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REVIEW 3 major objections 9 minor 44 references

Fast randomized Kronecker tensor decomposition: algorithms and error analysis

T0 review · 3 major / 9 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Randomized SVDs with oversampling and power iterations can replace the SVD chain in TTr1SVD, giving a Kronecker tensor decomposition with expected error bounded by tail energy times exponentially decaying factors and order-of-magnitude…

desk verdict A sensible randomized-TTr1SVD combination with useful experiments, but the advertised error bound in Theorem 4 is not proved and the paper overstates its speedups. read the letter →

arxiv 2412.02597 v3 pith:OENUDRQL submitted 2024-12-03 math.NA cs.NA

Salman Ahmadi-Asl , Naeim Rezaeian , Andre L. F. de Almeida , Yipeng Liu This is my paper · ORCID
classification math.NAcs.NA MSC 15A6946N4015A23
keywords KroneckertensordecompositionrandomizedSVDpoweriterationoversamplingTTr1SVDcompletioncompressionapproximation
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

Kronecker tensor decomposition (KTD) writes a large tensor as a short sum of Kronecker products of smaller normalized tensors, and the standard TTr1SVD method computes it by a chain of truncated SVDs that is too slow for large data. The paper's central claim is that each deterministic SVD in that chain can be replaced by a randomized SVD with oversampling and power iterations, producing a randomized KTD algorithm (R-KTD) with a controlled expected error and much lower cost. The advertised guarantee is Theorem 4: the expected squared Frobenius error of the randomized approximation is bounded by the tail singular-value energy (the squared norm of the singular values omitted by truncation) times a sum of factors, one per SVD step, each decaying exponentially with the number of power iterations. If the bound is right, large-scale tensor compression, completion, denoising, and super-resolution become practical without the deterministic algorithm's $O(I^{N+1})$ cost. The paper also gives pass-efficient and Tucker-compressed variants and reports several orders of magnitude acceleration on synthetic and real datasets.

What carries the argument

The load-bearing object is the randomized SVD used as a drop-in replacement at every node of the TTr1SVD tree (the chain of truncated SVDs on successive unfoldings and reshaped singular vectors). On an unfolding matrix $X$, the algorithm draws a Gaussian sketch $\Omega$ of width $R+P$, forms $Y=(XX^T)^q X\Omega$ for $q$ power iterations, orthonormalizes $Y$ to get $Q$, and then computes the SVD of the small matrix $Q^T X$. The error analysis is carried by the per-matrix bound $E\|X-QQ^TX\|_F^2 \le (1+\frac{R}{P-1}\tau^{4q})\sum_{j>R}\sigma_j^2$; Theorem 4 assembles these per-step bounds along the chain of unfoldings into a single inequality with tail energy $\sigma_{R'+1}^2+\cdots+\sigma_R^2$. The reshaping-and-permutation equivalence between KTD and CPD with orthonormal rank-1 terms is what makes a chain of matrix SVDs produce the tensor factors in the first place.

What would settle it

Run R-KTD on a synthetic tensor with known KTD rank and known singular spectra, over many random trials and for increasing numbers of SVD steps $T$, and compare the empirical mean-squared error with the Theorem 4 bound $(\sum_i\epsilon_i)(\sigma_{R'+1}^2+\cdots+\sigma_R^2)$; if the observed error grows faster than the bound as $T$ increases, the additive error-propagation assumption is false.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is a drop-in randomization of TTr1SVD: at every node of the sequential SVD tree, the deterministic truncated SVD is replaced by a randomized SVD with oversampling $P$ and power iterations $q$. The resulting rank-$R'$ approximation satisfies the expected-error inequality $$E\|X - \sum_{r=1}^{R'} \sigma_r $X_r^{{(1)}}$\otimes\cdots\otimes $X_r^{{(M)}}$\|$_F^{2}$ \le \left(\sum_{i=1}^T \epsilon_i\right)(\sigma_{R'+1}^2+\cdots+\$sigma_R^{2}$),$$ with $\epsilon_i = 1 + \frac{R}{P-1}\tau_i^{4q}$, where $\tau_i$ is the spectral gap at the $i$-th SVD step and $T$ is the number of randomized SVDs. The factors $\tau_i^{4q}$ decay exponentially with $q$, so the expected error stays within a modest multiple of the optimal tail energy. The paper claims this makes randomized KTD a near-optimal approximation in expectation, and the experiments show the same reconstruction quality as the deterministic KTD at a fraction of the runtime.

