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REVIEW 3 major objections 6 minor 56 references

Nonnegative Tensor Decomposition Via Collaborative Neurodynamic Optimization

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a swarm of collaborating recurrent neural networks, coordinated by particle swarm optimization, solves nonnegative canonical polyadic decomposition with almost-sure global convergence.

desk verdict Worth reading for the algorithm and experiments, but the global-convergence theorem is unproved and the discrete-time stability proof has a boundary error. read the letter →

arxiv 2411.18127 v2 pith:Z4MEA5SH submitted 2024-11-27 math.NA cs.NA

Salman Ahmadi-Asl , Valentin Leplat , Anh-Huy Phan , Andrzej Cichocki This is my paper · ORCID
classification math.NAcs.NA MSC 15A6965F9990C26
keywords nonnegativetensordecompositioncanonicalpolyadiccollaborativeneurodynamicoptimizationrecurrentneuralnetworkparticleswarmdiscrete-timeprojectionHessianpreconditioningglobalconvergence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Nonnegative canonical polyadic decomposition (CPD) is a nonconvex tensor factorization problem whose objective can have many local minima, and classical algorithms such as alternating least squares and HALS can stall on tensors with highly collinear factor matrices. This paper claims that a 'collaborative neurodynamic' model—a swarm of recurrent neural networks, each running a projected gradient-like differential equation, with particle swarm optimization (PSO) exchanging their best candidate solutions—solves this problem with a strong guarantee: the swarm's best iterate converges to a global minimum with probability one. A discrete-time version, based on an Euler discretization with adaptive step size, is shown to be Lyapunov stable and convergent to a partial optimum. If these convergence claims hold, the approach offers a way to push nonnegative tensor decompositions out of local minima without relying on tensor-specific structure, and the paper reports that it outperforms HALS, multiplicative updates, and other baselines on collinear and real-world data tensors, including hyperspectral unmixing tasks.

What carries the argument

The load-bearing object is the collaborative neurodynamic system: $q$ independent one-layer recurrent neural networks with dynamics $\epsilon_1 \dot A = -A + [A - \nabla_A F\, P_A^{-1}]_+$ (and analogous equations for $B,C$), where $[\,\cdot\,]_+$ is projection onto the nonnegative orthant and $P_A = (C^TC)*(B^TB)$ is the Hessian preconditioner for factor $A$. Around this sits the PSO loop (2.6)-(2.9), a diversity measure $\mathrm{DI}^{(k)}$ that triggers wavelet mutation (2.11)-(2.12) when particles cluster, and reinitialization that refreshes the swarm from the feasible region. The argument's logical joints are Lemma 4.1, identifying RNN equilibria with KKT points, and Lemma 4.2, the Solis–Wets random-search theorem that converts monotone non-increasing best values plus full-support sampling into almost-sure global convergence. The discrete counterpart is the Euler-based projection neural network (5.2)-(5.4) with Armijo backtracking, whose stability hinges on the step-size interval in Theorem 6.2.

What would settle it

Take a one-dimensional feasible interval $[l,u]$ and iterate the wavelet-mutation map (2.11)-(2.12) from a collection of current states; if the reachable set after any number of steps omits a positive-measure interval of $[l,u]$, then the claimed support $\mathcal{H}(\mu_k) = \Omega$ used in Theorem 4.3 is false for that $B$, and Assumption II is not satisfied by the implemented search.

