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arxiv: 2411.01986 · v3 · submitted 2024-11-04 · 🧮 math.NA · cs.NA

Randomized coupled decompositions

Erna Begovic , Anita Carevic , Ivana Sain Glibic This is my paper

Pith reviewed 2026-05-23 17:48 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords coupled matrix factorizationcoupled matrix-tensor factorizationrandomized algorithmssingular value decompositiondata fusionface recognitionlow-rank decompositionprojection subspace
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The pith

Coupled matrix and tensor factorizations reduce to a direct singular value decomposition when a common factor is shared.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that coupled matrix factorization, where two low-rank matrices share a common factor, and coupled matrix-tensor factorization, where a matrix and tensor share one, admit exact solutions computed via a single singular value decomposition instead of iterative optimization. For very large data the authors introduce randomized versions of these direct methods, including a new way to pick the random projection subspace that weights contributions from both input objects equally. Numerical experiments confirm the randomized versions run faster while preserving accuracy, and the same algorithms achieve high success rates when applied to face recognition. A reader would care because this replaces potentially slow, convergence-dependent iterations with a one-shot linear algebra step that scales to bigger fusion problems.

Core claim

When two matrices or a matrix and a tensor are generated from low-rank factors that include a shared common factor matrix, their joint factorization is recovered exactly by forming a combined data matrix and taking its singular value decomposition; the left and right singular vectors directly supply the unknown factors. The same construction extends to a randomized algorithm that projects onto a subspace chosen to balance the influence of each original matrix or tensor, yielding approximate factors at far lower cost.

What carries the argument

The direct SVD of a stacked or concatenated data matrix that encodes the shared-factor coupling, together with a balanced random projection subspace for the large-scale case.

If this is right

  • The method replaces iterative solvers with a single SVD whose cost is known and predictable.
  • Randomized variants with balanced subspace selection run in time linear in the data dimensions while retaining high accuracy.
  • The same direct and randomized pipelines apply without change to the face-recognition task and report high success rates.
  • Numerical tests on synthetic and real data confirm that the randomized algorithms match the accuracy of their deterministic counterparts at lower cost.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The balanced-subspace idea could be tested on other coupled problems that currently rely on alternating least squares.
  • Direct SVD recovery removes the need to monitor iteration convergence, which may simplify deployment in automated pipelines.
  • The face-recognition results indicate the approach may transfer to other multimodal identification tasks where one modality is a matrix and another a tensor.

Load-bearing premise

The input matrices and tensors must be generated exactly from low-rank factors that include one identical shared factor matrix, so that the coupling produces a structure whose singular vectors recover the factors without residual error.

What would settle it

Construct two small matrices known to share an exact common factor, run the direct SVD procedure, and verify that the recovered factors equal the originals up to scaling and column permutation with machine-precision error.

Figures

Figures reproduced from arXiv: 2411.01986 by Anita Carevic, Erna Begovic, Ivana Sain Glibic.

