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REVIEW 4 major objections 5 minor 46 references

Fast Tensor Completion via Approximate Richardson Iteration

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Tensor completion can be made fast by lifting the problem so that missing entries become free variables, then solving via approximate Richardson iteration with fast tensor-decomposition solvers as black boxes.

desk verdict Genuinely new theoretical framework for tensor completion, but the claimed running time has a missing β factor and the experiments test the exact solver, not the sublinear approximate one. read the letter →

arxiv 2502.09534 v2 pith:S6DULVOD submitted 2025-02-13 cs.DS cs.LGmath.STstat.TH

Mehrdad Ghadiri , Matthew Fahrbach , Yunbum Kook , Ali Jadbabaie This is my paper · ORCID
classification cs.DScs.LGmath.STstat.TH MSC 15A6965F1068W20
keywords tensorcompletiondecompositionalternatingleastsquaresRichardsoniterationleveragescoresamplingKroneckerregressionKhatri-Raoproductsublinear-timealgorithm
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

This paper argues that tensor completion is slow because the partially observed design matrix loses the Kronecker, Khatri-Rao, or tensor-train structure that makes tensor decomposition fast, and that this obstacle can be removed by a lifting trick. The unobserved entries are treated as free variables, restoring the full structured matrix, and the lifted least-squares problem is solved by alternating between filling in missing values and running a fast structured regression solver. The authors prove that this alternation is exactly a preconditioned Richardson iteration on the original problem, and that running the inner regression approximately with leverage-score sampling still converges at essentially the same rate provided a spectral ratio is small and the inner solver error is below a threshold. The payoff is concrete: CP, Tucker, and tensor-train completion algorithms whose per-round cost is sublinear in the number of observed entries, with empirical 100x speedups over direct ALS on real tensors.

What carries the argument

The machinery is the lifted problem plus the mini-ALS iteration: with $\mathbf{A}$ the full structured design matrix and $\mathbf{P}$ the row-restricted observed matrix, each outer step sets $\hat{\mathbf{q}}^{(k)} = \hat{\mathbf{q}} + (\mathbf{A} - \tilde{\mathbf{P}})\mathbf{x}^{(k)}$ and then approximately solves $\min_{\mathbf{x}} \|\mathbf{A}\mathbf{x} - \hat{\mathbf{q}}^{(k)}\|^2$ using a fast tensor-decomposition solver. The identity carrying the argument is Lemma 3.5: this alternation is exactly Richardson iteration preconditioned by $\mathbf{A}^\top\mathbf{A}$ on the original regression. The convergence engine is the spectral sandwich $\mathbf{P}^\top\mathbf{P} \preceq \mathbf{A}^\top\mathbf{A} \preceq \beta \mathbf{P}^\top\mathbf{P}$ together with the inner-solver tolerance $\hat\varepsilon < 1/\beta^2$, which says how much approximation error the outer iteration can absorb before the speedup degrades.

What would settle it

Construct a rank-$R$ tensor with a missing pattern that is not uniformly random, for instance observing only entries in a small set of slices or fibers, compute $\mathbf{P}^\top\mathbf{P}$ and $\mathbf{A}^\top\mathbf{A}$, and measure the iteration count of approximate-mini-ALS; the theorem predicts iteration count growing linearly in $\beta/(1-\sqrt{\hat\varepsilon}\,\beta)$, so a demonstration where measured iterations grow faster than $\beta$, or where leverage-score sampling with $\hat\varepsilon < 1/\beta^2$ fails to produce a $(1+\hat\varepsilon)$-optimal inner solve, would falsify the central convergence claim.

