Pith. sign in

REVIEW 4 major objections 6 minor 50 references

Incremental Hierarchical Tucker Decomposition

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read The paper presents the first incremental hierarchical Tucker decomposition algorithm for streaming tensor batches, with guaranteed error bounds and no full-data storage or reconstruction.

desk verdict Genuinely first incremental HT algorithm with a clever batch format, but the promised error bound is not actually proven; worth a serious referee with major revision. read the letter →

arxiv 2412.16544 v1 pith:O462CV64 submitted 2024-12-21 math.NA cs.NAeess.SP

Doruk Aksoy , Alex A. Gorodetsky This is my paper · ORCID
classification math.NAcs.NAeess.SP MSC 15A6965F55
keywords hierarchicalTuckerdecompositionincrementaltensoralgorithmsstreamingdatacompressionnetworkslow-rankapproximationonlinelatentrepresentationrank-adaptivestreams
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

Tensor streams — sequences of multidimensional arrays — are normally compressed with incremental versions of CP, Tucker, or tensor-train formats, but the hierarchical Tucker (HT) format had no incremental update rule. This paper claims to close that gap with HT-RISE, an online algorithm that updates an existing batch-HT representation by projecting each new batch onto the current cores, computing the residual, and appending only the missing directions revealed by that residual. Because the new directions are orthogonal to the existing cores, old data never needs to be stored or reconstructed, and previously streamed tensors keep their exact reconstructions after every update. The paper also introduces BHT-l2r, a one-shot batch decomposition that carries the batch dimension in the root core rather than as a leaf; experiments report up to 6.2x compression and 3.7x time savings over standard HT compression and up to 3.1x compression and 3.2x time savings over an incremental tensor-train baseline.

What carries the argument

The load-bearing object is the projection residual used to expand each HT core. When a new batch is contracted against the existing orthonormal cores, the part that is not representable is $R_{\ell,j} = \Pi_{\ell,j}^{k-1} C_{\ell,(j)}$ with $\Pi_{\ell,j}^{k-1} = I - U_{\ell,j}^{k-1}(U_{\ell,j}^{k-1})^T$. An error-truncated SVD of this residual yields new left singular vectors $U_R$, and the updated core is the concatenation $U^k = [U^{k-1}, U_R]$, which is orthonormal by construction; parent cores are zero-padded so tensor-network dimensions stay consistent. This residual-expansion step is what makes the update incremental (only missing directions are added), makes it memory-safe (the full stream is never formed), and lets the proof decompose the global error into layerwise SVD truncation errors.

What would settle it

Construct a small two-layer HT, stream a batch that has shared energy across two sibling nodes, truncate each residual at exactly the node-wise tolerance, and compare the true reconstruction error to the claimed bound. If the observed error exceeds the bound, or if an earlier tensor's reconstruction changes after a later update, the residual-orthogonality premise is violated.

Watch

Extended reading notes

Core claim

The paper's central claim is that a hierarchical Tucker decomposition can be maintained incrementally, updating all cores of the dimension tree as new batches arrive while keeping the accumulation tensor under a prescribed relative error tolerance. The batch hierarchical Tucker format is the key reformulation: instead of treating the batch index as one more leaf dimension, the batch is absorbed into the root core, so a batch of similar tensors shares the whole hierarchy of transfer cores and only the root grows with the sample count. HT-RISE updates layer by layer from leaves to root: it projects the new batch onto the existing orthonormal cores, forms the residual $R_{\ell,j}$, truncates its SVD at a node-wise tolerance, and concatenates the new left singular vectors onto the existing basis. The paper argues that this update adds only the genuinely new information, that the node-wise truncation errors combine into the same global bound as the one-shot decomposition, and that padding parent cores with zeros leaves the reconstruction of every previously streamed tensor unchanged.

Load-bearing premise

The error guarantee depends on the assumption that each node's SVD truncation error can be summed independently, because the residuals from projecting a new batch onto different cores are orthogonal; if truncating one layer's rank interferes with another layer's residual, the stated worst-case bound can fail.

Editorial extensions

If this is right

  • HT-RISE gives the HT format the same streaming capability that already existed for CP, Tucker, and tensor-train formats, so online applications can keep an HT representation under a bounded error.
  • Updates are non-destructive: reconstructing any tensor that arrived in an earlier batch from the current representation gives exactly the same result it gave before the new batch was seen.
  • Moving the batch dimension to the root core makes the batch format more economical than standard HT for batches of similar tensors; the reported gains reach 6.2x compression and 3.7x time reduction on image data.
  • Compared with an incremental tensor-train method, HT-RISE reaches the target test-set error after far fewer training batches on multi-scale data, and completes streams the baseline does not finish within the wall-time limit.
  • The resulting latent representation is a matrix slice in the root core, so its size is governed by the complexity of the data rather than capped by the number of accumulated tensors.

