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(MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Modeling both polynomial coefficients and feature embeddings as matrix product operators yields feature-order-independent polynomial regressors that beat existing tensor-decomposition baselines.

desk verdict The architecture is genuinely new and worth reviewers' time, but the headline 'improves over TT/MPS' claim currently rests on a benchmark where the TT baseline often fails to run and on an unverified order-independence assumption. read the letter →

arxiv 2607.15916 v1 pith:JXCCFRJB submitted 2026-07-17 cs.LG

Niccolò Ciolli , Anders Vestergaard N{o}rskov , Michael Kastoryano , Petr Taborsky , Morten M{o}rup This is my paper · ORCID
classification cs.LG
keywords tensornetworksmatrixproductoperatorspolynomialregressionmultivariatefeatureorderindependencedecompositionnaturalgradientsymmetrymask
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

The paper introduces (MPO)^2, a framework that parameterizes both the polynomial coefficient tensor and a learned linear transformation of the input features as matrix product operators (MPOs). Because the two MPO layers couple to every feature rather than to feature-specific blocks, the resulting polynomial is independent of feature ordering, unlike tensor-train/matrix-product-state baselines. The framework subsumes earlier polynomial-tensor models and adds structured MPOs for projections, convolutions, and symmetry masking. On tabular regression and classification benchmarks, (MPO)^2 outperforms existing CPD- and TT/MPS-based polynomial models on most datasets, while a convolutional variant achieves strong image-classification accuracy with fewer parameters.

What carries the argument

The matrix product operator (MPO) — a chain of tensors contracted along one shared 'bond' dimension. The polynomial coefficient tensor T is written as one MPO, and the learned feature embedding A as another, so a degree-N polynomial becomes a contraction of the two chains with the input tensor. The masking MPO, built from the Heaviside matrix Θ and the hyperdiagonal tensor I, enforces the ordered sum over non-decreasing feature indices, so each monomial is meant to receive exactly one coefficient.

What would settle it

Enumerate all monomials for a small configuration (e.g., 4 features, degree 3), construct the masking MPO from Θ and I, and check that contracting it with a generic coefficient tensor yields a distinct output index for every non-decreasing multi-index and zero elsewhere. A single duplicate or missing monomial refutes the central construction.

Watch

Extended reading notes

Core claim

The central claim is that an MPO formulation of the full polynomial regression problem — two MPOs, one for coefficients and one for an input-space linear map — is simultaneously more expressive than CPD, free of the feature-order dependence of TT/MPS, and general enough to absorb structured operators as MPO blocks. The authors construct MPO representations for convolutions and a monomial-symmetry mask, and empirically show that the resulting models outperform prior tensor-based polynomial regressors on most of the tested datasets.

Load-bearing premise

The masking MPO is asserted to give exactly one coefficient per unique monomial at general rank, but this one-to-one mapping is not proven; if it is off, the parameter-count and expressive-power advantages of the masked variant collapse.

Editorial extensions

If this is right

  • On the tabular benchmarks reported, (MPO)^2 achieves the best or near-best test metrics among all tensor-network polynomial models on most datasets, with CPD and TT/MPS variants clearly behind.
  • Feature order independence removes the need for manual feature-ordering heuristics that TT/MPS models require.
  • The framework supports three structured MPO blocks — linear projection, convolution, and symmetric masking — and thereby accommodates inductive biases such as translation invariance.
  • Both second-order (alternating natural gradient) and first-order (AdamW) optimizers are viable; the paper reports gradient descent as faster and more memory-efficient with comparable accuracy.
  • On MNIST and Fashion-MNIST, the convolutional (MPO)^2 reaches high test accuracy with substantially fewer parameters than compared tensor-network and CNN-MLP models.

Reading between the lines

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

  • If the masking-MPO bijection holds rigorously, the construction provides an exact, compact parameterization of the symmetric monomial space; that could be reused as a regularizing vocabulary in other polynomial-function learners, such as kernel methods or symbolic-regression priors.
  • The invariant ring, invariant to cyclic permutations, could serve as an inductive bias for exchangeable or rotation-symmetric data where full permutation symmetry is too expensive to enforce.
  • The two-layer MPO recipe generalizes beyond convolutions: any linear operation that commutes with the feature mode can likely be folded into the embedding MPO, suggesting extensions to graph filters, random features, or wavelet transforms.
  • A direct test of the expressivity claim: on a synthetic high-degree polynomial of known low rank, compare M-(MPO)^2 against an unconstrained MPO with the same parameter budget to see whether the mask restores accuracy without rank inflation.
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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