Figure 1
Figure 1. Figure 1: Graphical description of CMF. Y ∈ R m×n2 , their coupled rank-k approximation is given by X ≈ UV T and Y ≈ UWT , (1.1) where U ∈ R m×k , V ∈ R n1×k , W ∈ R n2×k . Big and complex data sets are often represented by tensors rather than matrices. Therefore, in addition to the matrix-matrix case, it is also important to consider tensor-matrix factorization, generalizing the problem (1.1). There are various alg… view at source ↗
Figure 2
Figure 2. Figure 2: It is clear that, for every generated example, our a [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of two different approaches for computing the projection matrix Q on 100 randomly generated examples. Algorithm 3 RSI CMF Input: X ∈ R m×n1 , Y ∈ R m×n2 , k < min{n1, n2} Output: U ∈ R m×k , V ∈ R n1×k , W ∈ R n2×k Generate random matrices Ω1 ∈ R n1×k and Ω2 ∈ R n2×k . for 1 ≤ i ≤ q do [Q1, ∼] = qr(XΩ1, 0) Ω1 = XTQ1 [Q2, ∼] = qr(Y Ω2, 0) Ω2 = Y TQ2 end for [Q, ∼] = qr([Q1, Q2], 0) [U, V, W b ] =… view at source ↗
Figure 3
Figure 3. Figure 3: Relative errors with the respect to the dimension of the projection subspace in the Algorithm 5 for SyntheticTest1. 3.2. SyntheticTest2. We consider n = 1000, r = 15, d = 2 and c = 50. We are looking for the low-rank approximation of order k = 50. Selected results for the Algorithms 2, 3, and 5 are presented in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative errors with the respect to the dimension of the projection subspace in the Algorithm 5 for SyntheticTest2. 3.3. SyntheticTest3. For this experiment, we consider m = 10000, n = 500 and r = 50. We are looking for the rank k = 30 approximation. Selected results are presented in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: RBKI with ℓ = 1 (on the left) and ℓ = 2 (on the right). Rank-revealing QR leads to a smaller oversampling parameter. Let us end this section with a remark on another well-known randomization technique — Generalized Nystrom (GN), [19]. Remark 3.1. Generalized Nystrom is a method that produces a rank-k approximation of any matrix X ∈ R m×n using random matrices. We combined stabilized GN with CMF and tested … view at source ↗
Figure 6
Figure 6. Figure 6: Graphical description of the tensor X from CMTF (4.7). Moreover, the Hadamard product, denoted by ∗, of two matrices A, B ∈ R m×n , is their element￾wise product, A ∗ B =      a11b11 a12b12 · · · a1nb1n a21b21 a22b22 · · · a2nb2n . . . . . . . . . . . . am1bm1 am2bm2 · · · amnbmn      ∈ R m×n . 4.2. Tensor in Tucker format. Let X ∈ R m×n2×n3 and Y ∈ R m×n . Joint rank-k approxima￾tion of X and Y … view at source ↗
Figure 7
Figure 7. Figure 7: Graphical description of the tensor X from CMTF (4.9). of the random Gaussian matrices created for the randomized CMTF will be larger. Instead of projection (2.3), we have Xb (1) = ΠX(1)ΩX(1), Yb = ΠY ΩY. That is, for X ∈ R m×n2×n3 and Y ∈ R m×n , Ω1 is n2n3 × k and Ω1 is n × k matrix. Except for this difference, the randomization process is the same. Numerical results of the randomized algorithms are disc… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of Algorithms 6 and 7 for CMTF for 100 randomly gen￾erated examples. 5.2. Second example — Synthetic tensor and matrix. This example is inspired by the matrix X from TestSynthetic2. We define a diagonal matrix S = diag(1, . . . , 1, d−2 , d−3 , . . . , d−(n−r+1)) ∈ R n×n and construct matrix Y ∈ R n×n and tensor X ∈ R n×n×n in the following way: • UY=orth(rand(n)); VY=orth(rand(n)); • Y = UY*S*V… view at source ↗
Figure 9
Figure 9. Figure 9: Relative errors with respect to number of iterations in RBKI algorithm for Matrix-Tensor decomposition in the Tucker format. Algorithm 1. The accuracy is determined by the relative errors, errX = kX − S ×1 U ×2 V2 ×3 V3k kX k (5.1) and errY as it is given in (3.1). The selected results, together with the running times, are presented in [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: An example of 15 images per person from the Georgia Tech database [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Images of five people used in the tests of our face recognition algorithms. We constructed two versions of the face recognition algorithm. The first one employs the coupled factorization of two matrices. The second one uses the coupled matrix and tensor fac￾torization. The approximation rank in CMF was set to k = 5. Randomization parameters are q = 2 for RSI and ℓ = 5, q = 2 for RBKI. The algorithm steps … view at source ↗
read the original abstract