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Extended reading notes

Core claim

The paper establishes that the unstructured least-squares problems arising in ALS for tensor completion can be solved by solving a larger, structured problem instead. Lifting introduces the missing entries as free response variables, so the design matrix becomes the same Kronecker, Khatri-Rao, or tensor-train product matrix as in the fully observed decomposition problem, and the original solution is recovered exactly (Lemma 3.1). Alternating minimization on the lifted problem, called mini-ALS, is shown in Lemma 3.5 to reproduce the preconditioned Richardson iteration for the original regression. The main theorem, Theorem 3.7, quantifies the permitted sloppiness: if the observed design $\mathbf{P}$ satisfies $\mathbf{P}^\top\mathbf{P} \preceq \mathbf{A}^\top\mathbf{A} \preceq \beta \mathbf{P}^\top\mathbf{P}$ and the inner structured solver returns a $(1+\hat\varepsilon)$-approximate solution with $\hat\varepsilon < 1/\beta^2$, then the outer loop reaches relative error $1 + O(\hat\varepsilon/\beta^2)$ plus an $\varepsilon$-term in $O\!\left(\frac{\beta}{1-\sqrt{\hat\varepsilon}\,\beta} \cdot T \log(\beta/\varepsilon)\right)$ time. Under uniform random observation and an incoherence condition, $\beta = 2/p$, yielding CP, Tucker, and TT completion algorithms whose per-round cost is sublinear in the number of observed entries and, on the paper's CP experiments, roughly 100x faster than direct ALS.

Load-bearing premise

The whole speed guarantee rests on the observed rows giving a good spectral approximation of the full tensor's design matrix; if missing data is adversarial or concentrated where the factors are small, the ratio $\beta$ is large and the inner solver must be so accurate that the sublinear-time guarantee collapses.

Editorial extensions

If this is right

  • Every sampling-based fast TD ALS algorithm becomes a tensor-completion algorithm, so future improvements in TD solvers transfer to completion automatically.
  • The per-round cost for CP, Tucker, and TT completion is independent of $|\Omega|$, giving sublinear-time ALS updates in the number of observed entries.
  • A smaller sample ratio $p$ means a larger $\beta$ and more outer iterations, while $p \to 1$ drives $\beta \to 1$ and the method reduces to ordinary fast tensor decomposition.
  • The Tomasi-Bro imputation method is recovered as a single mini-ALS step; running the loop to convergence, as here, gives Richardson-type guarantees rather than uncontrolled EM drift.
  • The required inner tolerance $\hat\varepsilon < 1/\beta^2$ is an explicit accuracy budget, quantifying when leverage-score sampling is safe to use as a blackbox.

Reading between the lines

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

  • Editorial: the same lifting-plus-approximate-Richardson wrapper should apply to any tensor-network decomposition with a sampling-based ALS solver, such as tensor ring, tensor wheel, or t-SVD, even though the paper only works out CP, Tucker, and TT.
  • Editorial: the acceleration heuristic, which extrapolates collinear iterates by a geometric series, is essentially heavy-ball momentum for a convex quadratic; a provable momentum variant may exist, and the paper leaves the analysis of the speedup open.
  • Editorial: a direct experimental test of the theory would measure the ratio $\beta$ from the observed mask and factors and check the predicted iteration count; the reported speedups on random masks suggest $\beta$ is small when missingness is benign.
  • Editorial: treating missing values as free variables connects tensor completion to missing-data imputation; the paper's guarantee shows when iterative imputation is safe, but adversarial missingness remains the practical danger zone where the spectral condition fails.
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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 / 5 minor

Summary. The paper proposes a lifting method for tensor completion (TC) least-squares subproblems. Instead of solving the masked regression min_x ||Px - q||^2 directly, it introduces the unobserved entries as free variables, obtaining min_x,b ||Ax - b||^2 with a structured full design matrix A. The alternating minimization over x and the missing entries is shown to be equivalent to a preconditioned Richardson iteration (Lemma 3.5). The main theoretical result, Theorem 3.7, gives a convergence guarantee for an approximate version of this iteration, where each structured regression is solved only to multiplicative error (1 + b_epsilon). This is then combined with leverage-score sampling algorithms for CP, Tucker, and tensor-train decompositions to obtain sampling-based TC algorithms. Experiments on a coupled matrix problem and on CP completion of real-world tensors are reported.