Load-bearing premise

The load-bearing assumption is that the errors from the successive randomized SVDs in the TTr1SVD tree combine additively into a single tail-energy term; the paper's proof of Theorem 4 says only 'using Theorem 3, the proof is straightforward' and does not show how one step's error flows into the next, so if errors amplify multiplicatively along the tree instead of adding, the advertised bound does not follow.

Editorial extensions

If this is right

  • If Theorem 4 is correct, the randomized KTD error is controlled by a known tail-energy term, so a user can choose KTD rank and power iteration before running the algorithm and have an expected-error guarantee.
  • The reported experiments show about an order of magnitude speedup on a $100\times100\times100\times100$ synthetic tensor and larger gains on a $1000\times1000\times1000$ tensor, making KTD practical for tensors that require gigabytes of memory.
  • In iterative tensor completion, replacing the deterministic low-rank operator with R-KTD reduced video-completion time from 235 seconds to 70 seconds on the Akiyo test while maintaining PSNR.
  • Compressing the full Kodak dataset dropped from 14.45 seconds to 3.10 seconds with $q=1$ at nearly identical PSNR, so batch image compression becomes much faster.

Reading between the lines

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

  • Editorial inference: Theorem 4's additive form suggests the early SVD steps dominate the error, so spending more power iterations on early tree levels could improve accuracy per pass; the paper does not test this adaptive allocation.
  • Editorial inference: because TTr1SVD is a sequential-unfolding scheme, the same randomized replacement should transfer to tensor-train and other hierarchical SVD decompositions, although the paper only claims it for KTD.
  • Editorial inference: the guarantee depends on per-step spectral gaps $\tau_i$, so tensors with slowly decaying singular values will need larger $q$; a gap-free bound based on stable rank would be a natural extension.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 9 minor

Summary. The paper proposes randomized algorithms for computing the Kronecker Tensor Decomposition (KTD), replacing the deterministic SVDs in the TTr1SVD pipeline with randomized SVDs that use oversampling and power iterations. Three variants are presented: R-KTD (randomized SVD), RF-KTD (pass-efficient randomized SVD), and PT-KTD (randomized Tucker compression followed by randomized KTD). The manuscript claims a theoretical guarantee in Theorem 4 that the expected Frobenius-norm error of the randomized KTD is bounded by a product of per-step factors times a single tail-energy term, with exponential decay in the number of power iterations. Extensive experiments on synthetic tensors, images, and videos report substantial speedups over the deterministic baseline while maintaining comparable accuracy in compression, completion, denoising, and super-resolution applications.

Significance. If the theoretical guarantee were established, the paper would provide a valuable practical contribution: randomized KTD with controlled expected error and large speedups would make KTD scalable to large tensors. The empirical results, particularly the order-of-magnitude speedups in the larger synthetic and video experiments, are plausible and potentially useful to practitioners. The paper also demonstrates that power iterations improve reconstruction quality, and it ships a pass-efficient variant that permits flexible pass budgets, which is a practically relevant design choice. However, the advertised theoretical contribution is not proven as written: the proof of Theorem 4 is a single sentence that does not address error propagation through the recursive TTr1SVD tree, and the stated bound is therefore unsupported. The empirical claims, while suggestive, are not presented with error bars or statistical significance tests, and some runtime comparisons are on small tensors where the speedups are modest.

