Pith. sign in

REVIEW 2 major objections 5 minor 81 references

Personalized Coupled Tensor Decomposition for Multimodal Data Fusion: Uniqueness and Algorithms

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A common tensor and its per-dataset distinct parts are uniquely recoverable under mild conditions.

desk verdict Central uniqueness claims hold up; a solid, genuinely more general coupled tensor model with interpretable conditions, minor experimental and proof-presentation gaps. read the letter →

arxiv 2412.01102 v2 pith:ZY6VON5I submitted 2024-12-02 cs.LG eess.SP

Ricardo Augusto Borsoi , Konstantin Usevich , David Brie , Tülay Adali This is my paper · ORCID
classification cs.LGeess.SP
keywords coupledtensordecompositionmultimodaldatafusionsharedanddistinctcomponentspersonalizedlearningcanonicalpolyadicuni-modeuniquenessidentifiabilityhyperspectralimage
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a personalized coupled tensor decomposition model in which each measured tensor is the sum of a common tensor passed through a multilinear degradation operator plus a tensor distinct to that dataset. It then gives deterministic and generic conditions under which the common and distinct tensors are uniquely recoverable up to the inherent permutation and scaling ambiguities of the decomposition. These conditions are interpretable: one measured tensor must be fully unique, and for each tensor mode there must be a measured tensor that is unique in that mode and whose degradation matrix has full column rank. The constructive proof yields a semi-algebraic algorithm, and an alternating least squares optimization method is also proposed. Experiments on synthetic data and on hyperspectral/multispectral images with cloud contamination show that accounting for dataset-specific components improves recovery compared with methods that model only shared structure.

What carries the argument

The machinery is the combination of uni-mode uniqueness of the individual CPDs with mode-by-mode alignment through pseudoinverses. A CPD (canonical polyadic decomposition) expresses a tensor as a sum of rank-one terms, and a tensor is uni-mode unique in mode $j$ when its mode-$j$ factor matrix is the unique part of the decomposition. The fully unique tensor $Y_\eta$ fixes the permutation and scaling of the common factor across all three modes, while each mode-$j$ unique tensor $Y_{\xi_j}$ with full-column-rank $P_{\xi_j,j}$ lets the common factor $C_j$ be extracted by left-multiplying with the pseudoinverse $P_{\xi_j,j}^\dagger$, then matched across datasets by solving an assignment problem. The Kruskal-rank condition in A3 ensures the ambiguity permutation is block diagonal, so common columns never mix with distinct columns.

What would settle it

Generate a noiseless instance of model (8)--(13) whose factors satisfy A1--A3, then search numerically for an alternative decomposition with a different permutation between common and distinct columns that still reproduces all $Y_k$. The theorem predicts that only trivial permutation-scaling ambiguities exist; finding any non-trivially different decomposition would refute it. A more targeted check is to construct an instance where the matrix in A3 has two proportional columns while all other conditions hold, and show that common and distinct parts can be swapped without changing the measurements.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3: under assumptions A1--A3, the common tensor $C$ and the distinct tensors $\{D_k\}_k$ in the model $Y_k = P_k(C) + D_k$ are uniquely recoverable from the measurements $\{Y_k\}_k$, where $P_k$ is a separable multilinear operator $P_k(C) = C \times_1 P_{k,1} \times_2 P_{k,2} \times_3 P_{k,3}$. The conditions are that one measured tensor $Y_\eta$ is fully unique (its whole CPD is unique up to permutation and scaling), that for each mode $j$ some measured tensor $Y_{\xi_j}$ is uni-mode unique in mode $j$ with $P_{\xi_j,j}$ of full column rank, and that at least one of these indices differs from $\eta$ while a Kruskal-rank condition prevents common and distinct columns from being confounded. Theorem 4 turns these into generic conditions expressed directly in dimensions and ranks of the degradation matrices, holding with probability one when the factors are drawn from an absolutely continuous distribution. The proof is constructive and motivates the semi-algebraic recovery algorithm.

Load-bearing premise

The proof hinges on having, for every mode of the tensors, at least one measurement whose degradation matrix in that mode has full column rank, so that the common factor can be recovered by inverting that matrix; if all degradation matrices are rank-deficient in some mode, the common and distinct parts cannot be separated in that mode.

Editorial extensions

If this is right

  • If Theorem 3 holds, the common tensor $C$ and each distinct tensor $D_k$ can be recovered from the measured tensors $\{Y_k\}_k$, giving an interpretable shared-versus-specific decomposition of multimodal data.
  • The generic inequalities of Theorem 4 let a practitioner check, from tensor sizes and the ranks of the degradation matrices alone, whether a planned acquisition setup will make the decomposition identifiable.
  • The semi-algebraic algorithm, derived directly from the proof, computes the decomposition from a small number of individual CPDs and assignment problems and supplies a principled initialization for the optimization method.
  • In the hyperspectral/multispectral imaging experiments, the model with distinct components achieves lower NRMSE than methods without them as soon as image-specific variability such as clouds is present.
  • The framework covers applications such as multi-task fMRI, where task-common activations $C$ and task-specific networks $D_k$ are separated from multiple measured tensors.