Reading between the lines

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

  • The residual-expansion update is a generic mechanism: the same project-residual-append step should transfer to n-ary dimension trees or to time-varying tree topologies, since each node update only needs its own orthonormal basis and its children's ranks.
  • Because each layer is updated independently after the previous layer's projection, the per-layer SVDs could be replaced by randomized or streaming SVDs to trade a little accuracy for much lower per-batch cost on very large streams.
  • The paper's own experiments suggest the advantage is domain-dependent: on simple, globally low-rank data the extra HT hierarchy is overhead, so the method is best matched to multi-scale or locally structured data.
  • A direct test of the invariance claim is to partition one fixed dataset into different batch orders and check whether each original tensor's slice reconstruction is identical across orderings; this would exercise the past-stream guarantee without needing ground truth.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes two algorithms for hierarchical Tucker (HT) decompositions of tensor streams: BHT-l2r, a leaves-to-root construction of a 'batch hierarchical Tucker' format in which the batch dimension is stored only in the root core, and HT-RISE, an incremental algorithm that updates an existing batch-HT representation when new batches arrive. The authors claim that HT-RISE is the first incremental HT algorithm, that both algorithms are rank-adaptive, and that HT-RISE provides guaranteed per-batch error bounds while never storing or reconstructing the full accumulation. The numerical section compares HT-RISE with the incremental tensor-train algorithm TT-ICE* on four datasets (PDEBench, self-oscillating gels, MineRL, BigEarthNet) and reports compression ratios, reduction ratios, wall time, and relative test error, with claims of up to 6.2x compression and 3.7x time reduction for BHT-l2r over one-shot HT, and up to 3.1x compression and 3.2x time reduction for HT-RISE over TT-ICE*.

Significance. If the stated theoretical guarantees were rigorously established, the paper would fill a real gap: incremental algorithms exist for CP, Tucker, and TT formats, and an incremental HT format with provable per-batch error control would be a useful contribution for streaming high-dimensional scientific and image data. The batch hierarchical Tucker modification is simple and clearly motivated, and the empirical study is broad, covering four realistic datasets and comparing against a strong TT baseline. The claim that HT-RISE generalizes with fewer training batches than TT-ICE* is interesting and supported by the reported test-error trajectories. However, the central proof of the incremental error guarantee is a sketch that omits the key inequality, and without Theorem 5 being formally stated and proved, the paper's main theoretical selling point is not yet established. The experimental sections also lack variance information despite averaging over seeds, which weakens the quantitative claims.