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Extended reading notes

Core claim

The paper's central claim is Theorem 4.3: the CNO-CPD algorithm is globally convergent with probability one, meaning the sequence of global-best positions $\{p^{(k)}_{\mathrm{best}}\}$ generated by the swarm satisfies $\lim_{k\to\infty} P(x^{(k)} \in \Omega_g) = 1$ for every initial point, where $\Omega_g$ is the set of global minimizers of $\|X - [[A,B,C]]\|_F^2$ over $A,B,C \ge 0$. The proof routes through two assumptions: the swarm's objective values never increase (Assumption I), and the stochastic search eventually visits any positive-measure region of the feasible set (Assumption II); then a random-search lemma gives almost-sure convergence. For the discrete model, Theorem 6.3 states that the backtracking DTPNN produces a monotonically non-increasing objective sequence that reaches an equilibrium point, and combining it with Lemma 4.2 gives the same probability-one guarantee for CNO-DTPNN. The paper also claims that Hessian preconditioning with $P_A = (C^TC) * (B^TB)$ accelerates both continuous and discrete dynamics.

Load-bearing premise

The guarantee collapses if Assumption II fails: the proof asserts, without deriving it from the formulas, that wavelet mutation and reinitialization make the swarm visit every positive-volume part of the feasible region infinitely often.

Editorial extensions

If this is right

  • For tensors with high collinearity, where HALS and other baselines converge very slowly, CNO-CPD reaches relative error $10^{-4}$ in about 20 seconds while the best baseline stays near $10^{-1}$ after 100 seconds (Example 2).
  • A population of $q$ RNNs improves the attainable relative error monotonically in the Monte Carlo experiments (Figure 8), so adding networks trades computation for escaping local minima.
  • The log-barrier variant removes the projection but yields worse relative errors than the classical projected ODE, so the projected dynamics rather than the barrier is the recommended formulation.
  • The discrete CNO-DTPNN, with semi-implicit updates and backtracking step sizes, gives the practical algorithm for large tensors where integrating the continuous ODE is expensive.
  • If Theorem 4.3 is correct, the algorithm carries a formal almost-sure global optimality property that classical CPD algorithms lack.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension — Theorem 4.3's assumption that wavelet mutation makes the search support cover all of $\Omega$ is not derived from the PSO and mutation formulas; a direct reachability check of (2.11)-(2.12) would determine whether the almost-sure guarantee genuinely holds for the implemented algorithm.
  • Extension — The same collaborative scheme could be transplanted to Tucker, tensor-train, and block-term decompositions, and to Kullback-Leibler or alpha-beta divergence objectives; the paper lists these as future work, but the convergence argument is largely objective-agnostic beyond smoothness and bounded-below conditions.
  • Extension — The comparison with HALS-PSO in Example 2 suggests the benefit is not merely PSO reinitialization; isolating the contribution of the ODE integrator versus the swarm communication in a controlled ablation would test that reading.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a collaborative neurodynamic optimization method for nonnegative canonical polyadic decomposition (CNO-CPD): each of several recurrent neural networks solves a projected-gradient-type continuous-time system, and the networks exchange information through particle swarm optimization with wavelet mutation. A discrete-time projection neural network (DTPNN) with backtracking steps and Hessian preconditioning is also introduced. The main theoretical claims are that CNO-CPD is globally convergent with probability one (Theorem 4.3), that the discrete-time model is Lyapunov stable and convergent (Theorems 6.2 and 6.3), and that the collaborative discrete version inherits the almost-sure global convergence guarantee. The paper also reports extensive numerical comparisons on synthetic collinear tensors, real image datasets, and hyperspectral data, with code made publicly available.

Significance. If the almost-sure global convergence claim were established, the paper would make a substantial contribution to nonnegative tensor decomposition by offering a distributed stochastic search method with a rigorous global guarantee in a nonconvex setting. The experimental work, particularly on tensors with highly collinear factors, suggests that the proposed scheme can be empirically competitive, and the availability of code is a practical strength. However, the mathematical core is not sound as stated: the proof of the headline global-convergence theorem rests on an unverified probabilistic assumption, and the equilibrium analysis of the discrete-time model contains a boundary-error that propagates into the stability and convergence claims. As a math.NA paper, these are load-bearing issues, so the theoretical contribution as written cannot be accepted.