Reading between the lines

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

  • As an extension the paper does not explore, the per-mode argument in Theorem 3 should carry over to higher-order tensors, provided each mode has a full-resolution uni-mode unique view; this would cover the fifth-order fMRI model mentioned in Remark 1.
  • The conditions suggest a design rule: if one modality can be made fully identifiable and every mode is observed at full resolution in at least one other modality, the personalized decomposition is guaranteed, which could guide sensor and acquisition design before data collection.
  • The assignment-based alignment step is the part of the semi-algebraic algorithm most likely to suffer under noise; replacing it with a robust matching procedure or a refinement step could improve low-SNR behavior, consistent with the paper's observation that the optimization method outperforms the semi-algebraic one at 20 dB.
  • Because the framework assumes known ranks $R$ and $L_k$, practical use requires a rank-selection procedure; the paper's ablation shows graceful degradation under mild rank misspecification, but a principled rank estimator remains an open problem.
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

2 major / 5 minor

Summary. The paper proposes a personalized coupled tensor decomposition (CTD) model in which each measured tensor takes the form Y_k = P_k(C) + D_k, where C is a common tensor and D_k are dataset-specific distinct tensors, all admitting canonical polyadic decompositions. The main theoretical contribution is a set of identifiability results: Theorem 3 gives deterministic conditions for unique recovery of C and {D_k} based on full uniqueness of one measured tensor, mode-wise (uni-mode) uniqueness with full-column-rank degradation operators, and a non-proportionality condition; Theorem 4 translates these into generic uniqueness conditions expressed through tensor dimensions, CP ranks, and ranks of the degradation matrices. The paper also proposes two algorithms, a semi-algebraic method derived from the constructive proof and an alternating least squares optimization method, and evaluates them on synthetic data and on a hyperspectral/multispectral image fusion problem with cloud contamination, comparing against STEREO, SCOTT, CT-STAR, and CB-STAR.

Significance. If the theorems are correct, the paper provides an interpretable and fairly general identifiability framework for shared/distinct component analysis that subsumes several earlier CTD models. The constructive nature of Theorem 3 is a genuine strength, since it directly motivates the semi-algebraic algorithm, and the generic conditions in Theorem 4 are stated in terms of easily checked quantities. The paper is also careful to acknowledge that the real-data cloud model is only an approximation to the proposed generative model. The central deterministic proof is internally consistent, and the generic theorem is plausible; however, the proof of Lemma 2 has a concrete gap that must be repaired before Theorem 4 is fully established. The experimental section includes meaningful baselines and ablations, though reporting of variability is sparse. Overall, the contribution is significant and appropriate for the journal, but the proof issue requires revision.

major comments (2)
  1. [Section IV-B, Lemma 2 (proof around Eqs. (28)-(29))] The proof of Lemma 2 is incomplete as written. In the case R ≤ M, the matrix rZ0 = [[I_T,0],[0,0]] is claimed to be of the block-triangular form (28); however, the top-left block X1 in (28) has size M×R, so a T×T identity can be placed there only if T ≤ R (and T ≤ M). Since T = min(N, R+L), T > R is possible (e.g., R=1, L=10, M=5, N=10 gives T=10 > R). Thus the existence argument does not establish that a matrix in the prescribed family has rank T, and the semicontinuity conclusion does not follow. This matters because Theorem 4, Step 2 uses Lemma 2 to control rank([P_ξj,j C_j, D_ξj,j]) in the uni-mode uniqueness argument. The lemma statement is true and the gap is repairable, for instance by an explicit construction of X1, X~2,1, and X~2,2 realizing rank T, but the proof needs correction.
  2. [Section V-A, Algorithm 1, line 13] The pseudocode returns D_k = Y_k − C_alg, but under the measurement model (8) the distinct tensor is D_k = Y_k − P_k(C_alg). As written, Algorithm 1 is not a correct solver of the model unless every P_k is the identity. The surrounding text in Section V-A, Step 5 gives the correct definition (with P_k applied to C_alg), so the pseudocode should be fixed to match the model.
minor comments (5)
  1. [Section IV-A, Theorem 3, proof Step 3 (Eq. (24))] The proof uses the necessary condition that essential uniqueness of the CPD of Y_η implies that P_η,j C_j has no proportional columns, citing [50] without a precise statement. Please state the condition explicitly and give a specific reference to the relevant result in [50].
  2. [Section IV, Theorems 3 and 4] The theorems state that C and {D_k} can be 'uniquely recovered', but the proof compares alternative decompositions with the same prescribed ranks R and L_k. Please state explicitly that uniqueness is within the model class with fixed ranks R and L_k, since otherwise the claim is too strong.
  3. [Appendix A, Eqs. (48)-(50)] There is a dimension mismatch in the normal equations: the data term should be J_{k,1}^T rY_kx1y^T P_{k,1}, not J_{k,1}^T rY_kx1y P_{k,1}; the same transpose is missing in the mode-2 and mode-3 equations (49) and (50).
  4. [Section V-A, Algorithm 1, line 9] The symbol Y_ηℓ appears to be a typo; it should be Y_ξℓ (or Y_{ξ_ℓ}) to match the surrounding text.
  5. [Section VI, Tables I and II] Please report standard deviations or confidence intervals for the Monte Carlo averages. In addition, the semi-algebraic NRMSE of 0.9088 in Table I at 30 dB deserves a sentence of explanation given the text's characterization of the semi-algebraic method as achieving low NRMSE at high SNR.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniqueness results are derived from external identifiability theorems and constructive proofs, with self-citations used only as context and baselines.

