Pith. sign in

REVIEW 3 major objections 6 minor 23 references

OCTANE frames autoencoder training as an optimal control problem, letting layer widths emerge from tensor rank truncation.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Autoencoders as continuous-time optimal control problems solved with rank-adaptive tensor compression, yielding memory savings and automatic layer-width profiles on MNIST denoising and deblurring.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A promising but under-supported combination of optimal control and tensor ranks; the numerical error analysis does not cover the actual experimental setup. the 3 major comments →

arxiv 2509.08169 v1 pith:4OW5R227 submitted 2025-09-09 math.OC cs.LG

OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case

Ratna Khatri , Anthony Kolshorn , Colin Olson , Harbir Antil This is my paper
classification math.OC cs.LG MSC 34H0537N4049K1549M4165K1068T0565Z05
keywords autoencodersoptimal controltensor-train decompositionrank-adaptive Euler schemearchitecture discoveryimage denoisingimage deblurringlow-rank tensor manifolds
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper is trying to establish that a deep autoencoder can be modeled as an optimal control problem — the encoder and decoder are coupled differential equations, and training means choosing time-dependent weights and biases to minimize a reconstruction loss. Its central claim is that when the state and adjoint equations are solved with a rank-adaptive explicit Euler integrator on low-rank tensor manifolds, the ranks selected at each step define the network's layer widths, so the architecture is discovered rather than hand-designed. This matters because autoencoder design is usually heuristic, and memory use in deep networks is a practical bottleneck. If the claim holds, compact autoencoders can be trained with far less memory and little data, with the structure of the network adapting to the task.

Core claim

On its own terms, the paper claims that compression in an autoencoder does not have to be imposed by a preset layer-width schedule. In the OCTANE formulation the encoder and decoder are the forward solutions of two coupled ODEs, and each explicit Euler step is rounded to a tensor-train manifold by truncation operators whose tolerances are set by local error bounds. The sequence of selected ranks is then read as the discovered architecture: the rank profile literally gives the layer widths and produces a butterfly-shaped structure. The paper also claims the resulting discrete optimality system is differentiable in the right sense — gradient checks against finite differences pass at the expect

What carries the argument

The load-bearing object is the rank-adaptive explicit Euler integrator applied to tensor-valued ODEs: at each step the right-hand side is first truncated by an operator T_s and the updated state is then truncated by T_r, with tolerances chosen through the error bounds (3a)-(3b) and (6a)-(6b). Working in the tensor-train format — a low-rank decomposition that stores a high-order array through a chain of small core tensors — this single mechanism performs three jobs at once: it discretizes the continuous optimal control ODEs, it keeps the state and adjoint trajectories low-rank, and it generates the rank profile that defines the autoencoder architecture. The reverse-time version of the same sc

Load-bearing premise

The whole scheme depends on the assumption that the rank-truncated explicit Euler steps stay first-order accurate for the coupled state and adjoint optimality system when truncation tolerances are chosen at the aggressive scales used in the experiments; the paper proves one-step local error bounds but not convergence of the full algorithm.

What would settle it

Fix T and run the full OCTANE training loop at decreasing τ (increasing layer count N), and compare the discrete gradient computed by the paper's training procedure to a finite-difference gradient of the same discrete loss. If the mismatch does not fall at roughly first order as τ shrinks, or plateaus because rank truncation dominates, the order-one consistency claim fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Layer widths stop being a hyperparameter: the truncation-selected ranks during forward integration define the discovered autoencoder architecture.
  • Memory use is tied to tensor ranks rather than full data dimensions; the experiments report average savings of about 7–16% for denoising and 47–57% for deblurring on the tested MNIST tasks.
  • Training is feasible with very little data: 20 training images sufficed in the reported denoising and deblurring runs.
  • A practical operating window for the step size and time horizon, roughly 0.3 ≤ τ ≤ 1.3 with T ∈ {10, 15}, is recommended for consistent performance at lower cost.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The rank-profile view suggests a cheap architecture-search alternative: instead of training many candidate widths, one could train once and read the widths off the rank curve; whether this profile is stable across random initializations is a direct test the paper does not report.
  • Because the integration scheme is described as agnostic to the tensor format, the framework should extend to Tucker or HOSVD decompositions; a testable prediction is that equivalent rank profiles and reconstruction quality would emerge in any fixed low-rank format.
  • The paper imposes a symmetric encoder-decoder by setting r_d = flip(r_e); relaxing that symmetry could expose tasks where the encoder needs more capacity than the decoder, and whether the butterfly shape is optimal is left open.
  • The memory savings are reported for the state variables; an end-to-end accounting that includes adjoint trajectories and optimizer state would clarify the true practical gain.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript proposes OCTANE, an autoencoder framework in which the encoder and decoder are defined by coupled ODEs and trained as an optimal control problem. A Lagrangian derivation gives first-order necessary conditions (state, adjoint, and design equations), and the state/adjoint systems are discretized with a rank-adaptive explicit Euler scheme on tensor-train manifolds. The algorithm stores encoder ranks during forward propagation, imposes decoder ranks as the reversal of encoder ranks, and trains with a gradient-based optimizer. Experiments on MNIST digit denoising and deblurring show reconstructions, rank profiles, memory comparisons, and a recommended step-size/layer-count interval.