major comments (3)
  1. [§4, Theorem 4.3 and Assumption II] The proof of Theorem 4.3 does not establish Assumption II. The argument states that wavelet mutation and re-initialization make the support of the search distribution cover the feasible region, and concludes that the product condition (4.1) holds. This conflates support coverage with the conditional-probability condition: Assumption II requires, for every positive-measure Borel subset B, that the infinite product of (1 - μ_k(B)) is zero, where μ_k(B) is the conditional probability of hitting B at iteration k. The paper gives no lower bound on these conditional probabilities. In Algorithm 3.1, re-initialization from the whole feasible region occurs only at the initial step; thereafter the states are moved by the PSO update (2.6)-(2.7) with β1 = β2 = 0.01, and wavelet mutation is applied only when the diversity falls below δ. The map (2.11)-(2.12) has bounded image within [l, u] and need not assign positive probability to every positive-measure Borel set, especially when the swarm is contracted near a current best. Thus Lemma 4.2 cannot be invoked, and the claim that CNO-CPD is globally convergent with probability one is unsupported.
  2. [§5, Lemma 5.1] The equivalence claimed in Lemma 5.1 is false on the boundary of the nonnegative orthant. The equilibrium condition -x + [x - ∇F(x)]_+ = 0 does not imply ∇F(x) = 0: if x_i = 0 and ∇_i F(x) < 0, then [x_i - ∇_i F(x)]_+ = 0 = x_i, so the fixed-point equation holds while the gradient component is nonzero. Consequently the continuous-time neurodynamic (3.5)-(3.7) and the discrete-time update (5.2)-(5.4) need not have the same equilibrium points. This error is load-bearing because the subsequent stability analysis in §6, including Theorem 6.2 and the use of equilibrium points in the proof of Theorem 6.3, and the conclusion that CNO-DTPNN inherits the global convergence of Theorem 4.3, all rely on the equilibrium characterization.
  3. [§6, Theorem 6.3] The proof of Theorem 6.3 asserts that because the sequence {f(x_k)} is monotone non-increasing and bounded, and because (5.11) holds with ∇f(x_k)^T (x_{k+1} - x_k) ≥ 0, it follows that ∇f(x_k) = 0 in the limit. This does not follow: monotone convergence and Armijo-type inequalities with variable step sizes do not by themselves imply convergence to a stationary point unless additional conditions on the step sizes or the descent direction are supplied. The paper does not provide such conditions. Together with the boundary error in Lemma 5.1, this leaves the convergence claim for DTPNN and hence the global-convergence claim for CNO-DTPNN unproved.
minor comments (6)
  1. [§1, Abstract and §8] The text refers to the 'continues CPD' and later to 'continues neurodynamic'; these should read 'continuous'.
  2. [Algorithm 5.1, lines 6, 9, 12] The backtracking updates are written as λ_k ← -β λ_k, which makes the step size negative. The surrounding text says the step size is multiplied by a positive constant 0 < β < 1, so the minus signs appear to be typographical errors; as written, the algorithm does not implement the described backtracking procedure.
  3. [§7, Example 1 and Figure 4] The comparison in Figure 4 reports a single run for several baselines (notably ANLS 'for a successful run'), and no details are given on how many restarts, initializations, or random seeds were used. This makes it difficult to assess whether the reported speedups are statistically meaningful.
  4. [§7, Example 2, Figure 7] The text states that the population size is q = 5 for the collinearity experiments, while the captions of Figure 7 report q = 30; the discrepancy should be resolved.
  5. [§7, Example 2, Case Study I] The description of Case Study I says one factor matrix is highly collinear while 'two other factor matrices (A(2), A(3))' have collinearity 0.4 ≤ μ ≤ 0.6; presumably the first of these should be A(1), since A(3) was already designated as the highly collinear factor. Please clarify.
  6. [§9, Acknowledgment] The acknowledgment thanking 'the editor and two reviewers' is inappropriate for a submitted manuscript and should be removed regardless of the review outcome.