Significance. If the main theorem is correct, the paper makes a useful conceptual contribution: it gives a principled way to convert an unstructured TC regression into a sequence of structured TD regressions, so that future fast TD solvers can be plugged into TC. The proof of Theorem 3.7 is self-contained and the iteration count is consistent with the contraction rate up to the stated constants. The paper also releases code and tests the lifting idea on real-world tensors, and the exact-mini-ALS variant shows substantial speedups over direct solves. The main caveats are that the spectral-ratio assumption beta is not shown to hold for the evolving ALS matrices, that the CP running-time corollary appears to miss a factor of beta, and that the CP experiments do not exercise the approximate-mini-ALS with leverage-score sampling that the theory is designed to justify.

major comments (4)
  1. [Section 3.3, 'Bounding beta'] The sublinear-time claim depends on the existence of a small spectral ratio beta satisfying eP^T eP <= A^T A <= beta eP^T eP. The paragraph in Section 3.3 derives beta = 2/p from uniform row sampling and an incoherence condition on A, but A is built from the current ALS factor matrices and changes at every outer iteration. The paper does not prove that these evolving Khatri-Rao, Kronecker, or TT-structured matrices remain coherent with a controlled parameter, nor does it handle the case where an entire direction is unobserved, in which case P^T P is singular and no finite beta exists. The ridge-regularization fix mentioned in the same paragraph is not analyzed for Theorem 3.7. A concrete test would be to track the empirical beta (or the leverage scores) over ALS iterations in the experiments; without this, the statement that beta = O(1) is an assumption rather than a derived consequence.
  2. [Section 4.1, Corollary 4.1] The claimed CP completion running time appears to be off by a factor of beta. The text before Eq. (10) sets b_epsilon = Theta(epsilon_1 / beta^2), so each approximate structured regression costs T_{b_epsilon} = O~(beta^2 / epsilon_1 * sum_n (I_n R^2 + N R^3)). Theorem 3.7 then requires O(beta / (1 - sqrt(b_epsilon) beta)) = O(beta) Richardson iterations per factor update, giving a total of O~(beta^3 / epsilon_1 * sum_n (I_n R^2 + N R^3) log(1/epsilon_2)) for one round, not the stated O~(beta^2 / epsilon_1 ...). If 'one round' is intended to mean something different, the terminology should be clarified; otherwise the corollary should be corrected. The corrected bound is still sublinear in |Omega|, but the advertised exponent changes.
  3. [Section 5.2, CP completion experiments] The CP experiments compare direct, parafac, mini-als, and accelerated-mini-als, where mini-als and accelerated-mini-als use exact inner solves (b_epsilon = 0). The approximate-mini-ALS algorithm with actual leverage-score sampling, which is the algorithm needed to realize the sublinear-time guarantee of Theorem 3.7, appears only in the coupled-matrix warm-up of Section 5.1. Consequently, the reported 100x speedup for CP completion on real-world tensors validates the lifting/Richardson idea but not the sampling-based approximate method that is the paper's principal theoretical contribution. Adding CP experiments with approximate-mini-ALS, or qualifying the empirical claim, is needed.
  4. [Section 3.3, spectral condition and Theorem 3.7] Theorem 3.7 requires b_epsilon < 1/beta^2 and the iteration count grows linearly with beta. The paragraph on bounding beta does not discuss the interaction between the observation rate p and this requirement: for small p, beta = 2/p is large, and the inner solver must be extremely accurate, which increases the sampling cost through the 1/b_epsilon factor. The paper should state explicitly that the sublinear-time guarantee degrades polynomially in beta and should give the resulting total cost as a function of p, not only of the fixed parameters R and epsilon_1.
minor comments (5)
  1. [Theorem 3.7 / Algorithm 1] The loop bound in Algorithm 1 and the time expression in Theorem 3.7 use two equivalent forms of the iteration count; please align the notation so the reader can easily match the pseudocode to the theorem.
  2. [Section 4.1, CP completion] The text says 'Applying Bharadwaj et al. (2023, Corollary 3.3), one round of ALS runs in time O~(epsilon^{-1} sum(...))' but does not state which epsilon parameter of that corollary is being used and how it maps to b_epsilon; making this mapping explicit would help verify Corollary 4.1.
  3. [Section 5.1, coupled matrix problem] The warm-up reports that approximate-mini-als samples 1% of rows per iteration, but it does not report the failure probability or the number of trials for the stochastic variant; adding this information would strengthen the comparison.
  4. [Section 5.2, experiments] The figure captions do not specify the exact epsilon values used for the mini-als and accelerated-mini-als curves, nor whether the reported times include the cost of computing the full tensor or only observed entries; these details are needed to interpret the '100x faster' claim.
  5. [Section 3.3, 'Bounding beta'] The sentence about alpha zeta^2-ridge leverage scores is telegraphic; it should state how alpha is chosen from the observation rate p and how the regularization term is incorporated into the lifted problem (5).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the lifting/Richardson equivalence and Theorem 3.7 are derived from stated spectral assumptions and external lemmas, not from the conclusion.

