Pith. sign in

REVIEW 4 major objections 5 minor 38 references

Multivariate Convolutional Sparse Coding with Low Rank Tensor

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Adding a low-rank tensor structure to convolutional sparse coding lets a single model encode multivariate signals with far fewer nonzero coefficients while maintaining or improving reconstruction accuracy.

desk verdict A genuinely new tensor-CSC model with a clean Kruskal-regression link, but the formal constraint set makes the rank-1 sparsity claim false and the algorithm solves a different problem. read the letter →

arxiv 1908.03367 v1 pith:YFWODAAL submitted 2019-08-09 stat.ML cs.LG

Pierre Humbert (CMLA) , Julien Audiffren (CMLA) , Laurent Oudre (L2TI) , Nicolas Vayatis (CMLA) This is my paper · ORCID
classification stat.MLcs.LG MSC 15A69
keywords convolutionalsparsecodingtensordecompositionCPlow-rankKruskalregressionmultivariatesignalalternatingoptimizationdictionarylearning
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 proposes K-CSC, a convolutional sparse coding model for multivariate (tensor) signals that constrains each activation tensor to be both element-wise sparse and of low CP-rank. The central claim is that this low-rank constraint makes the encoding far more efficient: the number of activation parameters drops from a product of mode sizes to a sum, and experiments on synthetic tensors, animated color images, and fMRI data show that the resulting AK-CSC algorithm reconstructs signals using far fewer nonzero coefficients than an unconstrained ADMM baseline at equal or better accuracy. If true, this gives a practical way to scale convolutional sparse coding to high-order, high-dimensional data such as video and neuroimaging, and links the model to Kruskal tensor regression, from which the paper draws theoretical support.

What carries the argument

The load-bearing object is the CP (Canonical Polyadic) decomposition of the activation tensors, written through the Kruskal operator, which represents each activation as a sum of $R$ outer products of one-dimensional factors: $Z_k = \sum_{r=1}^R z^{(1)}_{k,r} \circ \cdots \circ z^{(p)}_{k,r}$. This representation cuts the number of activation parameters from $K\prod_i m_i$ to $KR\sum_i m_i$, and it turns each multidimensional convolution with an activation into a separable convolution that can be evaluated through the FFT. The other key identity is the equivalence in Proposition 2, which rewrites the convolution as an inner product with a circulant tensor, showing K-CSC is a Kruskal tensor regression; this is what lets the Z-step be solved by standard multi-channel CSC solvers and gives the model its theoretical footing.

What would settle it

Run AK-CSC on a synthetic tensor generated with known CP-rank and a known dictionary, then check whether the recovered $Z$ factors have rows of unit norm at every iteration and whether reconstruction error drops sharply only when the user-supplied rank equals the true rank. If the unconstrained $Z$-step systematically produces rows with norm far from one, or if the error profile shows no transition at the true rank, the paper's central efficiency claim would fail.

Watch

Extended reading notes

Core claim

K-CSC writes the observed tensor signal as a sum over $K$ dictionary atoms convolved with activation tensors $Z_k$, and requires each $Z_k$ to factor as a CP decomposition into $R$ rank-one terms via the Kruskal operator. The paper shows that, under mild assumptions, this convolutional model is mathematically equivalent to a rank-$R$ Kruskal tensor regression, where each convolution with a rank-one activation becomes an inner product with a translated circulant tensor. On the algorithmic side, the paper introduces AK-CSC, an alternating scheme in which each mode's activation update reduces to a standard multi-channel convolutional sparse coding problem, and each dictionary update is a smooth convex problem. The paper's empirical claim is that this low-rank parametrization is what buys efficiency: with the same sparsity level, AK-CSC uses fewer nonzero coefficients than unconstrained ADMM and gives better or comparable reconstruction on synthetic tensors, a 30x30x3x20 animated color sequence, and a 31x37x31 fMRI volume.