major comments (4)
  1. [Appendix A, Theorem 5] Theorem 5 is presented as a heading with no formal statement: there is no stated bound, no specification of the norm or the tolerance, and no list of hypotheses. A theorem that is never stated cannot be verified. The heading 'Theorem 5 (HT-RISE approximation error)' is followed directly by a proof sketch, but the reader cannot tell what exactly is claimed (per-batch error? accumulated error? worst-case or expected?). This is a load-bearing omission because the abstract and Section 3.2 promise 'provable error bounds' for HT-RISE.
  2. [Appendix A, proof of Theorem 5, Eqs. (29)-(33)] The proof sketch asserts that the residual SVD errors accumulate via Theorem 3 (Grasedyck's Lemma 3.10), but Theorem 3 applies to a one-shot leaves-to-root HOSVD of a single tensor with orthonormal transfer cores. In HT-RISE the updates are greedy: new basis vectors computed from the residual of one node are appended to existing cores, and parent cores are zero-padded (Eqs. (14)-(17)). The residuals at different layers and nodes are not orthogonal in the full tensor space, and the proof never establishes the key inequality ∥Y^k - \tilde{Y}^k∥_F^2 ≤ Σ_{ℓ,j} ∥E_{R,ℓ,j}∥_F^2 for the final reconstructed batch. Eq. (29) only decomposes the mode-ℓ unfolding of the intermediate core; it does not show how the truncation errors propagate through the subsequent contractions up to the root concatenation (Eq. (19)). Without this step, the central claim that HT-RISE maintains a per-batch error bounded by ε_des is unsupported.
  3. [Algorithm 2, line 16; Theorem 5] The early-exit condition in Algorithm 2 (line 16) skips all core updates except the root when the projection error is below ε_des. The proof of Theorem 5 does not treat this branch. A complete proof must show that the early-exit update also satisfies the claimed error bound, since the root is updated with the projection C̄^k_1 (line 46) rather than with a full residual-corrected representation. The theorem should cover both the early-exit and the full-update paths.
  4. [Problem 2 and Appendix A] Problem 2 asks for a guaranteed bound on the accumulated tensor error ∥X^k - \hat{X}^k∥_F at every step k, but the theoretical discussion in Appendix A concerns only the error for a new batch Y^k relative to ε_des = ε_rel ∥Y^k∥_F. No theorem in the appendix states how per-batch guarantees combine into an accumulation-level bound. Theorem 6 only shows that past slices are reconstructed identically after later updates; it does not address whether the squared errors of all slices sum to a bound on the full accumulated error. The gap between Problem 2 and the provided theory should be closed or the problem statement should be amended to match what is actually proved.
minor comments (6)
  1. [Section 1 and various appendices] The manuscript contains numerous typos and spelling errors, including 'univerally' (Section 1), 'Summmary' (Section 4.3.2 and Appendix D), 'intersting' (Appendix D.1), 'normalziation' (Appendix D.5), and 'likekly' (Appendix D.2). A careful proofreading pass is needed.
  2. [Section 3.1 and Appendix A] The cross-referencing of results is inconsistent: Section 3.1 refers to 'Theorem 4' in Appendix A, but the appendix labels the statement as 'Corollary 4'. Additionally, Theorem 5 is numbered but has no statement, which is unusual and confusing; the numbering should be cleaned up as part of the revision.
  3. [Theorem 1 vs. Corollary 4] Theorem 1 (adapted from Kressner and Tobler) prescribes a node-wise tolerance ε_nw = ε_abs/√(2d−3), while Corollary 4 uses ε_nw = ε_abs/√(2d−2). The manuscript should clarify why the batch construction has one more truncating SVD than the standard HT construction, or correct the constants if they are inconsistent.
  4. [Section 4, Tables 2-4 and 6] The experiments are averaged over five seeds (PDEBench, MineRL, BigEarthNet), but the tables and figures report only point estimates with no standard deviations or confidence intervals. Since several comparisons are close (e.g., BigEarthNet at ε_rel = 0.10), reporting variance is important for assessing whether the observed differences are meaningful.
  5. [Figure 12 and Table 3] Figure 12's left panel y-axis label reads 'Total Time' while the text consistently refers to 'Compression Time' elsewhere; please make the terminology uniform. In Table 3, the entry '4581,78' should be '4581.78'.
  6. [Appendix A, proof of Theorem 6] The proof of Theorem 6 contains a confusing sentence: 'the entries of G^t_{1,1}(i1,i2,: r^k_{1,1}) are all zero for any i_j > r^k_{2,j}'. The condition should presumably involve the index range of the padded mode (i.e., i1 > r^k_{2,1} or i2 > r^k_{2,2}), not a generic i_j. This should be rephrased for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: HT-RISE's error guarantees are supported by external truncation lemmas and standard SVD bounds, not by definitions or fitted parameters.

full rationale

The derivation chain is not circular. Algorithm 1's error bound (Corollary 4) is obtained by setting a uniform node-wise tolerance εnw = εabs/sqrt(2d−2) and then applying Theorem 3, which the paper states as a direct consequence of Grasedyck (2010, Lemma 3.10); the resulting bound ||Y−Y~||F ≤ εabs is a standard sequential-truncation guarantee, not an equivalent restatement of the tolerance. Algorithm 2 likewise sets εnw = εdes/sqrt(2d−2) and bounds each residual SVD truncation by εnw, then invokes Theorem 3 to accumulate layerwise bounds; the proof sketch of Theorem 5 is abbreviated and its formal statement is missing, but that is a rigor/correctness gap, not a circular reduction, because the load-bearing inequality is imported from an external lemma rather than from HT-RISE's own output. Claim 1's projection-error identity is a direct orthogonality/Pythagorean argument, not a definitional tautology. The only self-references are TT-ICE* used as an experimental baseline and Theorem 6's proof being described as similar to the authors' previous Theorem 4; neither is load-bearing, and Theorem 6 is proved in-line. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors, and no known empirical pattern is repackaged as new under a change of coordinates. The empirical compression and timing claims are benchmark measurements, so they cannot be circular. Hence the paper deserves a circularity score of 0.