Circularity Check

2 steps flagged · score 5.0 of 10

Theorem 4.3's almost-sure global convergence is essentially Assumption II restated; the proof defines the search support to include Ω and then asserts (4.1), so the central guarantee is carried by its own input rather than by an analysis of the CNO dynamics.

  1. self definitional [Section 4, Assumption II (Eq. 4.1), Lemma 4.2, and Theorem 4.3]
    "Assumption II. Given an arbitrary Borel subset B of Ω (B ⊂ Ω), with a discrete Lebesgue measure µ(B) > 0, assume (4.1) ∏_{k=1}^∞ (1 − µ_k(B)) = 0 ... Assumption II means that with a probability of zero the CNO-CPD fails to find a point of B after infinite time. ... Theorem 4.3. The CNO-CPD is globally convergent with probability one. ... Due to using wavelet mutation and re-initializing the RNNs from the feasible region Ω, the actual support is H(µ_k) = ∪_{i=1}^q H(µ_i^k) ∪ Ω, which means that the search space S covers the feasible region. So Assumption II is also held."

    Lemma 4.2 (Solis–Wets) turns Assumptions I and II into the literal conclusion lim_k P(x^{(k)} ∈ Ω_g) = 1 of Theorem 4.3. Assumption II already states that the search fails to hit every positive-measure B with probability zero after infinite time, i.e., the almost-sure global-reaching property being proved. The proof verifies Assumption II by defining the support to include Ω ('H(µ_k) = ... ∪ Ω'), which only asserts coverage of Ω as a set, not the conditional-probability condition (4.1). PSO updates (2.6)–(2.7) contract particles toward p_best with β1 = β2 = 0.01 and wavelet mutation (2.11)–(2.12) has bounded deterministic-once-φ image, so (4.1) does not follow. The global-convergence theorem thus reduces to an assumption whose content is the theorem.

  2. other [Section 6, paragraph after Theorem 6.3]
    "Combining results presented in Theorems 4.3 and 6.3, we deduce that the CNO-DTPNN convergences to a global minimum with probability one."

    The discrete-time global guarantee is obtained by chaining the local equilibrium convergence of Theorem 6.3 to Theorem 4.3. Since Theorem 4.3's global w.p.1 statement is exactly the content of Assumption II, this second claim inherits the same assumption-as-conclusion structure: no independent argument establishes that the PSO/wavelet dynamics satisfy (4.1).

full rationale

The circularity is localized to the almost-sure global-convergence theorem. The paper's numerical comparisons against HALS, MUR, CCG, CGP, BFGSP, and GradP on random and real-world tensors are external benchmarks and are not circular; the PSO/wavelet implementation is a genuine algorithmic contribution. The cited convergence machinery (Solis–Wets [40], RNN equilibrium lemmas [50, 52]) is by other authors, so this is not a self-citation chain. However, Theorem 4.3 does not derive global convergence from the algorithm's dynamics: Lemma 4.2's Assumption II is a probabilistic restatement of the conclusion, and the proof's attempt to verify it by declaring the support to be ... ∪ Ω conflates set coverage with the conditional-probability product condition (4.1). The later CNO-DTPNN global claim inherits this gap. Other weaknesses, such as Lemma 5.1's boundary assertion that the projected fixed-point equation implies ∇F = 0, are correctness gaps rather than circularity, so they do not raise the circularity score further.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central convergence guarantee depends on the Solis-Wets stochastic-search theorem and on an unverified claim that the PSO-wavelet process samples every positive-measure subset of the feasible set. The discrete stability proof depends on a fixed-point characterization that is false at the nonnegative boundary. Performance comparisons depend on hand-set PSO parameters, swarm sizes, step sizes, and regularizers.