Load-bearing premise

The whole procedure depends on the claim that, despite the full problem being nonconvex, minimizing over each $Z$ block separately is a convex problem whose unconstrained solve still respects the unit-norm row constraints of the activation factors; the paper states this but supplies no proof or convergence theorem.

Editorial extensions

If this is right

  • For a tensor signal of order $p$, the activation parameter count falls from a product of mode sizes to a sum, so the model stays tractable as the number of modes grows.
  • The equivalence to Kruskal tensor regression means statistical guarantees and regularization strategies from low-rank tensor regression carry over to the convolutional model.
  • Because the low-rank form uses separable filters, each convolution can be computed via FFT, reducing filtering cost from $O(n_1 n_2 w_1 w_2)$ to $O(n_1 n_2 (w_1 + w_2))$ for two-dimensional images.
  • With $R=1$ or $p=1$ the model reduces to previously studied rank-one multivariate CSC and to standard univariate convolutional dictionary learning, so the framework unifies those methods.

Reading between the lines

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

  • The empirical flatness of the loss when $R$ exceeds the true rank suggests reconstruction error alone cannot identify the true rank; a rank-selection rule based on hold-out data or an extra penalty would be a natural next step.
  • Because the low-rank constraint already encodes strong structure, the same framework could plausibly serve as a denoiser or a completion method for partially observed multivariate signals, though the paper does not test this.
  • The paper's convergence evidence is empirical only; a formal guarantee that AK-CSC reaches a stationary point of problem (4) would be needed before relying on it in high-stakes applications.
  • If the model is correct, a signal whose true activation rank exceeds the user-supplied $R$ should reconstruct poorly; this is a direct prediction that synthetic experiments with known rank could test.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes K-CSC, a convolutional sparse coding model for multivariate tensor signals in which activation tensors are constrained to have low CP rank and elementwise sparsity. It proves an algebraic equivalence (Proposition 2) between the K-CSC model and a Kruskal tensor regression problem, and it introduces an alternating optimization algorithm (AK-CSC) that alternates between updating one CP factor block per mode and updating the dictionary. The paper evaluates AK-CSC on synthetic third-order tensors, an RGB animation tensor, and an fMRI tensor, comparing against an ADMM-based convolutional sparse coding solver. The central claims are that low-rank activation structure yields more parsimonious encodings with comparable or better reconstruction, and that the Kruskal-regression equivalence provides theoretical support for the model.

Significance. The algebraic reduction of multivariate convolutional sparse coding to Kruskal tensor regression is a useful conceptual bridge and is, as far as I can tell, correctly derived for unconstrained factor matrices. The parameter-count argument (K(R sum m_i) activations versus K prod m_i for the full tensor) is well motivated, and comparing against ADMM on real tensor data is a reasonable first evaluation strategy. However, the paper does not ship code, does not report error bars for most quantitative comparisons, and, more importantly, the formal model in Eq. (4) is not the problem solved by Algorithm 1. Because the central empirical claim is made for the rank-1 real-data experiments, where the stated row-normalization constraint would eliminate sparsity entirely, the main contribution needs substantial reworking before the results can be accepted.