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

The algorithms are rank-adaptive and take the relative error tolerance as a user input; no constants are fitted to data in the derivation. The experimental choices of tensor reshaping and axis ordering affect results but are not part of the theoretical claims. The paper introduces no new physical or mathematical entities.

assumptions (4)
  • standard math Layer-wise error bound of Grasedyck (2010, Lemma 3.8): the error incurred at one layer of the HT decomposition is bounded by the sum of the truncated SVD errors at that layer.
    Invoked in the proof of Lemma 2 and Theorem 3, and relied upon for the BHT-l2r error bound in Corollary 4.
  • standard math Total error bound of Grasedyck (2010, Lemma 3.10): the total HT approximation error is bounded by the sum of layer-wise errors.
    Used to derive the nodewise tolerance epsilon_nw = epsilon_abs / sqrt(2d - 2) and the per-batch error guarantees.
  • domain assumption Cores remain orthonormal under BHT-l2r and HT-RISE updates.
    This is used in Claim 1 to equate the projection error with the reconstruction error, and in Theorem 6 to show past tensors are unchanged. It holds only if the initial decomposition is orthonormal and the residual SVD basis vectors are orthogonal to the existing basis, which the paper argues but does not fully prove for all update paths.
  • domain assumption The residual SVD truncation at each node is independent and the truncation errors accumulate as in the one-shot HT decomposition.
    The proof sketch of Theorem 5 assumes that applying Theorem 3 to the updated cores with nodewise truncation tolerance yields the global bound, which requires that the new basis vectors added at one layer do not change the truncation error at other layers.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Incremental Hierarchical Tucker Decomposition." pith.science (2026). https://pith.science/paper/O462CV64

@misc{pith2026241216544,
  author       = {Pith},
  title        = {Pith review of: Incremental Hierarchical Tucker Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O462CV64}},
  note         = {Machine review of arXiv:2412.16544}
}
abstract

We present two new algorithms for approximating and updating the hierarchical Tucker decomposition of tensor streams. The first algorithm, Batch Hierarchical Tucker - leaf to root (BHT-l2r), proposes an alternative and more efficient way of approximating a batch of similar tensors in hierarchical Tucker format. The second algorithm, Hierarchical Tucker - Rapid Incremental Subspace Expansion (HT-RISE), updates the batch hierarchical Tucker representation of an accumulated tensor as new batches of tensors become available. The HT-RISE algorithm is suitable for the online setting and never requires full storage or reconstruction of all data while providing a solution to the incremental Tucker decomposition problem. We provide theoretical guarantees for both algorithms and demonstrate their effectiveness on physical and cyber-physical data. The proposed BHT-l2r algorithm and the batch hierarchical Tucker format offers up to $6.2\times$ compression and $3.7\times$ reduction in time over the hierarchical Tucker format. The proposed HT-RISE algorithm also offers up to $3.1\times$ compression and $3.2\times$ reduction in time over a state of the art incremental tensor train decomposition algorithm.

Figures

Figures reproduced from arXiv: 2412.16544 by the authors.