Coupled decompositions are a widely used tool for data fusion. As the volume of data increases, so does the dimensionality of matrices and tensors, highlighting the need for more efficient coupled decomposition algorithms. This paper studies the problem of coupled matrix factorization (CMF), where two matrices represented in low-rank form share a common factor. Additionally, it explores coupled matrix and tensor factorization (CMTF), where a matrix and a tensor are represented in low-rank form, also sharing a common factor matrix. We show that these problems can be solved using a direct approach with singular value decomposition (SVD), rather than relying on an iterative method. Knowing that matrices coming from real-world applications are often very large, the computational cost can be substantial. To address this issue and improve the efficiency, we propose new techniques for randomizing these algorithms. This includes a novel strategy for selecting a projection subspace that takes into account the contribution from both matrices involved in the decomposition equally. We present extensive results of numerical tests that confirm the efficiency of our algorithms. Furthermore, as a novel approach and with a high success rate, we apply our randomized algorithms to the face recognition problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that coupled matrix factorization (CMF) and coupled matrix-tensor factorization (CMTF) problems, in which matrices or a matrix and tensor share a common low-rank factor, admit exact direct solutions via singular value decomposition rather than iterative optimization. It further proposes randomized variants of these direct methods, with a novel projection-subspace selection rule that weights the contribution of both factors equally, and reports that numerical experiments confirm the efficiency of the randomized algorithms while also demonstrating a high success rate on a face-recognition task.

Significance. A rigorously supported direct SVD route for CMF/CMTF would eliminate the need for iterative solvers in a common data-fusion setting and could therefore reduce computational cost for large-scale problems. The proposed equal-contribution randomization strategy, if accompanied by error bounds that remain valid under disparate singular spectra, would extend the applicability of randomized SVD techniques to coupled settings. The face-recognition experiment supplies a concrete, falsifiable test case. At present these potential contributions are limited by the absence of supporting derivations and analysis.

major comments (3)
  1. [Abstract / direct-method section] Abstract and the section presenting the direct method: the claim that CMF and CMTF admit exact, non-iterative SVD solutions is asserted without an explicit derivation showing how the shared factor matrix is recovered from the SVD of the concatenated or coupled data; this derivation is load-bearing for the central claim that iteration can be avoided.
  2. [Randomized-algorithm section] Section describing the randomized algorithms: the novel equal-contribution projection-subspace selection rule is introduced without probabilistic approximation bounds or an analysis of bias when the two matrices possess singular spectra of markedly different scale; such bounds are required to substantiate that the balancing strategy preserves accuracy at scale.
  3. [Numerical-results section] Numerical-results section: the reported tests assert efficiency and high success on face recognition but supply neither conditioning numbers, scale-disparity metrics between the coupled factors, nor quantitative comparisons against standard randomized SVD baselines, leaving the practical reliability of the equal-weighting rule unverified.
minor comments (2)
  1. Notation for the common factor matrix should be introduced once and used consistently; several passages refer to it by different symbols.
  2. Standard references on randomized SVD (e.g., Halko et al., 2011) are missing from the bibliography; their inclusion would clarify how the new subspace rule differs from existing techniques.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. The comments highlight areas where the presentation and supporting analysis can be strengthened. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Abstract / direct-method section] Abstract and the section presenting the direct method: the claim that CMF and CMTF admit exact, non-iterative SVD solutions is asserted without an explicit derivation showing how the shared factor matrix is recovered from the SVD of the concatenated or coupled data; this derivation is load-bearing for the central claim that iteration can be avoided.

    Authors: We agree that an explicit derivation is essential to support the central claim. In the revised manuscript we will expand the direct-method section with a complete, step-by-step derivation that shows precisely how the shared factor matrix is recovered from the SVD of the coupled data. revision: yes

  2. Referee: [Randomized-algorithm section] Section describing the randomized algorithms: the novel equal-contribution projection-subspace selection rule is introduced without probabilistic approximation bounds or an analysis of bias when the two matrices possess singular spectra of markedly different scale; such bounds are required to substantiate that the balancing strategy preserves accuracy at scale.