4 major / 4 minor

Summary. The paper proposes (MPO)^2, a tensor-network parameterization for multivariate polynomial regression and classification. The model represents both the polynomial coefficient tensor and an input-feature transformation as matrix product operators (MPOs), with optional structured operators for linear projections, convolutions, and masking of monomial redundancies. The authors claim the resulting polynomial is feature-order independent, more expressive than CPD-based polynomial models, and empirically superior to existing CPD/TT/MPS-based tensor polynomial baselines. Experiments cover 10 regression and 10 classification UCI datasets, plus MNIST/Fashion-MNIST and CIFAR image experiments, with comparisons to TeMPO, CPD, TNML-P/TNML-F, GP, XGBoost, and MLP.

Significance. If the central claims hold, the paper is a useful contribution: it offers a flexible MPO-based architecture with learned feature transforms, a second-order blockwise optimization scheme, and a public codebase. The range of ablations and the inclusion of several structured MPO variants are strengths. However, the headline empirical claim of 'improvement over existing tensor decomposition based polynomial models' is weakened by the frequent failure of the TT/MPS baselines in the reported tables, and the claimed feature-order independence is never directly tested. The unproved masking and ring constructions affect advertised framework components and need rigorous justification.

major comments (4)
  1. [Table II and Section IV] The comparison against TNML baselines is incomplete. In Table II, six of ten TNML-P regression results are flagged 'F' and omitted, and TNML-F reports only three non-F values. Section IV discloses that missing/flagged results are 'due to training instability across all seeds for the model specifications we iterated over.' A baseline that fails to run on most datasets cannot support the conclusion that (MPO)^2 'improves over existing tensor decomposition based polynomial models.' Please report all seeded outcomes (including large negative values), or restrict the claim to datasets where the baseline is stable, and discuss why instability occurs.
  2. [Section I and Conclusions] Feature order independence is a central claimed advantage, but no permutation-ablation experiment is provided. The paper never permutes input feature order for TNML-P/TNML-F or for (MPO)^2 to demonstrate that the former is order-sensitive while the latter is not. The comparison across different datasets is indirect evidence at best. Add experiments that randomly permute feature columns and report test metrics for each ordering, ideally over several permutations.
  3. [Section II-D3, Eqs. (17)-(20)] The masking MPO construction is asserted rather than proved. The text states that the Heaviside/hyperdiagonal construction 'can be rewritten' and then 'extract[s] an MPO,' but the claimed one-to-one correspondence between ordered monomials and coefficient tensor entries is not proven for general MPO rank. If the mask duplicates or drops monomials, the parameter-count and expressivity claims for M-(MPO)^2 fail. Please provide a formal proof, or state precisely the conditions under which the construction works, and clarify the boundary bond indices (the current notation is ambiguous).
  4. [Section II-E, Eq. (22)] The claim that 'the coefficients of the optimal solution are permutation invariant with respect to a basis change, belonging to the fully symmetric space' is not justified. Equation (4) is not permutation invariant for arbitrary \tilde T; the optimal solution need not be fully symmetric unless the data/target has that symmetry, and the linearity of the model alone does not imply it. Moreover, the ring with identical blocks and periodic boundary conditions is only cyclic-invariant, not fully symmetric, and the additional condition that all matrices commute is stated without proof. This underpins the expressivity claim for the Ring variant and needs a corrected argument or an explicit assumption.
minor comments (4)
  1. [General] There are numerous typos and formatting errors, e.g., 'enalbing', 'whit', inconsistent use of r vs. R as ranks, and the notation in Eq. (15) uses b,n in a context where d,n would be expected. A careful proofreading is needed.
  2. [Section II-B, Eq. (2)] The definition of multivariate polynomial in Eq. (2) uses ordered index ranges d_N ≥ d_{N-1} ≥ ... ≥ d_1, but the subsequent Type I and Type II formulations in Eqs. (3)-(4) sum over all d(N). Clarify how the ordered form relates to the unordered tensor representation, especially when discussing symmetric coefficients.
  3. [Section II-G] The complexity analysis would benefit from a summary table with parameter counts for each variant, not just arithmetic costs. Currently the reader must reconstruct the block dimensions from the prose.
  4. [Figure 5] The image experiments report only average accuracy curves without error bars or standard deviations; reporting variability across seeds would strengthen the comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: (MPO)^2's central constructions are explicit and its comparisons are external, not self-fulfilling definitions or fitted predictions.