Significance. The core idea is attractive: a continuous optimal-control formulation of autoencoders, combined with low-rank tensor integration, could provide a principled way to obtain compact architectures. The Lagrangian derivation in Sections 3–4 is self-contained and standard, and the gradient test in Section 6.1.3 is a useful verification of the discrete gradients. The authors also give enough algorithmic detail to make the experiments reproducible. If the convergence and architecture-discovery claims can be repaired, OCTANE would be a valuable contribution. However, as written, the advertised mathematical rigor and the claim that the architecture is automatically discovered are not supported: the truncation tolerances used in all experiments invalidate the stated first-order consistency, and the decoder rank profile is imposed by symmetry rather than emergent.

major comments (3)
  1. [Section 2.1, Eqs. (3a)-(3b); Table 2; Section 6.4] The conditions (3a)-(3b) yield a first-order guarantee only when Ms and Mr are fixed constants independent of tau. In Table 2 the authors set Ms=tau^{-1} and Mr=tau^{-2}, so the right-hand sides of (3a) and (3b) become 1 and 1. Writing e_s=N-T_s(N) and e_r=Y_j+tau T_s(N)-T_r(...), the one-step error of (2) is -tau e_s - e_r + O(tau^2), bounded by Ms tau^2 + Mr tau^2 + O(tau^2)=tau+1+O(tau^2)=O(1). Thus the scheme is not shown to be consistent as tau->0, and the discrete optimality system (13)-(15) is not established as a faithful discretization of (10)-(12). Section 6.4 explicitly confirms that this choice decouples the tolerance from tau. Please take Ms,Mr to be O(1) constants independent of tau, or provide a convergence analysis of the full rank-adaptive scheme under the actual parameter choices, and rerun the experiments under conditions where the claimed order holds.
  2. [Section 5.1(c)-(e); Algorithm 5, line 7] The decoder rank profile is prescribed by r_d=flip(r_e) at Algorithm 5 line 7 and in Section 5.1(c). Consequently only the encoder ranks are selected by the truncation operators; the decoder widths are imposed by symmetry. This contradicts the statement in Section 5.1(e) that the resulting encoder and decoder ranks define the discovered architecture and that the ranks emerge naturally from the dynamics. Please qualify the architecture-discovery claim, or allow the decoder ranks to be chosen independently and test whether a symmetric profile actually emerges.
  3. [Section 6.4, Eq. (23)] The recommended interval tau_proposed=(0.3,1.3) is obtained by intersecting thresholds (alpha<1e-3, SSIM>=0.96, etc.) on the same experimental runs whose behavior it is meant to predict. This is a fit to the data, not a validated recommendation. Independent hyperparameter validation, such as a separate sweep or cross-validation, is needed before presenting (23)-(24) as guidance. In addition, the experiments report no baselines and no error bars, so the comparative claims of effectiveness and the memory-saving percentages in Sections 6.2-6.3 are not yet established.
minor comments (6)
  1. [Section 2.1] The phrase 'order one local truncation error' is imprecise: the underlying Euler step has second-order local truncation error and first-order global error. Please correct the terminology.
  2. [Figure 7 caption] Typo: 'delurring' should be 'deblurring'.
  3. [Figure 6] The leftmost subplot appears to lack a title or axis label indicating 'Reconstruction Error'; please add it.
  4. [Algorithms 1-4] The comments state 'Choose smallest s_j>0 s.t. (3a) holds' and 'If not found, break', but no search procedure or fallback is described. Specify how the ranks are determined when the condition is not met.
  5. [Section 4.1 and Algorithms 1-4] Notation conflict: u_i:=u(t_i) is introduced in Section 4.1, but u is later used for the intermediate update tensor in Algorithms 1-4. Rename one of them to avoid confusion.
  6. [Equation (22)] For Ms=tau^{-1}, Mr=tau^{-2}, the TT tolerances epsilon become independent of tau. This is directly related to Major Comment 1 and should at least be acknowledged as a limitation of the error control.