full rationale

The core derivation chain is self-contained against external benchmarks. Theorem 3 is a conditional identifiability statement proved constructively from explicit hypotheses A1-A3: A1 assumes one measured tensor Y_eta is fully unique, A2 assumes per-mode uni-mode uniqueness with full-column-rank degradation, and A3 prevents common/distinct column mixing through a Kruskal-rank condition on the concatenated factor matrix. None of these hypotheses is defined in terms of the conclusion that C and D_k are recoverable; they are genuine assumptions about individual tensors and measurement operators. The key algebraic step, equations (16)-(23), uses only the full-column-rank property of P_xi_j,j and the uniqueness of the selected tensors to align permutations and scalings, and the recovery of D_k then follows directly from the model equation Y_k = P_k(C) + D_k, which is a legitimate deduction rather than a tautology. Theorem 4 reduces to Theorem 3 through standard genericity lemmas (Lemmas 1 and 2) whose conditions (26)-(27) are dimension-rank bounds, not restatements of the desired recovery. The semi-algebraic algorithm is explicitly derived from the constructive proof of Theorem 3, so it is an algorithm motivated by the identifiability argument rather than a disguised prediction. The optimization algorithm is an ALS scheme with independently justified linear-algebra updates. Experimental comparisons are made against external baselines (STEREO, SCOTT, CT-STAR, CB-STAR, SDF), and hyperparameters such as ranks are tuned on a separate dataset. Self-citations ([10], [11], [13], and [45]) are used for context, as prior-work baselines, or as externally published peer-reviewed theorems; in particular, Theorem 2 of Guo et al. [45] is an external mathematical result whose stated assumptions do not include the present paper's target, so citing it is legitimate independent support even though one author overlaps. The paper also explicitly acknowledges limitations, such as real data not exactly following the model and the flexible-coupling extension needing separate uniqueness analysis, which supports a transparent rather than circular presentation. I find no step in which a prediction reduces by construction to a fitted parameter or to a self-citation chain.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The theoretical results rest on standard CPD uniqueness theorems and on the modeling assumption that each measured tensor decomposes into a common part under known multilinear degradations plus a distinct low-rank part. No new physical entities or fitted constants are central to the derivations. Rank values are model parameters, assumed known in the theory and selected via validation in the real-data experiments.

free parameters (3)
  • CP rank R of common tensor = 5 in synthetic experiments; tuned on validation set in real experiments
    The uniqueness theorems and algorithms take R as given; in real experiments the rank was chosen to maximize performance on a separate dataset, not on the test data.
  • CP ranks L_k of distinct tensors = 5 in synthetic experiments; tuned on validation set in real experiments
    Same as R: structural model ranks, assumed known in the theory, selected via validation in experiments.
  • Cloud cover level (real data) = CC 0% to 9.7%, CP 0% to 13.8%
    Experimental control parameter varied to test robustness, not part of the central theoretical claim.
assumptions (5)
  • standard math Kruskal's sufficient condition for CPD uniqueness (Theorem 1) and its refinements are used as external benchmarks.
    Section II-B, Theorem 1, and Theorem 2 from [45] are used in the proofs of Theorems 3 and 4.
  • domain assumption The measurement operators P_{k,j} have rank min(N_{k,j}, M_j) and are known exactly.
    Model (9) in Section III; the recovery algorithms require knowledge of P_k.
  • domain assumption In the generic theorem, factor matrices are drawn from a joint absolutely continuous distribution.
    Theorem 4, Section IV-B; standard genericity assumption avoiding measure-zero bad cases.
  • domain assumption The CPD of each measured tensor Y_k with factor matrices [P_{k,j} C_j, D_{k,j}] has rank exactly R+L_k and is the canonical decomposition for the fully unique / uni-mode unique tensors.
    Definitions in Section III and conditions A1/A2 in Theorem 3; if Y_k had a lower rank, uniqueness of the alternative decomposition would not follow.
  • ad hoc to paper For the cloud experiment, the cloud-corrupted images are approximated by the model (8)-(12); the paper admits this is an approximation.
    Section VI-B: the data do not really follow the model in (10), (11), leading to an approximation problem.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Personalized Coupled Tensor Decomposition for Multimodal Data Fusion: Uniqueness and Algorithms." pith.science (2026). https://pith.science/paper/ZY6VON5I

@misc{pith2026241201102,
  author       = {Pith},
  title        = {Pith review of: Personalized Coupled Tensor Decomposition for Multimodal Data Fusion: Uniqueness and Algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZY6VON5I}},
  note         = {Machine review of arXiv:2412.01102}
}
read the original abstract

Coupled tensor decompositions (CTDs) perform data fusion by linking factors from different datasets. Although many CTDs have been already proposed, current works do not address important challenges of data fusion, where: 1) the datasets are often heterogeneous, constituting different "views" of a given phenomena (multimodality); and 2) each dataset can contain personalized or dataset-specific information, constituting distinct factors that are not coupled with other datasets. In this work, we introduce a personalized CTD framework tackling these challenges. A flexible model is proposed where each dataset is represented as the sum of two components, one related to a common tensor through a multilinear measurement model, and another specific to each dataset. Both the common and distinct components are assumed to admit a polyadic decomposition. This generalizes several existing CTD models. We provide conditions for specific and generic uniqueness of the decomposition that are easy to interpret. These conditions employ uni-mode uniqueness of different individual datasets and properties of the measurement model. Two algorithms are proposed to compute the common and distinct components: a semi-algebraic one and a coordinate-descent optimization method. Experimental results illustrate the advantage of the proposed framework compared with the state of the art approaches.

Figures

Figures reproduced from arXiv: 2412.01102 by the authors.

Figure 1
Figure 1. Illustration of the model in (8). A latent tensor C is common in all measurements and acquired through an operator Pk while tensors Dk are distinct to each measurement, leading to a personalized decomposition [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the multilinear measurement model in ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. NRMSE for different SNRs for the example with synthetic data. [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: NRMSE (between Ckpαq and Cp) of the solutions from the optimization algorithm (init. 2) as a function of α. 3 7 0.1 4 6 0.15 NRMSE R Lk 5 5 0.2 6 4 7 3 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

81 extracted references · 75 canonical work pages

  1. [1]

    Structured data fusion,

    L. Sorber, M. Van Barel, and L. De Lathauwer, “Structured data fusion,” IEEE Journal of Selected Topics in Signal Processing , vol. 9, no. 4, pp. 586–600, 2015

    2015

  2. [2]