major comments (3)
  1. [Section 5, Eq. (16)] The proof of Theorem 4, stated as 'Using Theorem 3, the proof is straightforward,' does not justify the bound. Theorem 3 applies to a single randomized SVD of one fixed matrix and bounds its error by that matrix's own singular-value tail. In Algorithm 3, however, the second and subsequent randomized SVDs operate on matrices produced by the previous randomized SVDs; those intermediate matrices are random perturbations of the exact unfoldings of X. Their singular values, spectral gaps tau_i, and tail sums are not the same as the fixed (sigma^2_{R'+1}+...+sigma^2_R) appearing on the right-hand side of (16). The manuscript gives no argument that per-step errors combine additively, that perturbations are not amplified by subsequent truncations, or that the final reconstruction error is controlled by the sum of per-step factors times a single tail energy. A rigorous recursive bound would need to track how each intermediate error propagates through the remaining SVD tree, typically producing compounded or multiplicative terms and multiple tail energies. This missing composition argument is the load-bearing step for the paper's advertised theoretical contribution and, as written, the claim is unsupported.
  2. [Section 6] The complexity comparison is not sufficiently substantiated. The paper states that the deterministic KTD complexity is O(I^{N+1}) and the randomized KTD is O(I^N R), but no derivation is given for either expression, and the dependence on the oversampling parameter P, the power iteration q, and the intermediate tensor shapes is omitted. In particular, the randomized algorithm performs T = R^(N-2) types of randomized SVDs on reshaped right-singular vectors, and the sizes of these matrices depend on the block dimensions J_n^{(m)}; the stated complexity O(I^N R) does not transparently follow. Since the speedup claim is central to the paper, this complexity comparison needs a clearer derivation or at least a precise reference to the operations counted.
  3. [Section 5, 'It is also known...' paragraph] The claim that 'similar upper bounds as in Theorem (4) can be straightforwardly estimated for this new strategy' is not supported. The pass-efficient algorithm of Bjarkason [40] is cited for matrix approximation, but no adaptation to the tensor tree is described, and the error analysis of that algorithm is not coupled to the TTr1SVD recursion. Given that Theorem 4 itself is not proven, the extension to RF-KTD is even less justified, yet the simulations use RF-KTD as one of the proposed methods. Please provide either a derivation or explicitly mark this as an empirical heuristic.
minor comments (9)
  1. [Abstract] The abstract says the error is bounded by 'a sum of tail energies multiplied by factors that decay exponentially,' while Theorem 4 states a sum of epsilon_i multiplied by a single tail energy. These formulations are inconsistent: the abstract implies multiple tail energies, the theorem has one. Please reconcile.
  2. [Section 3] Equation (13) contains a typo in the denominator: it reads ||sigma^2_{R'+1}+...+sigma^2_R||_F, which is a scalar and not a Frobenius norm of a matrix; it should be sqrt(sigma^2_{R'+1}+...+sigma^2_R) divided by the appropriate norm of X. Also, the text refers to 'relative approximation error (16)' when the displayed equation is (13); please fix the cross-reference.
  3. [Section 3, Algorithm 1] In Algorithm 1, the input dimensions are denoted J_n^{(m)} but the reshape on line 1 uses these dimensions without specifying the ordering; also the permutation vector p is not defined in this algorithm. For reproducibility, please define p explicitly.
  4. [Section 5, Theorem 4] The theorem assumes X has exact KTD rank R, but the bound is stated for an approximation with KTD rank R' (presumably R' < R). The relationship between R, R', and the truncation ranks used in the inner SVDs of Algorithm 3 is not specified. The notation 'sigma_{R'+1}+...+sigma_R' also reuses R in two roles (the true KTD rank and the intermediate SVD rank). This ambiguity makes the theorem hard to parse.
  5. [Section 7, Example 1] The running-time comparisons in Figure 2 are reported for only a single run or the mean of 100 runs? The text says 100 Monte Carlo simulations for the errors, but the running-time figure is not described as averaged. Please clarify the variance of the timings; a speedup claim of 'several orders of magnitude' is not supported by Figure 2, which shows roughly one order of magnitude in the displayed range.
  6. [Section 7, Example 2] The statement 'a single image is not a large tensor' conflicts with the earlier claim that the randomized KTD gives several orders of magnitude speedup on large tensors. For the Kodak image, the reported speedup is about 4.5x (14.45 vs 3.10 seconds), not orders of magnitude. Please adjust the language to match the reported numbers.
  7. [Section 7, Example 3] In the video compression experiment, the text says patches of size 16x12x30 and 11x12x10, but the frame count is 300; the relationship between these patch sizes and the video dimensions 176x144x300 is not explained. Also, Figure 5 (left) is labeled 'Aikyo' while the text says 'Foreman'; please correct the labels.
  8. [Section 7, Example 5] The denoising and super-resolution experiments are presented without comparison to a baseline or to the deterministic KTD, so the reader cannot judge whether the proposed method is competitive in quality. Also, the PSNR gains are not reported numerically; please add quantitative metrics.
  9. [Section 2] The notation for the tensor Kronecker product is introduced, but the paper also uses the standard Kronecker product symbol in the preliminaries; the notation for the tensor Kronecker product (same symbol) may cause confusion. Consider using a distinct symbol or explicitly noting the difference in each use.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; Theorem 4's one-line proof is a derivation gap, not a self-referential reduction.