Figure 1
Figure 1. Flow of the proposed HT-RISE algorithm. The algorithm updates the hierarchical Tucker representation of an accumulated tensor in batch hierarchical Tucker form as new batches of tensors become available. Green represents the input data and red represents the output data. size of the data makes it infeasible or impossible to compute an approximation using a one￾shot tensor decomposition algorithm due to computational… view at source ↗
Figure 2
Figure 2. Three possible configurations of the dimensions for a five dimensional tensor with shape n1 × n2 × n3 × n4 × n5. Each dimension tree describes a different interaction between dimensions and therefore yields different compression performance. For a more detailed study on the effect of the axis reordering, please refer to Appendix D.3. Root, transfer, and leaf nodes are colored in red, blue, and green, respectively. F… view at source ↗
Figure 3
Figure 3. Tensor network diagrams of an 8D tensor and its representations in Tucker format, hierarchical Tucker format, and batch hierarchical Tucker format. The streaming/batch dimension is labeled N 3.1 Batch hierarchical Tucker In this section, we aim to provide a solution to Problem 1 by presenting a hierarchical Tucker decomposition algorithm for batches of tensors. One approach to compressing a batch of tensors in HT fo… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Step-by-step decomposition of an N-batch of 5-D tensors with the BHT-l2r algorithm (Al￾gorithm 1). The decomposition starts with the leaves of the last layer. Since the dimension tree is constructed beforehand, Algorithm 1 has the information about which dimensions’ le…
Figure 5
Figure 5. Figure 5: Example snapshots from scientific datasets. 4.2.1 Self-oscillating gel simulations The first scientific dataset arises from solutions of a parametric PDE that simulates the motion of a hexagonal sheet of self-oscillating gels. This dataset is used both in donwstream le…
Figure 6
Figure 6. Figure 6: Compression ratio (CR - left) and reduction ratio (RR - right) of the algorithms on the self-oscillating gel dataset. TT-ICE∗ offers 2.5 − 3.3× the CR and 4.6 − 9.5× the RR of HT-RISE. For more detailed comparisons, please refer to Figure D.12 for experiments with εrel…
Figure 7
Figure 7. Figure 7: Compression time (left) and Relative Test Error (right) of the algorithms on the self￾oscillating gel dataset. HT-RISE takes 2.8 − 3.2× the time it takes TT-ICE∗ to complete compressing the stream. Neither method struggles to reduce the RTE below the target ε for both …
Figure 8
Figure 8. Figure 8: Compression ratio (CR - left) and reduction ratio (RR- - right) of the algorithms on the PDEBench 3D turbulent Navier-Stokes dataset. HT-RISE offers 2.4 − 3.1× the CR of TT-ICE∗ but results in orders of magnitude lower RR. TT-ICE∗ does not complete the entire stream du…
Figure 9
Figure 9. Figure 9: Compression time (left) and Relative Test Error (right) of the algorithms on the PDEBench 3D turbulent Navier-Stokes dataset. Until TT-ICE∗ hits the maximum walltime limit, TT-ICE∗ takes 1.8 − 3.2× of the time it takes HT-RISE to complete compressing the stream. In add…
Figure 10
Figure 10. Figure 10: Example frames from image-based datasets. 4.3.1 Minecraft frames First person games are of interest for inverse reinforcement learning purposes. The MineRL Basalt competition dataset (Milani et al., 2024) contains gameplay episodes from the game Minecraft and is desig…
Figure 11
Figure 11. Figure 11: Compression ratio (CR - left) and reduction ratio (RR - right) of the algorithms on the Basalt MineRL dataset.TT-ICE∗ offers 1 − 2.9× the CR and 1.6 − 3× the RR of HT-RISE. The results are averaged over 5 seeds. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: Compression time (left) and Relative Test Error (right) of the algorithms on the Basalt MineRL competition dataset. Each target ε level is shown with dashed lines of their corresponding color. HT-RISE takes 2.8−3.2× the time it takes TT-ICE∗ to complete compressing th…
Figure 13
Figure 13. Figure 13: Reconstructed multispectral images from BigEarthNet’s training set, captured by the Sentinel-2 satellite, are displayed. Images compressed using HT-RISE are in the top row, while those compressed with TT-ICE∗ appear in the bottom row. Different columns represent recon…
Figure 14
Figure 14. Figure 14: Reconstructed multispectral images from BigEarthNet’s test set, captured by the Sentinel￾2 satellite, are displayed. Images compressed using HT-RISE are in the top row, while those com￾pressed with TT-ICE∗ appear in the bottom row. Different columns represent reconstr…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 39 canonical work pages

  1. [1]

    Inverse design of self-oscillatory gels through deep learning

    Doruk Aksoy, Silas Alben, Robert D Deegan, and Alex A Gorodetsky. Inverse design of self-oscillatory gels through deep learning. Neural Computing and Applications, 34 0 (9): 0 6879--6905, 2022

    work page 2022

  2. [2]

    An incremental tensor train decomposition algorithm

    Doruk Aksoy, David J Gorsich, Shravan Veerapaneni, and Alex A Gorodetsky. An incremental tensor train decomposition algorithm. SIAM Journal on Scientific Computing, 46 0 (2): 0 A1047--A1075, 2024 a

    work page 2024

  3. [3]

    Marks, Joshua D

    Doruk Aksoy, Sruti Vutukury, Thomas A. Marks, Joshua D. Eckels, and Alex A. Gorodetsky. Compressed analysis of electric propulsion simulations using low rank tensor networks. 2024 b

    work page 2024

  4. [4]

    Semi-implicit methods for the dynamics of elastic sheets

    Silas Alben, Alex A Gorodetsky, Donghak Kim, and Robert D Deegan. Semi-implicit methods for the dynamics of elastic sheets. Journal of Computational Physics, 399: 0 108952, 2019

    work page 2019

  5. [5]

    Video pretraining (vpt): Learning to act by watching unlabeled online videos

    Bowen Baker, Ilge Akkaya, Peter Zhokov, Joost Huizinga, Jie Tang, Adrien Ecoffet, Brandon Houghton, Raul Sampedro, and Jeff Clune. Video pretraining (vpt): Learning to act by watching unlabeled online videos. Advances in Neural Information Processing Systems, 35: 0 24639--24654, 2022