Circularity Check

1 steps flagged

Decoder rank profile is forced by r_d=flip(r_e), so the 'automated architecture discovery' claim is partly definitional; the core optimal-control derivation is self-contained.

specific steps
  1. self definitional [Section 5.1(c) and (e); Algorithm 5 line 7]
    "Define r_d := flip(r_e) to impose symmetry across the encoder-decoder interface. ... The resulting vectors of encoder and decoder ranks in forward integration, selected by the truncation operators, define the discovered autoencoder architecture. ... The resultant ranks are not learned via optimization; they emerge naturally from the dynamics of the problem and the input data."

    Algorithm 5 line 7 sets r_d <- flip(r_f e), so the decoder rank profile is, by construction, the reverse of the encoder rank profile. The decoder half of the claimed 'discovered autoencoder architecture' therefore carries no independent dynamical information; it is forced to mirror the encoder. The statement that ranks 'emerge naturally from the dynamics of the problem and the input data' is definitionally false for the decoder, making the architecture-discovery claim partially circular. The encoder ranks are still data-dependent, but the full published rank profile is not an emergent output as claimed.

full rationale

The main mathematical derivation is self-contained: the optimal-control formulation, Lagrangian, and state/adjoint/design equations are derived in Section 3 without relying on a load-bearing self-citation; the rank-adaptive Euler scheme is described in Section 2 and attributed to the independent reference [20]. No uniqueness theorem is smuggled in, and no known result is merely renamed. The only significant circularity is the forced decoder rank mirror in Section 5.1(c)/Algorithm 5, which partly undermines the 'automated architecture discovery' claim by defining half of the reported architecture in terms of the other half. The empirical tau-recommendation in Section 6.4 is explicitly presented as a heuristic derived from the plotted experiments and is checked on digit 4, so it is a post-hoc fit rather than a disguised prediction, though it should not be read as a validated general law. The M_s=tau^-1, M_r=tau^-2 choice raises a consistency/correctness issue for the Euler error bounds, but that is a numerical-analysis gap rather than a circular step. Overall score reflects one definitional reduction in a supporting claim, not circularity in the core derivation.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The central claims rest on the accuracy of the rank-adaptive Euler-TT scheme and on the heuristic choices of hyperparameters (tau, T, N, M_s, M_r, regularization weights). The core optimal control derivation is standard and does not require additional axioms beyond standard calculus of variations. No new physical entities are introduced; the rank profile is a computed output, not a postulated object.