full rationale

I walked the claimed derivation chain. The core model (Sec. II-C, Eqs. 6-7) is an explicit architectural definition: two MPO layers for coefficients and feature maps. Improved expressiveness is attributed to the external tensor-train result [31], not to the authors' own prior work. The masking MPO (Sec. II-D3, Eqs. 17-21) is a constructive gadget built from Heaviside and hyperdiagonal tensors; it is not a parameter fitted to the data, and any failure of the claimed one-to-one monomial mapping would be a correctness gap rather than a circular reduction. The ring structure (Sec. II-E) is introduced as a defined ansatz; the assertion that the optimal solution is permutation invariant is an unsupported modeling premise, not a prediction recovered from fitting, so it is not circularity. The only self-citation is the footnote disclosing a non-archival workshop version [1]; it is not load-bearing. All reported results are direct benchmark comparisons across external baselines; no fitted parameter is relabeled as a prediction. The acknowledged missing/unstable TNML results (Sec. IV, Table II) weaken the empirical case but do not make the contribution equivalent to its inputs. Overall, no step reduces, by the paper's own equations or by self-citation, to its own input.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim is empirical/architectural, so the main free parameters are model capacity choices (degree, ranks, projection dimension, regularization) tuned on validation; they are not fitted to enforce a target result. The main structural axioms are standard tensor-network expressivity imported from [31] and domain assumptions about optimizer convergence and polynomial symmetry. One ad hoc assumption is the exact monomial mask; if incorrect, only the M-MPO variant is affected, but the paper presents it as a general capability.

free parameters (5)
  • Polynomial degree N = varied in validation grid
    Model capacity hyperparameter; chosen via validation for each dataset.
  • MPO rank R (coefficient tensor T) = varied; e.g., pixel rank 2/8/16, patch rank 1/2/5
    Controls expressivity and parameter count; selected on validation for image and tabular experiments.
  • MPO rank R' (input transform A) = varied; set to 1 for L-MPO
    Controls capacity of the feature embedding; L-MPO uses rank 1 to impose independent subspace transforms.
  • Projection dimension D' = not specified numerically; D' << D
    L-MPO reduces input dimensionality; chosen by hand or validation.
  • Tikhonov regularization schedule (lambda_start, gamma) = 5.0, 0.25
    Stabilizes natural-gradient Hessian; hand-set and exponentially decayed per sweep.
assumptions (4)
  • standard math MPO decompositions can represent any tensor given sufficient rank and MPO expressivity exceeds CPD (Oseledets [31]).
    Relied on in Section II-C; imported from the cited external result.
  • domain assumption The optimal Type-II polynomial coefficient tensor can be assumed permutation invariant/symmetric.
    Used to justify the invariant ring in Section II-E; true in function space but not obviously for the restricted MPO ring parameterization without the commuting-block constraint.
  • domain assumption Blockwise natural gradient (ALS) with Tikhonov regularization converges to a good solution for all datasets.
    Required for the benchmark results; no convergence guarantees are provided.
  • ad hoc to paper The masking MPO construction (Eqs. 18-20) implements the exact symmetric polynomial without duplicating or dropping monomials.
    Central derivation in Section II-D3, asserted but not proved; if false the M-MPO variant loses its claimed parameter reduction.

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Cite this review

Pith. "Pith review of (MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators." pith.science (2026). https://pith.science/paper/JXCCFRJB

@misc{pith2026260715916,
  author       = {Pith},
  title        = {Pith review of: (MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JXCCFRJB}},
  note         = {Machine review of arXiv:2607.15916}
}
abstract

Central to machine learning and signal processing is the ability to perform universal function approximation and learn complex input-output relationships from limited numbers of observations. Multivariate polynomial models offer a natural way to express such relationships through multiplicative feature interactions, but their coefficient tensors grow exponentially in size with the polynomial degree. Existing tensorized polynomial models reduce this cost, yet canonical polyadic decompositions have rank-limited expressivity, and tensor train formulations are feature order dependent. We introduce Multivariate Polynomial Optimization based on Matrix Product Operators (MPO)$^2$, a framework that combines learned MPO feature embeddings with compact polynomial weight tensors. This yields feature order independent polynomial representations that can incorporate structured operators such as projections, convolutions, and masks for weight tensor symmetries. Across regression and classification benchmarks, (MPO)$^2$ improves over existing tensor decomposition based polynomial models and provides a flexible alternative for efficient polynomial function approximation.