    2022

  6. [6]

    eckart-young

    J Douglas Carroll and Jih-Jie Chang. Analysis of individual differences in multidimensional scaling via an n-way generalization of “eckart-young” decomposition. Psychometrika, 35 0 (3): 0 283--319, 1970

    1970

  7. [7]

    L1-norm higher-order orthogonal iterations for robust tensor analysis

    Dimitris G Chachlakis, Ashley Prater-Bennette, and Panos P Markopoulos. L1-norm higher-order orthogonal iterations for robust tensor analysis. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 4826--4830. IEEE, 2020

    work page 2020

  8. [8]

    Randomized algorithms for the approximations of tucker and the tensor train decompositions

    Maolin Che and Yimin Wei. Randomized algorithms for the approximations of tucker and the tensor train decompositions. Advances in Computational Mathematics, 45 0 (1): 0 395--428, 2019

    work page 2019

Show all 50 references
  1. [9]

    Low-rank tensor-network encodings for video-to-action behavioral cloning

    Brian Chen, Doruk Aksoy, David J Gorsich, Shravan Veerapaneni, and Alex Gorodetsky. Low-rank tensor-network encodings for video-to-action behavioral cloning. Transactions on Machine Learning Research, 2024. ISSN 2835-8856. URL https://openreview.net/forum?id=w4DXLzBPPw

    2024

  2. [10]

    Efficient computation of tucker decomposition for streaming scientific data compression

    Saibal De, Zitong Li, Hemanth Kolla, and Eric T Phipps. Efficient computation of tucker decomposition for streaming scientific data compression. arXiv preprint arXiv:2308.16395, 2023

    2023 arXiv

  3. [11]

    On the best rank-1 and rank-(r 1, r 2,..., rn) approximation of higher-order tensors

    Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle. On the best rank-1 and rank-(r 1, r 2,..., rn) approximation of higher-order tensors. SIAM journal on Matrix Analysis and Applications, 21 0 (4): 0 1324--1342, 2000 a

    2000

  4. [12]

    A multilinear singular value decomposition

    Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle. A multilinear singular value decomposition. SIAM journal on Matrix Analysis and Applications, 21 0 (4): 0 1253--1278, 2000 b

    2000

  5. [13]

    A newton--grassmann method for computing the best multilinear rank-(r\_1, r\_2, r\_3) approximation of a tensor

    Lars Eld \'e n and Berkant Savas. A newton--grassmann method for computing the best multilinear rank-(r\_1, r\_2, r\_3) approximation of a tensor. SIAM Journal on Matrix Analysis and applications, 31 0 (2): 0 248--271, 2009

    2009

  6. [14]

    Hierarchical singular value decomposition of tensors

    Lars Grasedyck. Hierarchical singular value decomposition of tensors. SIAM journal on matrix analysis and applications, 31 0 (4): 0 2029--2054, 2010

    2010

  7. [15]

    Harris, K

    Charles R. Harris, K. Jarrod Millman, St \' e fan J.van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaim...

    2020

  8. [16]

    explanatory

    Richard A Harshman et al. Foundations of the parafac procedure: Models and conditions for an “explanatory” multi-modal factor analysis. UCLA working papers in phonetics, 16 0 (1): 0 84, 1970

    1970

  9. [17]

    Multi-grid tensorized fourier neural operator for high-resolution pdes

    Jean Kossaifi, Nikola Kovachki, Kamyar Azizzadenesheli, and Anima Anandkumar. Multi-grid tensorized fourier neural operator for high-resolution pdes. arXiv preprint arXiv:2310.00120, 2023

    2023 arXiv

  10. [18]

    Recompression of hadamard products of tensors in tucker format

    Daniel Kressner and Lana Perisa. Recompression of hadamard products of tensors in tucker format. SIAM Journal on Scientific Computing, 39 0 (5): 0 A1879--A1902, 2017

    2017

  11. [19]

    Algorithm 941: Htucker---a matlab toolbox for tensors in hierarchical tucker format

    Daniel Kressner and Christine Tobler. Algorithm 941: Htucker---a matlab toolbox for tensors in hierarchical tucker format. ACM Transactions on Mathematical Software (TOMS), 40 0 (3): 0 1--22, 2014

    2014

  12. [20]

    Streaming tensor train approximation

    Daniel Kressner, Bart Vandereycken, and Rik Voorhaar. Streaming tensor train approximation. SIAM Journal on Scientific Computing, 45 0 (5): 0 A2610--A2631, 2023

    2023

  13. [21]

    Principal component analysis of three-mode data by means of alternating least squares algorithms