    Under- standing data fusion within the framework of coupled matrix and tensor factorizations,

    E. Acar, M. A. Rasmussen, F. Savorani, T. Næs, and R. Bro, “Under- standing data fusion within the framework of coupled matrix and tensor factorizations,” Chemometrics and Intelligent Laboratory Systems , vol. 129, pp. 53–63, 2013

    work page 2013

  3. [3]

    Multimodal data fusion: an overview of methods, challenges, and prospects,

    D. Lahat, T. Adali, and C. Jutten, “Multimodal data fusion: an overview of methods, challenges, and prospects,” Proceedings of the IEEE , vol. 103, no. 9, pp. 1449–1477, 2015

    2015

  4. [4]

    Structure-revealing data fusion,

    E. Acar, E. E. Papalexakis, G. Gürdeniz, M. A. Rasmussen, A. J. Lawaetz, M. Nilsson, and R. Bro, “Structure-revealing data fusion,” BMC bioinformatics, vol. 15, no. 1, pp. 1–17, 2014

    work page 2014

  5. [5]

    Chapter 10 - Coupled tensor decompositions for data fusion,

    C. Chatzichristos, S. Van Eyndhoven, E. Kofidis, and S. Van Huffel, “Chapter 10 - Coupled tensor decompositions for data fusion,” inTensors for data processing , Y . Liu, Ed. Elsevier, 2022, pp. 341–370

    work page 2022

  6. [6]

    Turbo-SMT: Accelerating coupled sparse matrix-tensor factorizations by 200x - Suplementary material,

    E. E. Papalexakis, C. Faloutsos, T. M. Mitchell, P. P. Talukdar, N. D. Sidiropoulos, and B. Murphy, “Turbo-SMT: Accelerating coupled sparse matrix-tensor factorizations by 200x - Suplementary material,” in Pro- ceedings of the 2014 SIAM International Conference on Data Mining . SIAM, 2014, pp. 118–126

    work page 2014

  7. [7]

    Link prediction in heterogeneous data via generalized coupled tensor factorization,

    B. Ermi¸ s, E. Acar, and A. T. Cemgil, “Link prediction in heterogeneous data via generalized coupled tensor factorization,” Data Mining and Knowledge Discovery, vol. 29, no. 1, pp. 203–236, 2015

    work page 2015

  8. [8]

    Hyperspectral super- resolution with coupled tucker approximation: Recoverability and SVD- based algorithms,

    C. Prévost, K. Usevich, P. Comon, and D. Brie, “Hyperspectral super- resolution with coupled tucker approximation: Recoverability and SVD- based algorithms,” IEEE Transactions on Signal Processing, vol. 68, pp. 931–946, 2020

    work page 2020

Show all 81 references
  1. [9]

    Hyperspec- tral super-resolution: A coupled tensor factorization approach,

    C. I. Kanatsoulis, X. Fu, N. D. Sidiropoulos, and W.-K. Ma, “Hyperspec- tral super-resolution: A coupled tensor factorization approach,” IEEE Transactions on Signal Processing, vol. 66, no. 24, pp. 6503–6517, 2018

    2018

  2. [10]

    Coupled tensor decomposition for hyperspectral and multispectral image fusion with inter-image variability,

    R. A. Borsoi, C. Prévost, K. Usevich, D. Brie, J. C. M. Bermudez, and C. Richard, “Coupled tensor decomposition for hyperspectral and multispectral image fusion with inter-image variability,” IEEE Journal of Selected Topics in Signal Processing , vol. 15, no. 3, pp. 702–717, 2021

    2021

  3. [11]

    Hyperspectral super-resolution accounting for spectral variability: Coupled tensor LL1-based recovery and blind unmixing of the unknown super-resolution image,

    C. Prévost, R. A. Borsoi, K. Usevich, D. Brie, J. C. M. Bermudez, and C. Richard, “Hyperspectral super-resolution accounting for spectral variability: Coupled tensor LL1-based recovery and blind unmixing of the unknown super-resolution image,” SIAM Journal on Imaging Sciences,...

    2022

  4. [12]

    Coupled canonical polyadic decomposition of multi-group fMRI data with spatial reference and orthonormality constraints,

    L.-D. Kuang, Z.-M. He, J. Zhang, and F. Li, “Coupled canonical polyadic decomposition of multi-group fMRI data with spatial reference and orthonormality constraints,” Biomedical Signal Processing and Control , vol. 80, p. 104232, 2023

    2023

  5. [13]

    Coupled CP tensor decomposition with shared and distinct components for multi-task fMRI data fusion,

    R. A. Borsoi, I. Lehmann, M. A. B. S. Akhonda, V . Calhoun, K. Usevich, D. Brie, and T. Adali, “Coupled CP tensor decomposition with shared and distinct components for multi-task fMRI data fusion,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processi...

    2023

  6. [15]

    Multidimensional harmonic retrieval via coupled canonical polyadic decomposition–Part II: Algorithm and multirate sampling,

    ——, “Multidimensional harmonic retrieval via coupled canonical polyadic decomposition–Part II: Algorithm and multirate sampling,” IEEE Transactions on Signal Processing , vol. 65, no. 2, pp. 528–539, 2016

    2016

  7. [16]

    Multiple invariance ESPRIT for nonuniform linear arrays: A cou- pled canonical polyadic decomposition approach,

    ——, “Multiple invariance ESPRIT for nonuniform linear arrays: A cou- pled canonical polyadic decomposition approach,” IEEE Transactions on Signal Processing, vol. 64, no. 14, pp. 3693–3704, 2016

    2016

  8. [17]

    Coupled canonical polyadic decompositions and multiple shift invariance in array process- ing,

