cal paths with two transitions (See Fig.S3 and Movie
+S3). Finally, when the system was taken apart and
+reassembled, we were able to reliably reproduce the
+transitions associated with the designed trajectories.
+DISCUSSION
+The
+experimental
+validation
+of
+this
+design
+paradigm with a butterfly bifurcation of 5 equilib-
+ria strongly supports the conjecture that this frame-
+work could be extended to design systems perform-
+ing increasingly sophisticated functions by operating
+surface that can instruct the design of a trajectory.
+For
+example the sign of the third derivative of the potential
+signals whether the state’s angle will increase or decrease as
+it bifurcates. Moreover, the merging of saddle node surfaces
+can also be delineated by plotting the cusp bifurcations.
+Here, we do not include this additional information for ease
+of viewing
+near bifurcations with a growing number of equilib-
+ria. Potential energies with these increasingly rare
+bifurcations can be found efficiently, because the gra-
+dient continuation algorithm follows a one dimen-
+sional search path. Moreover, the associated lever
+mechanisms provide a design feature where the op-
+eration of the machine will likely be confined to a
+small parameter volume, enabling the execution of
+these actions by realizable machines.
+Microscopic magneto-elastic machines could prove
+to be a useful instance of design instructed by bifur-
+cations of multiple equilibria: An important emerg-
+ing strategy for manufacturing microscopic and soft
+machines is fabricating them using two dimensional
+lithographic and printing techniques [18–22]. Such
+fabrication techniques, however, restrict the imple-
+mentation of compound mechanisms composed of
+springs, cogs, screws etc. that are used to achieve
+complex actions in traditional macroscale machines.
+These lever mechanisms could be replaced with mag-
+neto elastic mechanisms with lever advantages in-
+duced by bifurcations. Magnetic interactions are es-
+pecially well suited for this purpose since they are
+long ranged and not easily screened. This long range
+allows for global changes to the conformation in re-
+sponse to local actuation of system components.
+Importantly, since bifurcations of multiple equi-
+libria are notoriously sensitive to variations of pa-
+rameters, there is a concern that a machine oper-
+ating near such bifurcations will be very sensitive
+to environmental noise, such as thermal vibrations,
+as well as to fabrication precision. Indeed, close to
+a bifurcation the sensitivity of the system to varia-
+tions of certain combinations of the system param-
+eters grows exponentially as the number of asso-
+ciated equilibria increases.
+Mathematically this is
+captured by mapping the potential to a canonical
+normal form via a change of coordinates [4, 5, Sec.
+36.6] (see SI for derivation). Practically, however,
+this increased sensitivity is often blunted outside of
+the infinitesimal environment of the bifurcation. At
+a finite distance from the bifurcation the mapping
+to the normal form or its linearization will often
+cease to be valid because of other singularities of
+the potential or the nonlinear fall off in the poten-
+tial. This non-linearity is especially pronounced in
+keplerian potentials such as that of magnetic inter-
+actions. Critically, the saddle node manifolds coa-
+lescing at the bifurcation are generically preserved
+outside this radius of convergence as they are topo-
+logically protected and can only annihilate at a cusp
+or a bifurcation of more equilibria. Thus, operating
+a machine near a bifurcation of multiple equilibria,
+but at a finite distance from it, allows the design
+of trajectories that take advantage of the multiple
+saddle node transitions associated with it, and their
+
+6
+FIG. 4.
+3-state Cycle Near Butterfly Bifurcation Point (a.) Theory The saddle node surfaces of a magneto-
+elastic system with three active control parameters, x,y and z are plotted, their color denotes the angle θ at which
+the snap occurs. The system’s magnetic pattern is designed using the gradient continuation algorithm such that it
+operates near a butterfly bifurcation where multiple saddle node surfaces coalesce, enabling multiple snap-through
+transitions at the surfaces. A trajectory (colored tube with white arrows) is chosen such that the system snaps in
+cycles between three states Large (L), Medium (M) and Small (S) angles. The system’s predicted state is denoted by
+the tube’s color. At intersections of the trajectory with a surface where their colors match the system is predicted
+to snap to a new state. (b.) Experimental demonstration: The colored dots mark the experimental value of the
+system’s state as it follows the designed trajectory. We observe three distinct transitions as predicted.
+lever advantages, while avoiding the local exponen-
+tial sensitivity.
+Similarly, the sensitivity of a system designed near
+a bifurcation of multiple equilibria to external noise
+grows exponentially with the number of associated
+states.
+This growth in sensitivity arises from the
+decrease in the potential barriers between adjacent
+states. For example in a potential with k equilibria
+where all the potential barriers are of equal height,
+and the minima are equally deep (which is propor-
+tional to a Chebyshev polynomial of the first kind
+of order k + 1) the barrier heights decay as 2−k.
+This sensitivity seems prohibitive as we imagine im-
+plementing this design principle to create systems
+cycling between multiple states.
+Despite this in-
+creased sensitivity, however, we estimate that the
+strength of magnetic interactions assures that mag-
+neto elastic systems are robust to thermal noise at
+the microscale. Specifically, in magneto elastic sys-
+tems the potential is proportional to the dipole-
+dipole interaction strength µ0µ2L6/R3 of two mag-
+nets with magnetic dipole densities µ panel size L
+and typical distance between dipoles R.
+Thermal
+noise is then comparable to the magneto elastic po-
+tential barrier height when the number of equilibria
+k ∼ log2
+�
+µ0µ2L3/(R/L)3
+kbT
+�
+. The magnetic dipole den-
+sities µ are of order 106A/m at the microscale [24].
+The smallest two state door (equivalent to the de-
+vice in Fig. 1a) that is robust to thermal noise is
+then ∼ .1µm in size, approaching the size limit of
+30nm for fabricating stable magnetic domains [25].
+Conversely, a 100 µm machine will become sensitive
+to thermal noise near a bifurcation of ∼ 40 equilib-
+ria, that is 20 distinct states compressed in a span
+of 100 degrees.
+Finally, the designs that we have implemented in
+this paper assume operation in a low Reynolds num-
+ber regime where inertia can be neglected. In the
+macroscale implementation this was achieved by at-
+taching a damping panel immersed in a solution of
+glycerol. We expect our designs to work even bet-
+ter as these machines are implemented at smaller
+scales since the importance of inertia drops quadrat-
+ically with the system size. Operation of a 100 µm
+scale machine in water, for example, would enable
+the system to be in the low Re regime while operat-
+ing at rates that are 1000 fold faster than those in
+the macroscale experiment.
+CONCLUSIONS
+We have shown that the operation of multi-
+parameter machines near bifurcations of multiple
+equilibria allows them to efficiently and robustly cy-
+cle between multiple conformation.
+Moreover, we
+developed a generic step-by-step framework to de-
+sign and implement systems that operate near such
+bifurcations.
+Specifically, we: 1) created a search
+algorithm that optimizes over fabrication and other
+system parameters to enable operation near such bi-
+furcations; 2) mapped the manifold of saddle node
+bifurcations to determine a useful trajectory for the
+machine operation and; 3) demonstrated the ro-
+bustness of this approach by constructing and op-
+erating a magneto elastic machine that can cycle
+and robustly snap between multiple distinct con-
+figurations in response to small variations of a few
+control parameters.
+Importantly, this design ap-
+proach and step-by-step implementation is generic
+and could be applied to many complex systems with
+
+z [cm] - Control
+Parameter
+-0.2
+L
+-0.4 -
+M
+-0.4
+-0.2
+-0.6 
+-0.4 -
+-0.8
+2
+-0.8 
+-0.6 -
+-0.8 -
+-
+[rad]
+2.0-
+0
+1
+1.5
+MO
+-1.0
+-0.5
+y
+-0.5
+-1.0
+-0.5
+y [cm] - Control Parameter
+-3
+-1.0
+-2
+X
+-2.0
+1.5
+X7
+multiple interacting components ranging from arti-
+ficial proteins, where the interactions are electro-
+static, to neural networks (both biological and syn-
+thetic) where the interactions are governed by net-
+work topology.
+Cycling between transitions in mechanical imple-
+mentations of such systems can generate work or lo-
+comotion. If the system is over-damped, as is often
+the case in microscopic systems operating in fluids,
+work and locomotion can be achieved by coupling
+the system to mechanisms that break time reversal
+symmetry.
+These mechanisms include ratchets or
+cilia-like flexible rods [26]. In the case of the mag-
+neto elastic hinge described here, time reversal sym-
+metry is broken by combining the smooth transla-
+tions of the control panel with abrupt transitions in
+the state of the dynamic panel. In systems where
+the control variable is not a mechanical parameter
+time reversal symmetry can be broken by using the
+angle as an effective dynamical variable governing a
+system with multiple degrees of freedom such as is
+often used to parameterize robot locomotion.
+More broadly, it is interesting to consider the ex-
+tension of our work to systems with a larger number
+of dynamical variables (θ1, θ2, . . .). Here, we envision
+that by working near bifurcations of multiple vari-
+ables (e.g. elliptic umbilic bifurcations) one could
+organize snaps between states separated along mul-
+tiple variables. Such designs require extending our
+search algorithm to multiple variables while main-
+taining its low dimensional search path.
+Alterna-
+tively, one could design mechanisms based on multi-
+ple local bifurcations that are weakly coupled across
+the machine.
+For example, one bifurcation of n
+states could be used to control θ1 while a second
+bifurcation of m states organizes the dynamics of
+the variable θ2. By weakly coupling the panels, and
+hence the variables θ1 and θ2, the machine can trans-
+form between n × m states in a coordinated fashion.
+Indeed this approach is already being implemented
+for bifurcations with two states [27–29]. Increasing
+the number of states associated with each variable
+would enable a similarly rich landscape for machine
+design with far fewer mechanical elements or panels.
+Finally, it is interesting to consider whether this
+design paradigm can be used to understand natural
+systems beyond the Venus fly trap and humming-
+bird beak. For example, molecular machines such
+as proteins often transition between different config-
+urations. It is interesting to consider whether such
+transitions can be thought of as snaps organized by
+bifurcations of many states [3, 6]. As another ex-
+ample, bifurcation theory has been implemented to
+identify and explain epigenetic dynamics of cell dif-
+ferentiation [30–32]. These approaches often focus
+on consecutive 2-state bifurcations. The results pre-
+sented here however, suggest that a comparably sim-
+ple evolutionary pathway could entail development
+of multi-state bifurcations. Such a structures could
+allow the addition of new states while maintaining
+the existing configuration through an evolutionary
+process, similar to the path taken by the gradient
+continuation algorithm.
+I.
+MATERIALS AND METHODS:
+Construction of experimental hinge system
+Panel P1 is constrained to a set of linear trans-
+lation stages that allow its position to be adjusted
+manually to any x or y coordinates near the cusp.
+For experiments near the butterfly bifurcation point,
+an extra translation stage is attached to Panel P1 to
+allow adjustment of its z coordinate. Panel P2 is
+attached to an OVA friction-less thrust air bushing
+with a 13mm shaft. The air bushing is attached to
+a fixed metal housing to limit Panel P2 to its ro-
+tational degree of freedom. A T-shaped paddle is
+attached to the bottom of the shaft and immersed
+in glycerol to introduce damping to the system. Ad-
+ditionally, we position a Basler Ace aca3088-57um
+area scan camera above the center of the air bushing
+to take top-view images of the air bushing which are
+then used to calculate the angle response of Panel
+P2 to high precision.
+A.
+Panels for experiments near cusp point
+Each magnetic panel is constructed using two 1/16
+in thick laser-cut acrylic pieces and nine grade N48
+neodymium magnets of diameter 1/16 in and height
+1/8 in.
+Magnets are arranged in a 3-by-3 square
+lattice with lattice constant of 2.5 cm.
+B.
+Panels for experiments near butterfly point
+Each magnetic panel is constructed using two 1/16
+in thick laser-cut acrylic pieces and nine grade N48
+neodymium magnets of diameter 1/8 in and height
+1/8 in. Magnets are arranged in a 3-by-3 square lat-
+tice with lattice constant of 2.5 cm. In panel P1 the
+x, y position of two of the magnets is displaced ac-
+cording to the design determined by the search algo-
+rithm. The two magnets whose position is offset are
+the magnet in the bottom row on the right column,
+whose offsets are dx1 = 1.418cm, dy1 = −0.273cm,
+and the magnet in the middle row on the left column,
+with offsets dx2 = −0.826cm, dy2 = −0.986cm. A
+
+8
+technical drawing illustrating the panels used for the
+butterfly experiment is included in the SI.
+FIG. 5. Experimental Setup Sketch of the experimen-
+tal system used for demonstration of cycles and angle
+measurements. Panel P1 is attached to a set of transla-
+tion stages which allows us to implement the spatial con-
+trol parameters in all experiments. Panel P2 is attached
+to an air bushing that is fixed in space. An attachment
+submerged in glycerol is added to the base of Panel P2
+to introduce damping to the system.
+Angle measurements
+A marker is attached to the top of the air bush-
+ing, and a camera records the location of the marker
+during the experiment. At each given time, the mea-
+sured angle is the determined by three points: cur-
+rent marker location, location of the center of rota-
+tion, and marker location at θ = 0. We calibrate
+the system by recording the location of the pixel at
+θ = 0 and several other distinct angles. The pixel
+location corresponding to the center of rotation is
+obtained using a fitted circle through the calibra-
+tion data points.
+The resulting angle is then de-
+duced from the three measured points. This data
+collection process is conducted in MATLAB.
+Acknowledgments We thank Michael Brenner,
+Chrisy Xiyu Du, Yan Yang, Robert Distasio, and
+John Guckenheimer for inspiring discussions. This
+work was financially supported primarily by NSF
+Grant DMREF-89228, NSF Grant EFRI-1935252,
+NSF Grant CBET-2010118, Cornell Center for Ma-
+terials Research DMR-1719875, and by Air Force
+Office of Scientific Research Grant MURI: FA9550-
+16-1-0031. I.G was also supported by the Cornell
+Laboratory of Atomic and Solid State Physics. D.H
+was supported by an NSF Graduate Research Fel-
+lowship Grant No. DGE-2139899.
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+(Cambridge university press, 2007).
+Supplemental material - Bifurcation instructed design of multistate machines
+II.
+CALCULATION OF THE POTENTIAL ENERGY LANDSCAPE
+To model the dynamics of our experimental hinge system, we compute the potential energy landscape
+arising from the dipole-dipole interactions between the magnets embedded in each panel. The magnets used
+in our experiments are well approximated by perfect dipoles. Therefore, the potential energy for the system
+is a sum of dipole-dipole interaction energies
+V = −
+�
+i∈P 1
+�
+j∈P 2
+µ0m2
+4π|rij|3 [3( ˆmi · ˆrij)( ˆmj · ˆrij) − ˆmi · ˆmj] ,
+(S1)
+where µ0 is the vacuum permeability, m is the dipole strength (identical for all magnets), mi is the orientation
+of dipole i, and rij is the distance between magnets i and j. Note that the interaction energy for dipoles in
+the same panel is constant, so we can restrict the sum to pairs of dipoles in different panels.
+To derive the θ dependence of the energy landscape, we must write the dipole orientations and positions
+in terms of our control parameters x, y, and z and the dynamical variable θ. The dipoles on P1 are always
+oriented in the z-direction, while the dipoles on the rotating panel P2 have orientation that changes with θ:
+ˆmi = δiˆz
+ˆmj = δj{sin θ , 0, − cos θ},
+(S2)
+where δi = ±1 is the orientation of magnet i with respect to panel P1 (similar for δj). The positions of
+individual dipoles are given by
+ri = {xi, yi, 0} + {x, y, z}
+rj = Rθ{xj, yj, 0},
+(S3)
+leading to interdipole distance rij = ri − rj. Here xi and yi are the x − y positions of dipole i in panel
+P1 (similar for xj, yj), x, y, and z are the coordinates of the control panel, and Rθ is the rotation matrix
+corresponding to a rotation by angle θ about the y-axis.
+Together Eqs. (S1-S3) give the potential energy in terms of the hinge angle θ, our control parameters x,
+y, and z, and design parameters xi, yi, δi, xj, yj, and δj. Since the hinge experiment is heavily damped, θ
+follows gradient dynamics ˙θ = ∂θV and the stable equilibrium angles are given by the local minima of the
+potential landscape V .
+
+2
+III.
+CUSP EXPERIMENTS
+In the cusp experiments, Panel P1’s x and y positions are measured as displacements from their value
+when the panels are 180◦ open, and are aligned along z and y such that the panel’s backs and bottoms
+are parallel. The magnets closest to the hinge axis are removed from it by 0.75cm on both panels. The
+back of the cylindrical magnets are aligned with the panel’s backs. The damping paddle used in the cusp
+experiments have dimensions 1.5cm by 3.0cm.
+A.
+Experimental estimation of the cusp point
+We estimate the location of the cusp point as the bifurcation of the two measured saddle node curves. We
+map the saddle node curve by toggling x (y), for a given value of y (or x), so that the system snaps back
+and forth, and record the values of the control parameters x and y, and θ immediately after each transition
+(Fig. S2). Moreover, to verify the position of the cusp we record the angle θ of the system before and after
+snapping, and observe that the change in angle upon snapping disappears at the cusp point.
+Finally, we inspect all data collected along the bifurcation curves as shown in Fig. 2a in the main text,
+and use a spline fit for the saddle-node bifurcations from L to S and the saddle-node bifurcations from S to
+L. We define the cusp point as the intersection of the two splines.
+B.
+Single snap experiment
+The mangeto elastic potential calculated for the experiment predicts a cusp at a slightly removed para-
+metric position. The discrepency between the experimentally measured and theoretically predicted cusps
+could be due to fabrication errors. To effectively compare theory and experiment in this section only, we
+parameterize the system as a function of its displacement from the cusp for both theory and experiment
+using using dx and dy. We then follow the predicted path by controlling panel P1’s x and y positions using
+the translation stages. We begin the experiment by letting the system maintain its equilibrium at the initial
+dx, dy position. We then change the position of Panel P1 at a slow and steady rate. Angle measurements
+are recorded at various locations in the loop as shown in Fig. S1(a) (see also Fig. 1 in the main text), and
+the change in position is paused once the transition happens at point vi in order to let the system settle
+down and obtain an accurate angle measurement. We confirm that the system returns to its original state
+once we return to the starting dx, dy position.