    Pieter M Kroonenberg and Jan De Leeuw. Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45: 0 69--97, 1980

    1980

  14. [22]

    Multi-task fmri data fusion using iva and parafac2

    Isabell Lehmann, Evrim Acar, Tanuj Hasija, Vince D Calhoun, Peter J Schreier, and T \"u lay Adali. Multi-task fmri data fusion using iva and parafac2. In ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1466--1470. IEEE, 2022

    2022

  15. [23]

    An incremental tensor-train decomposition for cyber-physical-social big data

    Huazhong Liu, Laurence T Yang, Yimu Guo, Xia Xie, and Jianhua Ma. An incremental tensor-train decomposition for cyber-physical-social big data. IEEE Transactions on Big Data, 7 0 (2): 0 341--354, 2018

    2018

  16. [24]

    Trawl: Tensor reduced and approximated weights for large language models

    Yiran Luo, Het Patel, Yu Fu, Dawon Ahn, Jia Chen, Yue Dong, and Evangelos E Papalexakis. Trawl: Tensor reduced and approximated weights for large language models. arXiv preprint arXiv:2406.17261, 2024

    2024 arXiv

  17. [25]

    Statistical Methods for Tensor Data Analysis, pages 817--829

    Qing Mai and Xin Zhang. Statistical Methods for Tensor Data Analysis, pages 817--829. Springer London, London, 2023. ISBN 978-1-4471-7503-2. doi:10.1007/978-1-4471-7503-2_39. URL https://doi.org/10.1007/978-1-4471-7503-2_39

    2023 doi

  18. [26]

    Low-rank tucker decomposition of large tensors using tensorsketch

    Osman Asif Malik and Stephen Becker. Low-rank tucker decomposition of large tensors using tensorsketch. Advances in neural information processing systems, 31, 2018

    2018

  19. [27]

    Hall thruster simulations in warpx

    Thomas Marks and Alex Gorodetsky. Hall thruster simulations in warpx. 2024

    2024

  20. [28]

    Bedd: The minerl basalt evaluation and demonstrations dataset for training and benchmarking agents that solve fuzzy tasks

    Stephanie Milani, Anssi Kanervisto, Karolis Ramanauskas, Sander Schulhoff, Brandon Houghton, and Rohin Shah. Bedd: The minerl basalt evaluation and demonstrations dataset for training and benchmarking agents that solve fuzzy tasks. Advances in Neural Information Processing Sys...

    2024

  21. [29]

    Parallel randomized tucker decomposition algorithms

    Rachel Minster, Zitong Li, and Grey Ballard. Parallel randomized tucker decomposition algorithms. arXiv preprint arXiv:2211.13028, 2022

    2022 arXiv

  22. [30]

    Tensor-train decomposition

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

    2011

  23. [31]

    Breaking the curse of dimensionality, or how to use svd in many dimensions

    Ivan V Oseledets and Eugene E Tyrtyshnikov. Breaking the curse of dimensionality, or how to use svd in many dimensions. SIAM Journal on Scientific Computing, 31 0 (5): 0 3744--3759, 2009

    2009

  24. [32]

    Tensor methods in computer vision and deep learning

    Yannis Panagakis, Jean Kossaifi, Grigorios G Chrysos, James Oldfield, Mihalis A Nicolaou, Anima Anandkumar, and Stefanos Zafeiriou. Tensor methods in computer vision and deep learning. Proceedings of the IEEE, 109 0 (5): 0 863--890, 2021

    2021

  25. [33]

    Learning mesh-based simulation with graph networks

    Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalez, and Peter W Battaglia. Learning mesh-based simulation with graph networks. arXiv preprint arXiv:2010.03409, 2020

    2010 arXiv

  26. [34]

    Streaming tensor factorization for infinite data sources

    Shaden Smith, Kejun Huang, Nicholas D Sidiropoulos, and George Karypis. Streaming tensor factorization for infinite data sources. In Proceedings of the 2018 SIAM International Conference on Data Mining, pages 81--89. SIAM, 2018

    2018

  27. [35]

    Bigearthnet: A large-scale benchmark archive for remote sensing image understanding

    Gencer Sumbul, Marcela Charfuelan, Beg \"u m Demir, and Volker Markl. Bigearthnet: A large-scale benchmark archive for remote sensing image understanding. In IGARSS 2019-2019 IEEE International Geoscience and Remote Sensing Symposium, pages 5901--5904. IEEE, 2019

    2019

  28. [36]

    Bigearthnet-mm: A large-scale, multimodal, multilabel benchmark archive for remote sensing image classification and retrieval [software and data sets]

    Gencer Sumbul, Arne De Wall, Tristan Kreuziger, Filipe Marcelino, Hugo Costa, Pedro Benevides, Mario Caetano, Beg \"u m Demir, and Volker Markl. Bigearthnet-mm: A large-scale, multimodal, multilabel benchmark archive for remote sensing image classification and retrieval [softw...