free parameters (6)
  • PSO inertia weight alpha = 0.5
    Set by hand in Section 7; controls particle momentum and affects convergence speed and exploration.
  • PSO acceleration constants beta1, beta2 = 0.01
    Set by hand in Section 7; controls influence of personal and global best on particle velocities.
  • Swarm size q = 5 to 30 depending on experiment
    Chosen per experiment; larger q gives lower relative error in Figure 8 but increases computational cost.
  • Time constants epsilon1, epsilon2, epsilon3 and ODE solver settings = not reported
    Appear in the continuous dynamics (3.5)-(3.7); no values or solver tolerances are given, and real-data experiments use the discrete model.
  • Step size lambda and backtracking constants alpha, beta in Algorithm 5.1 = not fully specified
    Backtracking constants and initial step sizes are not reported, although the Armijo inequality (5.11) is used.
  • Preconditioner regularization delta and log-barrier parameter gamma = not reported
    delta is added in P_a=(C^T C)*(B^T B)+delta I in Section 5, and gamma appears in the log barrier formulation of Section 3.1; neither value is given.
assumptions (4)
  • standard math Equilibrium points of a one-layer projection RNN correspond one-to-one with KKT points of the constrained problem (Lemma 4.1, cited from [52]).
    Used to justify that the continuous neurodynamic stops at stationary points of problem (2.1).
  • ad hoc to paper The stochastic process generated by PSO with wavelet mutation satisfies Assumption II, i.e., a positive probability of hitting every positive-measure subset of the feasible set infinitely often.
    Used in the proof of Theorem 4.3; the proof only claims support coverage of Omega, not the required probability product condition.
  • ad hoc to paper A fixed point x = [x - nabla f]_+ implies nabla f = 0.
    Used in Lemma 5.1 and in the proof of Theorem 6.2; false at boundary points where x_i=0 and the gradient component is positive.
  • standard math Armijo backtracking line search (5.11) with bounded-below objective ensures convergence to a stationary point.
    Standard descent lemma used in Theorem 6.3, though the final step from convergence of f to zero gradient is not fully justified.

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Pith. "Pith review of Nonnegative Tensor Decomposition Via Collaborative Neurodynamic Optimization." pith.science (2026). https://pith.science/paper/Z4MEA5SH

@misc{pith2026241118127,
  author       = {Pith},
  title        = {Pith review of: Nonnegative Tensor Decomposition Via Collaborative Neurodynamic Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4MEA5SH}},
  note         = {Machine review of arXiv:2411.18127}
}
read the original abstract

This paper introduces a novel collaborative neurodynamic model for computing nonnegative Canonical Polyadic Decomposition (CPD). The model relies on a system of recurrent neural networks to solve the underlying nonconvex optimization problem associated with nonnegative CPD. Additionally, a discrete-time version of the continuous neural network is developed. To enhance the chances of reaching a potential global minimum, the recurrent neural networks are allowed to communicate and exchange information through particle swarm optimization (PSO). Convergence and stability analyses of both the continuous and discrete neurodynamic models are thoroughly examined. Experimental evaluations are conducted on random and real-world datasets to demonstrate the effectiveness of the proposed approach.

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Reference graph

Works this paper leans on

56 extracted references · 50 canonical work pages

  1. [15]

    F an and J

    J. F an and J. W ang, A collective neurodynamic optimization approach to nonnegative tensor decomposition, in International Symposium on Neural Networks, Springer, 2017, pp. 207– 213

    work page 2017

  2. [1]

    B. S. Alexandrov, V. G. Stanev, V. V. Vesselinov, and K. Ø. Rasmussen , Nonnegative tensor decomposition with custom clustering for microphase separation of block copolymers, Statistical Analysis and Data Mining: The ASA Data Science Journal, 12 (2019), pp. 302– 310

    work page 2019

  3. [2]

    M. G. Asante-Mensah, S. Ahmadi-Asl, and A. Cichocki , Matrix and tensor completion using tensor ring decomposition with sparse representation , Machine Learning: Science and Technology, 2 (2021), p. 035008

    work page 2021

  4. [3]