+C.
+Scaling experiment
+To fit the scaling relations, we use the the same section of the data set used for determining the location
+of the experimental cusp point. We neglect data in the nonlinear region of the saddle-node curves far away
+from the cusp point, as well as data too close to the cusp point, where the errors due to measurement noise
+are comparable to the distance to the cusp. The data points used for the scaling relations are highlighted in
+Fig. S2(a). The state parameter values used in the scaling analysis correspond to the angle measurements
+obtained at the points right after the snap through transitions.
+IV.
+ONE DIMENSIONAL BIFURCATIONS OF EQUILIBRIA: NORMAL FORM AND
+SCALING
+The ability to design magneto elastic machines and control parameter pathways that robustly lead to
+complex actions corroborates the validity of a new design paradigm: operation near bifurcations of multiple
+equilibria. The demonstrated trajectories take advantage of the structure of available dynamics near bifur-
+cations of equilibira. These bifurcations are the loci of multiple distinct coalescing saddle node manifolds,
+as illustrated for the idealized symmetric butterfly bifurcation (Fig 3b in the main text). By weaving a
+trajectory that crosses and avoids chosen saddle node bifurcations we design a pathway that leads to com-
+plex actions. The system then cycles through multiple states via small variations of the control parameters,
+
+3
+taking advantage of the multiple accessible lever mechanisms associated with these saddle node surfaces.
+The sensitivity of the realized design increases as the number of equilibria associated with the bifurcation
+grows.
+Butterfly, cusp and saddle node bifurcations are the first in a series of bifurcations of equilibria in
+one-dimensional gradient systems.
+More generally, in systems with a single degree of freedom x, bifur-
+cations of k equilibria are points in parameter space where the first k derivatives of the potential vanish,
+{dV/dx, d2V/dx2, . . . , dkV/dxk} = ⃗0. That is, they are equilibrium points satisfying k − 1 equations beyond
+that of mechanical equilibrium dV/dx = 0 and therefore lie on a manifold of co-dimension k − 1 within the
+equilibrium manifold. The sensitivity of a bifurcation of k equilibria to variation in its parameters can be
+estimated through the topological equivalence of the dynamics near it to those in a normal form potential
+�V = ϕk+1 +
+k−1
+�
+i=1
+aiϕi,
+(S4)
+where the variable ϕ(θ) and normal form parameters ai(p) are coordinate transformations of the angle θ and
+parameters p respectively. The normal form describes the unfolding of the Taylor expansion of the potential
+at the bifurcation V ∼ xk+1 by variations of the parameters [S1–S3]. The unfolded normal form potential
+demonstrates that the parameteric environment of a codimension k bifurcation includes domains with 1 to
+⌈(k + 1)/2⌉ minima delineated by k saddle-node manifolds which coalesce at the bifurcation. Moreover, it
+implies scaling relations between the variation in the system’s state upon a snap through transition induced
+by crossing a saddle node bifurcation associated with a codimension k − 1 bifurcation and the variation of a
+normal form parameter that causes the snap:
+δϕ ∝ a1/(k−m+1)
+m
+,
+m < k.
+(S5)
+Heuristically the scaling can be derived from the normal form by noting that near the bifurcation the kth
+derivative of the potential must still vanish, and so δϕ2 ∼ ak. Similarly the next k − 1 derivatives must
+progressively vanish, setting the scaling of am. An explicit proof is given in [S4] and summarized in [S5,
+Sec. 36.6]. These scaling relations carry over to the original variable and parameters near the bifurcation
+where the maps ϕ(x) and am(p) are approximately linear. Indeed, the scaling relations we experimentally
+observed near a the cusp bifurcations are those of the systems state with the normal form parameters near
+a bifurcation of three equilibria, i.e., a cusp [S4, S5].
+These scaling relations imply that the sensitivity of the system to variations of parameters grows expo-
+nentially with the number of associated equilibria. A system designed near a bifurcation of k equilibria can
+toggle its state between order unity separated states, δϕ ∼ 1/2, in response to variations of the linear normal
+form coefficient a1 of order 1/2k. That is, both the potential lever advantage and the sensitivity to noise in
+the parameters grow as the number of associated equilibria grows. However, the parametric domain in which
+the mapping to the normal form is linear is often very small. The nonlinearity of the mapping often blunts
+the sensitivity of the response. Thus, the increased lever advantage near bifurcations of multiple equilibria
+is often not experimentally accessible. Conversely the system is not so sensitive to parametric noise when
+operated at a small parametric distance from the bifurcation about which it is designed, as demonstrated
+by the reproducibility of the experimental three state system, which was easily constructed twice.
+V.
+CONTINUATION ALGORITHMS
+To find bifurcations of multiple equilibria in the dynamics of our model system and to map out the saddle
+node structure in the vicinity of the high-order point, we use a series of continuation algorithms. In one
+dimension, a codimension k bifurcation point is defined by the vanishing of the first k derivatives of the
+potential: ∂j
+θV (θ∗, {ξi}) = 0 for j = 1, 2, . . . , k. These constraints define a codimenion k manifold in the
+space of dynamical variables and parameters (θ∗, {ξi}).
+A.
+Traditional continuation
+Standard continuation algorithms compute bifurcation curves by varying a small number of parameters,
+and then projecting onto the bifurcation manifold [S1]. For example, suppose we have found a co-dimension
+
+4
+k bifurcation. This requires the first k derivatives of the potential vanish, fixing θ∗ and k − 1 parame-
+ters ξ1, ξ2, . . . , ξk−1. Varying an additional parameter ξk produces a line emanating from our initial point
+(θ∗, {ξ}) = p. The continuation algorithm maps out this line by (i) taking a step along the tangent vec-
+tor Tk(p) to the curve, which is the null-vector of the gradient of the first k derivatives of the potential
+Tk(p) ≡
+�
+⃗v ∈ Rk+1 | ∀j ∈ (1, 2, . . . , k), ⃗v · ∇θ,ξ1,ξ2,...,ξk∂j
+θV = 0
+�
+and (ii) correcting this step using a Newton-
+Raphson algorithm4 to search perpendicular to the step for a point where the first k derivatives of the po-
+tential vanish. This approach can be used to progressively search for higher order bifurcation points. For
+example, a fixed-point can be continued until ∂2V (θ∗, {ξi})/∂θ2 vanishes, indicating a saddle node bifurca-
+tion. Continuing the saddle-node can lead to a cusp bifurcation, which in turn might lead to a swallowtail
+bifurcation. In this way, progressively adding parameters and performing continuations of one-dimensional
+curves can lead toward high-codimension bifurcation points. Once we have found a high-order bifurcation
+point, we use this algorithm to map out the saddle node surfaces nearby. The surfaces can in turn be used
+to design cycles in control parameters that cause the system to perform desired snapping transitions.
+The standard continuation approach, however, has limitations for microscopic machine design. In partic-
+ular, it has limited utility for finding the high-order bifurcation points near which our machine will operate.
+In our model system we have many free parameters, including the positions of each of the magnets embed-
+ded in the panels. Varying a given experimental parameter does not guarantee we will find the next order
+bifurcation point. Instead we want to vary many parameters simultaneously, which greatly improves the
+likelihood that a higher-order bifurcation point is contained within the search space and allows for a more
+efficient approach toward that point. We have developed a gradient continuation algorithm to carry out this
+multi-parameter search.
+B.
+Design algorithm: Gradient continuation
+The gradient continuation algorithm works as follows. Suppose we have N parameters ξi in our system,
+plus the degree-of-freedom θ.
+A point, p, where the first k derivatives of the potential vanish belongs
+to a co-dimension k manifold in the full (N + 1)-dimensional augmented parameter space, composed of
+the equilibrium state and control parameters, (θ∗, {ξi}). Starting from the point p, take a step along the
+gradient of the k+1 derivative of the potential ∇θ,ξ1,ξ2,...,ξN ∂k+1
+θ
+V , projected onto the tangent surface to the
+manifold at p. The tangent surface is the null-space of the gradient of the first k derivatives of the potential5,
+Tk,N(p) ≡
+�
+⃗v ∈ RN+1 | ∀j ∈ (1, 2, . . . , k), ⃗v · ∇θ,ξ1,ξ2,...,ξN ∂j
+θV = 0
+�
+. This procedure finds the step within the
+co-dimension k manifold that maximizes the change in ∂k+1
+θ
+V , which we need to vanish in order to find the
+next order bifurcation. After the step, the algorithm performs a corrective Newton-Raphson search [S6],
+constrained to the hyperplane T ⊥
+k,N(p) perpendicular to the null-space, which returns to the codimension
+k manifold on which the first k derivatives of the potential vanish. As in the standard continuation, this
+approach is repeated to progressively find higher order bifurcation points. A visualization of the gradient
+search algorithm, applied to the potential V = θ6 + a4θ4 + a2θ2 + a1θ, is shown in Fig. 3b in the main text.
+VI.
+BUTTERFLY EXPERIMENTS
+A.
+Butterfly panels
+In the butterfly experiments, Panel P1’s x, y and z positions are measured as displacements from their
+value when the panels are 180◦ open, the magnets closest to the hinge axis are removed from it by 2.5cm
+on both panels, the panels are aligned vertically, and the back of the cylindrical magnets on Panel P1 are
+aligned with the center of the magnets on Panel P2. This small change in magnet alignment (compared
+4 Newton-Raphson(f, Ω, p) [S6] searches for the roots of the
+functions f over the space Ω starting at the point p.
+5 Notice that this algorithm uses all N parameters ξ1, . . . , ξN
+to search for a codimension k bifurcation, while the stan-
+dard continuation in the previous section only used k pa-
+rameters ξ1, . . . , ξk. The null-space Tk,N(p) has dimension
+(N − k + 1).
+
+5
+to the single snap experiment) is found to reduce the discrepancy between experiment and prediction. An
+illustration for the panels is shown in Fig. S3. The damping paddle has dimensions 8.0cm by 2.5cm for the
+butterfly experiments. The position of the magnets on panel P1 was changed such that the system operates
+next to a butterfly bifurcation, as specified in the main text and in the following sections.
+B.
+Application of the continuation algorithm
+To find an experimentally feasible path and magnetic pattern, we implement the continuation algorithm
+by first finding a butterfly point in parameter space, then validating the resulting pattern against known
+experimental constraints (e.g. we require physically realizable panel angles and magnet positions). Before
+each search using the continuation algorithm, we first randomly generate orientations of the 18 magnetic
+dipoles on the two panels. We also randomly select two magnets on Panel P1 to be displaced from their
+lattice positions, by (dx1, dy1) and (dx2, dy2) respectively. The search algorithm is always initialized with
+the values {θ, dx, dy, dz, dx1, dy1, dx2, dy2} = {1.1rad, 0.5cm, −0.25cm, 0, 0, 0, 0, 0}.
+Next,
+we
+let
+the
+algorithm
+try
+to
+find
+a
+butterfly
+bifurcation
+point.
+If
+no
+but-
+terfly
+point
+can
+be
+found,
+we
+repeat
+the
+initialization
+process
+and
+repeat
+the
+search
+with
+a
+new
+randomly
+generated
+magnetic
+pattern.
+The
+butterfly
+point
+corresponding
+to
+the
+pattern
+we
+used
+in
+our
+experiments
+is
+located
+at
+{θ, dx, dy, dz, dx1, dy1, dx2, dy2}
+=
+{2.131rad, −0.355cm, −0.304cm, −0.824cm, 0.918cm, −0.698cm, −0.326cm, −0.486cm}.
+If the butterfly point is found, we investigate the potential plots at various points in parameter space near
+the bifurcation point. Specifically, we offset one or more of the 6 search parameters by ±0.2 and find the
+number of minima that exist between 0 to 180 degrees at each of these locations. The potential plots at
+locations with three minima are then inspected to decide the experimental feasibility of the pattern. Ideally,
+all three minima are at least 5 degrees apart, and the smallest minimum is at least 5 degrees (for z = 0)
+to prevent the panels from touching during experiment. We also look for patterns with large triple-minima
+regions, for example if three visibly deep minima can be observed when at least one parameter is changed
+by ±0.5cm.
+After an experimentally feasible pattern is discovered, we manipulate the three experimentally controllable
+parameters (x,y,z) continuously around the point with deepest triple minima and observe changes in our
+model of the potential landscape. The design of the control path is guided by visualization of the saddle-node
+surfaces mapped out using the standard continuation algorithm detailed above. Several paths are tested in
+the model to obtain the desired sequence of bifurcations and to optimize various properties of the transitions
+(e.g. the magnitude of the snaps and depth of the minima).
+C.
+Experiments for trajectories near a butterfly point
+We set up the experiment by laser-cutting the holes for magnets at the exact locations corresponding to the
+found dx1, dy1, dx2, dy2 values, which were 1.418cm, −0.273cm, −0.826cm, and −0.986cm respectively. We
+also add a translation stage to control Panel P1’s z position. We begin the experiment by following the exact
+coordinates provided by the theoretically designed path. In the event that a predicted transition cannot be
+seen using the predicted path coordinates (due to fabrication or calibration errors shifting the surface), we
+translate the system further from the original predicted path to determine a more robust path that may
+account for some shifting in coordinates due to experimental errors (for example see Fig. 4b in the main
+text). Once an experimental path is shown to demonstrate the predicted behavior with the desired number
+of state transitions, we record the locations for state transitions in experiment, and repeat the experiment
+while slowing down the rate of change in x,y positions near the transitions to give the system enough time to
+respond in the presence of large damping. Those experiments show excellent qualitative agreement with the
+theoretically designed paths, although the locations at which transitions happen and the equilibrium angle
+of the panel are often shifted by a small amount due to experimental error.
+
+6
+D.
+Additional operation mode: double-snap trajectories
+The intricate saddle-node surface structure near the butterfly bifurcation enables a variety of snapping
+behaviors with the same panel design, beyond the 3-state cycle presented in the main text. Here we present
+a second snapping sequence that was measured experimentally.
+By using the same trajectory in parameter space as the three-snap sequence in the main text, but traverses
+the path in the reverse direction, we observe a two-snap sequence between small (S) and large (L) angles.
+Fig. S4a shows this trajectory together with the same saddle surfaces from Fig. 4a in the main text. The
+experimentally measured angles along this backward cycle are shown in Fig. S4b (see also Movie S2). Besides
+a minor systematic shift in the angles of the L state, we find excellent fidelity between the predicted and
+measured angles. The snapping transitions occur almost exactly at the predicted locations.
+Our example trajectories demonstrate that the saddle-node structure in the vicinity of a butterfly bifurca-
+tions enables a great deal of flexibility in controlling state transitions of a mechanical system. For practical
+applications, further fine-tuning of the control trajectory can be used to optimize features the system’s
+behavior (e.g., the positions of the steady states and their lifetimes in the presence of environmental noise).
+VII.
+GENERALIZATIONS: MULTIDIMENSIONAL BIFURCATIONS AND SUPPLEMENTAL
+SCALING BEHAVIOURS
+A.
+Stopping conditions in higher dimensions
+While our proof-of-concept experiment is limited to a hinge with a single degree of freedom (the opening
+angle), our approach and gradient continuation algorithm are straightforward to apply to systems with
+multiple degrees of freedom, e.g. a microscopic robot with multiple panels connected by elastic hinges. The
+cuspoidal bifurcations discussed in this paper also naturally appear in higher-dimensional gradient systems.
+However, the analytic criteria to classify them is somewhat more complicated: we can not simply search
+for points where higher order derivatives of the potential vanish. In this section we will discuss stopping
+criteria in higher dimensions, i.e. what quantities should we follow with the gradient continuation algorithm
+to search for bifurcations of increasing order?
+With two or more degrees of freedom, a saddle-node bifurcation occurs when a fixed-point (stable or
+unstable) collides with a saddle point, resulting in mutual annihilation. This occurs when an eigenvalue of
+the Hessian of the potential Aij = −∂θi∂θjV crosses 0 (here θi are the dynamical variables). For the purposes
+of applying gradient continuation starting from a fixed point, it is therefore convenient to use det A as the
+stopping criteria, since the determinant vanishes when an eigenvalue does.
+Near a saddle-node bifurcation, the state space can be decomposed (by the Center Manifold Theorem)
+into (i) the invariant center manifold emanating from the fixed point along the direction of the critical
+eigenvector (with eigenvalue 0) and (ii) a stable/unstable manifold on which the flows exponentially grow or
+decay (for the purposes of machine design we generally want only stable directions). Due to the vanishing
+eigenvalue, the dynamics on the center manifold are nonlinear at lowest order.
+These dynamics can be
+determined perturbatively by expanding the gradient of the potential, projecting onto the center manifold
+and enforcing the invariance of the center manifold [S1]. Higher-order bifurcations occur when the center
+manifold expansion coefficients vanish. For example, vanishing quadratic term indicates a cusp bifurcation,
+vanishing cubic term indicates a swallowtail, and so on. Thus these coefficients replace the higher-order
+derivatives of the potential as the stopping criteria in the gradient continuation algorithm. Below we give
+explicit expressions for these expansion coefficients.
+Suppose we have an n-dimensional system θ ∈ Rn that undergoes a saddle node bifurcation at θ = 0.
+Near this point, the dynamics can be expanded as follows,
+˙θ = A θ + F(θ),
+(S6)
+where A is the Hessian of the potential (which has a zero eigenvalue) and F(θ) collects all quadratic and
+
+7
+higher-order terms in multilinear forms,
+F(θ) = 1
+2B(θ, θ) + 1
+6C(θ, θ, θ) + 1
+24D(θ, θ, θ, θ) + O(||θ||5)
+= 1
+2
+n
+�
+i,j=1
+∂2F(φ)
+∂φi∂φj
+����
+φ=0
+θiθj + 1
+6
+n
+�
+i,j,k=1
+∂3F(φ)
+∂φi∂φj∂φk
+����
+φ=0
+θiθjθk
++ 1
+24
+n
+�
+i,j,k,l=1
+∂4F(φ)
+∂φi∂φj∂φk∂φl
+����
+φ=0
+θiθjθkθl + O(||θ||5).