full rationale

The paper's central derivation chain is self-contained. Lemma 3.1 proves equivalence of the lifted problem and the original TC regression directly by partitioning the squared norm over Ω and its complement. Lemma 3.5 derives the equivalence between mini-ALS and preconditioned Richardson iteration using only the normal equations and the orthogonality assumption (P−A)ᵀ[P q]=0. The convergence rate is inherited from an external result, Lee and Vempala, Lemma 6.1, and Theorem 3.7 extends it to approximate inner solves with a fully written induction; no step in that proof imports the conclusion as an assumption. The β bound in Section 3.3 is obtained from Cohen et al. (2015), an independent uniform-sampling lemma, together with an explicit incoherence condition, and the paper does not define β in terms of the final error guarantee. Citations to the authors' own prior work (Fahrbach et al. 2022; Ghadiri et al. 2023a) are used only to supply running times of published TD subroutines, which have their own external guarantees, and they are not used to justify the uniqueness or correctness of the lifting idea. The empirical warm-up uses approximate-mini-ALS with leverage-score sampling, while the CP experiments use exact mini-ALS; that is an evaluation-coverage gap rather than a circular reduction. Likewise, the sensitivity of the guarantees to large β and to evolving incoherence of the lifted design matrices is a substantive robustness concern, but it is not a case of the paper deriving a claim from itself. No fitted parameter is renamed as a prediction, no forbidden-uniqueness argument is invoked, and no known result is merely relabeled as a new contribution. Hence the appropriate circularity score is very low.

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

The central derivation is self-contained given the spectral assumption and the blackbox TD solvers. The only data-dependent object is β, which is not fitted but bounded by 2/p under random observation. No new latent variables, forces, or conserved quantities are introduced.

assumptions (4)
  • domain assumption Rows of A are observed uniformly at random with probability p ≥ cµs log s / I, and A satisfies the incoherence condition with parameter µ.
    Invoked in the 'Bounding β' paragraph to obtain A^T A ⪯ (2/p) P^T P; without this, the key spectral condition and the small-β convergence rate fail.
  • domain assumption The blackbox structured regression algorithms achieve (1+ε)-approximate least squares for any right-hand side within their stated running times.
    Theorems 3.7 and Corollaries 4.1 to 4.4 assume an approx-least-squares subroutine with these guarantees; the paper imports these algorithms from Cheng et al. (2016), Larsen and Kolda (2022), Bharadwaj et al. (2023), Fahrbach et al. (2022), and Bharadwaj et al. (2024).
  • standard math Preconditioned Richardson iteration contraction, Lemma 6.1 of Lee and Vempala (2024).
    Used in Lemma 3.5 and Theorem 3.7 to quantify convergence of exact mini-ALS. Standard textbook result.
  • standard math Exact arithmetic model for the stated complexity bounds.
    Section 3.2 states all algorithms use exact arithmetic; the paper notes all constants can be represented with O(log(κ/ε)) bits.

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Pith. "Pith review of Fast Tensor Completion via Approximate Richardson Iteration." pith.science (2026). https://pith.science/paper/S6DULVOD

@misc{pith2026250209534,
  author       = {Pith},
  title        = {Pith review of: Fast Tensor Completion via Approximate Richardson Iteration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6DULVOD}},
  note         = {Machine review of arXiv:2502.09534}
}
read the original abstract

We study tensor completion (TC) through the lens of low-rank tensor decomposition (TD). Many TD algorithms use fast alternating minimization methods to solve highly structured linear regression problems at each step (e.g., for CP, Tucker, and tensor-train decompositions). However, such algebraic structure is often lost in TC regression problems, making direct extensions unclear. This work proposes a novel lifting method for approximately solving TC regression problems using structured TD regression algorithms as blackbox subroutines, enabling sublinear-time methods. We analyze the convergence rate of our approximate Richardson iteration-based algorithm, and our empirical study shows that it can be 100x faster than direct methods for CP completion on real-world tensors.