    Authors: Deriving tight probabilistic error bounds that remain valid for arbitrary singular spectra in the coupled setting is a substantial theoretical task that lies beyond the scope of the present algorithmic contribution. We will nevertheless add a dedicated discussion of possible bias under scale disparity together with additional numerical diagnostics that quantify the effect of disparate spectra on the equal-contribution rule. revision: partial

  3. Referee: [Numerical-results section] Numerical-results section: the reported tests assert efficiency and high success on face recognition but supply neither conditioning numbers, scale-disparity metrics between the coupled factors, nor quantitative comparisons against standard randomized SVD baselines, leaving the practical reliability of the equal-weighting rule unverified.

    Authors: We will augment the numerical-results section with the requested quantities: condition numbers of the coupled factors, explicit scale-disparity metrics, and side-by-side comparisons against standard randomized SVD baselines. These additions will allow readers to assess the practical reliability of the equal-weighting strategy more directly. revision: yes

Circularity Check

0 steps flagged

No circularity: direct SVD reformulation and novel subspace selection presented as algorithmic contributions without reduction to inputs

full rationale

The paper's core claims are that CMF/CMTF admit an exact direct SVD solution (instead of iteration) and that a new equal-contribution projection subspace can be used for randomization. These are methodological proposals supported by numerical tests; the abstract and described structure contain no fitted parameters renamed as predictions, no self-citation chains invoked as uniqueness theorems, and no ansatz smuggled via prior work. The derivation chain is self-contained as an algorithmic reformulation rather than a closed loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the low-rank shared-factor assumption is implicit but not quantified.

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Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    E. Acar, R. Bro, and A. K. Smilde , Data fusion in metabolomics using coupled matrix and tensor factor- izations, Proceedings of the IEEE, 103 (2015), pp. 1602–1620

    work page 2015

  2. [2]

    E. Acar, D. M. Dunlavy, and T. G. Kolda , A scalable optimization approach for fitting canonical tens or decompositions, Journal of Chemometrics, 25 (2011), pp. 67–86

    work page 2011

  3. [3]

    E. Acar, D. M. Dunlavy, T. G. Kolda, and M. Mørup , Scalable tensor factorizations for incomplete data, Chemometrics and Intelligent Laboratory Systems, 106 (20 11), pp. 41–56. Multiway and Multiset Data Analysis

    work page

  4. [4]

    E. Acar, T. G. Kolda, and D. M. Dunlavy , All-at-once optimization for coupled matrix and tensor factorizations, in MLG’11: Proceedings of Mining and Learning with Graphs, August 2011

    work page 2011

  5. [5]

    Bagherian, R

    M. Bagherian, R. B. Kim, C. Jiang, M. A. Sartor, H. Derksen, an d K. Najarian , Coupled ma- trix–matrix and coupled tensor–matrix completion methods for predicting drug–target interactions , Briefings in Bioinformatics, 22 (2020), pp. 2161–2171

    work page 2020

  6. [6]

    R. A. Borsoi, I. Lehmann, M. A. B. S. Akhonda, V. D. Calhoun, K. Usevich, D. Brie, and T. Adali , Coupled cp tensor decomposition with shared and distinct co mponents for multi-task fmri data fusion , in ICASSP 2023 - 2023 IEEE International Conference on Acousti cs, Speech and Signal Processing (ICASSP), 2023, pp. 1–5

    work page 2023

  7. [7]

    Drineas and M

    P. Drineas and M. W. Mahoney , Randnla: randomized numerical linear algebra , Commun. ACM, 59 (2016), p. 80–90

    work page 2016

  8. [8]

    El Hachimi, K

    A. El Hachimi, K. Jbilou, A. Ratnani, and L. Reichel , A tensor bidiagonalization method for higher- order singular value decomposition with applications , Numerical Linear Algebra with Applications, 31 (2024), p. e2530

    work page 2024

  9. [9]

    Hached, K

    M. Hached, K. Jbilou, C. Koukouvinos, and M. Mitrouli , A multidimensional principal component analysis via the c-product golub–kahan–svd for classificat ion and face recognition , Mathematics, 9 (2021), p. 1249

    work page 2021

  10. [10]

    Halko, P

    N. Halko, P. G. Martinsson, and J. A. Tropp , Finding structure with randomness: probabilistic algorit hms for constructing approximate matrix decompositions , SIAM Rev., 53 (2011), pp. 217–288

    work page 2011

  11. [11]