+(S7)
+Let ψ and ϕ be the right and left eigenvectors corresponding to the zero eigenvalue: Aψ = 0 and AT ϕ = 0.
+The projection of θ onto the center manifold ϑ = ϕ · θ has dynamics
+˙ϑ = a2ϑ2 + a3ϑ3 + O(ϑ4).
+(S8)
+Following Kuznetsov, we derive the coefficients up to fourth order (third order is given in Ref. [S1]),
+a2 = 1
+2ϕ · B(ψ, ψ)
+a3 = 1
+6ϕ · C(ψ, ψ, ψ) + 1
+2ϕ · B(ψ, b2)
+a4 = 1
+24ϕ · D(ψ, ψ, ψ, ψ) + 1
+4ϕ · C(ψ, ψ, b2) + 1
+8ϕ · B(b2, b2) + 1
+6ϕ · B(ψ, b3),
+(S9)
+where
+b2 = A−1
+su
+�
+ψ[ϕ · B(ψ, ψ)] − B(ψ, ψ)
+�
+b3 = A−1
+su
+�
+ψ[ϕ · C(ψ, ψ, ψ) + 3ϕ · B(ψ, b2)] + 3b2[ϕ · B(q, q)] − C(ψ, ψ, ψ) − 3B(ψ, b2)
+�
+(S10)
+and A−1
+su is the inverse of A restricted to the stable/unstable subspace (which doesn’t have zero eigenvalues).
+As mentioned above, vanishing a2 indicates a cusp, if a3 also vanishes we have a swallowtail, and if all three
+coefficients are zero we have a butterfly. The vectors b2 and b3 describe the curvature of the center manifold
+in the full θ space, θ = qϑ + b2ϑ2/2 + b3ϑ3/6. While these bifurcations are one dimensional (they occur on
+the one-dimensional invariant center manifold), the curvature of the center manifold as we move further from
+the bifurcation point could allow snapping between states with reasonable separation in multiple dimensions.
+In principle, this would enable machines to carry out work cycles near a butterfly bifurcation.
+B.
+Scaling for the Thom’s seven: hyperbolic and elliptic umbilics
+Beyond the quasi-one-dimensional bifurcations there are also cuspoidal bifurcations that are genuinely mul-
+tidimensional. In two dimensions, for example, we have elliptic umbilic, hyperbolic umbilic, and parabolic
+umbilic catastrophes (these together with the four one-dimensional bifurcations saddle-node, cusp, swallow-
+tail, and butterfly make up the Thom seven). Like the cusp and butterfly bifurcations, the unfolding of the
+normal form predicts and intricate saddle-surface structure describing how fixed-points and saddle-points
+come together and collide in the vicinity of the bifurcation point. These higher-dimensional bifurcations also
+obey advantageous scaling laws, relating the changes in state to the variation of control parameters. For
+example, the normal form potentials for the elliptic and hyperbolic umbilics are
+Velliptic = θ3
+1
+3 − θ1θ2
+2 + a(θ2
+1 + θ2
+2) + bθ1 + cθ2
+Vhyperbolic = θ3
+1 + θ3
+2 + aθ1θ2 + bθ1 + cθ2
+(S11)
+from which the follow scaling can be derived [S4],
+δθ1, δθ2 ∼ a
+b, c ∼ a2.
+(S12)
+Increasing the dimension further leads to even more cuspoidal bifurcations; these have been enumerated
+by Arnold using an ADE classification [S7]. While the search criteria for such bifurcations is increasingly
+complicated, they provide a rich design space for multi-component machines.
+
+8
+C.
+Reynolds number scaling
+The magnetic decorations in our experiments are arranged in each panel about a square lattice with
+unit separation of 2.5cm. To explore over-damped, gradient dynamics, that are ubiquitous in microscopic
+mechanisms, the rotating panel is attached to a paddle moving through a glycerol bath. The results of our
+experiments then hold also for smaller systems in fluid with comparable kinematic viscosity. If the system
+is smaller by a factor Ω ≪ 1, the time ∆t it takes our macroscopic over-damped system, of typical size L,
+to traverse an angular expanse ∆θ is equal to the time it takes a microscopic system, of size ΩL to traverse
+the same angular expanse in the same liquid. This comes about because both the viscous drag force and
+the magnetic force between dipoles of magnetization M1 and M2, FDrag ∼ L2 ˙γ, Fdipole ∼ M1M2/R4, are
+quadratic in the typical system sizes. For over-damped dynamics this results in a length-scale independent
+strain-rate, ˙γ. The system is over-damped if its Reynolds number Re = L2 ˙γ/ν, is smaller then 1, where ν
+is the fluid’s kinematic viscosity. The Reynold’s number of a miniaturized system is therefore smaller by a
+factor of Ω2. Reducing the system’s size can compensate for changes in the system’s composition, such as
+embedding it in water rather than glycerol, or the growth of magnetic dipole strength density as the system
+size decreases.
+[S1] Y. A. Kuznetsov, “Topological equivalence, bifurcations, and structural stability of dynamical systems,” in
+Elements of Applied Bifurcation Theory (Springer New York, New York, NY, 2004)
+.
+[S2] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
+(Springer New York, New York, NY, 1983)
+.
+[S3] J. W. Bruce and P. J. Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory, 2nd
+ed. (Cambridge University Press, 1992)
+.
+[S4] M. V. Berry, Journal of Physics A: Mathematical and General 10, 2061 (1977)
+.
+[S5] DLMF, “NIST Digital Library of Mathematical Functions,” http://dlmf.nist.gov/, Release 1.1.4 of 2022-01-15
+(2022), f. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R.
+Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
+[S6] W.
+H.
+Press,
+S.
+A.
+Teukolsky,
+W.
+T.
+Vetterling,
+and
+B.
+P.
+Flannery,
+Numerical recipes 3rd edition: The art of scientific computing (Cambridge university press, 2007)
+.
+[S7] V. I. Arnol’d, “Bifurcations of equilibria,” in Dynamical Systems V, edited by V. I. Arnol’d (Springer Berlin
+Heidelberg, Berlin, Heidelberg, 1994) pp. 10–38
+.
+[S8] M. Smith, G. Yanega, and A. Ruina, Journal of theoretical biology 282, 41 (2011)
+.
+
+9
+FIG. S1. Single snap-through mechanism (a.) As we vary the control parameters along a loop around the cusp
+point as shown, we expect to see a single snap-through buckling behavior (point v to point vi) for each cycle, akin
+to how hummingbirds use their beak to capture prey [S8]. (b.) The predicted potential energy curves for points
+labeled from i to vi are presented. The saddle-node bifurcation occurs between v and vi as indicated by the arrow
+in v. (c.) We experimentally observe the predicted snap-through behavior. Due to experimental errors, the location
+of the cusp point is shifted, but we see excellent agreement between the theory and measurements after shifting the
+coordinates to align the theoretical and experimental cusp points.
+FIG. S2. Snap Through transitions near a cusp. These plots show the equilibrium angle recorded in experiments
+following a snap-through transition. The corresponding (x, y) denote the values of the control parameters at which
+the snap-through occurred. (a.) Highlights the the data points used to fit the cusp scaling. We exclude data far from
+the cusp, where higher order terms in the normal form are non-negligible, and close to the cusp, where measurement
+and fabrication error are comparable to the distance from the cusp. (b.) Highlights the data corresponding to the
+upper and lower saddle-node curves.
+
+a)
+b)
+c)
+ty
+i
+vi
+iii
++
+anel
+-y
+L 2.9
+-y
+ [rad] - System State
+Snap-through
+i:  = 2.606
+vi: 0 = 2.653
+c+↑
+Snap-through ↑
++c
+Snap-through
+v
+ii
+S
+2.5
+ii: θ = 2.841 
+v: θ = 2.830
+LSI
+h+ ↑
+-y 
+L!
+S
+h+↑
+yT
+Cusp Point
+iii
+iv
+-c
+-0.03
+0.3
+m
+0.15
+-0.06
+0
+dy [cm] - Control Parameter
+-0.15
+-0.09
+iii: 0 = 2.854 
+iv: 0 = 2.864a)
+b)
+·ScalingDataPoints
+· Upper Saddle Node Curve
+·UnusedDataPoints
+O
+·
+LowerSaddleNodeCurve
+·EstimatedCuspPoint
+·EstimatedCuspPoint
+2.85
+2.85
+[rad] - System State
+2.8
+[rad] - System State
+2.8
+2.75
+2.75
+2.7
+2.7.
+2.65
+2.65
+0
+2.6
+0.15
+2.6
+0.15
+0.2
+Q
+0.2
+2.55
+0.25
+2.55
+0.25
+0.8
+0.3
+0.8
+0.3
+0.35
+0.75
+,0.75
+0.35
+x [cm]
+x [cm]
+Control
+y [cm] -
+Control
+Parameter
+Parameter10
+FIG. S3. Butterfly panels: In the butterfly experiments, Panel P1’s x, y and z positions are measured as displacements
+from their value when the panels are 180◦ open, the magnets closest to the hinge axis are removed from it by 2.5cm
+on both panels, the panels are aligned vertically and the back of the cylindrical magnets on Panel P1 are aligned with
+the center of the magnets on Panel P2. This small change in magnet alignment is found to reduce the discrepancy
+between experiment and prediction.
+FIG. S4.
+2-state Cycle Near Butterfly Bifurcation Point (a.) Theory The saddle node surfaces of a magneto-
+elastic system with three active control parameters, x,y and z are plotted, their color denotes the angle θ at which
+the snap occurs. The system’s magnetic pattern is designed using the gradient continuation algorithm such that it
+operates near a butterfly bifurcation where multiple saddle node surfaces coalesce, enabling multiple snap-through
+transitions at the surfaces. A trajectory (colored tube with white arrows) is chosen such that the system snaps back
+and forth between two states with Large (L) and Small (S) angles. This trajectory is identical to that for the 3-state
+cycle in Fig. 4 in the main text, but the path is traversed in the opposite direction. The system’s predicted state is
+denoted by the tube’s color. At intersections of the trajectory with a surface where their colors match the system is
+predicted to snap to a new state. (b.) Experimental demonstration: The colored dots mark the experimental value
+of the system’s state as it follows the designed trajectory. We observe two distinct transitions as predicted.
+
+Hinge Axis
+11
+2..
+X
+dx2
+Z
+7.5
+5
+2.5
+0
+[cm]
+x [cm][cm] - Control
+Parameter
+-0.2 -
+-0.4
+-0.4 
+Stat
+-0.2
+7
+-0.6
+-0.4 
+tem
+-0.8 -
+-0.8
+-0.6 -
+-0.8 -
+0
+-2.0】
+rad
+-1.5
+.1
+-1.0 >
+2
+-0.5
+y
+-0.5
+S
+-1.0
+-0.5
+-1.0
+3
+y [cm] - C
+Parameter
+1.5
+x
+-2.0
+1.5
+x

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+How to Allocate your Label Budget? Choosing between
+Active Learning and Learning to Reject in Anomaly Detection
+Lorenzo Perini,1 Daniele Giannuzzi, Jesse Davis 1
+1 KU Leuven, Department of Computer Science, DTAI & Leuven.AI, B-3000 Leuven, Belgium
+lorenzo.perini@kuleuven.be, danielegiannuzzi1998@gmail.com, jesse.davis@kuleuven.be
+Abstract
+Anomaly detection attempts at finding examples that deviate
+from the expected behaviour. Usually, anomaly detection is
+tackled from an unsupervised perspective because anomalous
+labels are rare and difficult to acquire. However, the lack of
+labels makes the anomaly detector have high uncertainty in
+some regions, which usually results in poor predictive perfor-
+mance or low user trust in the predictions. One can reduce
+such uncertainty by collecting specific labels using Active
+Learning (AL), which targets examples close to the detec-
+tor’s decision boundary. Alternatively, one can increase the
+user trust by allowing the detector to abstain from making
+highly uncertain predictions, which is called Learning to Re-
+ject (LR). One way to do this is by thresholding the detector’s
+uncertainty based on where its performance is low, which re-
+quires labels to be evaluated. Although both AL and LR need
+labels, they work with different types of labels: AL seeks
+strategic labels, which are evidently biased, while LR requires
+i.i.d. labels to evaluate the detector’s performance and set the
+rejection threshold. Because one usually has a unique label
+budget, deciding how to optimally allocate it is challenging.
+In this paper, we propose a mixed strategy that, given a budget
+of labels, decides in multiple rounds whether to use the bud-
+get to collect AL labels or LR labels. The strategy is based
+on a reward function that measures the expected gain when
+allocating the budget to either side. We evaluate our strategy
+on 18 benchmark datasets and compare it to some baselines.
+Introduction
+Anomaly detection is the task of automatically detect-
+ing examples that do not follow expected patterns (Chan-
+dola, Banerjee, and Kumar 2009). These examples, named
+anomalies, are usually indicative of critical events such as
+water leaks in stores (Perini, Vercruyssen, and Davis 2022),
+breakdowns in gas turbines (Zhao, Wen, and Li 2016), or
+failures in the petroleum extraction (Mart´ı et al. 2015). Such
+critical events usually come along with elevated (mainte-
+nance) costs or with substantial natural damages (e.g., dis-
+persion of petroleum or gas). Thus, detecting anomalies in
+time is a relevant task that limits such resource waste.
+Collecting labels, especially for anomalies, is often a
+hard task because anomalies are costly events (e.g., ma-
+chine failures cannot be voluntarily induced), or simply
+time-consuming (e.g., you may need to label 100s of exam-
+ples before getting an anomaly). Thus, anomaly detection is
+often tackled from an unsupervised perspective. However,
+the lack of labels usually forces the unsupervised detector
+to have high uncertainty on specific regions of the example
+space (Perini, Vercruyssen, and Davis 2020). High uncer-
+tainty is undesirable because it is often associated with poor
+predictive performance or reduced trust in the predictions.
+This uncertainty can be tackled in two complementary
+ways. On the one hand, one can try to learn a more accu-
+rate detector by acquiring a limited number of labels using
+Active Learning (AL) (Abe, Zadrozny, and Langford 2006).
+On the other hand, it is possible to increase the user trust
+in the detector’s outputs by allowing the detector to abstain
+from making a prediction when it is highly uncertain, which
+is called Learning to Reject (LR) (Hendrickx et al. 2021;
+De Stefano, Sansone, and Vento 2000). One way to do this
+is to set a rejection threshold on the detector’s uncertainty
+based on where its performance is poor (Cortes, DeSalvo,
+and Mohri 2016). However, evaluating the detector perfor-
+mance requires labels.
+Both of these approaches rely on labeled data. However,
+the types of labels needed for each approach are quite differ-
+ent. Many AL strategies rely on biased sampling strategies
+such as explicitly targeting acquiring labels, for example, for
+which the detector is highly uncertain (i.e., near the detec-
+tor’s current decision boundary) as these are known to yield
+better performance (Pimentel et al. 2020; Culotta and Mc-
+Callum 2005). Alas, using such labels to evaluate the detec-
+tor’s performance, as required when setting the threshold in
+LR, will yield a biased performance estimate and hence a
+sub-optimal threshold (Marrocco, Molinara, and Tortorella
+2007). Thus, if a user has a fixed budget for acquiring la-
+bels there is a tension between collecting (a) strategic labels
+that can be used to train a better detector, or (b) i.i.d. labels
+that can be used to evaluate performance and set a proper
+rejection threshold. Therefore, a data scientist is confronted
+with the challenging question of how they should optimally
+allocate their label budget between these two purposes.
+In this paper, we assume that the label budget can be
+split and allocated in multiple rounds. We introduce BAL-
+LAD (Budget allocation for Active Learning and Learning to
+reject in Anomaly Detection) a novel adaptive strategy that,
+in each allocation round, (1) measures the potential reward
+obtained by assigning the budget to either AL or LR, and (2)
+chooses the highest reward option to collect the labels.
+arXiv:2301.02909v1  [cs.LG]  7 Jan 2023
+
+Preliminaries and Related Work
+Anomaly Detection.
+Let X be a d−dimensional random
+variable with unknown p(X). We are given a dataset D =
+{x1, . . . , xn} with n examples and d features is drawn i.i.d.
+from p(X). Let V = {xn+1, . . . , xm} ∼i.i.d p(X), m >
+n, be a validation set. Let Y be the label random variable,
+such that Y |X = x indicates the class label (1 if anomaly,
+0 if normal) for x ∈ Rd. An anomaly detection problem
+is the task of finding an anomaly score function h: Rd →
+R and a threshold t ∈ R such that Y = h(t)(X), where
+h(t)(x) = 1 if h(x) ≥ t, 0 otherwise. Usually, one sets t
+based on the contamination factor γ, i.e. the proportion of
+anomalies (Perini, Buerkner, and Klami 2022).
+Pool-based Active Learning (AL).
+The goal of pool-
+based AL strategies is to reduce the detector’s uncertainty
+by selecting the most informative training instances. The
+AL approaches can be classified into 3 categories (Monarch
+2021): uncertainty-based sampling strategies aim to select
+the unlabeled data samples with the highest uncertainty (Ha-
+cohen, Dekel, and Weinshall 2022), diversity strategies cap-
+ture the diversity among the training data (Abe, Zadrozny,
+and Langford 2006; Dagan and Engelson 1995), combined
+strategies integrate the advantages of uncertainty-based and
+diversity-based criteria (Ebert, Fritz, and Schiele 2012).
+Learning to Reject (LR).