major comments (4)
  1. [Section 3, Eq. (4) and Algorithm 1] The feasible set S in Eq. (4) requires ||(Z_{k,l})_{i,:}||_F = 1 for every row of every factor matrix. For R=1, which is the setting used in the real-data experiments of Section 5 (Figures 3 and 4), each row is a single scalar, so every entry of every Z_{k,l} has modulus one; the l1 penalty in (4) is then constant and cannot induce sparsity. Algorithm 1, line 14, solves an unconstrained multichannel CSC problem without the unit-row constraint. The method evaluated in the experiments is therefore not a solver for the stated model (4), and the reported advantage in the number of nonzero coefficients cannot be attributed to the proposed formalism.
  2. [Section 3, paragraph before Algorithm 1] The sentence 'The non-convex problem (4) is convex with respect to each Z block' is not justified as written. For fixed values of all other blocks, the data-fidelity term is indeed convex in Z_{k,l}, but the unit-row normalization constraint in S is nonconvex, so the constrained block subproblem is not a convex problem. Either the constraints should be removed or replaced by a convex surrogate, or the convexity claim must be explicitly restricted to the unconstrained relaxation actually solved in (5).
  3. [Section 2.3, Proposition 2] Proposition 2 establishes an algebraic equivalence with Kruskal regression for unconstrained factor matrices, but the paper's actual constrained model (4) is defined on S with row-normalized Z_{k,l}. The proposition does not cover the normalized feasible set, so it does not, by itself, justify the 'interesting theoretical guarantees' claimed in the abstract and in Section 2.3. No convergence or recovery theorem is supplied for Algorithm 1 either; Figure 2A is an empirical loss curve, not a proof that the alternating scheme converges.
  4. [Section 3, Algorithm 1 and Section 5] The paper claims to provide an efficient optimization algorithm for the K-CSC model, but it gives no convergence analysis for the alternating scheme. Given that the objective is nonconvex and that the algorithm is inconsistent with the stated constraints, the status of the iterates is unclear. In addition, the quantitative comparison in Section 5 is incomplete: the real-data evaluations in Figures 3B and 4 are qualitative, and no code is provided, which limits reproducibility of the central empirical claim.
minor comments (5)
  1. [Section 3, Eq. (5) and Algorithm 1 line 14] Equation (5) includes the ridge penalty beta_l ||Z_{k,l}||_F^2, but Algorithm 1 line 14 solves the subproblem without the corresponding beta_m term; the algorithm and the stated objective should be aligned.
  2. [Sections 8.2 and 8.3] The appendix contains unresolved placeholder cross-references ('Figure??') in Sections 8.2 and 8.3, and the learned-atom figure is referred to but not displayed with a stable number.
  3. [Section 5] The real-data experiments report qualitative visual comparisons rather than quantitative reconstruction errors with standard deviations; a table of l2 errors and nonzero-count statistics across trials would strengthen the claimed advantage.
  4. [Section 4, multivariate CSC paragraph] The phrase 'R = +infinity' is informal: R is defined as a CP rank in Proposition 1, and the intended meaning 'no low-rank constraint' should be stated explicitly rather than through an infinite value.
  5. [Appendix, Proposition 4] Proposition 4 in the appendix duplicates Proposition 3 but does not mention the normalization constraints of Eq. (4), which further highlights the mismatch between the theoretical development and the constrained model.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the core derivation is an algebraic equivalence, and the experimental comparison is against an external ADMM baseline.

full rationale

The paper's derivation chain is self-contained in the relevant sense. Proposition 2 rewrites the convolutional K-CSC model as a Kruskal tensor regression through a circulant-tensor inner-product identity; this is an algebraic equivalence, not a parameter fitted to an outcome. The optimization algorithm reduces each Z-block update to a standard multi-channel CSC subproblem solved with external solvers (Garcia-Cardona and Wohlberg, 2018), and the experiments compare reconstruction against an external ADMM baseline (Wohlberg, 2016b). The reported 'fewer non-zero coefficients' are measured from the low-rank parameterization, not obtained by fitting a parameter to a subset of data and then predicting that same subset. There is one self-citation: Richard et al. (2012), coauthored by Vayatis, is used to support the general remark that sparse and low-rank regularizations can have adversarial influence; this is not load-bearing for the model, algorithm, or experimental claims. The paper does contain correctness risks that are outside the circularity definition: the claim that (4) is convex in each Z block is unsupported and likely false because the feasible set S imposes nonconvex unit-norm row constraints; Algorithm 1 line 14 solves an unconstrained problem without those constraints; and for R=1 the unit-row normalization makes l1 sparsity ineffective. These are consistency and validity concerns, not instances where a result is equivalent to its inputs by construction.