full rationale

The claimed derivation chain is not circular. The central theoretical result, Theorem 4, is presented as following from Theorem 3, which is an external one-matrix randomized-SVD bound with oversampling and power iteration. Theorem 3 is not derived from Theorem 4, nor from the KTD construction; it is quoted from prior literature (reference [12]) and its assumptions do not include the target tensor-decomposition result. The weakness in Theorem 4 is that its proof is only one sentence, "Using Theorem 3, the proof is straightforward," and the error-composition through the TTr1SVD tree is never actually derived. That is an omitted or unsupported step, not a circular one: the right-hand side of (16) involves a sum of per-step factors times a single tail energy, whereas a genuine recursive proof would need to show how perturbations from earlier randomized SVDs propagate through later ones, which is absent. No fitted parameters are renamed as predictions: oversampling P, power iteration q, and ranks are inputs to the algorithm, and the experiments report measured errors and running times rather than postdicting a fitted quantity. The paper does contain self-citations, notably references [4], [5], [24], and [25] by the author team, but these are used for background on randomized tensor algorithms and for the completion/application framework, not as load-bearing justification for Theorem 4 or for the KTD error bound. Thus the central advertised contribution has independent content, and the main concern is a soundness/completeness gap in the proof of Theorem 4 rather than circular reasoning.

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

The central claim depends on user-chosen parameters (rank, oversampling, power iterations) and on four assumptions: a quoted randomized SVD bound, the deterministic TTr1SVD error formula, an unproven additive error propagation through the tree, and an assumed extension to pass-efficient variants. No new physical or mathematical entities are introduced.

free parameters (4)
  • oversampling P = not reported
    User-specified; the error bound in Theorem 4 contains the factor R/(P-1), but the simulations do not state the value of P used.
  • power iteration q = q=0,1,2 in experiments
    User-specified; the error bound decays as tau^{4q}, and the paper recommends q=1 or 2, but the cost grows linearly with q.
  • KTD rank R = varies per experiment (10 to 50)
    User-specified truncation rank; it is central to all error and complexity claims and is chosen differently across experiments.
  • pass budget for RF-KTD = three passes in Example 1
    User-specified for the pass-efficient variant; the paper claims similar error bounds without proof.
assumptions (4)
  • domain assumption The randomized SVD error bound in Theorem 3, quoted from prior literature, holds for each intermediate matrix in the TTr1SVD tree.
    The paper uses this bound as the starting point of Theorem 4. The bound is attributed to reference [12] and is not re-derived or validated for the specific unfoldings encountered in TTr1SVD.
  • domain assumption The TTr1SVD decomposition has approximation error equal to the tail sum of squared singular values of the appropriate unfolding, as in equation (13).
    This is taken from the deterministic TTr1SVD theory of Batselier et al. [17]. The paper extends this formula to the randomized setting without proving that the same tail energy applies.
  • ad hoc to paper Errors propagate additively through the sequence of randomized SVDs in the tree, so the total expected error is bounded by (sum_i epsilon_i) times the tail energy.
    This is the unproven content of Theorem 4. The proof is a single sentence and no composition argument for the tree structure is provided.
  • ad hoc to paper The pass-efficient randomized algorithm from [40] yields similar error bounds for the randomized KTD.
    The paper states that 'similar upper bounds as in Theorem 4 can be straightforwardly estimated for this new strategy' without any proof or additional justification.

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Cite this review

Pith. "Pith review of Fast randomized Kronecker tensor decomposition: algorithms and error analysis." pith.science (2026). https://pith.science/paper/OENUDRQL

@misc{pith2026241202597,
  author       = {Pith},
  title        = {Pith review of: Fast randomized Kronecker tensor decomposition: algorithms and error analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OENUDRQL}},
  note         = {Machine review of arXiv:2412.02597}
}
read the original abstract

This paper proposes fast randomized algorithms for computing the Kronecker Tensor Decomposition (KTD) by replacing the sequence of deterministic SVDs in the TTr1SVD framework with randomized SVDs incorporating oversampling and power iterations. The proposed algorithms can decompose a given tensor into the KTD format significantly faster than existing state-of-the-art deterministic methods. Our principal idea is to use randomization to reduce computational complexity while maintaining controlled accuracy. A detailed theoretical analysis is presented, including a recursive error bound that accounts for error propagation through the TTr1SVD tree structure. We prove that the expected Frobenius norm error is bounded by a sum of tail energies multiplied by factors that decay exponentially with the number of power iterations. Extensive simulations on synthetic and real-world datasets demonstrate several orders of magnitude acceleration compared to the deterministic approach, with applications to tensor completion, video/image compression, image denoising, and image super-resolution.