Figures

Figures reproduced from arXiv: 2607.15916 by the authors.

Figure 1
Figure 1. Graphical representation of (a) a tensor, (b) the product of two matrices and (c) the matrix product operator (MPO). [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Existing tensor network modeling procedures for multivariate polynomial regression based on (a) the CPD decomposition, (b) the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. (a) The convolution (MPO)2 network, with the top layer of coefficients T acting on the convolved inputs Xn through the kernels Gn. This form is the one used in the implementation of the code, since it requires contractions between smaller spaces. (b) The same network rewritten as an (MPO)2 : the middle layer is composed, for each column, of an identity block (I ⊗ I) which connects horizontal legs of the MPO independ… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (a) Diagrammatic representation of the masking MPO in Equation 18. (b) Diagrammatic representation of a block of the masking [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Accuracy on the test set for MNIST and Fashion MNIST classification tasks as function of parameters. TeMPO as described in [21] [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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Reference graph

Works this paper leans on

55 extracted references · 1 canonical work pages

  1. [1]

    (MPO)ˆ2: Multivariate polynomial optimization based on matrix product operators,

    N. Ciolli, A. V. Nørskov, M. Kastoryano, P. Taborsky, and M. Mørup, “(MPO)ˆ2: Multivariate polynomial optimization based on matrix product operators,” 2026. [Online]. Available: https://openreview.net/forum?id=LPXdN9s1JA

    2026

  2. [2]

    Multilayer feedfor- ward networks are universal approximators,

    K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedfor- ward networks are universal approximators,”Neural networks, vol. 2, no. 5, pp. 359–366, 1989

    1989

  3. [3]

    C. K. Williams and C. E. Rasmussen,Gaussian processes for machine learning. MIT press Cambridge, MA, 2006, vol. 2, no. 3

    2006

  4. [4]

    The variational gaussian process,

    D. Tran, R. Ranganath, and D. M. Blei, “The variational gaussian process,”Iclr, 2016

    2016

  5. [5]

    Attention is all you need,

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, /suppress L. Kaiser, and I. Polosukhin, “Attention is all you need,” Advances in neural information processing systems, vol. 30, 2017

    2017

  6. [6]

    Long short-term memory,

    S. Hochreiter and J. Schmidhuber, “Long short-term memory,” Neural computation, vol. 9, no. 8, pp. 1735–1780, 1997

    1997

  7. [7]

    Language modeling with gated convolutional networks,

    Y. N. Dauphin, A. Fan, M. Auli, and D. Grangier, “Language modeling with gated convolutional networks,” inInternational conference on machine learning. Pmlr, 2017, pp. 933–941

    2017

  8. [8]

    Multiplicative interactions and where to find them,

    S. M. Jayakumar, W. M. Czarnecki, J. Menick, J. Schwarz, J. Rae, S. Osindero, Y. W. Teh, T. Harley, and R. Pascanu, “Multiplicative interactions and where to find them,” inInterna- tional conference on learning representations, 2020

    2020

Show all 55 references
  1. [9]

    Why does deep and cheap learning work so well?

    H. W. Lin, M. Tegmark, and D. Rolnick, “Why does deep and cheap learning work so well?”Journal of Statistical Physics, vol. 168, pp. 1223–1247, 2017

    2017

  2. [10]

    The pi-sigma network: An efficient higher- order neural network for pattern classification and function ap- proximation,

    Y. Shin and J. Ghosh, “The pi-sigma network: An efficient higher- order neural network for pattern classification and function ap- proximation,” inIJCNN-91-Seattle international joint conference on neural networks, vol. 1. Ieee, 1991, pp. 13–18

    1991

  3. [11]

    Ridge polynomial networks,

    ——, “Ridge polynomial networks,”IEEE Transactions on neural networks, vol. 6, no. 3, pp. 610–622, 1995

    1995

  4. [12]

    A sigma-pi-sigma neural network (spsnn),

    C.-K. Li, “A sigma-pi-sigma neural network (spsnn),”Neural Processing Letters, vol. 17, pp. 1–19, 2003

    2003

  5. [13]