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

The central contribution rests on a handful of modeling and optimization assumptions. The CP-low-rank form of activation tensors is a domain assumption, not derived. The proof of per-block convexity and the equivalence between the constrained problem (4) and the unconstrained Z-step (5) used by Algorithm 1 are asserted rather than proved. Hyperparameters R, alpha_l, beta_l, and K are selected by hand or by grid search in the experiments, so the empirical performance depends on tuning choices that are only partially reported.

free parameters (4)
  • CP rank R = R*=4 in synthetic data; R=1 in Mario and fMRI experiments; overestimated values tested in Figure 2
    The low-rank CP-Rank is a central model input chosen by hand or from knowledge of the true rank; no rank-selection criterion is provided.
  • Sparsity penalties alpha_l = Grid-searched in [0,100]^3
    These weights control the number of nonzero coefficients and the reconstruction error; Figures 2C and 3A show sensitivity to them.
  • Ridge penalties beta_l = Not reported
    The ridge terms are introduced in (5) to stabilize the updates, but their values are never specified and they are omitted from Algorithm 1.
  • Dictionary size K and atom sizes = K=10 with atom size R2x4x8 in synthetic data; K=20 with atom sizes 20x10x3x3 and 10x10x10 in real data
    The number of atoms and their shapes are user-chosen capacity parameters that affect the model and the reported runtimes.
assumptions (5)
  • standard math Every finite tensor has an exact CP decomposition with some rank R.
    Proposition 1 states existence; the model then assumes a small fixed R for the activations.
  • domain assumption Each activation tensor Z_k has CP-rank exactly R.
    Remark 1 explicitly fixes CP-Rank(Z_k)=R; this is the low-rank structure the method exploits.
  • domain assumption The noise E has independent, centered subgaussian components.
    Section 2.2 states this to justify the negative log-likelihood objective in (4).
  • ad hoc to paper Problem (4) is convex in each Z block, so block coordinate descent is valid.
    Section 3 asserts this without proof and gives no convergence guarantee; Algorithm 1 omits the row-normalization constraints from S.
  • domain assumption Guarantees for Kruskal tensor regression transfer to the K-CSC setting.
    Used to claim theoretical guarantees; no theorem in this paper proves the transfer.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Multivariate Convolutional Sparse Coding with Low Rank Tensor." pith.science (2026). https://pith.science/paper/YFWODAAL

@misc{pith2026190803367,
  author       = {Pith},
  title        = {Pith review of: Multivariate Convolutional Sparse Coding with Low Rank Tensor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFWODAAL}},
  note         = {Machine review of arXiv:1908.03367}
}
read the original abstract

This paper introduces a new multivariate convolutional sparse coding based on tensor algebra with a general model enforcing both element-wise sparsity and low-rankness of the activations tensors. By using the CP decomposition, this model achieves a significantly more efficient encoding of the multivariate signal-particularly in the high order/ dimension setting-resulting in better performance. We prove that our model is closely related to the Kruskal tensor regression problem, offering interesting theoretical guarantees to our setting. Furthermore, we provide an efficient optimization algorithm based on alternating optimization to solve this model. Finally, we evaluate our algorithm with a large range of experiments, highlighting its advantages and limitations.

Figures

Figures reproduced from arXiv: 1908.03367 by the authors.