Figures

Figures reproduced from arXiv: 2412.02597 by the authors.

Figure 1
Figure 1. Fast CPD with a fast prior Tucker compression. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. (Left) Running time comparison of the proposed randomized algorithms and the de [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Comparing the image quality obtained by the deterministic KTD and the proposed R [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: (left) Running time comparison for compressing the kodim23 using the deterministic [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Comparing the PSNRs achieved by the deterministic KTD and the proposed R-KTD for [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Some reconstructed frames of the Foreman and Aikyo videos using the KTD and pro [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Comparing the running times of the deterministic KTD and the proposed R-KTD for [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: The recovered images using the proposed randomized KTD and the deterministic KTD [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Comparing the recovered images using the proposed randomized KTD algorithm for [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: The PSNR comparison of all reconstructed frames of the Aikyo video using the pro [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: Some reconstructed frames of the Aikyo video using the KTD and proposed R-KTD [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: The results of denoising for Kodim23 and using different noise types, the first row is [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: The super-resolution results for three color images. The KTD rank [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

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Reference graph

Works this paper leans on

44 extracted references · 41 canonical work pages

  1. [40]

    E. K. Bjarkason, Pass-efficient randomized algorithms for low-rank matrix approximation using any number of views, SIAM Journal on Scientific Com- puting 41 (4) (2019) A2355–A2383

    work page 2019

  2. [1]

    Comon, X

    P. Comon, X. Luciani, A. L. F. de Almeida, Tensor decompositions, alter- nating least squares and other tales, Journal of Chemometrics: A Journal of the Chemometrics Society 23 (7-8) (2009) 393–405. 24

    work page 2009

  3. [2]

    Favier, A

    G. Favier, A. L. F. de Almeida, Overview of constrained PARAFAC models, EURASIP Journal of Advances in Signal Processing 2014 (142) (2014) 1– 25

    work page 2014

  4. [3]

    Cichocki, N

    A. Cichocki, N. Lee, I. Oseledets, A.-H. Phan, Q. Zhao, D. P. Mandic, et al., Tensor networks for dimensionality reduction and large-scale optimization: Part 1 low-rank tensor decompositions, Foundations and Trends® in Ma- chine Learning 9 (4-5) (2016) 249–429

    work page 2016

  5. [4]

    M. G. Asante-Mensah, S. Ahmadi-Asl, A. Cichocki, Matrix and tensor com- pletion using tensor ring decomposition with sparse representation, Machine Learning: Science and Technology 2 (3) (2021) 035008

    work page 2021

  6. [5]

    Ahmadi-Asl, M

    S. Ahmadi-Asl, M. G. Asante-Mensah, A. Cichocki, A. H. Phan, I. Os- eledets, J. Wang, Fast cross tensor approximation for image and video com- pletion, Signal Processing (2023) 109121

    work page 2023

  7. [6]

    N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E. Papalexakis, C. Faloutsos, Tensor decomposition for signal processing and machine learn- ing, IEEE Transactions on Signal Processing 65 (13) (2017) 3551–3582. doi:10.1109/TSP.2017.2690524

    arXiv 2017

  8. [7]

    Eld ´en, S

    L. Eld ´en, S. Ahmadi-Asl, Solving bilinear tensor least squares problems and application to hammerstein identification, Numerical Linear Algebra with Applications 26 (2) (2019) e2226

    work page 2019

Show all 44 references
  1. [8]

    F. L. Hitchcock, The expression of a tensor or a polyadic as a sum of prod- ucts, Journal of Mathematics and Physics 6 (1-4) (1927) 164–189

    1927

  2. [9]

    De Lathauwer, B

    L. De Lathauwer, B. De Moor, J. Vandewalle, A multilinear singular value decomposition, SIAM journal on Matrix Analysis and Applications 21 (4) (2000) 1253–1278

    2000

  3. [10]

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

    2011

  4. [11]