    R. L. Burden and J. D. F aires, Numerical analysis, Brooks Cole, 1997

    work page 1997

  5. [4]

    Che and J

    H. Che and J. W ang, A nonnegative matrix factorization algorithm based on a discrete-time projection neural network, Neural Networks, 103 (2018), pp. 63–71

    work page 2018

  6. [5]

    H. Che, J. W ang, and A. Cichocki , Sparse signal reconstruction via collaborative neurody- namic optimization , Neural Networks, (2022)

    work page 2022

  7. [6]

    Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations , The Annals of Mathematical Statistics, (1952), pp

    H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations , The Annals of Mathematical Statistics, (1952), pp. 493–507

    work page 1952

  8. [7]

    Cichocki and A.-H

    A. Cichocki and A.-H. Phan , Fast local algorithms for large scale nonnegative matrix and tensor factorizations , IEICE transactions on fundamentals of electronics, communications and computer sciences, 92 (2009), pp. 708–721

    work page 2009

Show all 56 references
  1. [8]

    Cichocki, R

    A. Cichocki, R. Zdunek, and S.-i. Amari , Hierarchical ALS algorithms for nonnegative matrix and 3d tensor factorization, in International Conference on Independent Component Analysis and Signal Separation, Springer, 2007, pp. 169–176

    2007

  2. [9]

    Cichocki, R

    A. Cichocki, R. Zdunek, A. H. Phan, and S.-i. Amari, Nonnegative matrix and tensor factor- izations: applications to exploratory multi-way data analysis and blind source separation , John Wiley & Sons, 2009

    2009

  3. [10]

    Comon , Tensor toolbox

    P. Comon , Tensor toolbox . http://www.gipsa-lab.fr/ ∼pierre.comon/TensorPackage/ tensorPackage.html

  4. [11]

    De Silva and L.-H

    V. De Silva and L.-H. Lim , Tensor rank and the ill-posedness of the best low-rank approxi- mation problem, SIAM Journal on Matrix Analysis and Applications, 30 (2008), pp. 1084– 1127

    2008

  5. [12]

    Eguchi and Y

    S. Eguchi and Y. Kano , Robustifying maximum likelihood estimation , Tokyo Institute of Statistical Mathematics, Tokyo, Japan, Tech. Rep, (2001)

    2001

  6. [13]

    Eld´en and S

    L. Eld´en and S. Ahmadi-Asl , Solving bilinear tensor least squares problems and application to hammerstein identification , Numerical Linear Algebra with Applications, 26 (2019), p. e2226

    2019

  7. [14]

    F an and J

    J. F an and J. W ang, A collective neurodynamic optimization approach to nonnegative ma- This manuscript is for review purposes only. NONNEGATIVE TENSOR DECOMPOSITION VIA CNO 21 trix factorization , IEEE transactions on neural networks and learning systems, 28 (2016), pp. 2344–2356

    2016

  8. [16]

    R. C. F arias, P. Comon, and R. Redon, Data mining by nonnegative tensor approximation , in 2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP), IEEE, 2014, pp. 1–6

    2014

  9. [17]

    F ´evotte and N

    C. F ´evotte and N. Dobigeon , Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization , arXiv preprint arXiv:1401.5649, (2014)

    2014 arXiv

  10. [18]

    G. H. Golub and C. F. V an Loan, Matrix computations, JHU press, 2013

    2013

  11. [19]

    F. L. Hitchcock , The expression of a tensor or a polyadic as a sum of products , Journal of Mathematics and Physics, 6 (1927), pp. 164–189

    1927

  12. [20]

    F. L. Hitchcock, Multiple invariants and generalized rank of a p-way matrix or tensor, Journal of Mathematics and Physics, 7 (1928), pp. 39–79

    1928

  13. [21]