+The goal of a detector’s re-
+ject option is to abstain from making a prediction when
+a detector is too uncertain about predicting a test exam-
+ple (Hendrickx et al. 2021; Cortes, DeSalvo, and Mohri
+2016). Our goal is to develop a detector-agnostic strategy
+that does ambiguity rejection, as novelty rejection would
+reject all anomalies. Thus, we use a dependent rejector ar-
+chitecture (Chow 1970). We indicate by C(x) the detector’s
+confidence for predicting x ∈ V , and with τ ∈ [0, 1] the re-
+jection threshold. If the confidence is below τ, the prediction
+is rejected ht(x) = ®. Note that for appropriate inference,
+we need to collect validation labels randomly (i.i.d.).
+A strategy to allocate the label budget
+This paper tackles the following problem:
+Given: initially unlabeled training set D and validation set
+V , the dataset’s contamination factor γ, an anomaly de-
+tector h, and a label budget B;
+Do: decide whether, in each allocation round k, to acquire
+labels for D (AL) or for V (LR).
+Both training the detector with more labels (AL) and learn-
+ing a threshold using larger validation data (LR) improve the
+detector’s performance. However, choosing the side to max-
+imize such improvement is challenging for multiple reasons.
+First, it requires measuring the reward of either side, i.e. the
+expected gain in terms of the detector’s improvement. Sec-
+ond, the rewards need to be on a similar scale such that nei-
+ther side is privileged during the process. Third, comparing
+a standard detector to one with the reject option is challeng-
+ing because the latter needs ad-hoc metrics to overcome the
+problem of predicting three classes (anomaly, normal, re-
+ject) (Nadeem, Zucker, and Hanczar 2009).
+In this paper, we introduce BALLAD, a strategy that mea-
+sures the reward of allocating the budget for AL, i.e. collect-
+ing strategic labels on the training set, and for LR, i.e., col-
+lecting random labels on the validation set. Let B = k·b ∈ N
+be our labelling budget. We perform k rounds and the la-
+bels of b examples are queried in each round. We initialize
+the problem by (1) training the detector with no labels and
+setting a default rejection threshold, and (2) collecting b ran-
+dom labels for V (LR) and for D (AL) for a total of 2b labels.
+This allows us to compute the initial rewards by measuring
+how the detector varies from (1) to (2): for LR, we mea-
+sure the variation after re-setting the validation threshold;
+for AL, we measure the variation after re-training the detec-
+tor with the new labels. Then, we start the allocation loop. In
+each round, we allocate the budget to the option (LR or AL)
+with the highest reward, and we update the reward using the
+new labels. We propose two alternative reward functions: 1)
+the entropy reward looks at the detector’s probabilities, ei-
+ther for prediction (AL), or for rejection (LR); 2) the cosine
+reward considers the predicted class labels, either anomaly
+yes/no (AL), or rejection yes/no (LR).
+Measuring the reward
+Because we do not know how beneficial the next label al-
+location would be for the detector, we look at the past and
+measure the effect of the last allocation round. Our challenge
+is to design a reward function that reflects the gain when
+querying the labels. We use the following methods to derive
+the reward for both AL and LR, by using as detector’s prob-
+abilities either the probability of predicting anomaly (AL),
+or the probability of rejecting the example (LR). Similarly
+to Vercruyssen et al. (2022), we consider two scenarios:
+Entropy.
+Adding more labels has the ability to decrease
+the overall uncertainty of the anomaly detector. Thus, we
+measure the variation of the detector’s probabilities as:
+Re(k) = Ex∼X [|H(hk(x)) − H(hk−1(x))|] ,
+(1)
+where H(h(x)) = −p log2 p is the entropy of the detector’s
+probabilities p, and the subscript indicates the query round
+(for k > 2). A large difference in entropy means a large
+detector variation, which indicates a large impact of the new
+labels and, in turn, a large reward Re.
+Cosine.
+More directly, one can measure the impact of the
+labels in terms of variation of class predictions. Given the
+detector’s probabilities, we threshold them at 0.5 and assign
+value 1 to higher probabilities and 0 to lower ones. Thus, we
+measure the cosine similarity between different outputs as
+Rc(k) = ED∼X
+�
+1 −
+hk(D) · hk−1(D)
+∥hk(D)∥ · ∥hk−1(D)∥
+�
+,
+(2)
+where h(D) is a vector containing the outputs (0 or 1) by the
+detector h, and ∥·∥ is the Euclidean norm. This metric is less
+sensitive to little variations in the detector and discriminates
+more in case the new labels change the predicted class.
+
+Deriving the detector’s probabilities
+Measuring the reward needs some probabilities, which are
+not easy to derive due to the partially supervised setting. For
+both prediction and rejection, we exploit the squashing func-
+tion: given a positive real score s ∈ R+ and a threshold
+λ ∈ R+, the squashing function
+Sλ : R+ → (0, 1),
+Sλ(s) = 1 − 2− s2
+λ2
+maps s to a probability > 0.5 if s > λ, and ≤ 0.5 other-
+wise. Roughly speaking, Sλ calibrates the probabilities by
+centering λ as the decision threshold.
+Detector’s posterior probabilities.
+Given the contamina-
+tion factor γ, a common approach to set the threshold t is
+by forcing the detector to have a training positive class rate
+equal to γ. Thus, one can center the probabilities to t by
+transforming the anomaly scores h(x) through the squash-
+ing function:
+P(ht(x) = 1) = St(h(x)) s. t. t = Qh(1 − γ).
+We set t as the 1−γth quantile of the score distribution such
+that only a proportion of γ scores have P(ht(x) = 1) ≥ 0.5.
+Rejection probabilities.
+Given a validation set with some
+labels, we (1) set a specific detector confidence C(x), and (2)
+set the rejection threshold τ ∈ [0, 1]. For the former, we use
+the detector’s posterior probabilities:
+C(x) = 2 ×
+��P(ht(x) = 1) − 0.5
+�� ∈ [0, 1].
+Thus, the closer P(ht(x) = 1) is to 0.5 (high uncertainty),
+the lower the detector confidence. For the latter, we opti-
+mize the threshold τ over the validation set (only the la-
+beled examples) by minimizing a cost function M(ht). Fi-
+nally, we compute the rejection probabilities by centering 1-
+confidence values to the rejection threshold, i.e. by applying
+the squashing function
+P(ht(x) = ®) = Sτ(1 − C(x)).
+The cost-based evaluation metric
+Given a detector with a reject option and a detector without
+it, we cannot compare their performance on the non-rejected
+examples, as they would have different test sets. Thus, we
+introduce a cost-based evaluation metric. Formally, given a
+rejection cost cr > 0, a false positive cost cfp > 0, and a
+false negative cost cfn > 0, the detector is evaluated as:
+Mh = cr · P(ht(X) = ®) + cfp · P(ht(X) = 1|Y = 0)
++ cfn · P(ht(X) = 0|Y = 1).
+Note that we assume cost null for the correct predictions,
+while every misprediction as well as the rejection gets pe-
+nalized. Because rejecting is assumed to be less costly than
+mispredicting, the rejection cost needs to satisfy the inequal-
+ity cr ≤ min{cfp × (1 − γ), cfn × γ}, otherwise one could
+predict either always normal and pay an expected cost of
+cfn × γ, or always anomaly and pay cfp × (1 − γ).
+Experiments
+We experimentally answer the following questions:
+Q1. Does BALLAD result in lower costs when compared to
+using only AL or LR?
+Q2. Which reward metric is better?
+Q3. Is the reward function on a similar scale for AL and LR?
+Q4. How does our strategy behave when varying cfp, cfn ?
+Experimental setup
+Methods.
+We compare BALLAD1 to two baselines: ALL-
+IN-AL allocates all the budget for active learning and sets
+the rejection threshold using the (biased) training labels; on
+the contrary, ALL-IN-LR allocates all the budget for learn-
+ing to reject and uses an unlabeled training set.
+Table 1: Properties of the 18 datasets used.
+Dataset
+# Examples
+# Features
+γ
+ALOI
+12384
+27
+0.0304
+Annthyroid
+7129
+21
+0.0749
+Arrhythmia
+271
+259
+0.0996
+Cardiotocography
+1734
+21
+0.0496
+Glass
+214
+7
+0.0421
+InternetAds
+1682
+1555
+0.0499
+KDDCup99
+48113
+40
+0.0042
+PageBlocks
+5473
+10
+0.1023
+PenDigits
+9868
+16
+0.0020
+Pima
+526
+8
+0.0494
+Shuttle
+1013
+9
+0.0128
+SpamBase
+2661
+57
+0.0499
+Stamps
+340
+9
+0.0912
+WBC
+223
+9
+0.0448
+WDBC
+367
+30
+0.0272
+WPBC
+160
+33
+0.0562
+Waveform
+3443
+21
+0.0290
+Wilt
+4655
+5
+0.0199
+Data.
+We carry out our study on 18 publicly available
+benchmark datasets, which are widely used in the litera-
+ture (Campos et al. 2016). See Table 1 for the properties.
+Setup.
+For each of the 18 benchmark datasets, we go as
+follows: (i) we split the dataset into training, validation and
+test sets using the proportions 40 − 40 − 20 (we have a large
+validation set to better measure the impact of rejection); (ii)
+we fit the anomaly detector on the unlabeled dataset and set
+the rejection threshold to the default value of 0.1; (ii) we
+allocate a budget b to LR and AL by randomly selecting
+the initial examples; (iii) we optimize the rejection thresh-
+old and measure the LR reward; (iv) we train the anomaly
+detector on the partially labeled training set and measure the
+AL reward; (v) we allocate the next round budget b to the
+option with the highest reward and repeat (iii) or (iv) until
+the whole budget B is used. During each of the steps, we
+measure the detector performance on the test set using our
+1Code available at https://github.com/Lorenzo-Perini/Ballad
+
+cost function. We set B to the 30% of the training set’s size,
+and b to 2% of it, such that we run 15 allocation rounds. We
+repeat (i - v) 10 times and report the average results. In total
+we run 18 × 15 × 10 = 2700 experiments.
+Costs and hyperparameters.
+We set cfp = cfn = 1 and
+cr = γ, following the cost inequality. SSDO (Vercruyssen
+et al. 2018) with its default parameters is used as the semi-
+supervised anomaly detector (Soenen et al. 2021). We use
+IFOREST (Liu, Ting, and Zhou 2008) as its unsupervised
+prior. We use Uncertainty Sampling as the active learning
+strategy (Zhan et al. 2021), and the entropy as default re-
+ward. For setting the rejection threshold, we use Bayesian
+Optimization (GP MINIMIZE implemented in SKOPT) with
+20 calls (Frazier 2018) and limit the rejection rate on the
+validation set to 50%.
+Experimental results
+Q1. Comparing BALLAD to the ALL-IN strategies. Fig-
+ure 1 shows the comparison between BALLAD with the en-
+tropy reward and the ALL-IN strategies on the 18 bench-
+mark datasets. On 8 datasets (Arrhythmia, Glass, Kdd-
+Cup99, Pima, SpamBase, Wbc, Wdbc, Wpbc), BALLAD re-
+sults in evident lower costs, although sometimes the differ-
+ence is small. On 5 datasets (Cardiotocography, InternetAds,
+PageBlocks, Stamps, Waveform) BALLAD performs simi-
+lar/worse than ALL-IN-AL. This happens because SSDO has
+an overall high performance and a contained uncertainty in
+the predictions. On the other hand, in 3 cases (Aloi, Annthy-
+roid, Wilt), allocating all the budget for LR has a lower cost.
+This is due to the detector being inaccurate and unable to
+learn from the training labels, which makes learning an opti-
+mal threshold more convenient. As support for this intuition,
+we analyze the plain test AUC of SSDO on the whole test
+set (no rejection) for each of the three previous cases. By ag-
+gregating over the rounds, SSDO obtains an average AUC
+equal to 0.86, 0.88, and 0.57 when the best strategy is, re-
+spectively, BALLAD, ALL-IN-AL, and ALL-IN-LR. Finally,
+BALLAD obtains an overall average cost of 0.043, which is
+≈ 20% lower than the baselines’ average cost (0.055 for
+ALL-IN-AL, 0.054 for ALL-IN-LR).
+Q2. Which reward function works better? We analyze
+both types of reward functions that we introduced in Eq. 1
+and Eq. 2. Table 2 shows the mean and standard deviation
+of the cost, divided by allocation round. Overall, using the
+cosine reward builds a strategy that produces on average low
+costs for little budget (≤ 10%), whereas, for a higher bud-
+get, the entropy reward obtains better average costs. This is
+due to the highly imbalanced choices made by the cosine re-
+ward: the strategy opts for AL in 93% of the cases, which
+usually improves a lot the detector’s performance with few
+labels but tends to produce little effect when enough labels
+are given. On the other hand, the entropy reward is more bal-
+anced and opts for AL in 63% of the cases. This allows the
+detector to keep decreasing the costs while learning during
+the allocation rounds and obtain more steady performance.
+Q3. Is the entropy reward balanced for AL and LR? Fig-
+ure 2 shows the distribution of the difference between AL
+and LR entropy rewards over all the 2700 experiments. Neg-
+Budget
+Entropy Re
+Cosine Rc
+2%
+0.0536 ± 0.0401
+0.0399 ± 0.0303
+4%
+0.0465 ± 0.0330
+0.0411 ± 0.0334
+6%
+0.0443 ± 0.0284
+0.0398 ± 0.0321
+8%
+0.0436 ± 0.0303
+0.0399 ± 0.0306
+10%
+0.0420 ± 0.0299
+0.0411 ± 0.0325
+12%
+0.0416 ± 0.0303
+0.0433 ± 0.0347
+14%
+0.0413 ± 0.0306
+0.0448 ± 0.0367
+16%
+0.0408 ± 0.0288
+0.0456 ± 0.0372
+18%
+0.0403 ± 0.0290
+0.0457 ± 0.0355
+20%
+0.0412 ± 0.0297
+0.0451 ± 0.0363
+22%
+0.0407 ± 0.0301
+0.0451 ± 0.0359
+24%
+0.0421 ± 0.0325
+0.0438 ± 0.0361
+26%
+0.0417 ± 0.0345
+0.0438 ± 0.0363
+28%
+0.0416 ± 0.0332
+0.0438 ± 0.0380
+30%
+0.0418 ± 0.0345
+0.0427 ± 0.0354
+Table 2: Average (± std) cost per test example over the
+datasets grouped by allocation round for each of the two re-
+ward functions. For low budgets, the cosine reward obtains
+lower costs, while not being competitive for high budgets.
+ative values indicate that the LR reward is higher than AL’s
+one, while the opposite holds for positive values. Overall,
+the median is close to 0, which means that there is no clearly
+predominant strategy. Because the left tail of the density is
+larger than the right one, we conclude that LR rewards have
+higher variability (std = 0.07 vs 0.03).
+Q4. The impact of varying cfp, and cfn. In this experi-
+ment, we penalize more false positives and false negatives
+by setting, one at a time, cfp and cfn to 10. We compare
+BALLAD to the two ALL-IN baselines. For cfp = 10, our
+strategy is still the best for low budgets (< 15%), reduc-
+ing the relative cost by between 5% and 25% with respect
+to the runner-up ALL-IN-LR. However, for higher budgets
+(> 15%), ALL-IN-LR becomes the best strategy as it re-
+duces BALLAD’s cost by around 20% and ALL-IN-AL’s cost
+by more than 40%. This happens because the anomaly de-
+tector produces too many false positives, which, if rejected,
+allow us to reduce the cost. For cfn = 10, BALLAD performs
+much better than the baselines, reducing their cost by around
+20% (vs ALL-IN-LR) and 24% (vs ALL-IN-AL).
+Conclusion
+We proposed BALLAD, a novel strategy to decide whether
+to allocate the budget for Active Learning (AL), i.e. labeling
+strategic training instances, or for Learning to Reject (LR),
+i.e. labeling a random validation set. Our key insight is that
+we can measure the expected reward when labeling either set
+and allocate the label in the next round to the option with the
+highest reward. We proposed two reward functions (entropy
+and cosine similarity based). Experimentally, we evaluated
+BALLAD on 18 datasets, and show that it performs better
+than simply allocating all the labels to either AL or LR.
+
+0.03
+0.06
+0.09
+0.12
+Aloi
+0.03
+0.06
+0.09
+0.12
+Ann
+0.03
+0.06
+0.09
+0.12
+Arr
+AL-LR
+All-in AL
+All-in LR
+0.03
+0.06
+0.09
+0.12
+Car
+0.03
+0.06
+0.09
+0.12
+Glass
+0.03
+0.06
+0.09
+0.12
+Int
+0.03
+0.06
+0.09
+0.12
+Cost per test example
+Kdd
+0.03
+0.06
+0.09
+0.12
+Page
+0.03
+0.06
+0.09
+0.12
+Pen
+0.03
+0.06
+0.09
+0.12
+Pima
+0.03
+0.06
+0.09
+0.12
+Shu
+0.03
+0.06
+0.09
+0.12
+Spam
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Stam
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wbc
+4
+8
+12
+16
+20
+24
+28
+Allocated budget (% of labeled examples)
+0.03
+0.06
+0.09
+0.12
+Wdbc
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wpbc
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wave
+4
+8
+12
+16
+20
+24
+28
+0.03
+0.06
+0.09
+0.12
+Wilt
+Figure 1: Comparison between BALLAD and the ALL-IN strategies on the 18 benchmarks. The x-axis reports the 15 rounds of
+2% labels each. The y-axis shows the average cost per test example. BALLAD obtains lower costs in the majority of cases.
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+0.3
+AL reward - LR reward
+Density
+Median
+Figure 2: Distribution of the difference between AL’s and
+LR’s entropy reward. The median close to 0 indicates the
+absence of a predominant strategy.
+Acknowledgements.
+This work was presented at the 1st
+AAAI Workshop on Uncertainty Reasoning and Quantifica-
+tion in Decision Making (UDM23).
+This research is supported by an FB Ph.D. fellowship by
+FWO-Vlaanderen (grant 1166222N) [LP], the Flemish Gov-
+ernment under the “Onderzoeksprogramma Artifici¨ele Intel-
+ligentie (AI) Vlaanderen” programme [JD], and KUL Re-
+search Fund iBOF/21/075 [JD].