Figure 1
Figure 1. Illustration of the multidimensional convolution with [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A) Loss (4) as a function of the rank parameter R and the number of full loops (Z and D steps). B) `2-distance after convergence as a function of the rank parameter R (the true one being R∗ = 4) on 8 trials C) Heatmap of the `2-distance for several hyperparameters values (R = R∗). without a low rank setting – a key component of our approach. In [Jiang et al., 2018], the authors introduce a new formulation based on a… view at source ↗
Figure 3
Figure 3. A) `2-distance and number of non-zero coefficients as a function of the sparsity level α on 20 trials. B) First 5 frames of: Original data (top), recon￾struction with AK-CSC (middle) and, reconstruction with ADMM (bottom) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Original data (top), reconstruction with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Heatmap of the `2-distance between Y and its reconstruction to tune hyperparameters when considering third order tensor and R∗ = 4. where D˜ k;r,j1,i2··· ,ip =  Dk;j1,:,··· ,: ?2,··· ,p z (2) k,r ◦ · · · ◦ z (p) k,r i2,··· ,ip . 8 Additional results 8.1 Synthetic dat…
Figure 6
Figure 6. Figure 6: Learned atoms for the fMRI. On the left, the atoms from our method. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Mean and standard deviation of the evolution of the loss function with [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Illustration of the true (on the top) and reconstruct (on the bottom) [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 34 canonical work pages

  1. [1]

    Supervised dictionary learning

    Julien Mairal, Jean Ponce, Guillermo Sapiro, Andrew Zisserman, and Francis R Bach. Supervised dictionary learning. In Advances in neural information processing systems, pages 1033--1040, 2009

    work page 2009

  2. [2]

    Sparse representation for signal classification

    Ke Huang and Selin Aviyente. Sparse representation for signal classification. In Advances in neural information processing systems, pages 609--616, 2007

    work page 2007

  3. [3]

    rm k -svd: An algorithm for designing overcomplete dictionaries for sparse representation

    Michal Aharon, Michael Elad, and Alfred Bruckstein. rm k -svd: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on signal processing, 54 0 (11): 0 4311--4322, 2006

    work page 2006

  4. [4]

    Online learning for matrix factorization and sparse coding

    Julien Mairal, Francis Bach, Jean Ponce, and Guillermo Sapiro. Online learning for matrix factorization and sparse coding. Journal of Machine Learning Research, 11 0 (Jan): 0 19--60, 2010

    work page 2010

  5. [5]

    Convolutional Dictionary Learning: A Comparative Review and New Algorithms

    Cristina Garcia-Cardona and Brendt Wohlberg. Convolutional dictionary learning. arXiv preprint arXiv:1709.02893, 2017

    work page Pith review arXiv 2017

  6. [6]

    Tensor regression with applications in neuroimaging data analysis

    Hua Zhou, Lexin Li, and Hongtu Zhu. Tensor regression with applications in neuroimaging data analysis. Journal of the American Statistical Association, 108 0 (502): 0 540--552, 2013

    work page 2013

  7. [7]

    Tensor decompositions for signal processing applications: From two-way to multiway component analysis

    Andrzej Cichocki, Danilo Mandic, Lieven De Lathauwer, Guoxu Zhou, Qibin Zhao, Cesar Caiafa, and Huy Anh Phan. Tensor decompositions for signal processing applications: From two-way to multiway component analysis. IEEE Signal Processing Magazine, 32 0 (2): 0 145--163, 2015

    work page 2015

  8. [8]

    Multivariate multilinear regression

    Ya Su, Xinbo Gao, Xuelong Li, and Dacheng Tao. Multivariate multilinear regression. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 42 0 (6): 0 1560--1573, 2012

    work page 2012

Show all 38 references
  1. [9]

    Low-rank regression with tensor responses

    Guillaume Rabusseau and Hachem Kadri. Low-rank regression with tensor responses. In Advances in Neural Information Processing Systems, pages 1867--1875, 2016

    2016

  2. [10]

    Near optimal sketching of low-rank tensor regression

    Xingguo Li, Jarvis Haupt, and David Woodruff. Near optimal sketching of low-rank tensor regression. In Advances in Neural Information Processing Systems, pages 3466--3476, 2017

    2017

  3. [11]