    Hu and J

    X. Hu and J. W ang, Solving pseudomonotone variational inequalities and pseudoconvex op- timization problems using the projection neural network , IEEE Transactions on Neural Networks, 17 (2006), pp. 1487–1499

    2006

  14. [22]

    Kennedy and R

    J. Kennedy and R. Eberhart , Particle swarm optimization , in Proceedings of ICNN’95- international conference on neural networks, vol. 4, IEEE, 1995, pp. 1942–1948

    1995

  15. [23]

    Kinderlehrer and G

    D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, SIAM, 2000

    2000

  16. [24]

    T. G. Kolda and B. W. Bader , Tensor decompositions and applications , SIAM review, 51 (2009), pp. 455–500

    2009

  17. [25]

    J. P. La Salle , The stability of dynamical systems , SIAM, 1976

    1976

  18. [26]

    Lee and H

    D. Lee and H. S. Seung, Algorithms for non-negative matrix factorization, Advances in neural information processing systems, 13 (2000)

    2000

  19. [27]

    G. Li, Z. Yan, and J. W ang, A one-layer recurrent neural network for constrained nonconvex optimization, Neural Networks, 61 (2015), pp. 10–21

    2015

  20. [28]

    Li and B

    N. Li and B. Li , Tensor completion for on-board compression of hyperspectral images, in 2010 IEEE International Conference on Image Processing, IEEE, 2010, pp. 517–520

    2010

  21. [29]

    X. Li, J. W ang, and S. Kwong , A discrete-time neurodynamic approach to sparsity- constrained nonnegative matrix factorization , Neural Computation, 32 (2020), pp. 1531– 1562

    2020

  22. [30]

    X. Li, J. W ang, and S. Kwong, Boolean matrix factorization based on collaborative neurody- namic optimization with boltzmann machines , Neural Networks, (2022)

    2022

  23. [31]

    Lim and P

    L.-H. Lim and P. Comon , Nonnegative approximations of nonnegative tensors , Journal of Chemometrics: A Journal of the Chemometrics Society, 23 (2009), pp. 432–441

    2009

  24. [32]

    Mihoko and S

    M. Mihoko and S. Eguchi , Robust blind source separation by beta divergence, Neural compu- tation, 14 (2002), pp. 1859–1886

    2002

  25. [33]

    Nesterov and B

    Y. Nesterov and B. T. Polyak , Cubic regularization of newton method and its global per- formance, Mathematical Programming, 108 (2006), pp. 177–205

    2006

  26. [34]

    I. V. Oseledets , Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), pp. 2295–2317

    2011

  27. [35]

    A. H. Phan and A. Cichocki , Extended HALS algorithm for nonnegative Tucker decompo- sition and its applications for multiway analysis and classification , Neurocomputing, 74 (2011), pp. 1956–1969

    2011

  28. [36]

    A.-H. Phan, K. Sobolev, K. Sozykin, D. Ermilov, J. Gusak, P. Tichavsk `y, V. Glukhov, I. Oseledets, and A. Cichocki , Stable low-rank tensor decomposition for compression of convolutional neural network , in European Conference on Computer Vision, Springer, 2020, pp. 522–539

    2020

  29. [37]

    A.-H. Phan, P. Tichavsky, and A. Cichocki , Low complexity damped gauss–newton algo- rithms for candecomp/parafac , SIAM Journal on Matrix Analysis and Applications, 34 (2013), pp. 126–147

    2013

  30. [38]

    Y. Qiu, G. Zhou, Y. Zhang, and S. Xie, Graph regularized nonnegative Tucker decomposition for tensor data representation , in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), IEEE, 2019, pp. 8613–8617

    2019

  31. [39]

    Royer, N

    J.-P. Royer, N. Thirion-Moreau, and P. Comon , Computing the polyadic decomposition of nonnegative third order tensors , Signal Processing, 91 (2011), pp. 2159–2171

    2011

  32. [40]

    F. J. Solis and R. J.-B. Wets , Minimization by random search techniques , Mathematics of operations research, 6 (1981), pp. 19–30