    A new sigma-pi-sigma neural network based on l1 and l2 regularization and applications,

    J. Jiao and K. Su, “A new sigma-pi-sigma neural network based on l1 and l2 regularization and applications,”AIMS Mathematics, vol. 9, no. 3, pp. 5995–6012, 2024. [Online]. Available: https://www.aimspress.com/aimspress-data/math/ 2024/3/PDF/math-09-03-293.pdf

    2024

  6. [14]

    A recurrent sigma- pi-sigma neural network,

    F. Deng, S. Liang, K. Qian, J. Yu, and X. Li, “A recurrent sigma- pi-sigma neural network,”Scientific Reports, vol. 14, p. 84299, 2024

    2024

  7. [15]

    Training sigma-pi neural networks with the grey wolf optimization algorithm,

    C. Sarıkaya, E. Bas, and E. Egrioglu, “Training sigma-pi neural networks with the grey wolf optimization algorithm,”Granular Computing, vol. 8, no. 5, pp. 981–989, 2023

    2023

  8. [16]

    Exploring the approximation capabilities of multiplicative neural networks for smooth functions,

    I. Ben-Shaul, T. Galanti, and S. Dekel, “Exploring the approximation capabilities of multiplicative neural networks for smooth functions,”arXiv preprint arXiv:2301.04605, 2023. [Online]. Available: https://arxiv.org/abs/2301.04605

    2023 arXiv

  9. [17]

    Multiplicative couplings facilitate rapid learning and information gating in recurrent neural networks,

    X. Zhang, M. Altrabulsi, W. Xu, R. Wimmer, M. M. Halassa, and Z. S. Chen, “Multiplicative couplings facilitate rapid learning and information gating in recurrent neural networks,”bioRxiv, 2025

    2025

  10. [18]

    Algebraic and optimization based algorithms for multivariate regression using symmetric tensor decomposition,

    S. Hendrikx, M. Bouss´ e, N. Vervliet, and L. De Lathauwer, “Algebraic and optimization based algorithms for multivariate regression using symmetric tensor decomposition,” in2019 IEEE 8th International Workshop on Computational Advances in Multi- Sensor Adaptive Processing (CA...

    2019

  11. [19]

    Tensor-based multivariate polynomial optimization with application in blind identification,

    M. Ayvaz and L. D. Lathauwer, “Tensor-based multivariate polynomial optimization with application in blind identification,” inProc. European Signal Processing Conference (EUSIPCO), 2021, pp. 1080–1084

    2021

  12. [20]

    Regres- sion and classification with spline-based separable expansions,

    N. Govindarajan, N. Vervliet, and L. De Lathauwer, “Regres- sion and classification with spline-based separable expansions,” Frontiers in big Data, vol. 5, p. 688496, 2022

    2022

  13. [21]

    CPD-Structured Multivariate Polynomial Optimization,

    M. Ayvaz and L. De Lathauwer, “CPD-Structured Multivariate Polynomial Optimization,”Frontiers in Applied Mathematics and Statistics, vol. 8, 3 2022. [Online]. Available: https: //doi.org/10.3389%2Ffams.2022.836433

    2022

  14. [22]

    Interpretable bayesian tensor network kernel machines with automatic rank and feature selection,

    A. Kilic and K. Batselier, “Interpretable bayesian tensor network kernel machines with automatic rank and feature selection,”

  15. [23]

    Deep polynomial neural networks,

    G. G. Chrysos, S. Moschoglou, G. Bouritsas, J. Deng, Y. Pana- gakis, and S. Zafeiriou, “Deep polynomial neural networks,” IEEE Transactions on Pattern Analysis and Machine Intelli- gence, vol. 44, no. 8, pp. 4021–4034, 2022

    2022

  16. [24]

    Augmenting deep classifiers with polynomial neural networks,

    G. G. Chrysos, M. Georgopoulos, J. Deng, J. Kossaifi, Y. Pana- gakis, and A. Anandkumar, “Augmenting deep classifiers with polynomial neural networks,” inEuropean Conference on Com- puter Vision. Springer, 2022, pp. 692–716

    2022

  17. [25]

    Supervised learning with tensor networks,

    E. Stoudenmire and D. J. Schwab, “Supervised learning with tensor networks,”Advances in neural information processing systems, vol. 29, 2016

    2016

  18. [26]

    A block-sparse tensor train format for sample-efficient high-dimensional polynomial regression,