    M. Sørensen, I. Domanov, and L. De Lathauwer, “Coupled canonical polyadic decompositions and multiple shift invariance in array process- ing,” IEEE Transactions on Signal Processing, vol. 66, no. 14, pp. 3665– 3680, 2018

    2018

  9. [18]

    Coupled coarray tensor CPD for DOA estimation with coprime L-shaped array,

    H. Zheng, Z. Shi, C. Zhou, M. Haardt, and J. Chen, “Coupled coarray tensor CPD for DOA estimation with coprime L-shaped array,” IEEE Signal Processing Letters , vol. 28, pp. 1545–1549, 2021

    2021

  10. [19]

    Coupled and incomplete tensors in blind system identification,

    F. Van Eeghem, O. Debals, N. Vervliet, and L. De Lathauwer, “Coupled and incomplete tensors in blind system identification,” IEEE Transac- tions on Signal Processing , vol. 66, no. 23, pp. 6137–6147, 2018

    2018

  11. [20]

    Double cou- pled canonical polyadic decomposition for joint blind source separation,

    X.-F. Gong, Q.-H. Lin, F.-Y . Cong, and L. De Lathauwer, “Double cou- pled canonical polyadic decomposition for joint blind source separation,” IEEE Transactions on Signal Processing, vol. 66, no. 13, pp. 3475–3490, 2018

    2018

  12. [21]

    PARAFAC-based blind estimation of pos- sibly underdetermined convolutive MIMO systems,

    Y . Yu and A. P. Petropulu, “PARAFAC-based blind estimation of pos- sibly underdetermined convolutive MIMO systems,” IEEE Transactions on Signal Processing , vol. 56, no. 1, pp. 111–124, 2007

    2007

  13. [22]

    Data fusion in metabolomics using coupled matrix and tensor factorizations,

    E. Acar, R. Bro, and A. K. Smilde, “Data fusion in metabolomics using coupled matrix and tensor factorizations,” Proceedings of the IEEE, vol. 103, no. 9, pp. 1602–1620, 2015

    2015

  14. [23]

    Tensor analysis and fusion of multimodal brain images,

    E. Karahan, P. A. Rojas-Lopez, M. L. Bringas-Vega, P. A. Valdés- Hernández, and P. A. Valdes-Sosa, “Tensor analysis and fusion of multimodal brain images,” Proceedings of the IEEE , vol. 103, no. 9, pp. 1531–1559, 2015

    2015

  15. [24]

    Early soft and flexible fusion of electroencephalography and functional magnetic resonance imaging via double coupled matrix tensor factorization for multisubject group analysis,

    C. Chatzichristos, E. Kofidis, W. Van Paesschen, L. De Lathauwer, S. Theodoridis, and S. Van Huffel, “Early soft and flexible fusion of electroencephalography and functional magnetic resonance imaging via double coupled matrix tensor factorization for multisubject group analys...

    2022

  16. [25]

    Learning spatiotemporal brain dynamics in adolescents via multimodal MEG and fMRI data fusion using joint tensor/matrix decomposition,

    I. Belyaeva, B. Gabrielson, Y .-P. Wang, T. W. Wilson, V . D. Calhoun, J. M. Stephen, and T. Adali, “Learning spatiotemporal brain dynamics in adolescents via multimodal MEG and fMRI data fusion using joint tensor/matrix decomposition,” IEEE Transactions on Biomedical Engineer...

    2024

  17. [26]

    ICA and IV A for data fusion: An overview and a new approach based on disjoint subspaces,

    T. Adali, M. Akhonda, and V . D. Calhoun, “ICA and IV A for data fusion: An overview and a new approach based on disjoint subspaces,” IEEE Sensors Letters, vol. 3, no. 1, pp. 1–4, 2019

    2019

  18. [27]

    Coupled matrix-tensor factorizations– the case of partially shared factors,

    L. De Lathauwer and E. Kofidis, “Coupled matrix-tensor factorizations– the case of partially shared factors,” in Proc. 51st Asilomar Conference on Signals, Systems, and Computers . IEEE, 2017, pp. 711–715

    2017

  19. [28]

    Common and distinct components in data fusion,

    A. K. Smilde, I. Måge, T. Naes, T. Hankemeier, M. A. Lips, H. A. Kiers, E. Acar, and R. Bro, “Common and distinct components in data fusion,” Journal of Chemometrics , vol. 31, no. 7, p. e2900, 2017

    2017

  20. [29]

    Unraveling diagnostic biomarkers of schizophrenia through structure- revealing fusion of multi-modal neuroimaging data,

    E. Acar, C. Schenker, Y . Levin-Schwartz, V . D. Calhoun, and T. Adali, “Unraveling diagnostic biomarkers of schizophrenia through structure- revealing fusion of multi-modal neuroimaging data,” Frontiers in neu- roscience, vol. 13, p. 416, 2019

    2019

  21. [30]

    Probabilistic linear discriminant analysis for inferences about identity,

    S. J. Prince and J. H. Elder, “Probabilistic linear discriminant analysis for inferences about identity,” in Proc. 11th international conference on computer vision (ICCV) . IEEE, 2007, pp. 1–8

    2007

  22. [31]

    Deep hyperspectral and multispectral image fusion with inter-image variability,

    X. Wang, R. A. Borsoi, C. Richard, and J. Chen, “Deep hyperspectral and multispectral image fusion with inter-image variability,” IEEE Transactions on Geoscience and Remote Sensing , vol. 61, 2023

    2023

  23. [32]

    Mining labelled tensors by discovering both their common and discriminative subspaces,

    W. Liu, J. Chan, J. Bailey, C. Leckie, and K. Ramamohanarao, “Mining labelled tensors by discovering both their common and discriminative subspaces,” in Proceedings of the 2013 SIAM International Conference on Data Mining . SIAM, 2013, pp. 614–622

    2013

  24. [33]