    Boosted sparse and low-rank tensor regression

    Lifang He, Kun Chen, Wanwan Xu, Jiayu Zhou, and Fei Wang. Boosted sparse and low-rank tensor regression. In Advances in Neural Information Processing Systems, pages 1009--1018, 2018

    2018

  4. [12]

    Tensor rank is np-complete

    Johan H stad. Tensor rank is np-complete. Journal of Algorithms, 11 0 (4): 0 644--654, 1990

    1990

  5. [13]

    Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics

    Joseph B Kruskal. Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear algebra and its applications, 18 0 (2): 0 95--138, 1977

    1977

  6. [14]

    Multilinear operators for higher-order decompositions

    Tamara Gibson Kolda. Multilinear operators for higher-order decompositions. Technical report, Sandia National Laboratories, 2006

    2006

  7. [15]

    Tensor decompositions and applications

    Tamara G Kolda and Brett W Bader. Tensor decompositions and applications. SIAM review, 51 0 (3): 0 455--500, 2009

    2009

  8. [16]

    Tensor decomposition for signal processing and machine learning

    Nicholas D Sidiropoulos, Lieven De Lathauwer, Xiao Fu, Kejun Huang, Evangelos E Papalexakis, and Christos Faloutsos. Tensor decomposition for signal processing and machine learning. IEEE Transactions on Signal Processing, 65 0 (13): 0 3551--3582, 2017

    2017

  9. [17]

    Tensor learning for regression

    Weiwei Guo, Irene Kotsia, and Ioannis Patras. Tensor learning for regression. IEEE Transactions on Image Processing, 21 0 (2): 0 816--827, 2012

    2012

  10. [18]

    Tensor completion for estimating missing values in visual data

    Ji Liu, Przemyslaw Musialski, Peter Wonka, and Jieping Ye. Tensor completion for estimating missing values in visual data. IEEE transactions on pattern analysis and machine intelligence, 35 0 (1): 0 208--220, 2013

    2013

  11. [19]

    Convolutional dictionary learning: A comparative review and new algorithms

    Cristina Garcia-Cardona and Brendt Wohlberg. Convolutional dictionary learning: A comparative review and new algorithms. IEEE Transactions on Computational Imaging, 4 0 (3): 0 366--381, 2018

    2018

  12. [20]

    A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

    Samuel Burer and Renato DC Monteiro. A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Mathematical Programming, 95 0 (2): 0 329--357, 2003

    2003

  13. [21]

    Rank, decomposition, and uniqueness for 3-way and n-way arrays

    Joseph B Kruskal. Rank, decomposition, and uniqueness for 3-way and n-way arrays. Multiway data analysis, pages 7--18, 1989

    1989

  14. [22]

    Estimation of simultaneously sparse and low rank matrices

    Emile Richard, Pierre-Andr \'e Savalle, and Nicolas Vayatis. Estimation of simultaneously sparse and low rank matrices. arXiv preprint arXiv:1206.6474, 2012

    2012 arXiv

  15. [23]

    Convolutional sparse representation of color images

    Brendt Wohlberg. Convolutional sparse representation of color images. In 2016 IEEE Southwest Symposium on Image Analysis and Interpretation (SSIAI), pages 57--60. IEEE, 2016 a

    2016

  16. [24]

    Blood cell detection and counting in holographic lens-free imaging by convolutional sparse dictionary learning and coding

    Florence Yellin, Benjamin D Haeffele, and Ren \'e Vidal. Blood cell detection and counting in holographic lens-free imaging by convolutional sparse dictionary learning and coding. In 2017 IEEE 14th International Symposium on Biomedical Imaging (ISBI 2017), pages 650--653. IEEE, 2017

    2017

  17. [25]

    A fast proximal method for convolutional sparse coding

    Rakesh Chalasani, Jose C Principe, and Naveen Ramakrishnan. A fast proximal method for convolutional sparse coding. In The 2013 International Joint Conference on Neural Networks (IJCNN), pages 1--5. IEEE, 2013