Figures

Figures reproduced from arXiv: 2502.09534 by the authors.

Figure 1
Figure 1. Coupled matrix results for E ∈ R n×n , X, Y ∈ R d×d with n = 2000 and d = 10 that compare the direct method, mini￾ALS, and approximate-mini-ALS via leverage score sampling. does not necessarily attain the minimal error in the matrix’s kernel space. i.e., the first term on the right-hand side of (10) can exceed the minimum error by a factor of (1 + ε), which allows it to follow a different convergence path and achiev… view at source ↗
Figure 2
Figure 2. Algorithm comparison for a low-rank CP completion task on the CARDIAC-MRI and HYPERSPECTRAL tensor datasets. The first column illustrates convergence rates and relative reconstruction error (RRE) of the ALS algorithm for various CP-decomposition ranks. The second column plots RRE over ALS steps for direct, parafac, and our method (mini-als) under different ε values. The third column shows total running time of these… view at source ↗
Figure 3
Figure 3. Algorithm comparison for a low-rank CP completion task on the RANDOM-CP and RANDOM-TUCKER tensor datasets. D.1.2. ACCELERATED METHODS We explain how to accelerate the Richardson iteration. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: In the one-variable case, our approach proceeds in the following order: (x (0), b(0) Ω ) → (x (1), b(0) Ω ) → (x (1), b(1) Ω ) → · · · . 1. For odd iterations (e.g., the first iteration), run mini-ALS normally. 2. For even iterations, compute xb (k+1) using normal mini…

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Works this paper leans on

46 extracted references · 44 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

  2. [2]

    M., Kolda, T

    Acar, E., Dunlavy, D. M., Kolda, T. G., and M rup, M. Scalable tensor factorizations for incomplete data. Chemometrics and Intelligent Laboratory Systems, 106 0 (1): 0 41--56, 2011

    work page 2011

  3. [3]

    Bader, B. W. and Kolda, T. G. Tensor toolbox for MATLAB , version 3.6. https://www.tensortoolbox.org/, 2023

    work page 2023

  4. [4]

    and Kala, R

    Baksalary, J. and Kala, R. The matrix equation AXB + CYD = E . Linear Algebra and its Applications, 30: 0 141--147, 1980

    work page 1980

  5. [5]

    and Moitra, A

    Barak, B. and Moitra, A. Noisy tensor completion via the sum-of-squares hierarchy. In Conference on Learning Theory, pp.\ 417--445. PMLR, 2016

    work page 2016

  6. [6]

    A., Murray, R., Grigori, L., Buluc, A., and Demmel, J

    Bharadwaj, V., Malik, O. A., Murray, R., Grigori, L., Buluc, A., and Demmel, J. Fast exact leverage score sampling from K hatri- R ao products with applications to tensor decomposition. In Advances in Neural Information Processing Systems, volume 36, pp.\ 47874--47901, 2023

    work page 2023

  7. [7]

    T., Malik, O

    Bharadwaj, V., Rakhshan, B. T., Malik, O. A., and Rabusseau, G. Efficient leverage score sampling for tensor train decomposition. In Advances in Neural Information Processing Systems, 2024

    work page 2024

  8. [8]

    and Recht, B

    Candes, E. and Recht, B. Exact matrix completion via convex optimization. Communications of the ACM, 55 0 (6): 0 111--119, 2012

    work page 2012

Show all 46 references
  1. [9]

    Incoherence-optimal matrix completion

    Chen, Y. Incoherence-optimal matrix completion. IEEE Transactions on Information Theory, 61 0 (5): 0 2909--2923, 2015

    2015

  2. [10]

    SPALS : F ast alternating least squares via implicit leverage scores sampling

    Cheng, D., Peng, R., Liu, Y., and Perros, I. SPALS : F ast alternating least squares via implicit leverage scores sampling. Advances in Neural Information Processing Systems, 29, 2016

    2016

  3. [11]

    B., Lee, Y

    Cohen, M. B., Lee, Y. T., Musco, C., Musco, C., Peng, R., and Sidford, A. Uniform sampling for matrix approximation. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, pp.\ 181--190, 2015