    Pairfac: Event analytics through discriminant tensor factorization,

    X. Wen, Y .-R. Lin, and K. Pelechrinis, “Pairfac: Event analytics through discriminant tensor factorization,” in Proceedings of the 25th ACM In- ternational on Conference on Information and Knowledge Management , 2016, pp. 519–528

    2016

  25. [34]

    Multi-set low-rank factorizations with shared and unshared components,

    M. Sørensen and N. D. Sidiropoulos, “Multi-set low-rank factorizations with shared and unshared components,” IEEE Transactions on Signal Processing, vol. 68, pp. 5122–5137, 2020

    2020

  26. [35]

    A flexible optimization frame- work for regularized matrix-tensor factorizations with linear couplings,

    C. Schenker, J. E. Cohen, and E. Acar, “A flexible optimization frame- work for regularized matrix-tensor factorizations with linear couplings,” IEEE Journal of Selected Topics in Signal Processing , vol. 15, no. 3, pp. 506–521, 2020

    2020

  27. [36]

    Exploring multimodal data fusion through joint decompositions with flexible couplings,

    R. C. Farias, J. E. Cohen, and P. Comon, “Exploring multimodal data fusion through joint decompositions with flexible couplings,” IEEE Transactions on Signal Processing, vol. 64, no. 18, pp. 4830–4844, 2016

    2016

  28. [37]

    Reproducibility in matrix and tensor decompositions: Focus on model match, interpretability, and uniqueness,

    T. Adali, F. Kantar, M. A. B. S. Akhonda, S. Strother, V . D. Calhoun, and E. Acar, “Reproducibility in matrix and tensor decompositions: Focus on model match, interpretability, and uniqueness,” IEEE Signal Processing Magazine, vol. 39, no. 4, pp. 8–24, 2022

    2022

  29. [38]

    Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank- (Lr,n,Lr,n,1) terms–part I: Uniqueness,

    M. Sørensen and L. D. De Lathauwer, “Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank- (Lr,n,Lr,n,1) terms–part I: Uniqueness,” SIAM Journal on Matrix Analysis and Applications , vol. 36, no. 2, pp. 496–522, 2015

    2015

  30. [39]

    Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank-(Lr,n,Lr,n,1) terms–part II: Algorithms,

    M. Sørensen, I. Domanov, and L. De Lathauwer, “Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank-(Lr,n,Lr,n,1) terms–part II: Algorithms,” SIAM Journal on Matrix Analysis and Applications , vol. 36, no. 3, pp. 1015–1045, 2015

    2015

  31. [40]

    Tensor completion from regular sub-Nyquist samples,

    C. I. Kanatsoulis, X. Fu, N. D. Sidiropoulos, and M. Akçakaya, “Tensor completion from regular sub-Nyquist samples,” IEEE Transactions on Signal Processing, vol. 68, pp. 1–16, 2019

    2019

  32. [41]

    Spectrum cartog- raphy via coupled block-term tensor decomposition,

    G. Zhang, X. Fu, J. Wang, X.-L. Zhao, and M. Hong, “Spectrum cartog- raphy via coupled block-term tensor decomposition,” IEEE Transactions on Signal Processing , vol. 68, pp. 3660–3675, 2020

    2020

  33. [42]

    Personalized PCA: Decoupling shared and unique features,

    N. Shi and R. A. Kontar, “Personalized PCA: Decoupling shared and unique features,” Journal of machine learning research , vol. 25, pp. 1– 82, 2024

    2024

  34. [43]

    Personalized dictionary learning for heterogeneous datasets,

    G. Liang, N. Shi, R. A. Kontar, and S. Fattahi, “Personalized dictionary learning for heterogeneous datasets,” in Thirty-seventh Conference on Neural Information Processing Systems , 2023

    2023

  35. [44]

    Heterogeneous matrix factorization: When features differ by datasets,

    N. Shi, R. A. Kontar, and S. Fattahi, “Heterogeneous matrix factorization: When features differ by datasets,” arXiv preprint arXiv:2305.17744, 2023

    2023 arXiv

  36. [45]

    Uni-mode and partial uniqueness conditions for CANDECOMP/PARAFAC of three-way ar- rays with linearly dependent loadings,

    X. Guo, S. Miron, D. Brie, and A. Stegeman, “Uni-mode and partial uniqueness conditions for CANDECOMP/PARAFAC of three-way ar- rays with linearly dependent loadings,”SIAM Journal on Matrix Analysis and Applications, vol. 33, no. 1, pp. 111–129, 2012

    2012

  37. [46]

    On the uniqueness of the canonical polyadic decomposition of third-order tensors–Part I: Basic results and uniqueness of one factor matrix,

    I. Domanov and L. De Lathauwer, “On the uniqueness of the canonical polyadic decomposition of third-order tensors–Part I: Basic results and uniqueness of one factor matrix,” SIAM Journal on Matrix Analysis and Applications, vol. 34, no. 3, pp. 855–875, 2013

    2013

  38. [47]

    Tensor decompositions and applications,

    T. G. Kolda and B. W. Bader, “Tensor decompositions and applications,” SIAM review, vol. 51, no. 3, pp. 455–500, 2009

    2009

  39. [48]

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

    J. B. Kruskal, “Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics,” Linear algebra and its applications , vol. 18, no. 2, pp. 95–138, 1977

    1977

  40. [49]

    On Kruskal’s uniqueness condi- tion for the Candecomp/Parafac decomposition,

    A. Stegeman and N. D. Sidiropoulos, “On Kruskal’s uniqueness condi- tion for the Candecomp/Parafac decomposition,” Linear Algebra and its applications, vol. 420, no. 2-3, pp. 540–552, 2007

    2007

  41. [50]