    Jia, S.-T

    Z.-G. Jia, S.-T. Ling, and M.-X. Zhao , Color two-dimensional principal component analysis for fa ce recog- nition based on quaternion model , in Intelligent Computing Theories and Application: 13th I nternational Conference, ICIC 2017, Liverpool, UK, August 7-10, 2017, Pr oceedings, Part I 13, Springer, 2017, pp. 177– 189

    work page 2017

  12. [12]

    H. A. L. Kiers , Towards a standardized notation and terminology in multiwa y analysis , Journal of Chemo- metrics, 14 (2000), pp. 105–122

    work page 2000

  13. [13]

    T. G. Kolda and B. W. Bader , Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455–500

    work page 2009

  14. [14]

    L. Li, S. Yan, D. Horner, M. Rasmussen, A. Smilde, and E. Acar A taman, Revealing static and dynamic biomarkers from postprandial metabolomics data th rough coupled matrix and tensor factorizations , Metabolomics, 20 (2024)

    work page 2024

  15. [15]

    Martinsson, Randomized methods for matrix computations, IAS/Park City Mathematics Series, (2016)

    P.-G. Martinsson, Randomized methods for matrix computations, IAS/Park City Mathematics Series, (2016)

    work page 2016

  16. [16]

    Martinsson and J

    P.-G. Martinsson and J. A. Tropp , Randomized numerical linear algebra: Foundations and algo rithms, Acta Numerica, 29 (2020), p. 403–572

    work page 2020

  17. [17]

    Meyer, C

    R. Meyer, C. Musco, and C. Musco , On the unreasonable effectiveness of single vector krylov met hods for low-rank approximation, in Proceedings of the 2024 Annual ACM-SIAM Symposium on Dis crete Algorithms (SODA), SIAM, 2024, pp. 811–845

    work page 2024

  18. [18]

    Murray, J

    R. Murray, J. Demmel, M. W. Mahoney, N. B. Erichson, M. Melnic henko, O. A. Malik, L. Grigori, P. Luszczek, M. Derezinski, M. E. Lopes, T. Liang, H. Luo, and J. Dongarra , Randomized numerical linear algebra : A perspective on the field with an eye to softw are, arXiv:2302.11474 [math.NA], (2023)

    work page arXiv 2023

  19. [19]

    Nakatsukasa , Fast and stable randomized low-rank matrix approximation , arXiv:2009.11392 [math.NA], (2020)

    Y. Nakatsukasa , Fast and stable randomized low-rank matrix approximation , arXiv:2009.11392 [math.NA], (2020)

    work page arXiv 2009

  20. [20]

    A. V. Nefian , Georgia Tech face database . http://www.anefian.com/research/face_reco.htm

    work page

  21. [21]

    A. K. Saibaba and A. Miedlar , Randomized low-rank approximations beyond gaussian rando m matrices , arXiv:2308.05814 [math.NA], (2023)

    work page arXiv 2023

  22. [22]

    Schenker, J

    C. Schenker, J. E. Cohen, and E. Acar , A flexible optimization framework for regularized matrix-t ensor factorizations with linear couplings , IEEE Journal of Selected Topics in Signal Processing, 15 (2 021), pp. 506– 521. 24 ERNA BEGOVI ´C KOV AˇC, ANITA CAREVI ´C, AND IV ANA ˇSAIN GLIBI ´C

    work page

  23. [23]

    , An optimization framework for regularized linearly couple d matrix-tensor factorization , in 2020 28th European Signal Processing Conference (EUSIPCO), 2021, pp . 985–989

    work page 2020

  24. [24]

    A. P. Singh and G. J. Gordon , Relational learning via collective matrix factorization , in Proceedings of the 14th ACM SIGKDD International Conference on Knowledge D iscovery and Data Mining, KDD ’08, New York, NY, USA, 2008, Association for Computing Machinery, p . 650–658

    work page 2008

  25. [25]