    Zniyed, R

    Y . Zniyed, R. Boyer, A. L. F. de Almeida, G. Favier, A TT-based hierarchical framework for decomposing high-order tensors, SIAM Journal on Scientific Computing 42 (2) (2020) A822–A848. 25

    2020

  5. [12]

    Q. Zhao, G. Zhou, S. Xie, L. Zhang, A. Cichocki, Tensor ring decomposi- tion, arXiv preprint arXiv:1606.05535 (2016)

    2016 arXiv

  6. [13]

    De Lathauwer, Decompositions of a higher-order tensor in block terms—part ii: Definitions and uniqueness, SIAM Journal on Matrix Analy- sis and Applications 30 (3) (2008) 1033–1066

    L. De Lathauwer, Decompositions of a higher-order tensor in block terms—part ii: Definitions and uniqueness, SIAM Journal on Matrix Analy- sis and Applications 30 (3) (2008) 1033–1066

    2008

  7. [14]

    A. L. F. de Almeida, G. Favier, J. C. M. Mota, A constrained factor decom- position with application to MIMO antenna systems, IEEE Transactions on Signal Processing 56 (6) (2008) 2429–2442

    2008

  8. [15]

    Stegeman, A

    A. Stegeman, A. L. F. de Almeida, Uniqueness conditions for constrained three-way factor decompositions with linearly dependent loadings, SIAM J. Mat. Anal. Appl. 31 (3) (2010) 1469–1490

    2010

  9. [16]

    A. H. Phan, A. Cichocki, P. Tichavsk `y, D. P. Mandic, K. Matsuoka, On revealing replicating structures in multiway data: A novel tensor decompo- sition approach, in: Latent Variable Analysis and Signal Separation: 10th International Conference, LV A/ICA 2012, Tel Aviv, Israel...

    2012

  10. [17]

    Batselier, N

    K. Batselier, N. Wong, A constructive arbitrary-degree kronecker prod- uct decomposition of tensors, Numerical Linear Algebra with Applications 24 (5) (2017) e2097

    2017

  11. [18]

    A.-H. Phan, A. Cichocki, P. Tichavsk`y, G. Luta, A. Brockmeier, Tensor com- pletion throughmultiple kronecker product decomposition, in: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, IEEE, 2013, pp. 3233–3237

    2013

  12. [19]

    A. H. Phan, A. Cichocki, P. Tichavsk `y, R. Zdunek, S. Lehky, From ba- sis components to complex structural patterns, in: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, IEEE, 2013, pp. 3228–3232

    2013

  13. [20]

    Pickard, C

    J. Pickard, C. Stansbury, C. Chen, A. Surana, A. Bloch, I. Rajapakse, Kro- necker product of tensors and hypergraphs, arXiv preprint arXiv:2305.03875 (2023)

    2023 arXiv

  14. [21]

    Tahaei, E

    M. Tahaei, E. Charlaix, V . Nia, A. Ghodsi, M. Rezagholizadeh, Kro- neckerbert: Significant compression of pre-trained language models through 26 kronecker decomposition and knowledge distillation, in: Proceedings of the 2022 Conference of the North American Chapter of the Ass...

    2022

  15. [22]

    Edalati, M

    A. Edalati, M. Tahaei, I. Kobyzev, V . P. Nia, J. J. Clark, M. Rezagholizadeh, Krona: Parameter efficient tuning with kronecker adapter, arXiv preprint arXiv:2212.10650 (2022)

    2022 arXiv

  16. [23]

    D. Wang, B. Wu, G. Zhao, M. Yao, H. Chen, L. Deng, T. Yan, G. Li, Kronecker cp decomposition with fast multiplication for compressing rnns, IEEE Transactions on Neural Networks and Learning Systems (2021)

    2021

  17. [24]

    Ahmadi-Asl, A

    S. Ahmadi-Asl, A. Cichocki, A. H. Phan, M. G. Asante-Mensah, M. M. Ghazani, T. Tanaka, I. Oseledets, Randomized algorithms for fast computa- tion of low rank tensor ring model, Machine Learning: Science and Tech- nology 2 (1) (2020) 011001

    2020

  18. [25]

    Ahmadi-Asl, S

    S. Ahmadi-Asl, S. Abukhovich, M. G. Asante-Mensah, A. Cichocki, A. H. Phan, T. Tanaka, I. Oseledets, Randomized algorithms for computation of Tucker decomposition and higher order SVD (HOSVD), IEEE Access 9 (2021) 28684–28706