    On the uniqueness of the canonical polyadic decomposition of third-order tensors–Part II: Uniqueness of the overall decomposition,

    I. Domanov and L. De Lathauwer, “On the uniqueness of the canonical polyadic decomposition of third-order tensors–Part II: Uniqueness of the overall decomposition,” SIAM Journal on Matrix Analysis and Applications, vol. 34, no. 3, pp. 876–903, 2013

    2013

  42. [51]

    On generic identifiability of 3-tensors of small rank,

    L. Chiantini and G. Ottaviani, “On generic identifiability of 3-tensors of small rank,” SIAM Journal on Matrix Analysis and Applications, vol. 33, no. 3, pp. 1018–1037, 2012

    2012

  43. [52]

    Canonical polyadic decomposition of third-order tensors: Reduction to generalized eigenvalue decomposition,

    I. Domanov and L. D. Lathauwer, “Canonical polyadic decomposition of third-order tensors: Reduction to generalized eigenvalue decomposition,” SIAM Journal on Matrix Analysis and Applications , vol. 35, no. 2, pp. 636–660, 2014

    2014

  44. [53]

    Decompositions of a higher-order tensor in block terms–part II: Definitions and uniqueness,

    L. De Lathauwer, “Decompositions of a higher-order tensor in block terms–part II: Definitions and uniqueness,” SIAM Journal on Matrix Analysis and Applications , vol. 30, no. 3, pp. 1033–1066, 2008

    2008

  45. [54]

    CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters,

    J. Douglas Carroll, S. Pruzansky, and J. B. Kruskal, “CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters,” Psychometrika, vol. 45, pp. 3– 24, 1980

    1980

  46. [55]

    Quantifying the interaction and contribution of multiple datasets in fusion: Application to the detection of schizophrenia,

    Y . Levin-Schwartz, V . D. Calhoun, and T. Adalı, “Quantifying the interaction and contribution of multiple datasets in fusion: Application to the detection of schizophrenia,” IEEE transactions on medical imaging , vol. 36, no. 7, pp. 1385–1395, 2017

    2017

  47. [56]

    Blind fMRI source unmixing via higher-order tensor decompositions,

    C. Chatzichristos, E. Kofidis, M. Morante, and S. Theodoridis, “Blind fMRI source unmixing via higher-order tensor decompositions,” Journal of neuroscience methods , vol. 315, pp. 17–47, 2019

    2019

  48. [57]

    Non-local tensor sparse coding for multi- image super-resolution in magnetic resonance imaging,

    C. Prévost and F. Odille, “Non-local tensor sparse coding for multi- image super-resolution in magnetic resonance imaging,” 2023

    2023

  49. [58]

    Multi-frame super-resolution MRI using coupled low-rank tucker approximation,

    ——, “Multi-frame super-resolution MRI using coupled low-rank tucker approximation,” in Proc. 30th European Signal Processing Conference (EUSIPCO). IEEE, 2022, pp. 1263–1267. PUBLISHED AT THE IEEE TRANSACTIONS ON SIGNAL PROCESSING, DOI: 10.1109/TSP.2024.3510680. COPYRIGHT BELON...

    2022

  50. [59]

    AFFIRM: affinity fusion-based framework for iteratively random motion correction of multi-slice fetal brain MRI,

    W. Shi, H. Xu, C. Sun, J. Sun, Y . Li, X. Xu, T. Zheng, Y . Zhang, G. Wang, and D. Wu, “AFFIRM: affinity fusion-based framework for iteratively random motion correction of multi-slice fetal brain MRI,” IEEE Transactions on Medical Imaging , vol. 42, no. 1, pp. 209–219, 2022

    2022

  51. [60]

    Spectral variability in hyperspectral data unmixing: A comprehensive review,

    R. A. Borsoi, T. Imbiriba, J. C. M. Bermudez, C. Richard, J. Chanussot, L. Drumetz, J.-Y . Tourneret, A. Zare, and C. Jutten, “Spectral variability in hyperspectral data unmixing: A comprehensive review,” IEEE geo- science and remote sensing magazine , vol. 9, no. 4, pp. 223–270, 2021

    2021

  52. [61]

    Tensor- based fusion of EEG and FMRI to understand neurological changes in schizophrenia,

    E. Acar, Y . Levin-Schwartz, V . D. Calhoun, and T. Adali, “Tensor- based fusion of EEG and FMRI to understand neurological changes in schizophrenia,” in Proc. IEEE international symposium on circuits and systems (ISCAS) . IEEE, 2017, pp. 1–4

    2017

  53. [62]

    Longitudinal metabolomics data analysis informed by mechanistic models,

    L. Li, H. C. Hoefsloot, B. M. Bakker, D. Horner, M. A. Ras- mussen, A. Smilde, and E. Acar, “Longitudinal metabolomics data analysis informed by mechanistic models,” bioRxiv, 2024, https://doi.org/10.1101/2024.08.13.607724

    2024 doi

  54. [63]

    D. S. Bernstein, Matrix mathematics: theory, facts, and formulas, 2nd ed. Princeton university press, 2009

    2009

  55. [64]

    Nonlinear multiview analysis: Identifiability and neural network-assisted implementation,

    Q. Lyu and X. Fu, “Nonlinear multiview analysis: Identifiability and neural network-assisted implementation,” IEEE Transactions on Signal Processing, vol. 68, pp. 2697–2712, 2020

    2020

  56. [65]

    The zero set of a polynomial,

    R. Caron and T. Traynor, “The zero set of a polynomial,” University of Windsor, Windsor, ON, Canada, Tech. Rep. WSMR 05-02, 2005

    2005

  57. [66]

    Comput- ing large-scale matrix and tensor decomposition with structured factors: A unified nonconvex optimization perspective,