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+ 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+1 
+ 
+I. INTRODUCTION 
+     Hair, made of keratin protein, pertains to beauty and 
+masculinity. Approximately 5 million hair follicles are 
+present throughout our body [1]. Scalp Hair maintains body 
+temperature and protects the brain from external heat. A 
+typical hair growth cycle runs for 2-7 years, according to 
+Patel et al. [2] and Wolff, Fischer, and Blume-Peytavi [3]. A 
+healthy human has 100,000 hairs on the scalp, and 50-100 
+hair loss per day is considered normal. Hair loss is not a 
+present-day issue. The hair-loss treatment was found in 
+ancient Ayurveda scriptures 6000 years ago [2]. However, 
+Hair and scalp-related issues are gaining more recognition 
+nowadays compared to earlier years due to certain factors, 
+such as environmental pollution, hormonal imbalance, 
+autoimmune disease, gut microbiota alteration, elevated 
+physical and mental stress levels in human lifestyle, seasonal 
+change, unhealthy diet, micronutrient deficiency, genetic 
+predisposition, and side-effects of drugs [2], [3]. According 
+to Peyravian et al., 80 million Americans have hair loss-
+related issues to some extent [4]. Although most hair loss 
+diseases are localized, some can spread to other locations. 
+Some 
+diseases 
+require 
+prescribed 
+drugs 
+and 
+hair 
+transplantation. Some diseases are caused by bacterial or 
+fungal infections and require antibiotic treatment. Often, 
+there are genetic and sexual predispositions in hair-scalp 
+diseases. 
+     Alopecia, folliculitis, and psoriasis are some common 
+causes of hair loss. There is a difference between regular hair 
+fall and alopecia; the latter develops coin-sized bald patches 
+all over the scalp area. Alopecia or patchy hair loss can be of 
+different types. Androgenetic alopecia or male-pattern 
+baldness (MPB) is the most common form of alopecia where 
+the hairline starts to recede, following a pattern where the 
+frontal and temple area are most affected. 70% of men and 
+40% of women get this type of hair loss and thinning issue 
+[3]. According to Liu et al., MPB is an X-linked polygenic 
+disease, and males are more genetically prone to develop 
+baldness at a mature age [5]. Topical minoxidil solution 
+thickens the hair by 50% [3]. On the other hand, Alopecia 
+areata (AA) is an autoimmune disease affecting individuals 
+irrespective of age and sex. Primarily affecting the scalp area, 
+AA can also spread in the beard, eyelashes, and eyebrows. In 
+this case, the body’s immune cells cannot recognize hair 
+follicles as ‘self.’ Instead, they consider these follicles as 
+‘foreign,’ which ultimately causes the hair follicles to be 
+Hair and Scalp Disease Detection using Machine 
+Learning and Image Processing 
+ 
+Mrinmoy Roy, Anica Tasnim Protity 
+ABSTRACT 
+ 
+Almost 80 million Americans suffer from hair loss due to aging, 
+stress, medication, or genetic makeup. Hair and scalp-related 
+diseases often go unnoticed in the beginning. Sometimes, a patient 
+cannot differentiate between hair loss and regular hair fall. 
+Diagnosing hair-related diseases is time-consuming as it requires 
+professional dermatologists to perform visual and medical tests. 
+Because of that, the overall diagnosis gets delayed, which worsens 
+the severity of the illness. Due to the image-processing ability, neural 
+network-based applications are used in various sectors, especially 
+healthcare and health informatics, to predict deadly diseases like 
+cancers and tumors. These applications assist clinicians and patients 
+and provide an initial insight into early-stage symptoms. In this 
+study, we used a deep learning approach that successfully predicts 
+three main types of hair loss and scalp-related diseases: alopecia, 
+psoriasis, and folliculitis. However, limited study in this area, 
+unavailability of a proper dataset, and degree of variety among the 
+images scattered over the internet made the task challenging. 150 
+images were obtained from various sources and then preprocessed 
+by denoising, image equalization, enhancement, and data balancing, 
+thereby minimizing the error rate. After feeding the processed data 
+into the 2D convolutional neural network (CNN) model, we obtained 
+overall training accuracy of 96.2%, with a validation accuracy of 
+91.1%. The precision and recall score of alopecia, psoriasis, and 
+folliculitis are 0.895, 0.846, and 1.0, respectively. We also created a 
+dataset of the scalp images for future prospective researchers. 
+Keywords:  Deep Learning, Health Informatics, Machine Learning, 
+Scalp/ Hair Diseases. 
+ 
+ 
+ Published Online:  
+ISSN: 2736-5492 
+DOI 10.24018/ejcompute.YEAR.Vol.Issue.ID 
+ 
+Mrinmoy Roy  
+Department 
+of 
+Computer 
+Science, 
+Northern Illinois University, USA.  
+(e-mail: mrinmoy.cs10 gmail.com)  
+Anica Tasnim Protity  
+Department of Biological Sciences, 
+Northern Illinois University, USA. 
+(e-mail: 
+protity.microbiology@gmail.com) 
+ 
+*Corresponding Author 
+@ 
+
+ 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+2 
+ 
+targeted and destroyed by the immune cells. It is an example 
+of a hereditary disease. The study from Benigno et al. 
+reported that, in the US alone, 700,000 individuals suffer 
+from AA [6]. This disease, if diagnosed early, might resolve 
+spontaneously. In severe cases, topical corticosteroid or 
+immune therapy is used [3]. 
+    Sometimes, the hair follicles might get inflamed because 
+of the action of bacterial accumulation. This follicle 
+inflammation 
+is 
+called 
+folliculitis 
+decalvans. 
+The 
+bacterium Staphylococcus aureus damages the follicle and 
+prevents hair growth. Staphylococcus aureus uses hair tufts 
+to 
+enter 
+underneath 
+the 
+follicle, 
+causing 
+chronic 
+inflammation, redness, swelling, scarring, itching, and hair 
+loss. Antibiotic treatment combined with surgical removal of 
+hair tufts and corticosteroids for reducing inflammation are 
+the 
+prescribed 
+treatment 
+for 
+Folliculitis 
+decalvans 
+[3]. Psoriasis is another form of common scalp skin disease. 
+According to [7], 54% of 5600 psoriasis patients had scalp 
+psoriasis. Severe psoriasis may cause significant itching, 
+scaling, and redness in the scalp. The application of topical 
+shampoo and corticosteroids are the treatment options by 
+Chan et al. [8].    
+    Some scalp infections may be treatable if diagnosed 
+early. Some but not all diseases may go on their own. Only 
+an expert physician can detect the illness by visual 
+observation. In some cases, early disease detection is 
+beneficial for dermatologists to initiate the treatment. An 
+early scalp inspection includes a dermatoscopic examination 
+of the scalp for inflammation, itching, localized lesion, 
+dandruff, follicular flakes, louse eggs (nits), and a scalp 
+biopsy. Besides visual observation, the patient can undergo 
+blood and hormone tests to detect the exact disease. 
+Unfortunately, most hair and scalp diseases are diagnosed in 
+advanced stages, which complicate the treatment options. All 
+these factors lengthen the diagnosis and treatment process. 
+Therefore, researchers are putting more effort into developing 
+different mechanisms for the early detection of hair and scalp 
+diseases. 
+    In the 21st century, with all the advancements in 
+computational technology, extensive application of machine 
+learning has made our daily lives simple, comfortable, and 
+secure. The increasing popularity of machine learning and its 
+nature to extract patterns from data are directing researchers 
+to incorporate several machine learning algorithms into 
+health informatics. Especially during the Covid-19 pandemic 
+era, different applications like restraining people from covid-
+19 spread [9], SARS-CoV-2 screening and treatment [10], 
+lock-down control in case of high dimensional input [11] 
+came into play, which made machine learning and healthcare 
+systems inseparable. Overall, adapting, integrating, and 
+developing deep learning-based applications on patients’ 
+information, medical reports, and audio-video feedback make 
+the diagnosis process faster. Nowadays, patients can get at 
+least the initial idea of disease detection by themselves using 
+easily accessible smart devices. All these applications clear 
+their confusion and help them make health-related decisions 
+independently.  
+    The high computational capability of neural networks is, 
+therefore, a breakthrough in healthcare and medical 
+diagnostic organizations. Convolutional neural networks 
+(CNN) have brought revolutionary success in detecting 
+deadly diseases. To date, neural networks are assisting 
+healthcare professionals in the early detection of different 
+types of tumors and cancers, such as skin cancer (melanoma) 
+[12], stomach cancer (adenocarcinoma) [13], and brain 
+tumors (glioblastoma) [14]. Neural networks are applicable 
+in detecting life-threatening dengue fever [15] and covid-19 
+[16] as well. In one study, CNN was used to extract complex 
+temporal dynamic features from heart rate variability (HRV) 
+signals, developing an algorithm that facilitated the early 
+detection of diabetics [17]. Using the image processing ability 
+of the neural networks, we can extract features from hair, skin 
+and scalp images to classify and categorize numerous hair and 
+scalp-related diseases. In this work, due to the importance of 
+early-stage hair disease detection, we applied convolutional 
+neural networks to 3 types of hair diseases and developed a 
+model to detect them successfully. 
+ 
+II. CHALLENGES AND CONTRIBUTIONS 
+    A classic application of computer vision is to detect 
+disease using digital images. Researchers can exploit a pool 
+of digital images obtained from one or more datasets, 
+preprocess the images, feed the images into the neural 
+network, and develop a model to detect the disease. 
+Unfortunately, minimal research has been performed on the 
+machine-learning approach for scalp disease detection. There 
+are several unique challenges behind this. First and foremost, 
+hair diseases are not localized and can spread to different 
+regions of the scalp, beard, eyebrows, eyelashes, and pubic 
+area. Second, every image needs different types of 
+preprocessing before feeding to neural networks. Different 
+scalp skin tones, hair colors, and types around the detection 
+zones make the imaging process more complicated. Third, no 
+proper dataset for scalp diseases is available over the internet, 
+and images taken from the internet differ in size and 
+resolution. Moreover, one must be conscious of minimalizing 
+and correcting the error in disease detection; otherwise, the 
+high false-positive and false-negative rates result in 
+misdiagnosis of the disease and worsening hair loss. 
+    To overcome the challenges, we developed a model 
+which can successfully classify the alopecia, folliculitis, and 
+psoriasis diseases with a minimal false-positive and false-
+negative rate. Though it is challenging to collect images for 
+the diseases from the internet, and the images are varied in 
+color, 
+shape, 
+and 
+resolution, 
+we 
+applied 
+various 
+preprocessing, such as denoising, resizing, enhancement and 
+created a dataset that might help in further scalp diseases 
+research. 
+ 
+III. RELATED WORKS 
+    Disease detection using machine learning approaches is 
+gaining popularity in health informatics. Many skin and 
+scalp-related diseases can be detected using images of 
+infected regions within a few seconds. In one study by 
+Choudhary et al. [18], a framework is developed to 
+differentiate alopecia areata from healthy hair. They obtained 
+200 healthy hair images from the figaro1K dataset and 68 
+alopecia areata hair images from DermNet. After a series of 
+enhancement and segmentation, three key features were 
+
+ 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+3 
+ 
+extracted from the images: texture, shape, and color. The 
+researchers divided the dataset into 70%-30% train-test-split 
+and applied a support vector machine (SNM) and k-nearest 
+neighbor (KNN) for the classification task. Overall, they 
+achieved 91.4% and 88.9% accuracy using SVM and KNN, 
+respectively, with a 10-fold cross-validation approach. 
+However, using other machine learning algorithms might 
+increase in the accuracy rate, which should have been 
+discussed. Besides, the application of Histogram Equalization 
+(HE) for image enhancement complicated the process of 
+getting accurate texture features from distorted images, as HE 
+itself adds noise to the output image, distorting the signals. 
+Moreover, this study only shed light on alopecia areata 
+disease, ignoring the inter-class differences between other 
+similar type diseases, which increased the likelihood of 
+inaccurate prediction of other diseases as alopecia areata, 
+thereby making this framework less reliable. 
+    Another study [19] proposed a model for early alopecia 
+detection. They used 100 samples for this research, with 80% 
+as training data and the other 20% as testing data. They 
+looked for four attributes, length of the hair, nail brittleness, 
+amount of damage made to the hair, and hair follicle. Two-
+layer feed-forward network with a back propagation 
+technique was used for detection purposes. The proposed 
+model system consisting of 4 input neurons, 10 hidden 
+neurons, and a linear output neuron, achieved 91% training 
+accuracy with 86.7% validation accuracy. It showed the best 
+performance at epoch 4 with a 0.059687 gradient. However, 
+the study has some pitfalls, too, as they did not mention their 
+data source or differentiate data classes with their respective 
+sample sizes. Also, no image pre-processing was performed 
+on the collected images. Although there is a possibility of 
+overfitting without a proper data balancing technique, this 
+report did not discuss the data balancing between the two 
+classes. Furthermore, they did not calculate the model’s false-
+positive and false-negative rates, which is crucial for a model 
+specially developed for the healthcare system. 
+    Related work [20] was performed on skin disease 
+detection, where machine learning was used to analyze the 
+digital image of the affected skin area for identifying eczema, 
+melanoma, and psoriasis. Their dataset consists of 80 images 
+from different websites specific to skin diseases. By using a 
+convolutional neural network for feature extraction and 
+applying multiclass SVM on those features, they achieved 
+100% accuracy in disease classification. However, they did 
+not explore other essential model performance matrices and 
+overfitting issues. In another skin disease detection-based 
+article [21], the authors proposed a scheme to classify skin 
+lesions into five categories: healthy, acne, eczema, benign, 
+and malignant melanoma, using a pre-trained CNN model, 
+AlexNET for feature extraction and error correcting output 
+codes support vector machine for classification. The dataset 
+consists of 9144 images from different sources and achieved 
+84.21% accuracy using a 10-fold cross-validation technique. 
+    Overall, we observed very few works on hair diseases. 
+The recent related works lack at least one of the following 
+categories – discussion over false positive and false negative 
+rates, ignoring inter-class differences, model reliability, and 
+overfitting problem. In this work, we have attempted to fill 
+these gaps by leveraging a convolutional neural network 
+algorithm on hair disease images while maintaining high 
+accuracy with good precision and recall scores. 
+ 
+IV. DATA DESCRIPTION & DEVICE 
+A. Data Collection 
+    The most challenging part of using visual images for 
+disease prediction and disease classification is data 
+collection. Often, one can get fewer appropriate images for 
+a specific illness found. Moreover, the pictures are scattered 
+over the internet. In this study, the authors extracted the 
+images from different websites, such as DermQuest, 
+DermNet, MedicineNet, DermnetNZ, and various medical 
+professionals. 
+ 
+TABLE I: IMAGES PER DISEASE 
+Disease 
+Quantity 
+Alopecia 
+65 
+Psoriasis 
+45 
+Folliculitis 
+40 
+ 
+ 
+Fig. 1. Image subset of each disease category. 
+ 
+    The image quantity is different for each category. We 
+found more alopecia-related images than other diseases 
+because alopecia is more frequent and severe among the 
+human population. The number of samples in each type of 
+disease is listed in Table I. Randomly selected images in 
+each category are graphically represented in Fig 1. Our 
+dataset is made publicly available on Github [22].  
+B. Device 
+    The research was conducted on Dell Latitude 5520 
+laptop device having 11th generation Intel Core i5 (8 MB 
+cache, 4 cores, 8 threads, up to 4.40 GHz Turbo) and 
+running on Windows 10 Pro operating system. The device 
+has 16 GB, 1 x 16 GB, DDR4, 3200 MHz random access 
+memory (RAM), and 256 GB, M.2 PCIe NVMe, SSD, 
+Class 35 (NVRAM). For the classification of images, we 
+utilized the integrated Intel Iris XE graphics capable with a 
+thunderbolt for I5-1145G7 vPro processor. For the data 
+
+Alopecia
+Psoriasis
+folliculitis 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+4 
+ 
+collection, we used iPhone-13 Pro Max having Hexa-core 
+(2x3.23 GHz Avalanche + 4x1.82 GHz Blizzard) CPU and 
+Apple GPU (5-core graphics). We used a mobile device 
+with 128GB 6GB RAM, and a 12 MP triple main camera 
+for the image collection. 
+V. PROPOSED MODEL 
+    In this section, we introduce the system workflow of our 
+model and explain the functions of each module in details. As 
+shown in Fig. 2, first, the captured image is sent to 
+preprocessing steps which are divided into three parts: image 
+equalization, image enhancement, and data balancing. 
+ 
+ 
+Fig. 2. System workflow of hair disease detection model. 
+ 
+Among these three, the first two parts are mainly for 
+increasing image quality, and the last part is for model 
+versatility. After the preprocessing steps, the image is passed 
+to the Neural Network model for the classification task. We 
+used a convolutional neural network that classifies an image 
+successfully into three different classes: alopecia, folliculitis, 
+and psoriasis. 
+A. Denoising 
+ 
+ 
+Fig. 3. Left original image & right non-local means denoised image. 
+ 
+    Noise is the degradation of image signals caused by 
+external sources [23]. Noise introduces random variations of 
+brightness or color information in the captured images. Most 
+of the time, images on the internet have some noise associated 
+with them. As we have collected most of the data samples 
+from different dermatology websites, the noise in our dataset 
+is not homogeneously distributed, which made it more 
+complex. Therefore, we applied additional filters for 
+denoising the collected images. We started with the gaussian 
+filter for a better image classification process. However, after 
+using the gaussian filter, the images become completely 
+blurred, which leads to the loss of important information and 
+damage to the edges. We then applied the median filter, 
+which worked better than the gaussian filter with kernel_size 
+= 3. Though we achieved better accuracy using the bilateral 
+filter, we got the best results while applying the non-local 
+means filter with patch_size = 3 and patch_distance = 5. This 
+non-local means filter preserved all the edges and reduced the 
+noise better than the other filters for our application which is 
+shown in Fig. 3. 