free parameters (6)
  • Step size tau = T/N; explored 0.18-3.75; recommended range (0.3,1.3) fitted to data
    Controls the Euler discretization and rank-adaptive tolerance; central to all numerical results.
  • Terminal time T = 10 in main experiments; recommended {10,15}
    Final time of the ODE system; interacts with N to set tau.
  • Number of layers N = 4 to 30 in scans
    Network depth equals number of time steps; recommended even N from tau range.
  • Error constants M_s, M_r = tau^-1, tau^-2 (Table 2)
    Chosen ad hoc to 'decouple' rank reduction from tau, but they void the O(tau) truncation error bounds.
  • Regularization weights lambda1..lambda4 = 1e-5, 1e-5, 1, 1 (denoising); 0, 0, 0.1, 0.1 (deblurring)
    Selected heuristically in (17) and Table 2; no sensitivity analysis.
  • Optimization iterations m1, m2 = 3, 30
    Mini-batch outer iterations and BFGS max iterations; chosen for experimental convenience.
axioms (6)
  • domain assumption The rank-adaptive Euler scheme from [20] provides a stable, first-order accurate approximation of (1) under conditions (3),(6).
    Borrowed from prior work; used without reproof in all discrete state/adjoint equations.
  • standard math Tensor-train rounding satisfies ||a - T(a)|| <= eps ||a|| (from [17, Cor. 2.4]).
    External bound used to set truncation tolerances in (22).
  • standard math The optimal control problem (7) has a stationary point satisfying the first-order necessary conditions (10)-(12).
    Assumes smoothness of tanh and L2 loss; standard calculus of variations.
  • domain assumption Using equal numbers of encoder and decoder layers (N_e = N/2) is sufficient for the considered tasks.
    Imposed in Section 4.1; no analysis of asymmetric splits.
  • domain assumption BFGS with Armijo line search converges to a stationary point of the discretized problem.
    Standard optimizer assumption; no convergence proof for this nonconvex objective.
  • ad hoc to paper The loose truncation tolerances implied by M_s=tau^-1, M_r=tau^-2 still yield sufficiently accurate solutions and meaningful ranks.
    No theory supports this; the experiments are the only evidence, and this is the paper's weakest numerical premise.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case." pith.science (2026). https://pith.science/paper/4OW5R227

@misc{pith2026250908169,
  author       = {Pith},
  title        = {Pith review of: OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4OW5R227}},
  note         = {Machine review of arXiv:2509.08169}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

This paper presents a novel, mathematically rigorous framework for autoencoder-type deep neural networks that combines optimal control theory and low-rank tensor methods to yield memory-efficient training and automated architecture discovery. The learning task is formulated as an optimization problem constrained by differential equations representing the encoder and decoder components of the network and the corresponding optimality conditions are derived via a Lagrangian approach. Efficient memory compression is enabled by approximating differential equation solutions on low-rank tensor manifolds using an adaptive explicit integration scheme. These concepts are combined to form OCTANE (Optimal Control for Tensor-based Autoencoder Network Emergence) -- a unified training framework that yields compact autoencoder architectures, reduces memory usage, and enables effective learning, even with limited training data. The framework's utility is illustrated with application to image denoising and deblurring tasks and recommendations regarding governing hyperparameters are provided.

Figures

Figures reproduced from arXiv: 2509.08169 by Anthony Kolshorn, Colin Olson, Harbir Antil, Ratna Khatri.