    M. G¨ otte, R. Schneider, and P. Trunschke, “A block-sparse tensor train format for sample-efficient high-dimensional polynomial regression,” 2021. [Online]. Available: https://arxiv.org/abs/ 2104.14255

    2021 arXiv

  19. [28]

    Y. Liu, J. Liu, Z. Long, C. Zhu, Y. Liu, J. Liu, Z. Long, and C. Zhu,Tensor regression. Springer, 2022

    2022

  20. [29]

    Tensor methods in computer vision and deep learning,

    Y. Panagakis, J. Kossaifi, G. G. Chrysos, J. Oldfield, M. A. Nicolaou, A. Anandkumar, and S. Zafeiriou, “Tensor methods in computer vision and deep learning,”Proceedings of the IEEE, vol. 109, no. 5, pp. 863–890, 2021

    2021

  21. [30]

    Tensornetwork for machine learning,

    S. Efthymiou, J. Hidary, and S. Leichenauer, “Tensornetwork for machine learning,” 2019. [Online]. Available: https://arxiv. org/abs/1906.06329

    2019 arXiv

  22. [31]

    Tensor-train decomposition,

    I. V. Oseledets, “Tensor-train decomposition,”SIAM Journal on Scientific Computing, vol. 33, no. 5, pp. 2295–2317, 2011

    2011

  23. [32]

    Random feature maps for dot product kernels,

    P. Kar and H. Karnick, “Random feature maps for dot product kernels,” inAistats, ser. Jmlr: W&cp, vol. 22, 2012, pp. 583–591. [Online]. Available: https://proceedings.mlr.press/v22/kar12/ kar12.pdf

    2012

  24. [33]

    Emergence of simple-cell receptive field properties by learning a sparse code for natural images,

    B. A. Olshausen and D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,”Nature, vol. 381, no. 6583, pp. 607–609, 1996

    1996

  25. [34]

    The alternating linear scheme for tensor optimization in the tensor train format,

    S. Holtz, T. Rohwedder, and R. Schneider, “The alternating linear scheme for tensor optimization in the tensor train format,” SIAM J. Sci. Comput., vol. 34, no. 2, p. 683–713, Mar. 2012. [Online]. Available: https://doi.org/10.1137/100818893

    2012 doi

  26. [35]

    The density-matrix renormalization group in the age of matrix product states,

    U. Schollw¨ ock, “The density-matrix renormalization group in the age of matrix product states,”Annals of Physics, vol. 326, no. 1, pp. 96–192, 2011, january 2011 Special Issue. [Online]. 13 Available: https://www.sciencedirect.com/science/article/pii/ S0003491610001752

    2011

  27. [36]

    Exponentially-convergent strategies for defeating the runge phenomenon for the approximation of non-periodic functions, part i: single-interval schemes,

    J. P. Boyd and J. R. Ong, “Exponentially-convergent strategies for defeating the runge phenomenon for the approximation of non-periodic functions, part i: single-interval schemes,”Comput. Phys, vol. 5, no. 2-4, pp. 484–497, 2009

    2009

  28. [37]

    L. N. Trefethen,Approximation theory and approximation practice, extended edition. Siam, 2019

    2019

  29. [38]

    The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format,

    S. Holtz, T. Rohwedder, and R. Schneider, “The Alternating Linear Scheme for Tensor Optimization in the Tensor Train Format,”SIAM Journal on Scientific Computing, vol. 34, no. 2, pp. A683–a713, 1 2012. [Online]. Available: https://doi.org/10.1137%2F100818893

    2012

  30. [39]

    Xgboost: A scalable tree boosting system,

    T. Chen and C. Guestrin, “Xgboost: A scalable tree boosting system,” inProceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, 2016, pp. 785–794

    2016

  31. [40]

    Scikit-learn: Machine learning in Python,

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,” Journal of Machine Learni...