+B. Image Equalization 
+    Often the captured image doesn’t reflect the natural view 
+and needs contrast enhancement to meet the level of realistic 
+view [24]. Especially images with high color depth and after 
+denoising effects need normalization for a better realistic 
+view [25]. First, we applied histogram equalization (HE). 
+However, the HE increases the contrast of the background 
+when used in images with low color depth, and information 
+is lost as the histogram is not confined to the local region. To 
+overcome the problem, we applied CLAHE (Contrast 
+Limited Adaptive Histogram Equalization) by dividing an 
+image into equal size non-overlapping areas and computing a 
+histogram for each region. After clipping the histogram, we 
+distributed the clipped value over the histogram equalization, 
+which gives us control of the over-amplification of the 
+contrast and generates the resultant image shown in Fig. 4. 
+ 
+ 
+Fig. 4. Image equalization using CLAHE. 
+ 
+C. Data Balancing 
+    The overall performance of a machine learning model 
+depends on the balanced dataset because, without it, minority 
+class detection becomes difficult. Balancing a dataset reduces 
+the risk of skewing towards the majority. Imbalanced data 
+might achieve high accuracy, but the results are biased toward 
+the majority class. As alopecia is a common disease, we have 
+more alopecia images than other diseases, which creates an 
+imbalanced dataset for our model. For balancing the dataset, 
+we used data augmentation techniques (re-scaling, random 
+rotating, cropping, vertical and horizontal flipping) and 
+oversampled the infrequent class. 
+D. Neural Network Model 
+    Neural network is the most applied model for visual data 
+analysis. Neural network needs limited human assistance and 
+
+Image
+Image
+Data
+Denoising
+Enhancement
+Augmentation
+Convolutional
+Diseaseclass
+Neural
+Detected
+Network
+Model 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+5 
+ 
+can identify complex non-linear relationship between input 
+and output. From global or local scale modeling [26] to 
+diagnosis by medical image classification, neural network is 
+using extensively. Moreover, Facial recognition, image 
+labeling, accurate video subtitles, assisting call centers, 
+automated virtual agents all these things are using neural 
+network. There are 3 types of neural network available: 
+Artificial Neural Networks (ANN), Convolution Neural 
+Networks (CNN) and Recurrent Neural Networks (RNN). 
+Each neural network has mainly 3 components: an input 
+layer, a processing layer, and an output layer.  
+ 
+ 
+Fig. 5. Neural Network Model. 
+ 
+   In this study, CNN is utilized for classification because it 
+takes image’s raw pixel data, trains a model, and extracts the 
+features automatically for better detection. We used autokeras 
+to find the best model for this problem. After trying 25 
+different combinations, we selected 3 hidden layers with 1 
+input and 1 output layer as our final model which is shown in 
+Fig. 5. For training the model, we used batch_size = 16 with 
+50 epochs for each batch. The preprocessed data is divided 
+into 70-30 train-test-split for training and validation purpose. 
+Our model consists of 256 inputs, 3 x 3 square kernel, 3 
+output units and a softmax output. We used ReLU as our 
+activation function to prevent the exponential growth of 
+required computation and to explore the non-linear 
+relationship between input and output variables. After each 
+convolutional layer, input goes through the pooling layer 
+having 2 x 2 kernel size to reduce the dimensions of the 
+features map. Pooling layer summarizes the presented 
+features in a region and helps to prevent the over-fitting 
+problem by down sampling. We also used dropout layer after 
+each pooling layer to prevent neurons in a layer from 
+synchronously optimizing their weights and converging to the 
+same goal. Our model’s dropout rate is 0.3, which means 30% 
+of the neurons of this layer will be randomly dropped in each 
+epoch. 
+    All the resultant 2-D arrays from pooled features map 
+passes through the flatten layer and converted to single 
+dimensional long continuous linear vector in the transition 
+towards the fully connected layer as in Fig. 5. In the fully 
+connected layer, every single output pixel from the 
+convolutional layers is connected to 3 output classes. Though 
+dense layer is computationally expensive, we used 2 dense 
+layers for our classification task. Finally, we used softmax 
+activation function to transform the 3 units of fully connected 
+layer to a probability distribution which is represented by a 
+vector of 3 elements, and the highest probability element 
+selected as the final class. We leveraged adam optimizer for 
+learning purpose and reducing the overall loss by changing 
+the weights and learning rates. We used adam because it can 
+handle sparse gradients on noisy problems and combines the 
+best properties of AgaGrad and RMSProp algorithms. 
+ 
+VI. RESULTS 
+    We trained our CNN model using the optimal 
+hyperparameters selected from the grid search. These 
+hyperparameters are listed in Table II. We divided the dataset 
+into 70%-30% train-test-split where 105 randomly selected 
+images are used for training and 45 random images for 
+testing. After applying the preprocessing steps, we used the 
+training dataset to train the CNN model and evaluated the test 
+dataset using the model. 
+ 
+TABLE II: HYPERPARAMETERS OF CNN MODEL 
+Hyperparameters 
+Values 
+Batch Size 
+16 
+Epoch 
+50 
+Kernel Size 
+3 x 3 
+Optimizer 
+Adam 
+Dropout Rate 
+0.3 
+Pooling Size 
+2 x 2 
+Activation Function 
+ReLU 
+ 
+ 
+Fig. 6. Training and Validation loss for CNN. 
+ 
+
+Conv 1
+Max-Pooling
+Conv 2
+Max-Pooling
+Conv 3
+Max-Pooling
+Convolution
+(2 x 2)
+Convolution
+(2 x 2) Kernel
+Convolution
+(2 x 2) Kernel
+Fully-Connected
+(3 x 3)
+Kernel
+(3 x 3)
+Dropout 0.3
+(3 × 3) Kernel
+Dropout 0.3
+Input
+OutputTraining and validation loss
+1.25
+Training loss
+1.00
+Validation loss
+0.50
+0.25
+0.00
+0
+10
+20
+30
+40
+50
+Epochs 
+RESEARCH ARTICLE 
+European Journal of Information Technologies and Computer Science 
+www.ej-compute.org 
+ 
+ 
+ 
+  
+DOI: http://dx.doi.org/10.24018/ejcompute.YEAR.VOL.ISSUE.ID   
+Vol X | Issue Y | Month Year 
+6 
+ 
+ 
+Fig. 7. Training and Validation Accuracy for CNN. 
+    Our system achieved 96.2% training accuracy and 
+91.1% validation accuracy on the unseen data. Validation and 
+training losses for every epoch are shown in Fig 6. The 
+training losses decreased from 1.1685 to 0.1017, and the 
+validation losses decrease from 1.1260 to 0.3438 while going 
+from epoch 1 to epoch 50. At the same time, Training 
+accuracy and validation accuracy increased to 96.2% and 
+91.1%, respectively, from epoch 1 to epoch 50, shown in Fig. 
+7. 
+ 
+ 
+Fig. 8. Confusion Matrix of Our Model. 
+ 
+ 
+Fig. 9. Fractional Incorrect Prediction of Our Model. 
+ 
+    The confusion matrix in Fig. 8 shows the correct and 
+wrong classification for each category with inter-class 
+classification. Among 45 test images, alopecia (label 0) has 
+19 images, psoriasis (label 1) has 13 images, and folliculitis 
+(label 2) has 13 images. A total of 17 alopecia images were 
+classified as alopecia and the other 2 were incorrectly 
+classified as psoriasis. Again, 11 psoriasis images are 
+classified as psoriasis, but 2 psoriasis images were incorrectly 
+classified as alopecia. All 13 folliculitis images are classified 
+correctly. The fractional incorrect prediction for each class is 
+shown in Fig 9. Our model achieved the precision and recall 
+score of 0.895 for the alopecia disease class, 0.846 for the 
+psoriasis disease class, and 1.0 for the folliculitis disease 
+class. As the precision and recall scores are same in each 
+class, F1 scores are also similar to their respective precision 
+and recall values. 
+ 
+VII. CONCLUSION 
+    Although early-stage detection of hair and scalp-related 
+diseases is the key to the treatment process, hair loss and scalp 
+diseases can often go undetected due to a lack of awareness 
+and a lengthy diagnosis test. An AI-based application might 
+pave the way to facilitate early disease detection. In this 
+study, we developed a machine learning model to accurately 
+predict three hair and scalp-related diseases: alopecia, 
+folliculitis, and psoriasis by feeding 150 preprocessed image 
+data into a 2-D convolutional neural network model. After 
+using 70% of the data to train the model, we analyzed 
+remaining 30% of images for testing our model. After 
+subsequent training, the model gave an overall 96.2% training 
+accuracy on the training data and 91.1% validation accuracy 
+for the test data, with a high precision and recall scores for 
+each disease type. We have also provided our dataset with this 
+study. Our proposed system would assist dermatologists and 
+patients with a better understanding of disease classification 
+and initiating early treatment options for the three most 
+frequently occurred hair and scalp diseases. 
+ 
+FUNDING 
+    No funding was used to write this research paper. 
+ 
+CONFLICT OF INTEREST 
+The authors declare that they do not have any conflicts of 
+interest. 
+ 
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+SELF-SUPERVISED LEARNING FOR A NONLINEAR INVERSE
+PROBLEM WITH FORWARD OPERATOR INVOLVING AN
+UNKNOWN FUNCTION ARISING IN PHOTOACOUSTIC
+TOMOGRAPHY
+Gyeongha Hwang1, Gihyeon Jeon2∗, Sunghwan Moon3
+1 Department of Mathematics, Yeungnam University, Gyeongsan 38541, Republic of Korea
+2 School of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
+3 Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
+*Corresponding author E-mails: rydbr6709@knu.ac.kr
+ABSTRACT. In this article, we concern with a nonlinear inverse problem with forward opera-
+tor involving an unknown function. The problem arises in diverse applications and is challenging
+by the presence of the unknown function, which makes it ill-posed. Additionally, the nonlinear
+nature of the problem makes it difficult to use traditional methods and thus the study has
+addressed a simplified version of the problem by either linearizing it or assuming knowledge of
+the unknown function. Here, we propose a self-supervised learning to directly tackle a nonlinear
+inverse problem involving an unknown function. In particular, we focus on an inverse problem
+derived in Photoacoustic Tomograpy (PAT) which is a hybrid medical imaging with high res-
+olution and contrast. PAT can be modelled based on the wave equation. The measured data
+is the solution of the equation restricted to the surface and the initial pressure of the equation
+contains the biological information on the object of interest. The speed of sound wave in the
+equation is unknown. Our goal is to determine the initial pressure and the speed of sound wave
+simultaneously. Under a simple assumption that the sound speed is a function of the initial
+pressure, the problem becomes a nonlinear inverse problem involving an unknown function. The
+experimental results demonstrate that the proposed algorithm performs successfully.
+1
+Introduction
+Inverse problem is to find the cause factor from the observed data, which has applications
+in many fields such as optics, radar, acoustics, communication theory, signal processing,
+medical imaging, computer vision, geophysics, oceanography, and astronomy because it
+tells us about what we cannot directly observe. The forward operator, the inverse of
+the inverse problem, can be modelled as an (non)-linear system and often involves an
+unknown function. Due to the nature of the inverse problem, it is usually very hard to
+know the cause factor. For example, in medical imaging the cause factor is the human
+body section and in seismology, we never know the structure of the earth’s interior.
+In this article, we concern with a nonlinear inverse problem with forward operator involv-
+ing an unknown function. Our goal is to find the unknown function and the inverse opera-
+tor simultaneously from the measurements. The problem is generally ill-posed because of
+1
+arXiv:2301.08693v1  [math.NA]  20 Jan 2023
+
+the unknown function. Additionally, the nonlinearity in the problem makes conventional
+methods difficult to use. To handle the problem, one may simplify the problem linearly
+or assume knowledge of the unknown function. Here we propose a self-supervised frame-
+work to directly tackle a nonlinear inverse problem involving an unknown function. In
+particular, we address an inverse problem derived in Photoacoustic Tomography (PAT).
+Although our proposed framework has been proposed to solve the problem arising in
+PAT, it is generic and can be extended to handle a nonlinear inverse problem involving
+an unknown function.
+The rest of the section is devoted to an introduction of PAT. In section 2, we formulate
+an inverse problem arising in PAT, which is nonlinear and also involves an unknown
+function. The structure and learning method of the proposed framework for the problem
+are described in section 3. The numerical simulation results in section 4 demonstrate that
+the proposed algorithm performs successfully.
+1.1
+Photoacoustic Tomography
+PAT is a hybrid medical imaging that combines the high contrast of optical imaging with
+the high spatial resolution of ultrasound images [1, 2, 3]. The physical basis of PAT is
+the photoacoustic effect discovered by Bell in 1881 [4]. In PAT, when an non-destructive
+testing target object absorbs a non-ionizing laser pulse, it thermally expands and emits
+acoustic waves. The emitted ultrasound contains biological information on the target
+object and is measured by an detector placed around it. The internal image of the target
+object is reconstructed from this measured data. The advantage of PAT is that it is
+economical and less harmful because of non-ionizing radiation use [5].
+The propagation of the emitted ultrasound p(x, t) can be described by the wave equation
+∂2
+t p(x, t) = c(x)2∆xp(x, t)
+on R2 × [0, ∞)
+(1)
+with the initial conditions
+p(x, 0) = f(x)
+∂tp(x, 0) = 0
+on R2.
+(2)
+Here c is the speed of waves and f is the initial pressure which contains biological in-
+formation such as the location of a cancer cells in a physically small tissue. It is natural
+assumption that f has compact support in the bounded domain Ω and the detectors are
+located on the boundary ∂Ω of the domain. Regarding the measurement procedure, the
+point-shaped detector measures the average pressure above ∂Ω where the detectors are
+located and this average pressure is the value of a pressure wave p(x, t). Therefore, one of
+mathematical problems in PAT is reconstructing f from the measured data p|∂Ω×[0,∞),
+which implies obtaining an internal image of the target object.
+It is well-known that given the initial pressure f and the speed c, the solution p is
+determined uniquely. Let us define the wave forward operator W as
+W : (f, c) �→ p|∂Ω×[0,∞),
+i.e.,
+W(f, c) = p|∂Ω×[0,∞).
+Reconstructing problem for f from W(f, c) is studied when speed c is constant [6, 7].
+Okanen, Stefanov and Uhlmann study the explicit reconstruction when the sound speed
+2
+
+is known
+[8, 9]. If c depends on space variable x, the problem become much more
+difficult. A few of researchers have studied the problem with a given variable sound
+speed [10, 11, 12, 13]. Liu and Uhlmann figure out the sufficient conditions for recovering
+f and c [14].
+Recently, the application of deep learning in an medical imaging including PAT has been
+investigated extensively. Roles of deep learning in tomography include forward and in-
+verse operator approximation, image reconstruction from sparse data, and artifact/noise
+removal from reconstructed images [15, 16, 17, 18, 19, 20, 21]. There are also studies on
+limited-view data (see [22, 23]). H. Shan et al. propose an iterative optimization algo-
+rithm which reconstructs f and c simultaneously via a supervised learning [24]. However,
+most works deal with linear inverse problems or inverse problems without involving an
+unknown function [25].
+Many studies on PAT with deep learning are based on a supervised learning. A supervised
+learning exploits a collection of paired data of the boundary data and the initial pres-
+sure. In practical applications, it is difficult to obtain the initial pressure, because initial
+pressure represents internal human body. Therefore, it is necessary to study a learning
+method exploiting the boundary data only. One such method is a self-supervised learning
+which exploits supervised signals that are generated from the input data by leveraging
+its structure [26, 27].
+2
+Problem formulation
+In this section, we formulate the problem precisely. For this, we will make several as-
+sumptions. First we assume f has compact support, since a target object is finite. Sec-
+ondly, c is assumed to be a function of f, namely c(x)2 = Γ(f(x)) for some function
+Γ : [0, 1] → [0, ∞), because the wave speed c depends on the medium. Lastly, we assume
+that Γ(0) and Γ(1) are known, namely Γ(0) = c0 and Γ(1) = c1. The last assumption is
+reasonable because Γ(0) and Γ(1) represent the wave speeds in the air and the highest
+thermal expansion coefficient respectively. Then the equation (1) is rewritten as:
+∂2
+t p(x, t) = Γ(f(x))∆xp(x, t)
+on R2 × [0, ∞).
+(3)
+Let us define WΓ by WΓ(f) = p|∂Ω×[0,∞) where p is the solution of (3) with initial
+conditions (2). Then the inverse problem can be formulated as determining unknown Γ
+and f from a given WΓ(f). However, this problem is ill-posed: for any Γ′ satisfying
+� Γ = Γ′ on Im(f)
+Γ ̸= Γ′ on Dom(Γ) \ Im(f),
+we have WΓ(f) = WΓ′(f). Hence Γ can not be uniquely determined from WΓ(f). There
+is also a possibility that there exist Γ1, Γ2, f1 and f2 such that Γ1 ̸= Γ2, f1 ̸= f2 and
+WΓ1(f1) = WΓ2(f2). Instead, we consider the following inverse problem:
+Problem 1. Let the collection of boundary data BΓ := {WΓ(f) | Γ : [0, 1] → [0, ∞), Γ(0) =
+c0, Γ(1) = c1 and f ∈ L2(R2) has compact support} be given.
+3
+
+1. Determine unknown Γ from BΓ.
+2. For all WΓ(f) ∈ BΓ, determine f.
+Then the uniqueness statements for Problem 1 are
+Hypothesis 1. If Γ1 ̸= Γ2, then BΓ1 ̸= BΓ2.
+and
+Hypothesis 2. For fixed Γ, if f1 ̸= f2, then WΓ(f1) ̸= WΓ(f2).
+In this article, we aim to solve Problem 1 under Hypothesis 1 and 2. The problem is
+difficult to solve because of
+1. (3) involves unknown Γ.
+2. (3) is not linear.