    1981

  33. [41]

    L. R. Tucker , Some mathematical notes on three-mode factor analysis , Psychometrika, 31 This manuscript is for review purposes only. 22 (1966), pp. 279–311

    1966

  34. [42]

    L. R. Tucker et al. , The extension of factor analysis to three-dimensional matrices , Contri- butions to mathematical psychology, 110119 (1964)

    1964

  35. [43]

    Uryasev and P

    S. Uryasev and P. M. Pardalos , Stochastic optimization: algorithms and applications , vol. 54, Springer Science & Business Media, 2013

    2013

  36. [44]

    V an der Maaten and G

    L. V an der Maaten and G. Hinton, Visualizing data using t-sne., Journal of machine learning research, 9 (2008)

    2008

  37. [45]

    W ang, Recurrent neural networks for solving linear matrix equations , Computers & Math- ematics with Applications, 26 (1993), pp

    J. W ang, Recurrent neural networks for solving linear matrix equations , Computers & Math- ematics with Applications, 26 (1993), pp. 23–34

    1993

  38. [46]

    W ang, Recurrent neural networks for computing pseudoinverses of rank-deficient matrices , SIAM Journal on Scientific Computing, 18 (1997), pp

    J. W ang, Recurrent neural networks for computing pseudoinverses of rank-deficient matrices , SIAM Journal on Scientific Computing, 18 (1997), pp. 1479–1493

    1997

  39. [47]

    W ang and G

    J. W ang and G. Wu, Recurrent neural networks for lu decomposition and cholesky factoriza- tion, Mathematical and computer modelling, 18 (1993), pp. 1–8

    1993

  40. [48]

    Z. Wei, H. Zhao, L. Zhao, and H. Yan , Multiscale co-clustering for tensor data based on canonical polyadic decomposition and slice-wise factorization , Information Sciences, 503 (2019), pp. 72–91

    2019

  41. [49]

    S. J. Wright , Numerical optimization, 2006

    2006

  42. [50]

    Xia and J

    Y. Xia and J. W ang , On the stability of globally projected dynamical systems , Journal of Optimization Theory and Applications, 106 (2000), pp. 129–150

    2000

  43. [51]

    Z. Yan, J. F an, and J. W ang, A collective neurodynamic approach to constrained global opti- mization, IEEE transactions on neural networks and learning systems, 28 (2016), pp. 1206– 1215

    2016

  44. [52]

    Z. Yan, J. W ang, and G. Li , A collective neurodynamic optimization approach to bound- constrained nonconvex optimization, Neural networks, 55 (2014), pp. 20–29

    2014

  45. [53]

    Yokota, Q

    T. Yokota, Q. Zhao, and A. Cichocki, Smooth parafac decomposition for tensor completion, IEEE Transactions on Signal Processing, 64 (2016), pp. 5423–5436

    2016

  46. [54]

    Zhang, D

    Y. Zhang, D. Jiang, and J. W ang, A recurrent neural network for solving sylvester equation with time-varying coefficients, IEEE Transactions on Neural Networks, 13 (2002), pp. 1053– 1063

    2002

  47. [55]

    G. Zhou, A. Cichocki, and S. Xie , Decomposition of big tensors with low multilinear rank , arXiv preprint arXiv:1412.1885, (2014)

    2014 arXiv

  48. [56]

    Proof of Theorem 5.3

    appendix. Proof of Theorem 5.3. To prove this theorem, we should consider the following three cases: • Case I. If li ≤ qi k,a ≤ ui, li ≤ qi k,b ≤ ui and li ≤ qi k,c ≤ ui, then since PΩ(qi k,a) = qi k,a, PΩ(qi k,b) = qi k,b, PΩ(qi k,c) = qi k,c, we have    ai k+1 = ai k − ...

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Reviewed August 12, 2026 · model on record in the stance chip above.