    2015

  4. [12]

    Optimal sketching for K ronecker product regression and low rank approximation

    Diao, H., Jayaram, R., Song, Z., Sun, W., and Woodruff, D. Optimal sketching for K ronecker product regression and low rank approximation. Advances in Neural Information Processing Systems, 32, 2019

    2019

  5. [13]

    Subquadratic K ronecker regression with applications to tensor decomposition

    Fahrbach, M., Fu, G., and Ghadiri, M. Subquadratic K ronecker regression with applications to tensor decomposition. Advances in Neural Information Processing Systems, 35: 0 28776--28789, 2022

    2022

  6. [14]

    Matrix Rank Minimization with Applications

    Fazel, M. Matrix Rank Minimization with Applications. PhD thesis, Stanford University, 2002

    2002

  7. [15]

    and Juki \'c , A

    Filipovi \'c , M. and Juki \'c , A. Tucker factorization with missing data with application to low- n -rank tensor completion. Multidimensional Systems and Signal Processing, 26 0 (3): 0 677--692, 2015

    2015

  8. [16]

    Approximately optimal core shapes for tensor decompositions

    Ghadiri, M., Fahrbach, M., Fu, G., and Mirrokni, V. Approximately optimal core shapes for tensor decompositions. In International Conference on Machine Learning, pp.\ 11237--11254. PMLR, 2023 a

    2023

  9. [17]

    Ghadiri, M., Peng, R., and Vempala, S. S. The bit complexity of efficient continuous optimization. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, pp.\ 2059--2070. IEEE, 2023 b

    2023

  10. [18]

    T., Padmanabhan, S., Swartworth, W., Woodruff, D

    Ghadiri, M., Lee, Y. T., Padmanabhan, S., Swartworth, W., Woodruff, D. P., and Ye, G. Improving the bit complexity of communication for distributed convex optimization. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pp.\ 1130--1140, 2024

    2024

  11. [19]

    Golub, G. H. and Overton, M. L. The convergence of inexact C hebyshev and R ichardson iterative methods for solving linear systems. Numerische Mathematik, 53 0 (5): 0 571--593, 1988

    1988

  12. [20]

    Golub, G. H. and van der Vorst, H. A. Closer to the solution: I terative linear solvers. In Institute of Mathematics and its Applications Conference Series, volume 63, pp.\ 63--92. Oxford University Press, 1997

    1997

  13. [21]

    R., Millman, K

    Harris, C. R., Millman, K. J., van der Walt, S. J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N. J., Kern, R., Picus, M., Hoyer, S., van Kerkwijk, M. H., Brett, M., Haldane, A., del R \' i o, J. F., Wiebe, M., Peterson, P., G \' e rard...

    2020

  14. [22]

    and Westmacott, M

    Healy, M. and Westmacott, M. Missing values in experiments analysed on automatic computers. Journal of the Royal Statistical Society: Series C (Applied Statistics), 5 0 (3): 0 203--206, 1956

    1956

  15. [23]

    Hillar, C. J. and Lim, L.-H. Most tensor problems are NP -hard. Journal of the ACM, 60 0 (6): 0 1--39, 2013

    2013

  16. [24]

    and Oh, S

    Jain, P. and Oh, S. Provable tensor factorization with missing data. Advances in Neural Information Processing Systems, 27, 2014

    2014

  17. [25]

    Kilmer, M. E. and Martin, C. D. Factorization strategies for third-order tensors. Linear Algebra and its Applications, 435 0 (3): 0 641--658, 2011

    2011

  18. [26]

    Kolda, T. G. and Bader, B. W. Tensor decompositions and applications. SIAM Review, 51 0 (3): 0 455--500, 2009

    2009

  19. [27]

    TensorLy : T ensor learning in P ython

    Kossaifi, J., Panagakis, Y., Anandkumar, A., and Pantic, M. TensorLy : T ensor learning in P ython. Journal of Machine Learning Research (JMLR), 20 0 (26), 2019

    2019

  20. [28]

    Larsen, B. W. and Kolda, T. G. Practical leverage-based sampling for low-rank tensor decomposition. SIAM Journal on Matrix Analysis and Applications, 43 0 (3): 0 1488--1517, 2022