+We are going to solve Problem 1 by exploiting a deep neural network (DNN). Since DNN
+can only handle with finite data, we address the following inverse problem.
+Problem 2. For given {WΓ(fi) | Γ : [0, 1] → [0, ∞), Γ(0) = c0, Γ(1) = c1 and fi ∈
+L2(R2) has compact support, i = 1, · · · , N}, determine Γ and {fi|i = 1, · · · , N}.
+3
+Network Design
+Figure 1: The network architecture
+We propose a self-supervised learning for the problem formulated in section 2. Our goal
+is simultaneously reconstructing {fi}N
+i=1 and Γ from given collection {WΓ(fi)}N
+i=1. The
+proposed framework is depicted in Figure 1. It consists of three components:
+4
+
+input
+output
+Wr(f)
+W (R(Wr(f)))
+R(Wr(f))
+M(R(Wr(f)))
+LOSS = MSEWr(f),W(RWr(f)))1. Reconstruction network R
+2. Mapping network M
+3. Wave forward operator W.
+The reconstruction network R learns to reconstruct the initial data from the measured
+data. The mapping network M approximates the function Γ : [0, 1] → [0, ∞) satisfying
+c(x)2 = Γ(f(x)). The forward operator W assigns to the initial data and the wave
+speed the measured data. Here we adopt the k-space method. If every component in the
+framework functions properly, then output should be same to the input. Thus we define
+the loss function as the difference between the input and the output:
+L = 1
+N
+N
+�
+i=1
+∥WΓ(fi) − WM(R(WΓ(fi))∥2
+∥WΓ(fi)∥2
+.
+Remark. Our method estimates Γ and the inverse operator W−1
+Γ . The estimated inverse
+operator can be used for the fast inference of the initial pressure from the boundary
+measurement.
+Remark. The proposed framework is generic and can be extended to handle a nonlinear
+inverse problem involving an unknown function.
+Now, the detailed structures of each component in the framework are described below.
+3.1
+Reconstruction network R
+The reconstruction network R is a network that reconstruct f from input data WΓ(f).
+Indeed, it approximates the inverse map W−1
+Γ
+: WΓ(f) �→ f. If the speed Γ of the wave
+is constant, it is well-known that the inverse map of (3) is linear [7, 28, 29]. Inspired by
+this fact, we propose the reconstruction network as a perturbation of a linear map:
+R := T1 + U ◦ T2,
+(4)
+where T1, T2 : Rm×m → Rm×m are linear and U : Rm×m → Rm×m is the U-net described
+in Figure 2. U-net is a type of convolutional neural network (CNN) introduced in [30] and
+is used widely in medical imaging. U-net consists of a contracting path and an expansive
+path. The contacting path has a typical CNN structure, where the input data is extracted
+into feature map with small size and large channel. In the expansive path, the size of
+the feature map increases again, and the number of channels decreases. In the end of R,
+since the range of f is [0, 1], we used the clamp function which rounds up values smaller
+than the minimum and round down values larger than the maximum.
+5
+
+Figure 2: U-net architecture for 64 × 64 size
+The proposed reconstruction network show a high performance for the low resolution
+data like 64 × 64 (see section 4.3.1). In case of high resolution data, however the linear
+operators T1 and T2 in the reconstruction network R make some problems, because
+they contains too many parameters. It causes lots of critical points which impede the
+convergence to the global minimum. It also makes a hardware issue and thus for the high
+resolution data, we employ Pixel Shuffle and Pixel Unshuffle which reduce the number of
+parameters contained in linear operators [31]. The Pixel Unshuffle splits one image into
+several images and the Pixel Shuffle merges several images into one image, as illustrated
+in Figure 3. Instead of applying the linear operators (T1 and T2) directly to the high
+resolution data, we process the data as follows (see Figure 4) :
+1. Split the high resolution data (m × m) into four low resolution data ( m
+2 × m
+2 ) by
+exploiting the Pixel Unshuffle.
+2. Apply four different linear operators to each low resolution data.
+3. Merge the outputs of the linear operators by using the Pixel Shuffle.
+6
+
+Figure 3: Pixel Shuffle and Pixel Unshuffle
+Figure 4: Architecture of alternative map to linear for high resolution data
+3.2
+Mapping network M
+We use multilayer perceptron (MLP) to approximate unknown Γ, because MLP can
+approximate any continuous function (a universal approximation theorem, see [32, 33]).
+The proposed network is a simple structure containing only three hidden layers of 10
+nodes. To satisfy the assumption that Γ(0) = c0 and Γ(1) = c1, the output of MLP is
+slightly manipulated as
+M(f) = MLP(f) − MLP(0) ∗ (1 − f) − MLP(1) ∗ f + ((c1 − c0)f + c0),
+7
+
+Linear map
+Linearmap2
+Pixel
+Pixel
+Unshuffle
+Shuffle
+Linear map 3
+Linear map 4so that
+M(0) = c0 and M(1) = c1.
+(5)
+3.3
+Forward problem
+A solution of the initial value problem (3) can be computed by the k-space method
+[34, 35]. The k-space method is a numerical method for computing solution of acoustic
+wave propagation, which uses information in the frequency space to obtain a solution for
+the next time step. For calculating propagation of p(x, t), let w(x, t) =
+1
+Γ(f(x))p(x, t) be
+an auxiliary field. Then we have,
+∂2
+t w(x, t) = ∆x [Γ(f(x))w(x, t)] .
+Taking the Fourier transform Fx for w with respect to x yields
+∂2
+t Fxw(k, t) = −|k|2Fx
+�
+Γ(f(·))w(·, t)
+�
+(k).
+(6)
+Meanwhile, the numerical approximation of the second derivative of Fxw is
+∂2
+t Fxw(k, t) ≈ Fxw(k, t + ∆t) − 2Fxw(k, t) + Fxw(k, t − ∆t)
+(∆t)2
+,
+(7)
+where ∆t is the time step. Then, by combining (6) and (7), we have
+Fxw(k, t + ∆t) = 2Fxw(k, t) − Fxw(k, t − ∆t) − (∆t)2|k|2Fx
+�
+Γ(f(·))w(·, t)
+�
+(k).
+By taking the inverse Fourier transform F−1
+k , we obtain
+w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) ��� F−1
+k
+�
+(∆t)2|·|2Fx
+�
+Γ(f)w
+�
+(·, t)
+�
+(x).
+Here, replacing (∆t)2|k|2 in the third term with 4 sin2 �
+(∆t)|k|
+2
+�
+provides more accurate
+discretization (see [34, 35]). Finally, we have wave propagation formula:
+w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) − F−1
+k
+�
+4 sin2
+�(∆t)| · |
+2
+�
+Fx
+�
+Γ(f)w
+�
+(·, t)
+�
+(x),
+or equivalently,
+p(x, t + ∆t) = 2p(x, t) − p(x, t − ∆t) − Γ(f)F−1
+k
+�
+4 sin2
+�(∆t)| · |
+2
+�
+Fx
+�
+p
+�
+(·, t)
+�
+(x).
+4
+Numerical Simulations
+In this section, we present the details of implementation and experimental results when
+Ω is the unit ball.
+8
+
+4.1
+Datasets
+The Shepp-Logan phantom, an artificial image that describes a cross section of the brain
+commonly used for simulation in tomography, contains 10 ellipses [36]. Each ellipse is
+created with 6 parameters: major axis, minor axis, the x-coordinate and the y-coordinate
+of center, rotation angle, and intensity value. The data set of the initial condition f defined
+on [−1.0, 1.0]2 ⊂ R2 is generated by slightly changing these 6 parameters with
+supp(f) ⊂
+�
+(x, y) ∈ R2 :
+x2
+0.692 +
+y2
+0.922 ≤ 1
+�
+.
+We create a set of 2,688 phantoms P = {fi}2688
+i=1 . For Γ, we consider four cases: linear,
+square root, square, and constant
+1. Γ1(f) = 0.3f + 0.7
+2. Γ2(f) = 0.3√f + 0.7
+3. Γ3(f) = 0.3f 2 + 0.7
+4. Γ4(f) = 0.7.
+For 1 ≤ j ≤ 4, we make the collection of data {WΓjfi}2688
+i=1 by using the forward operator
+for P and Γj. Of the 2,688 data, we use 2,048 data for training, 128 data for validation
+and 512 data for testing respectively.
+Figure 5: Examples of phantoms
+9
+
+4.2
+Training
+We use the Adam optimizer based on stochastic gradient descent and adaptive moment
+estimation to train the network
+[37]. There are two neural networks in the proposed
+frameworks: the reconstruction network R and the mapping network M. The learning
+rates for linear term of R, perturbation term of R, and M are chosen to be 10−4,
+10−3, and 10−3, respectively. Momentum parameters of the Adam optimizer were set at
+β1 = 0.9 and β2 = 0.999, respectively.
+We specifically put the batch size to 2. For general tasks, a moderately large batch
+size reduces training time. However, in this problem, a small batch size is advantageous
+because our model must be able to reconstruct an exact image for each data rather than
+an average result.
+4.3
+Results
+In this section, we illustrate experimental results. The overall results are displayed in
+Figure 6, Table 1, Figure 7, Figure 8, Table 2 and Figure 9. Here the losses for f and
+WΓ(f) are respectively defined by
+loss for f = 1
+N
+N
+�
+i=1
+∥fi − R(WΓ(fi))∥2
+∥fi∥2
+,
+and
+loss for WΓf = 1
+N
+N
+�
+i=1
+∥WΓ(fi) − WM(R(WΓ(fi))∥2
+∥WΓ(fi)∥2
+.
+4.3.1
+Low resolution data
+We conduct the simulation utilizing a dataset of images with a size of 64×64. The results
+for the mapping networks are shown in Figure 6. We see that the mapping networks
+accurately approximates Γ. When Γ3 = 0.3f 2 + 0.7, there is a difference between the
+plot of the mapping network M and the plot of Γ. This is because the values of f ∈ P
+almost belong to [0, 0.3]∪1 and so it has little effect on WΓ(f). In all cases, the process of
+training the mapping networks requires approximately 103 iterations. The results of the
+reconstruction networks are illustrated in Table 1 and Figure 7. Table 1 shows the test
+errors. So it can be concluded that the reconstruction networks accurately approximate
+the inverse maps in each case. The training of the reconstruction networks necessitates
+approximately 105 iterations.
+Remark. The assumption on Γ, (5) is crucial. If constraint (5) is not given in M, it
+may take a long time to approximate Γ, or it may fail to find Γ. Under the constraint, M
+can quickly determine Γ. Early determination of Γ helps the learning of reconstruction
+networks.
+10
+
+Figure 6: Comparison of the mapping network M and ground truth Γ for 64 × 64 data
+Assumption
+loss for f
+loss for WΓ(f)
+Γ1 = 0.3f + 0.7
+0.00504
+0.00702
+Γ2 = 0.3√f + 0.7
+0.00537
+0.00947
+Γ3 = 0.3f 2 + 0.7
+0.00557
+0.00634
+Γ4 = 0.7
+0.01373
+0.00456
+Table 1: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 64 × 64
+data
+11
+
+[2 = 0.3Vf + 0.7
+[1 = 0.3f + 0.7
+1.0
+ground truth
+1.0
+mapping network
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+[3 = 0.3f2 + 0.7
+[4 = 0.7
+1.0
+1.0
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0Figure 7: Reconstruction results according to Γ for 64 × 64 data
+4.3.2
+High resolution data
+In the simulation for high resolution data, two linear operators T1 and T2 in the recon-
+struction network (4) are replaced by alternative map described in Figure 4. The dataset
+is prepared with images of a size of 96 × 96. Similarly to the low resolution case, the
+mapping network M approximates Γ accurately within 103 iterations (Figure 8). On the
+other hand, the reconstruction networks for each Γ exhibit a slight decrease in perfor-
+mance but still acceptable (Table 2 and Figure 9). We surmise that the slight decrease
+in performance is a result of the reduction in parameters brought about by the Pixel
+Unshuffle and Pixel Shuffle operations.
+12
+
+[1 = 0.3f + 0.7
+[2 = 0.3V f + 0.7
+[3 = 0.3f2 + 0.7
+4 = 0.7
+ground truth
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0Figure 8: Comparison of the mapping network M and ground truth Γ for 96 × 96 data
+Assumption
+loss for f
+loss for WΓ(f)
+Γ1 = 0.3f + 0.7
+0.00860
+0.01293
+Γ2 = 0.3√f + 0.7
+0.01023
+0.01679
+Γ3 = 0.3f 2 + 0.7
+0.00710
+0.01132
+Γ4 = 0.7
+0.00689
+0.69511
+Table 2: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 96 × 96
+data
+13
+
+[2 = 0.3Vf + 0.7
+[1 = 0.3f + 0.7
+1.0
+ground truth
+1.0
+mapping network
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+[3 = 0.3f2 + 0.7
+[4 = 0.7
+1.0
+1.0
+0.7
+0.7
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0Figure 9: Reconstruction results according to Γ for 96 × 96 data
+5
+Conclusions
+Here, we propose a self-supervised learning for a nonlinear inverse problem with forward
+operator involving an unknown function. In medical imaging such as PAT, the initial
+pressure is mostly untrackable for the measured data. Moreover it is difficult to know the
+wave speed. So it is necessary to reconstruct the initial pressure f and the wave speed
+simultaneously. Under the simple assumption, the problem becomes a nonlinear inverse
+problem involving an unknown function. The experimental results demonstrate the high
+performance of the proposed algorithm. Our framework can be extended to a nonlinear
+inverse problem involving an unknown function, formulated under more complicated
+situations. This can be an interesting line of future research.