    A. K. Smilde, J. A. Westerhuis, and R. Boqu ´e, Multiway multiblock component and covariates regression models, Journal of Chemometrics, 14 (2000), pp. 301–331

    work page 2000

  26. [26]

    Sorber, M

    L. Sorber, M. V an Barel, and L. De Lathauwer , Structured data fusion , IEEE Journal of Selected Topics in Signal Processing, 9 (2015), pp. 586–600

    work page 2015

  27. [27]

    Sørensen, I

    M. Sørensen, I. Domanov, and L. De Lathauwer , Coupled canonical polyadic decompositions and (cou- pled) decompositions in multilinear rank- (Lr,n, Lr,n, 1) terms—Part II: Algorithms , SIAM J. Matrix Anal. Appl., 36 (2015), pp. 1015–1045

    work page 2015

  28. [28]

    M. D. Sorochan Armstrong, J. L. Hinrich, A. P. de la Mata, and J . J. Harynuk , Parafac2×n: Coupled decomposition of multi-modal data with drift in n mo des, Analytica Chimica Acta, 1249 (2023), p. 340909

    work page 2023

  29. [29]

    Sørensen and L

    M. Sørensen and L. De Lathauwer , Coupled tensor decompositions for applications in array si gnal pro- cessing, in 2013 5th IEEE International Workshop on Computational A dvances in Multi-Sensor Adaptive Processing (CAMSAP), 2013, pp. 228–231

    work page 2013

  30. [30]

    J. A. Tropp and R. J. Webber , Randomized algorithms for low-rank matrix approximation: Design, anal- ysis, and applications , arXiv:2306.12418 [math.NA], (2023)

    work page arXiv 2023

  31. [31]

    L. R. Tucker , Some mathematical notes on three-mode factor analysis , Psychometrika, 31 (1966), pp. 279– 311

    work page 1966

  32. [32]

    V an Eeghem and L

    F. V an Eeghem and L. De Lathauwer , Tensor similarity in chemometrics , in Comprehensive Chemomet- rics (Second Edition), S. Brown, R. Tauler, and B. Walczak, e ds., Oxford, 2020, Elsevier, pp. 337–354

    work page 2020

  33. [33]

    V an Mechelen and A

    I. V an Mechelen and A. K. Smilde , A generic linked-mode decomposition model for data fusion , Chemo- metrics and Intelligent Laboratory Systems, 104 (2010), pp . 83–94. OMICS

    work page 2010

  34. [34]

    M. A. O. V asilescu and D. Terzopoulos , Multilinear analysis of image ensembles: Tensorfaces , in Com- puter Vision—ECCV 2002: 7th European Conference on Compute r Vision Copenhagen, Denmark, May 28–31, 2002 Proceedings, Part I 7, Springer, 2002, pp. 447–460

    work page 2002

  35. [35]

    Y. Wei, P. Xie, and L. Zhang , Tikhonov regularization and randomized gsvd , SIAM Journal on Matrix Analysis and Applications, 37 (2016), pp. 649–675

    work page 2016

  36. [36]

    D. P. Woodruff , Sketching as a tool for numerical linear algebra , Found. Trends Theor. Comput. Sci., 10 (2014), p. 1–157

    work page 2014

  37. [37]

    Xiao and Y

    X. Xiao and Y. Zhou , Two-dimensional quaternion pca and sparse pca , IEEE transactions on neural networks and learning systems, 30 (2018), pp. 2028–2042

    work page 2018

  38. [38]

    S. Yan, L. Li, D. Horner, P. Ebrahimi, B. Chawes, L. Dragsted, M. Rasmussen, A. Smilde, and E. Acar Ataman , Characterizing human postprandial metabolic response usi ng multiway data analy- sis, Metabolomics, 20 (2024)

    work page 2024

  39. [39]

    J. Yang, D. Zhang, A. F. Frangi, and J.-y. Yang , Two-dimensional pca: a new approach to appearance- based face representation and recognition , IEEE transactions on pattern analysis and machine intelli gence, 26 (2004), pp. 131–137

    work page 2004