    2022

  21. [29]

    Lee, Y. T. and Vempala, S. Techniques in Optimization and Sampling. 2024. URL https://github.com/YinTat/optimizationbook/blob/main/main.pdf

    2024

  22. [30]

    Little, R. J. and Rubin, D. B. Statistical Analysis with Missing Data, volume 793. John Wiley & Sons, 2019

    2019

  23. [31]

    and Moitra, A

    Liu, A. and Moitra, A. Tensor completion made practical. Advances in Neural Information Processing Systems, 33: 0 18905--18916, 2020

    2020

  24. [32]

    Malik, O. A. and Becker, S. A sampling-based method for tensor ring decomposition. In International Conference on Machine Learning, pp.\ 7400--7411. PMLR, 2021

    2021

  25. [33]

    A., Bharadwaj, V., and Murray, R

    Malik, O. A., Bharadwaj, V., and Murray, R. Sampling-based decomposition algorithms for arbitrary tensor networks. arXiv preprint arXiv:2210.03828, 2022

    2022 arXiv

  26. [34]

    and Sun, N

    Montanari, A. and Sun, N. Spectral algorithms for tensor completion. Communications on Pure and Applied Mathematics, 71 0 (11): 0 2381--2425, 2018

    2018

  27. [35]

    M., Amano, K., and Foster, D

    Nascimento, S. M., Amano, K., and Foster, D. H. Spatial distributions of local illumination color in natural scenes. Vision Research, 120: 0 39--44, 2016

    2016

  28. [36]

    Oseledets, I. V. Tensor-train decomposition. SIAM Journal on Scientific Computing, 33 0 (5): 0 2295--2317, 2011

    2011

  29. [37]

    Richardson, L. F. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of ...

    1911

  30. [38]

    and Yu, C

    Shah, D. and Yu, C. L. Iterative collaborative filtering for sparse noisy tensor estimation. In 2019 IEEE International Symposium on Information Theory, pp.\ 41--45. IEEE, 2019

    2019

  31. [39]

    and Yu, C

    Shah, D. and Yu, C. L. Robust max entrywise error bounds for tensor estimation from sparse observations via similarity-based collaborative filtering. IEEE Transactions on Information Theory, 69 0 (5): 0 3121--3149, 2023

    2023

  32. [40]

    Tensor completion algorithms in big data analytics

    Song, Q., Ge, H., Caverlee, J., and Hu, X. Tensor completion algorithms in big data analytics. ACM Transactions on Knowledge Discovery from Data, 13 0 (1): 0 1--48, 2019

    2019

  33. [41]

    Tarzanagh, D. A. and Michailidis, G. Fast randomized algorithms for t-product based tensor operations and decompositions with applications to imaging data. SIAM Journal on Imaging Sciences, 11 0 (4): 0 2629--2664, 2018

    2018

  34. [42]

    and Bro, R

    Tomasi, G. and Bro, R. PARAFAC and missing values. Chemometrics and Intelligent Laboratory Systems, 75 0 (2): 0 163--180, 2005

    2005

  35. [43]

    Randomized tensor wheel decomposition

    Wang, M., Yu, Y., and Li, H. Randomized tensor wheel decomposition. SIAM Journal on Scientific Computing, 46 0 (3): 0 A1714--A1746, 2024

    2024

  36. [44]

    Tensor wheel decomposition and its tensor completion application

    Wu, Z.-C., Huang, T.-Z., Deng, L.-J., Dou, H.-X., and Meng, D. Tensor wheel decomposition and its tensor completion application. Advances in Neural Information Processing Systems, 35: 0 27008--27020, 2022

    2022

  37. [45]

    Tensor ring decomposition

    Zhao, Q., Zhou, G., Xie, S., Zhang, L., and Cichocki, A. Tensor ring decomposition. arXiv preprint arXiv:1606.05535, 2016

    2016 arXiv

  38. [46]

    Fully-connected tensor network decomposition and its application to higher-order tensor completion

    Zheng, Y.-B., Huang, T.-Z., Zhao, X.-L., Zhao, Q., and Jiang, T.-X. Fully-connected tensor network decomposition and its application to higher-order tensor completion. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp.\ 11071--11078, 2021

    2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.