    2011

  32. [41]

    The uci machine learning repository,

    M. Kelly, R. Longjohn, and K. Nottingham, “The uci machine learning repository,” https://archive.ics.uci.edu, 2019

    2019

  33. [42]

    Fixing weight decay regularization in adam,

    I. Loshchilov and F. Hutter, “Fixing weight decay regularization in adam,”CoRR, vol. abs/1711.05101, 2017. [Online]. Available: http://arxiv.org/abs/1711.05101

    2017 arXiv

  34. [43]

    Supervised learning with quantum-inspired tensor networks,

    E. M. Stoudenmire and D. J. Schwab, “Supervised learning with quantum-inspired tensor networks,” inAdvances in Neural Information Processing Systems (NeurIPS), vol. 29, 2016, pp. 4799–4807. [Online]. Available: https://arxiv.org/abs/1605.05775

    2016 arXiv

  35. [44]

    Adam: A method for stochastic optimization,

    D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”arXiv preprint arXiv:1412.6980, 2014

    2014 arXiv

  36. [45]

    Ciolli, “Mpo2,” https://git.kosmon.org/nicco/MPO2, 2026

    N. Ciolli, “Mpo2,” https://git.kosmon.org/nicco/MPO2, 2026

    2026

  37. [46]

    Optimization of cement-slag-based stabilizer proportions and macro-micro properties research of solidified soil,

    Q. Zhang, S. Chen, Y. Chen, S. Yu, B. Feng, and W. Gao, “Optimization of cement-slag-based stabilizer proportions and macro-micro properties research of solidified soil,”Processes, vol. 13, no. 12, 2025. [Online]. Available: https://www.mdpi. com/2227-9717/13/12/3855

    2025

  38. [47]

    Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions,

    B. Bahmani, H. S. Suh, and W. Sun, “Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions,”Computer Methods in Applied Mechanics and Engineering, vol. 422, p. 116827, Mar. 2024. [Online]. Available: http://dx.doi.org/10...

    2024

  39. [48]

    Characterization of hybrid composites with polyester waste fibers, olive root fibers and coir pith micro-particles using mixture design analysis for structural applications,

    M. R. Tufail, H. Jamshaid, R. Mishra, U. Hussain, M. Tichy, and M. Muller, “Characterization of hybrid composites with polyester waste fibers, olive root fibers and coir pith micro-particles using mixture design analysis for structural applications,”Polymers, vol. 13, no. 14, ...

    2021

  40. [49]

    Multi-objective optimisation of the mechanical properties of rice husk ash–modified lateritic concrete,

    O. S. Ogbo, E. O. Momoh, E. E. Ndububa, B. I. Abdulkarim, A. G. Amuda, and O. O. Afolayan, “Multi-objective optimisation of the mechanical properties of rice husk ash–modified lateritic concrete,”European Journal of Environmental and Civil Engineering, vol. 30, no. 1, p. 26634...

    2026

  41. [50]

    Using response surface models to analyze drug combinations,

    N. R. Twarog, N. E. Martinez, J. Gartrell, J. Xie, C. L. Tinkle, and A. A. Shelat, “Using response surface models to analyze drug combinations,”Drug Discovery Today, vol. 26, no. 8, pp. 2014–2024, 2021. [Online]. Available: https://www.sciencedirect. com/science/article/pii/S1...

    2014

  42. [51]

    Oil-recovery predictions for surfactant polymer flooding,

    K. Rai, R. T. Johns, M. Delshad, L. W. Lake, and A. Goudarzi, “Oil-recovery predictions for surfactant polymer flooding,” Journal of Petroleum Science and Engineering, vol. 112, pp. 341–350, 2013. [Online]. Available: https://www.sciencedirect. com/science/article/pii/S0920410...

    2013

  43. [52]

    The mnist database of handwritten digit images for machine learning research [best of the web],

    L. Deng, “The mnist database of handwritten digit images for machine learning research [best of the web],”IEEE signal processing magazine, vol. 29, no. 6, pp. 141–142, 2012

    2012

  44. [53]

    Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms,

    H. Xiao, K. Rasul, and R. Vollgraf, “Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms,” arXiv preprint arXiv:1708.07747, 2017

    2017 arXiv

  45. [54]

    The probabilistic tensor decomposition toolbox,

    J. L. Hinrich, K. H. Madsen, and M. Mørup, “The probabilistic tensor decomposition toolbox,”Machine Learning: Science and Technology, vol. 1, no. 2, p. 025011, jun 2020. [Online]. Available: https://dx.doi.org/10.1088/2632-2153/ab8241

    2020 doi

  46. [2015]

    Available: https://arxiv.org/abs/1504.01697

    [Online]. Available: https://arxiv.org/abs/1504.01697

    arXiv

  47. [2025]

    Available: https://arxiv.org/abs/2507.11136

    [Online]. Available: https://arxiv.org/abs/2507.11136

    arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.