    2021

  29. [37]

    Low-rank tucker approximation of a tensor from streaming data

    Yiming Sun, Yang Guo, Charlene Luo, Joel Tropp, and Madeleine Udell. Low-rank tucker approximation of a tensor from streaming data. SIAM Journal on Mathematics of Data Science, 2 0 (4): 0 1123--1150, 2020

    2020

  30. [38]

    Pdebench: An extensive benchmark for scientific machine learning

    Makoto Takamoto, Timothy Praditia, Raphael Leiteritz, Daniel MacKinlay, Francesco Alesiani, Dirk Pfl \"u ger, and Mathias Niepert. Pdebench: An extensive benchmark for scientific machine learning. Advances in Neural Information Processing Systems, 35: 0 1596--1611, 2022

    2022

  31. [39]

    Tensor bm-decomposition for compression and analysis of spatio-temporal third-order data

    Fan Tian, Misha E Kilmer, Eric Miller, and Abani Patra. Tensor bm-decomposition for compression and analysis of spatio-temporal third-order data. arXiv preprint arXiv:2306.09201, 2023

    2023 arXiv

  32. [40]

    Mach: Fast randomized tensor decompositions

    Charalampos E Tsourakakis. Mach: Fast randomized tensor decompositions. In Proceedings of the 2010 SIAM international conference on data mining, pages 689--700. SIAM, 2010

    2010

  33. [41]

    Implications of factor analysis of three-way matrices for measurement of change

    Ledyard R Tucker. Implications of factor analysis of three-way matrices for measurement of change. Problems in measuring change, 15 0 (122-137): 0 3, 1963

    1963

  34. [42]

    Some mathematical notes on three-mode factor analysis

    Ledyard R Tucker. Some mathematical notes on three-mode factor analysis. Psychometrika, 31 0 (3): 0 279--311, 1966

    1966

  35. [43]

    The extension of factor analysis to three-dimensional matrices

    Ledyard R Tucker et al. The extension of factor analysis to three-dimensional matrices. Contributions to mathematical psychology, 110119, 1964

    1964

  36. [44]

    A new truncation strategy for the higher-order singular value decomposition

    Nick Vannieuwenhoven, Raf Vandebril, and Karl Meerbergen. A new truncation strategy for the higher-order singular value decomposition. SIAM Journal on Scientific Computing, 34 0 (2): 0 A1027--A1052, 2012

    2012

  37. [45]

    Robust multi-dimensional harmonic retrieval using iteratively reweighted hosvd

    Fuxi Wen and Hing Cheung So. Robust multi-dimensional harmonic retrieval using iteratively reweighted hosvd. IEEE Signal Processing Letters, 22 0 (12): 0 2464--2468, 2015

    2015

  38. [46]

    Loretta: Low-rank economic tensor-train adaptation for ultra-low-parameter fine-tuning of large language models

    Yifan Yang, Jiajun Zhou, Ngai Wong, and Zheng Zhang. Loretta: Low-rank economic tensor-train adaptation for ultra-low-parameter fine-tuning of large language models. In Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Lingui...

    2024

  39. [47]

    Incremental cp tensor decomposition by alternating minimization method

    Chao Zeng and Michael K Ng. Incremental cp tensor decomposition by alternating minimization method. SIAM Journal on Matrix Analysis and Applications, 42 0 (2): 0 832--858, 2021

    2021

  40. [48]

    A randomized tensor singular value decomposition based on the t-product

    Jiani Zhang, Arvind K Saibaba, Misha E Kilmer, and Shuchin Aeron. A randomized tensor singular value decomposition based on the t-product. Numerical Linear Algebra with Applications, 25 0 (5): 0 e2179, 2018

    2018

  41. [49]

    Tensor network factorizations: Relationships between brain structural connectomes and traits

    Zhengwu Zhang, Genevera I Allen, Hongtu Zhu, and David Dunson. Tensor network factorizations: Relationships between brain structural connectomes and traits. Neuroimage, 197: 0 330--343, 2019

    2019

  42. [50]

    Decomposition of big tensors with low multilinear rank

    Guoxu Zhou, Andrzej Cichocki, and Shengli Xie. Decomposition of big tensors with low multilinear rank. arXiv preprint arXiv:1412.1885, 2014

    2014 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.