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+17
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf,len=469
+page_content='SELF-SUPERVISED LEARNING FOR A NONLINEAR INVERSE  PROBLEM WITH FORWARD OPERATOR INVOLVING AN  UNKNOWN FUNCTION ARISING IN PHOTOACOUSTIC  TOMOGRAPHY  Gyeongha Hwang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Gihyeon Jeon2∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Sunghwan Moon3  1 Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Yeungnam University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Gyeongsan 38541,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Republic of Korea  2 School of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Kyungpook National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Daegu 41566,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Republic of Korea  3 Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Kyungpook National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Daegu 41566,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Republic of Korea  Corresponding author E-mails: rydbr6709@knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='kr  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In this article, we concern with a nonlinear inverse problem with forward opera-  tor involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The problem arises in diverse applications and is challenging  by the presence of the unknown function, which makes it ill-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Additionally, the nonlinear  nature of the problem makes it difficult to use traditional methods and thus the study has  addressed a simplified version of the problem by either linearizing it or assuming knowledge of  the unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here, we propose a self-supervised learning to directly tackle a nonlinear  inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In particular, we focus on an inverse problem  derived in Photoacoustic Tomograpy (PAT) which is a hybrid medical imaging with high res-  olution and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' PAT can be modelled based on the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The measured data  is the solution of the equation restricted to the surface and the initial pressure of the equation  contains the biological information on the object of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The speed of sound wave in the  equation is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our goal is to determine the initial pressure and the speed of sound wave  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Under a simple assumption that the sound speed is a function of the initial  pressure, the problem becomes a nonlinear inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The  experimental results demonstrate that the proposed algorithm performs successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  1  Introduction  Inverse problem is to find the cause factor from the observed data, which has applications  in many fields such as optics, radar, acoustics, communication theory, signal processing,  medical imaging, computer vision, geophysics, oceanography, and astronomy because it  tells us about what we cannot directly observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The forward operator, the inverse of  the inverse problem, can be modelled as an (non)-linear system and often involves an  unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Due to the nature of the inverse problem, it is usually very hard to  know the cause factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For example, in medical imaging the cause factor is the human  body section and in seismology, we never know the structure of the earth’s interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  In this article, we concern with a nonlinear inverse problem with forward operator involv-  ing an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our goal is to find the unknown function and the inverse opera-  tor simultaneously from the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The problem is generally ill-posed because of  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='08693v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='NA]  20 Jan 2023  the unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Additionally, the nonlinearity in the problem makes conventional  methods difficult to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' To handle the problem, one may simplify the problem linearly  or assume knowledge of the unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here we propose a self-supervised frame-  work to directly tackle a nonlinear inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In  particular, we address an inverse problem derived in Photoacoustic Tomography (PAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Although our proposed framework has been proposed to solve the problem arising in  PAT, it is generic and can be extended to handle a nonlinear inverse problem involving  an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The rest of the section is devoted to an introduction of PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In section 2, we formulate  an inverse problem arising in PAT, which is nonlinear and also involves an unknown  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The structure and learning method of the proposed framework for the problem  are described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The numerical simulation results in section 4 demonstrate that  the proposed algorithm performs successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Photoacoustic Tomography  PAT is a hybrid medical imaging that combines the high contrast of optical imaging with  the high spatial resolution of ultrasound images [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The physical basis of PAT is  the photoacoustic effect discovered by Bell in 1881 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In PAT, when an non-destructive  testing target object absorbs a non-ionizing laser pulse, it thermally expands and emits  acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The emitted ultrasound contains biological information on the target  object and is measured by an detector placed around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The internal image of the target  object is reconstructed from this measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The advantage of PAT is that it is  economical and less harmful because of non-ionizing radiation use [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The propagation of the emitted ultrasound p(x, t) can be described by the wave equation  ∂2  t p(x, t) = c(x)2∆xp(x, t)  on R2 × [0, ∞)  (1)  with the initial conditions  p(x, 0) = f(x)  ∂tp(x, 0) = 0  on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (2)  Here c is the speed of waves and f is the initial pressure which contains biological in-  formation such as the location of a cancer cells in a physically small tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It is natural  assumption that f has compact support in the bounded domain Ω and the detectors are  located on the boundary ∂Ω of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Regarding the measurement procedure, the  point-shaped detector measures the average pressure above ∂Ω where the detectors are  located and this average pressure is the value of a pressure wave p(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Therefore, one of  mathematical problems in PAT is reconstructing f from the measured data p|∂Ω×[0,∞),  which implies obtaining an internal image of the target object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  It is well-known that given the initial pressure f and the speed c, the solution p is  determined uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Let us define the wave forward operator W as  W : (f, c) �→ p|∂Ω×[0,∞),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=',  W(f, c) = p|∂Ω×[0,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Reconstructing problem for f from W(f, c) is studied when speed c is constant [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Okanen, Stefanov and Uhlmann study the explicit reconstruction when the sound speed  2  is known  [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If c depends on space variable x, the problem become much more  difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' A few of researchers have studied the problem with a given variable sound  speed [10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Liu and Uhlmann figure out the sufficient conditions for recovering  f and c [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Recently, the application of deep learning in an medical imaging including PAT has been  investigated extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Roles of deep learning in tomography include forward and in-  verse operator approximation, image reconstruction from sparse data, and artifact/noise  removal from reconstructed images [15, 16, 17, 18, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' There are also studies on  limited-view data (see [22, 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' propose an iterative optimization algo-  rithm which reconstructs f and c simultaneously via a supervised learning [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' However,  most works deal with linear inverse problems or inverse problems without involving an  unknown function [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Many studies on PAT with deep learning are based on a supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' A supervised  learning exploits a collection of paired data of the boundary data and the initial pres-  sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In practical applications, it is difficult to obtain the initial pressure, because initial  pressure represents internal human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Therefore, it is necessary to study a learning  method exploiting the boundary data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' One such method is a self-supervised learning  which exploits supervised signals that are generated from the input data by leveraging  its structure [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2  Problem formulation  In this section, we formulate the problem precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For this, we will make several as-  sumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' First we assume f has compact support, since a target object is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Sec-  ondly, c is assumed to be a function of f, namely c(x)2 = Γ(f(x)) for some function  Γ : [0, 1] → [0, ∞), because the wave speed c depends on the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Lastly, we assume  that Γ(0) and Γ(1) are known, namely Γ(0) = c0 and Γ(1) = c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The last assumption is  reasonable because Γ(0) and Γ(1) represent the wave speeds in the air and the highest  thermal expansion coefficient respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then the equation (1) is rewritten as:  ∂2  t p(x, t) = Γ(f(x))∆xp(x, t)  on R2 × [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (3)  Let us define WΓ by WΓ(f) = p|∂Ω×[0,∞) where p is the solution of (3) with initial  conditions (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then the inverse problem can be formulated as determining unknown Γ  and f from a given WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' However, this problem is ill-posed: for any Γ′ satisfying  � Γ = Γ′ on Im(f)  Γ ̸= Γ′ on Dom(Γ) \\ Im(f),  we have WΓ(f) = WΓ′(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Hence Γ can not be uniquely determined from WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' There  is also a possibility that there exist Γ1, Γ2, f1 and f2 such that Γ1 ̸= Γ2, f1 ̸= f2 and  WΓ1(f1) = WΓ2(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Instead, we consider the following inverse problem:  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Let the collection of boundary data BΓ := {WΓ(f) | Γ : [0, 1] → [0, ∞), Γ(0) =  c0, Γ(1) = c1 and f ∈ L2(R2) has compact support} be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Determine unknown Γ from BΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For all WΓ(f) ∈ BΓ, determine f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Then the uniqueness statements for Problem 1 are  Hypothesis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If Γ1 ̸= Γ2, then BΓ1 ̸= BΓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  and  Hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For fixed Γ, if f1 ̸= f2, then WΓ(f1) ̸= WΓ(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  In this article, we aim to solve Problem 1 under Hypothesis 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The problem is  difficult to solve because of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' (3) involves unknown Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' (3) is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  We are going to solve Problem 1 by exploiting a deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Since DNN  can only handle with finite data, we address the following inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For given {WΓ(fi) | Γ : [0, 1] → [0, ∞), Γ(0) = c0, Γ(1) = c1 and fi ∈  L2(R2) has compact support, i = 1, · · · , N}, determine Γ and {fi|i = 1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3  Network Design  Figure 1: The network architecture  We propose a self-supervised learning for the problem formulated in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our goal  is simultaneously reconstructing {fi}N  i=1 and Γ from given collection {WΓ(fi)}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The  proposed framework is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It consists of three components:  4  input  output  Wr(f)  W (R(Wr(f)))  R(Wr(f))  M(R(Wr(f)))  LOSS = MSEWr(f),W(RWr(f)))1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Reconstruction network R  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Mapping network M  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Wave forward operator W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The reconstruction network R learns to reconstruct the initial data from the measured  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The mapping network M approximates the function Γ : [0, 1] → [0, ∞) satisfying  c(x)2 = Γ(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The forward operator W assigns to the initial data and the wave  speed the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here we adopt the k-space method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If every component in the  framework functions properly, then output should be same to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Thus we define  the loss function as the difference between the input and the output:  L = 1  N  N  �  i=1  ∥WΓ(fi) − WM(R(WΓ(fi))∥2  ∥WΓ(fi)∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our method estimates Γ and the inverse operator W−1  Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The estimated inverse  operator can be used for the fast inference of the initial pressure from the boundary  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The proposed framework is generic and can be extended to handle a nonlinear  inverse problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Now, the detailed structures of each component in the framework are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Reconstruction network R  The reconstruction network R is a network that reconstruct f from input data WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Indeed, it approximates the inverse map W−1  Γ  : WΓ(f) �→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If the speed Γ of the wave  is constant, it is well-known that the inverse map of (3) is linear [7, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Inspired by  this fact, we propose the reconstruction network as a perturbation of a linear map:  R := T1 + U ◦ T2,  (4)  where T1, T2 : Rm×m → Rm×m are linear and U : Rm×m → Rm×m is the U-net described  in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' U-net is a type of convolutional neural network (CNN) introduced in [30] and  is used widely in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' U-net consists of a contracting path and an expansive  path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The contacting path has a typical CNN structure, where the input data is extracted  into feature map with small size and large channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In the expansive path, the size of  the feature map increases again, and the number of channels decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In the end of R,  since the range of f is [0, 1], we used the clamp function which rounds up values smaller  than the minimum and round down values larger than the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  5  Figure 2: U-net architecture for 64 × 64 size  The proposed reconstruction network show a high performance for the low resolution  data like 64 × 64 (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In case of high resolution data, however the linear  operators T1 and T2 in the reconstruction network R make some problems, because  they contains too many parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It causes lots of critical points which impede the  convergence to the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' It also makes a hardware issue and thus for the high  resolution data, we employ Pixel Shuffle and Pixel Unshuffle which reduce the number of  parameters contained in linear operators [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The Pixel Unshuffle splits one image into  several images and the Pixel Shuffle merges several images into one image, as illustrated  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Instead of applying the linear operators (T1 and T2) directly to the high  resolution data, we process the data as follows (see Figure 4) :  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Split the high resolution data (m × m) into four low resolution data ( m  2 × m  2 ) by  exploiting the Pixel Unshuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Apply four different linear operators to each low resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Merge the outputs of the linear operators by using the Pixel Shuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  6  Figure 3: Pixel Shuffle and Pixel Unshuffle  Figure 4: Architecture of alternative map to linear for high resolution data  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  Mapping network M  We use multilayer perceptron (MLP) to approximate unknown Γ, because MLP can  approximate any continuous function (a universal approximation theorem, see [32, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  The proposed network is a simple structure containing only three hidden layers of 10  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' To satisfy the assumption that Γ(0) = c0 and Γ(1) = c1, the output of MLP is  slightly manipulated as  M(f) = MLP(f) − MLP(0) ∗ (1 − f) − MLP(1) ∗ f + ((c1 − c0)f + c0),  7  Linear map  Linearmap2  Pixel  Pixel  Unshuffle  Shuffle  Linear map 3  Linear map 4so that  M(0) = c0 and M(1) = c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (5)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3  Forward problem  A solution of the initial value problem (3) can be computed by the k-space method  [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The k-space method is a numerical method for computing solution of acoustic  wave propagation, which uses information in the frequency space to obtain a solution for  the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For calculating propagation of p(x, t), let w(x, t) =  1  Γ(f(x))p(x, t) be  an auxiliary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then we have,  ∂2  t w(x, t) = ∆x [Γ(f(x))w(x, t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Taking the Fourier transform Fx for w with respect to x yields  ∂2  t Fxw(k, t) = −|k|2Fx  �  Γ(f(·))w(·, t)  �  (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  (6)  Meanwhile, the numerical approximation of the second derivative of Fxw is  ∂2  t Fxw(k, t) ≈ Fxw(k, t + ∆t) − 2Fxw(k, t) + Fxw(k, t − ∆t)  (∆t)2  ,  (7)  where ∆t is the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Then, by combining (6) and (7), we have  Fxw(k, t + ∆t) = 2Fxw(k, t) − Fxw(k, t − ∆t) − (∆t)2|k|2Fx  �  Γ(f(·))w(·, t)  �  (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  By taking the inverse Fourier transform F−1  k , we obtain  w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) − F−1  k  �  (∆t)2|·|2Fx  �  Γ(f)w  �  (·, t)  �  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Here, replacing (∆t)2|k|2 in the third term with 4 sin2 �  (∆t)|k|  2  �  provides more accurate  discretization (see [34, 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Finally, we have wave propagation formula:  w(x, t + ∆t) = 2w(x, t) − w(x, t − ∆t) − F−1  k  �  4 sin2  �(∆t)| · |  2  �  Fx  �  Γ(f)w  �  (·, t)  �  (x),  or equivalently,  p(x, t + ∆t) = 2p(x, t) − p(x, t − ∆t) − Γ(f)F−1  k  �  4 sin2  �(∆t)| · |  2  �  Fx  �  p  �  (·, t)  �  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  4  Numerical Simulations  In this section, we present the details of implementation and experimental results when  Ω is the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Datasets  The Shepp-Logan phantom, an artificial image that describes a cross section of the brain  commonly used for simulation in tomography, contains 10 ellipses [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Each ellipse is  created with 6 parameters: major axis, minor axis, the x-coordinate and the y-coordinate  of center, rotation angle, and intensity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The data set of the initial condition f defined  on [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0]2 ⊂ R2 is generated by slightly changing these 6 parameters with  supp(f) ⊂  �  (x, y) ∈ R2 :  x2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='692 +  y2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='922 ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  We create a set of 2,688 phantoms P = {fi}2688  i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For Γ, we consider four cases: linear,  square root, square, and constant  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ1(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ2(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3√f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ3(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Γ4(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  For 1 ≤ j ≤ 4, we make the collection of data {WΓjfi}2688  i=1 by using the forward operator  for P and Γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Of the 2,688 data, we use 2,048 data for training, 128 data for validation  and 512 data for testing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Figure 5: Examples of phantoms  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  Training  We use the Adam optimizer based on stochastic gradient descent and adaptive moment  estimation to train the network  [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' There are two neural networks in the proposed  frameworks: the reconstruction network R and the mapping network M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The learning  rates for linear term of R, perturbation term of R, and M are chosen to be 10−4,  10−3, and 10−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Momentum parameters of the Adam optimizer were set at  β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='9 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  We specifically put the batch size to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' For general tasks, a moderately large batch  size reduces training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' However, in this problem, a small batch size is advantageous  because our model must be able to reconstruct an exact image for each data rather than  an average result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3  Results  In this section, we illustrate experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The overall results are displayed in  Figure 6, Table 1, Figure 7, Figure 8, Table 2 and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Here the losses for f and  WΓ(f) are respectively defined by  loss for f = 1  N  N  �  i=1  ∥fi − R(WΓ(fi))∥2  ∥fi∥2  ,  and  loss for WΓf = 1  N  N  �  i=1  ∥WΓ(fi) − WM(R(WΓ(fi))∥2  ∥WΓ(fi)∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='1  Low resolution data  We conduct the simulation utilizing a dataset of images with a size of 64×64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The results  for the mapping networks are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' We see that the mapping networks  accurately approximates Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' When Γ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7, there is a difference between the  plot of the mapping network M and the plot of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' This is because the values of f ∈ P  almost belong to [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3]∪1 and so it has little effect on WΓ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In all cases, the process of  training the mapping networks requires approximately 103 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The results of the  reconstruction networks are illustrated in Table 1 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Table 1 shows the test  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' So it can be concluded that the reconstruction networks accurately approximate  the inverse maps in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The training of the reconstruction networks necessitates  approximately 105 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The assumption on Γ, (5) is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' If constraint (5) is not given in M, it  may take a long time to approximate Γ, or it may fail to find Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Under the constraint, M  can quickly determine Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Early determination of Γ helps the learning of reconstruction  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  10  Figure 6: Comparison of the mapping network M and ground truth Γ for 64 × 64 data  Assumption  loss for f  loss for WΓ(f)  Γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00504  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00702  Γ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3√f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00537  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00947  Γ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00557  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00634  Γ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='01373  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='00456  Table 1: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 64 × 64  data  11  [2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3Vf + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  [1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  ground truth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  mapping network  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  [3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  [4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0Figure 7: Reconstruction results according to Γ for 64 × 64 data  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  High resolution data  In the simulation for high resolution data, two linear operators T1 and T2 in the recon-  struction network (4) are replaced by alternative map described in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The dataset  is prepared with images of a size of 96 × 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Similarly to the low resolution case, the  mapping network M approximates Γ accurately within 103 iterations (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' On the  other hand, the reconstruction networks for each Γ exhibit a slight decrease in perfor-  mance but still acceptable (Table 2 and Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' We surmise that the slight decrease  in performance is a result of the reduction in parameters brought about by the Pixel  Unshuffle and Pixel Shuffle operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  12  [1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  [2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3V f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  [3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  ground truth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0Figure 8: Comparison of the mapping network M and ground truth Γ for 96 × 96 data  Assumption  loss for f  loss for WΓ(f)  Γ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='69511  Table 2: Test errors for f and WΓ(f) according to Γ after 102,400 iterations for 96 × 96  data  13  [2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='0Figure 9: Reconstruction results according to Γ for 96 × 96 data  5  Conclusions  Here, we propose a self-supervised learning for a nonlinear inverse problem with forward  operator involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In medical imaging such as PAT, the initial  pressure is mostly untrackable for the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Moreover it is difficult to know the  wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' So it is necessary to reconstruct the initial pressure f and the wave speed  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Under the simple assumption, the problem becomes a nonlinear inverse  problem involving an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The experimental results demonstrate the high  performance of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Our framework can be extended to a nonlinear  inverse problem involving an unknown function, formulated under more complicated  situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' This can be an interesting line of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  References  [1] Huabei Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustic tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' CRC Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [2] Jun Xia, Junjie Yao, and Lihong V Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustic tomography: principles  and advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Electromagnetic waves (Cambridge, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' ), 147:1, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [3] Peter Kuchment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' The Radon transform and medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' SIAM, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  14  [1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='7  [2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='3V f + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='7  ground truth  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content='0[4] Alexander Graham Bell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' On the production and reproduction of sound by light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' In  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=', volume 29, pages 115–136, 1881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [5] Idan Steinberg, David M Huland, Ophir Vermesh, Hadas E Frostig, Willemieke S  Tummers, and Sanjiv S Gambhir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustic clinical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustics,  14:77–98, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [6] Gerhard Zangerl, Sunghwan Moon, and Markus Haltmeier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustic tomogra-  phy with direction dependent data: An exact series reconstruction approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Inverse  Problems, 35(11):114005, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [7] Minghua Xu and Lihong V Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Universal back-projection algorithm for photoa-  coustic computed tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Physical Review E, 71(1):016706, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [8] Lauri Oksanen and Gunther Uhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustic and thermoacoustic tomog-  raphy with an uncertain wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Mathematical Research Letters, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [9] Plamen Stefanov and Gunther Uhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Thermoacoustic tomography with variable  sound speed Inverse Problems, 25(7):075011, 16, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [10] Jianliang Qian, Plamen Stefanov, Gunther Uhlmann, and Hongkai Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' An efficient  neumann series–based algorithm for thermoacoustic and photoacoustic tomography  with variable sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' SIAM Journal on Imaging Sciences, 4(3):850–883, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [11] Zakaria Belhachmi, Thomas Glatz, and Otmar Scherzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  A direct method for  photoacoustic tomography with inhomogeneous sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Inverse Problems,  32(4):045005, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [12] Yulia Hristova, Peter Kuchment, and Linh Nguyen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Reconstruction and time rever-  sal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous  media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Inverse problems, 24(5):055006, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [13] Minam Moon, Injo Hur, and Sunghwan Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Singular value decomposition  of the wave forward operator with radial variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  arXiv preprint  arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='10793, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [14] Hongyu Liu and Gunther Uhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Determining both sound speed and internal  source in thermo-and photo-acoustic tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Inverse Problems, 31(10):105005,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [15] Stephan Antholzer, Markus Haltmeier, Robert Nuster, and Johannes Schwab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Pho-  toacoustic image reconstruction via deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  In Photons Plus Ultrasound:  Imaging and Sensing 2018, volume 10494, pages 433–442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' SPIE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  [16] Gregory Ongie, Ajil Jalal, Christopher A Metzler, Richard G Baraniuk, Alexan-  dros G Dimakis, and Rebecca Willett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Deep learning techniques for inverse prob-  lems in imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' IEEE Journal on Selected Areas in Information Theory, 1(1):39–56,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Deep learning for tomographic image  reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Deep learn-  ing for biomedical photoacoustic imaging: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustics, 22:100241, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Review of deep learning  for photoacoustic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Photoacoustics, 21:100215, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Deep learning for  photoacoustic tomography from sparse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Inverse problems in science and  engineering, 27(7):987–1005, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Photoacoustic microscopy with sparse data by convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content='  Photoacoustics, 22:100242, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' Limited-  view and sparse photoacoustic tomography for neuroimaging with deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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+page_content=' A new deep learning network for mitigating limited-view and  under-sampling artifacts in ring-shaped photoacoustic tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
+page_content=' Computerized  Medical Imaging and Graphics, 84:101720, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFAT4oBgHgl3EQfyh6B/content/2301.08693v1.pdf'}
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