Figure 1
Figure 1. Figure 1: Illustration of forward (left) and backward (right) propagation in a 4- layer OCTANE (i.e. 2 encoder layers and 2 decoder layers). The differential equa￾tions are solved using tensor-based, rank-adaptive explicit forward Euler scheme. Note that the reduction in layer size is by virtue of layer rank. map. However, it can be readily adapted to supervised tasks (e.g., classification or regression) by replacin… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of discretized gradients with finite difference approxima￾tions. The black line shows the error, and blue and red are reference lines with slopes 1 and 2, respectively. The expected rate of convergence is obtained. 6.1.4. Tensor Format and Toolbox. We do not restrict the choice of tensor representation or com￾putational toolbox; any format and library may be used. In our experiments, we adopt th… view at source ↗
Figure 3
Figure 3. Figure 3: The rank at layer 0 corresponds to the input [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstructions of representative samples from training (row 1), valida￾tion (row 2) and testing (row 3) data showing successful image denoising performed using OCTANE algorithm with N = {6, 12, 20} layers (columns 2-4), noisy input data as the initial condition (column 1) and reference solution (column 5) [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Rank distributions obtained via OCTANE algorithm for various number of layers N = 6, N = 12, and N = 20 in the image denoising task. Solid lines are encoder ranks and dotted lines are decoder ranks. Note that compression in the network is in the context of rank reduction. computed using MATLAB’s built-in functions. Training and validation metrics are averaged over 20 samples; testing over 1000 samples. Fro… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of memory size in bytes utilized by data in each layer stored as a tensor vs. MATLAB array (non-tensor version) for various number of layers for the image denoising task. Note that the OCTANE architecture saves an average of 7.10% (N = 6 layers), 14.35% (N = 12 layers), and 16.21% (N = 20) memory for each data type (training/validation/testing). Owing to rank-reduction, solving differential equa… view at source ↗
Figure 6
Figure 6. Figure 6: Plots of the reconstruction error, PSNR, and SSIM against the number of layers in the image denoising experiment. More layers generally corresponds to improvement in reconstruction error, but the marginal improvement comes at a relatively greater computational cost. SSIM asserts larger number of layers may not be necessary. PSNR follows the same trend as the reconstruction error. observations suggest that … view at source ↗
Figure 7
Figure 7. Figure 7: Reconstructions of representative samples from training (row 1), valida￾tion (row 2) and testing (row 3) data showing successful image delurring performed using OCTANE algorithm with N = {6, 12, 20} layers (Columns 2-4), blurred input data as the initial condition (column 1) and reference solution (column 5) [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: displays the rank distributions during forward propagation for the deblurring task with N = {6, 12, 20}. Compared to denoising, the ranks drop even further (as low as 9), indicating that OCTANE adapts its architecture based on the task [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Rank distributions obtained via OCTANE algorithm for N = 6, N = 12, and N = 20 in the image deblurring task. Solid lines are encoder ranks and dotted lines are decoder ranks. Owing to rank-reduction, solving differential equation on the low-rank manifold saves significant memory. To assess the impact of layer count in deblurring, [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Plots of the Reconstruction Errors, PSNR, and SSIM for various number of layers in the deblurring experiments. The best results are achieved at N = 20 layers. The testing data generally follows the same trend as training data. Overall, there is minimal improvement in the metrics around N = 10 to N = 20 (relatively expensive), therefore lesser layers are sufficient due to memory saving benefit. 6.4. The in… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of reconstruction errors (zoomed-out (left) and zoomed-in (center )) and SSIM (right) against the number of layers N for various values of final time T, with corresponding τ values mentioned on the plots for each image denoising experiment. These plots demonstrate that 0.3 ≤ τ ≤ 1.3 typically gives the best results, even for small N. 0 4 8 12 16 20 24 28 Number of Layers (N) 0.005 0.01 0.015 0.… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of reconstruction errors (zoomed-out (left) and zoomed￾in (center )) and SSIM (right) against the number of layers N for various values of final time T, with corresponding τ values mentioned on the plots for each image deblurring experiment. These plots demonstrate that 0.3 ≤ τ ≤ 1.3 typically gives the best results, even for small N. The reconstruction error plots (left/center in Figures 11 an… view at source ↗
Figure 13
Figure 13. Figure 13: Reconstructions of representative samples of digit 2 for the image de￾noising (top row) and deblurring (bottom row) experiments, for τ = 0.18 < 0.3 (column 2), τ = 0.63 ∈ τproposed (column 3) and τ = 3.75 > 1.3 (column 4), along with initial condition (column 1) and reference solutions (column 5). Observe that the reconstructions are poor for τ /∈ τproposed, and successful for τ ∈ τproposed [PITH_FULL_IM… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 20 canonical work pages · 3 internal anchors

  1. [1]

    Antil, T

    H. Antil, T. Brown, R. Khatri, A. Onwunta, D. Verma, and M. Warma. Optimal control, numerics, and applica- tions of fractional pdes. In E. Tr´ elat and E. Zuazua, editors,Numerical Control: Part A, volume 23 ofHandbook of Numerical Analysis, pages 87–114. Elsevier, Amsterdam, the Netherlands, 2022

    work page 2022

  2. [2]

    Antil, Z

    H. Antil, Z. Di, and R. Khatri. Bilevel optimization, deep learning and fractional laplacian regularization with applications in tomography.Inverse Problems, 2020

    work page 2020

  3. [3]

    Novel Deep neural networks for solving Bayesian statistical inverse

    H. Antil, H. C. Elman, A. Onwunta, and D. Verma. Novel deep neural networks for solving bayesian statistical inverse.arXiv preprint arXiv:2102.03974, 2021

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  4. [4]

    Antil, R

    H. Antil, R. Khatri, R. L¨ ohner, and D. Verma. Fractional deep neural network via constrained optimization. Machine Learning: Science and Technology, 2(1):015003, Dec 2020

    work page 2020

  5. [5]

    Tensor Decompositions in Deep Learning

    D. Bacciu and D. P. Mandic. Tensor decompositions in deep learning.arXiv preprint arXiv:2002.11835, 2020

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  6. [6]

    D. Bank, N. Koenigstein, and R. Giryes. Autoencoders.arXiv preprint arXiv:2003.05991, 2021

    Pith/arXiv arXiv 2003

  7. [7]