    2021

  19. [26]

    M. Che, Y . Wei, Randomized algorithms for the approximations of Tucker and the tensor train decompositions, Advances in Computational Mathemat- ics 45 (1) (2019) 395–428

    2019

  20. [27]

    Minster, A

    R. Minster, A. K. Saibaba, M. E. Kilmer, Randomized algorithms for low- rank tensor decompositions in the Tucker format, SIAM Journal on Mathe- matics of Data Science 2 (1) (2020) 189–215

    2020

  21. [28]

    Zhang, A

    J. Zhang, A. K. Saibaba, M. E. Kilmer, S. Aeron, A randomized tensor sin- gular value decomposition based on the t-product, Numerical Linear Algebra with Applications 25 (5) (2018) e2179

    2018

  22. [29]

    Batselier, H

    K. Batselier, H. Liu, N. Wong, A constructive algorithm for decomposing a tensor into a finite sum of orthonormal rank-1 terms, SIAM Journal on Matrix Analysis and Applications 36 (3) (2015) 1315–1337

    2015

  23. [30]

    C. F. Van Loan, N. Pitsianis, Approximation with Kronecker products, Springer, 1993. 27

    1993

  24. [31]

    Salmi, A

    J. Salmi, A. Richter, V . Koivunen, Sequential unfolding svd for tensors with applications in array signal processing, IEEE Transactions on Signal Pro- cessing 57 (12) (2009) 4719–4733

    2009

  25. [32]

    R. A. Harshman, et al., Foundations of the parafac procedure: Models and conditions for an” explanatory” multimodal factor analysis (1970)

    1970

  26. [33]

    eckart-young

    J. D. Carroll, J.-J. Chang, Analysis of individual differences in multidimen- sional scaling via an n-way generalization of “eckart-young” decomposition, Psychometrika 35 (3) (1970) 283–319

    1970

  27. [34]

    Rajih, P

    M. Rajih, P. Comon, R. A. Harshman, Enhanced line search: A novel method to accelerate parafac, SIAM journal on matrix analysis and applications 30 (3) (2008) 1128–1147

    2008

  28. [35]

    Tomasi, Practical and Computational Aspects in Chemometric Data Anal- ysis: Ph

    G. Tomasi, Practical and Computational Aspects in Chemometric Data Anal- ysis: Ph. D. Dissertation, Department of Food Science, Royal Veterinary and Agricultural University, 2006

    2006

  29. [36]

    Y . Chen, D. Han, L. Qi, New als methods with extrapolating search direc- tions and optimal step size for complex-valued tensor decompositions, IEEE Transactions on Signal Processing 59 (12) (2011) 5888–5898

    2011

  30. [37]

    H. A. Kiers, A three–step algorithm for candecomp/parafac analysis of large data sets with multicollinearity, Journal of Chemometrics: A Journal of the Chemometrics Society 12 (3) (1998) 155–171

    1998

  31. [38]

    Battaglino, G

    C. Battaglino, G. Ballard, T. G. Kolda, A practical randomized cp tensor decomposition, SIAM Journal on Matrix Analysis and Applications 39 (2) (2018) 876–901

    2018

  32. [39]

    Vervliet, L

    N. Vervliet, L. De Lathauwer, A randomized block sampling approach to canonical polyadic decomposition of large-scale tensors, IEEE Journal of Selected Topics in Signal Processing 10 (2) (2015) 284–295

    2015

  33. [41]

    Bro, Multi-way analysis in the food industry, model, algorithms and ap- plications, Doctoral Thesis, University of Amsterdam (1998)

    R. Bro, Multi-way analysis in the food industry, model, algorithms and ap- plications, Doctoral Thesis, University of Amsterdam (1998). 28

    1998

  34. [42]

    S. A. Goreinov, E. E. Tyrtyshnikov, N. L. Zamarashkin, A theory of pseu- doskeleton approximations, Linear algebra and its applications 261 (1-3) (1997) 1–21

    1997

  35. [43]

    Drineas, R

    P. Drineas, R. Kannan, M. W. Mahoney, Fast monte carlo algorithms for ma- trices i: Approximating matrix multiplication, SIAM Journal on Computing 36 (1) (2006) 132–157. 29

    2006

  36. [2012]

    Proceedings 10, Springer, 2012, pp. 297–305

    2012

Pith tools

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