    2013

  18. [26]

    A fast iterative shrinkage-thresholding algorithm for linear inverse problems

    Amir Beck and Marc Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM journal on imaging sciences, 2 0 (1): 0 183--202, 2009

    2009

  19. [27]

    Distributed optimization and statistical learning via the alternating direction method of multipliers

    Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine learning , 3 0 (1): 0 1--122, 2011

    2011

  20. [28]

    Multivariate convolutional sparse coding for electromagnetic brain signals

    Tom Dupr \'e La Tour, Thomas Moreau, Mainak Jas, and Alexandre Gramfort. Multivariate convolutional sparse coding for electromagnetic brain signals. In Advances in Neural Information Processing Systems, pages 3292--3302, 2018

    2018

  21. [29]

    Efficient multi-dimensional tensor sparse coding using t-linear combination

    Fei Jiang, Xiao-Yang Liu, Hongtao Lu, and Ruimin Shen. Efficient multi-dimensional tensor sparse coding using t-linear combination. In AAAI, 2018

    2018

  22. [30]

    High order tensor formulation for convolutional sparse coding

    Adel Bibi and Bernard Ghanem. High order tensor formulation for convolutional sparse coding. In 2017 IEEE International Conference on Computer Vision (ICCV), pages 1790--1798. IEEE, 2017

    2017

  23. [31]

    Tensor-based dictionary learning for spectral ct reconstruction

    Yanbo Zhang, Xuanqin Mou, Ge Wang, and Hengyong Yu. Tensor-based dictionary learning for spectral ct reconstruction. IEEE transactions on medical imaging, 36 0 (1): 0 142--154, 2017

    2017

  24. [32]

    Tensor-based dictionary learning for dynamic tomographic reconstruction

    Shengqi Tan, Yanbo Zhang, Ge Wang, Xuanqin Mou, Guohua Cao, Zhifang Wu, and Hengyong Yu. Tensor-based dictionary learning for dynamic tomographic reconstruction. Physics in Medicine & Biology, 60 0 (7): 0 2803, 2015

    2015

  25. [33]

    Convolutional dictionary learning through tensor factorization

    Furong Huang and Animashree Anandkumar. Convolutional dictionary learning through tensor factorization. In Feature Extraction: Modern Questions and Challenges, pages 116--129, 2015

    2015

  26. [34]

    Tensorly: Tensor learning in python

    Jean Kossaifi, Yannis Panagakis, Anima Anandkumar, and Maja Pantic. Tensorly: Tensor learning in python. Journal of Machine Learning Research, 20 0 (26): 0 1--6, 2019. URL http://jmlr.org/papers/v20/18-277.html

    2019

  27. [35]

    Sporco: A python package for standard and convolutional sparse representations

    Brendt Wohlberg. Sporco: A python package for standard and convolutional sparse representations. In Proceedings of the 15th Python in Science Conference, Austin, TX, USA, pages 1--8, 2017

    2017

  28. [36]

    Statistical learning with sparsity: the lasso and generalizations

    Robert Tibshirani, Martin Wainwright, and Trevor Hastie. Statistical learning with sparsity: the lasso and generalizations. Chapman and Hall/CRC, 2015

    2015

  29. [37]

    Efficient algorithms for convolutional sparse representations

    Brendt Wohlberg. Efficient algorithms for convolutional sparse representations. IEEE Transactions on Image Processing, 25 0 (1): 0 301--315, 2016 b

    2016

  30. [38]

    Functional connectivity of human striatum: a resting state fmri study

    Adriana Di Martino, Anouk Scheres, Daniel S Margulies, AMC Kelly, Lucina Q Uddin, Zarrar Shehzad, B Biswal, Judith R Walters, F Xavier Castellanos, and Michael P Milham. Functional connectivity of human striatum: a resting state fmri study. Cerebral cortex, 18 0 (12): 0 2735--...

    2008

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.