    Neural Architecture Search via Bregman Iterations

    L. Bungert, T. Roith, D. Tenbrinck, and M. Burger. Neural architecture search via bregman iterations.arXiv preprint arXiv:2106.02479, 2021

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  8. [8]

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud. Neural ordinary differential equations.Advances in Neural Information Processing Systems, 2018

    work page 2018

  9. [9]

    Glorot and Y

    X. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. InProceed- ings of the Thirteenth International Conference on Artificial Intelligence and Statistics, volume 9 ofProceedings of Machine Learning Research, pages 249–256, Chia Laguna Resort, Sardinia, Italy, 13–15 May 2010. PMLR

    work page 2010

  10. [10]

    Goodfellow, Y

    I. Goodfellow, Y. Bengio, and A. Courville.Deep Learning. MIT Press, Massachusetts, USA, 2016.http://www. deeplearningbook.org

    work page 2016

  11. [11]

    G¨ unther, L

    S. G¨ unther, L. Ruthotto, J. Schroder, E. Cyr, and N. Gauger. Layer-parallel training of deep residual neural networks.SIAM Journal on Mathematics of Data Science, 2:1–23, 01 2020

    work page 2020

  12. [12]

    C. T. Kelley.Iterative methods for optimization. Frontiers in applied mathematics. SIAM, Philadelphia, USA, 1999. 22 R. KHATRI, A. KOLSHORN, C. OLSON, AND H. ANTIL

    work page 1999

  13. [13]

    Lecun, L

    Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998

    1998

  14. [14]

    Merrill and A

    N. Merrill and A. Eskandarian. Modified autoencoder training and scoring for robust unsupervised anomaly detection in deep learning.IEEE Access, 8:101824–101833, 2020

    work page 2020

  15. [15]

    Merrill and C

    N. Merrill and C. C. Olson. A new autoencoder training paradigm for unsupervised hyperspectral anomaly detection. InIGARSS 2020 - 2020 IEEE International Geoscience and Remote Sensing Symposium, pages 3967– 3970, 2020

    work page 2020

  16. [16]

    Novikov, D

    A. Novikov, D. Podoprikhin, A. Osokin, and D. P Vetrov. Tensorizing neural networks. In C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, editors,Advances in Neural Information Processing Systems, volume 28, Montr´ eal, Canada, 2015. Curran Associates, Inc

    work page 2015

  17. [17]

    I. V. Oseledets. Tensor-train decomposition.SIAM Journal on Scientific Computing, 33(5):2295–2317, 2011

    2011

  18. [18]

    I. V. Oseledets, S. Dolgov, V. Kazeev, O. Lebedeva, and T. Mach. Tt-toolbox.https://github.com/oseledets/TT- Toolbox, 2014

    work page 2014

  19. [19]

    P. Peng, S. Jalali, and X. Yuan. Solving inverse problems via auto-encoders.IEEE Journal on Selected Areas in Information Theory, 1(1):312–323, 2020

    work page 2020

  20. [20]

    Rodgers, A

    A. Rodgers, A. Dektor, and D. Venturi. Adaptive integration of nonlinear evolution equations on tensor manifolds. Journal of Scientific Computing, 92(2), jun 2022

    work page 2022

  21. [21]

    Salakhutdinov and G

    R. Salakhutdinov and G. Hinton. Semantic hashing.International Journal of Approximate Reasoning, 50(7):969– 978, July 2009

    work page 2009

  22. [22]

    Wilkinson

    A. Wilkinson. Application of the tensor train decomposition in machine learning - a study and tradeoffs, 2019

    work page 2019

  23. [23]

    Zhavoronkov, Y

    A. Zhavoronkov, Y. A. Ivanenkov, A. Aliper, M. S. Veselov, V. A. Aladinskiy, A. V. Aladinskaya, V. A. Terentiev, D. A. Polykovskiy, M. D. Kuznetsov, A. Asadulaev, Y. Volkov, A. Zholus, R. R. Shayakhmetov, A. Zhebrak, L. I. Minaeva, B. A. Zagribelnyy, L. H. Lee, R. Soll, D. Madge, L. Xing, T. Guo, and A. Aspuru-Guzik. Deep learning enables rapid identifica...

    work page 2019

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.