    X. Fu, N. Vervliet, L. De Lathauwer, K. Huang, and N. Gillis, “Comput- ing large-scale matrix and tensor decomposition with structured factors: A unified nonconvex optimization perspective,” IEEE Signal Processing Magazine, vol. 37, no. 5, pp. 78–94, 2020

    2020

  58. [67]

    Enhanced line search: A novel method to accelerate PARAFAC,

    M. Rajih, P. Comon, and R. A. Harshman, “Enhanced line search: A novel method to accelerate PARAFAC,”SIAM journal on matrix analysis and applications, vol. 30, no. 3, pp. 1128–1147, 2008

    2008

  59. [68]

    On minimum-cost assignments in unbalanced bipartite graphs,

    L. Ramshaw and R. E. Tarjan, “On minimum-cost assignments in unbalanced bipartite graphs,” HP Labs, Palo Alto, CA, USA, Tech. Rep. HPL-2012-40R1, Tech. Rep., 2012

    2012

  60. [69]

    Generalized canonical polyadic tensor decomposition,

    D. Hong, T. G. Kolda, and J. A. Duersch, “Generalized canonical polyadic tensor decomposition,” SIAM Review, vol. 62, no. 1, pp. 133– 163, 2020

    2020

  61. [70]

    Stochastic mirror descent for low-rank tensor decomposition under non-euclidean losses,

    W. Pu, S. Ibrahim, X. Fu, and M. Hong, “Stochastic mirror descent for low-rank tensor decomposition under non-euclidean losses,” IEEE Transactions on Signal Processing , vol. 70, pp. 1803–1818, 2022

    2022

  62. [71]

    Nonnegative block-term decomposition with the β-divergence: Joint data fusion and blind spectral unmixing,

    C. Prévost and V . Leplat, “Nonnegative block-term decomposition with the β-divergence: Joint data fusion and blind spectral unmixing,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Rhodes Island, Greece: IEEE, 2023, pp. 1–5

    2023

  63. [72]

    Tensor decompositions, alternating least squares and other tales,

    P. Comon, X. Luciani, and A. L. De Almeida, “Tensor decompositions, alternating least squares and other tales,” Journal of Chemometrics: A Journal of the Chemometrics Society , vol. 23, no. 7-8, pp. 393–405, 2009

    2009

  64. [73]

    Vervliet, O

    N. Vervliet, O. Debals, L. Sorber, M. Van Barel, and L. De Lathauwer. (2016, Mar.) Tensorlab 3.0. Available online. [Online]. Available: https://www.tensorlab.net

    2016

  65. [74]

    Cloud detection over the Arctic region using airborne imaging spectrometer data during the daytime,

    B.-C. Gao, W. Han, S. C. Tsay, and N. F. Larsen, “Cloud detection over the Arctic region using airborne imaging spectrometer data during the daytime,” Journal of Applied Meteorology , vol. 37, no. 11, pp. 1421– 1429, 1998

    1998

  66. [75]

    Super-resolution for hyperspectral and multispectral image fusion accounting for seasonal spectral variability,

    R. A. Borsoi, T. Imbiriba, and J. C. M. Bermudez, “Super-resolution for hyperspectral and multispectral image fusion accounting for seasonal spectral variability,”IEEE transactions on image processing, vol. 29, pp. 116–127, 2019

    2019

  67. [76]

    Disjoint subspaces for common and distinct component analysis: Ap- plication to the fusion of multi-task fMRI data,

    M. Akhonda, B. Gabrielson, S. Bhinge, V . D. Calhoun, and T. Adali, “Disjoint subspaces for common and distinct component analysis: Ap- plication to the fusion of multi-task fMRI data,” Journal of neuroscience methods, vol. 358, p. 109214, 2021

    2021

  68. [77]

    Estimation of common subspace order across multiple datasets: Application to multi-subject fMRI data,

    S. Bhinge, Y . Levin-Schwartz, and T. Adalı, “Estimation of common subspace order across multiple datasets: Application to multi-subject fMRI data,” in Proc. 51st Annual Conference on Information Sciences and Systems (CISS) . IEEE, 2017, pp. 1–5

    2017

  69. [78]

    Determining the dimension and structure of the subspace correlated across multiple data sets,

    T. Hasija, T. Marrinan, C. Lameiro, and P. J. Schreier, “Determining the dimension and structure of the subspace correlated across multiple data sets,” Signal Processing, vol. 176, p. 107613, 2020

    2020

  70. [79]

    Solution of the matrix equation AX+XB=C,

    R. H. Bartels and G. W. Stewart, “Solution of the matrix equation AX+XB=C,” Communications of the ACM , vol. 15, no. 9, pp. 820–826, 1972

    1972

  71. [80]

    Solution of the Sylvester matrix equation axb t+ cxd t= e,

    J. D. Gardiner, A. J. Laub, J. J. Amato, and C. B. Moler, “Solution of the Sylvester matrix equation axb t+ cxd t= e,” ACM Transactions on Mathematical Software (TOMS) , vol. 18, no. 2, pp. 223–231, 1992

    1992

  72. [81]

    A Hessenberg-Schur method for the problem ax+xb=c,

    G. Golub, S. Nash, and C. Van Loan, “A Hessenberg-Schur method for the problem ax+xb=c,” IEEE Transactions on Automatic Control , vol. 24, no. 6, pp. 909–913, 1979

    1979

  73. [82]

    Computational methods for linear matrix equations,

    V . Simoncini, “Computational methods for linear matrix equations,” SIAM Review, vol. 58, no. 3, pp. 377–441, 2016. Ricardo Borsoi (Member, IEEE) received the Doc- torate degree from the Federal University of Santa Catarina, Brazil, and from the Côte d’Azur Univer- sity, Franc...

    2016

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.