{"id":"2c695eb5-7b7f-44c3-91c6-dd6dc0f5c2b3","arxiv_id":"2412.01102","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Introduces a personalized coupled tensor decomposition with uniqueness guarantees based on uni-mode uniqueness and a multilinear measurement model, with semi-algebraic and ALS algorithms.","lead":"The paper proposes a new tensor decomposition model that separates shared information (common to all datasets) from dataset-specific information, with mathematical conditions guaranteeing that the separation is unique, plus two algorithms to compute it. Tensor methods like these are used to fuse brain scans, hyperspectral images, or other multimodal measurements where each view has its own noise or artifacts.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's weakest-assumption identification (full column rank of the selected P_{ξ_j,j}) is a genuine and important identifiability condition, but it is an explicit hypothesis of Theorem 3, not a hidden or unproven step. The proof uses it exactly where it is needed: to left-invert P_{ξ_j,j} in equation (16) and to transfer the mode-j uniqueness to the common factor in equation (23). I agree with the reader that this is the most delicate assumption, but I do not see a flaw in the argument that depends on it. The proof of Theorem 3 is sound: Step 2 correctly shows that the permutation matrices are block diagonal using A3, and Step 3 shows that the permutations agree across modes using the uniqueness of Y_η plus the absence of proportional columns in P_{η,j}C_j, which follows from A3 for the relevant modes. The generic theorem Theorem 4 verifies A1–A3 with probability one under the stated rank conditions, and Lemmas 1–2 provide the needed lower bounds. The remaining issues in the paper (no error bars, no shared code, approximate real-data match) are valid but concern experimental reproducibility, not the central theoretical claim. Therefore I would not change the reader's CONDITIONAL verdict; the conditions already reflect these reproducibility concerns. My agreement is partial because I fully concur that the full-column-rank assumption is crucial, but I do not treat it as a weakness of the proof—it is a stated sufficient condition that is also close to necessary for identifiability.","tokens_in":29395,"tokens_out":32681,"duration_ms":270896,"concrete_test":"Run the semi-algebraic algorithm on a noiseless synthetic dataset satisfying A1–A3 (e.g., the configuration of Example 4) and verify that the recovered common tensor C and distinct tensors D_k match the ground truth up to numerical precision, over at least 100 random seeds. This would test the constructive proof end-to-end and would catch any hidden permutation-mixing failure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined the proof of Theorem 3 and the generic theorem Theorem 4 carefully. The central claim is conditional on A1–A3, and the proof is internally consistent. The key step where the common factor is separated from distinct factors uses A2 (full column rank of P_{ξ_j,j}) and A3 (Kruskal rank of the concatenated matrix > 1). A2 is genuinely necessary: if no dataset measures mode j with a full-column-rank operator, the component of C_j in the null space of every P_{k,j} is unobservable, so identifiability cannot hold. A3 prevents the permutation from mixing common and distinct columns: any such mixing would create a proportionality between two columns of the concatenated matrix, which is excluded by Kruskal rank > 1. I also checked the generic proof: Lemmas 1 and 2 are standard genericity arguments, and the lower bounds on Kruskal rank and rank used in conditions (26)–(27) follow. The paper explicitly acknowledges that the real data do not exactly match the model, and the experiments are approximate. The reader's concerns about missing error bars and code are about reproducibility and presentation, not about the validity of the central uniqueness claim. I found no gap that would invalidate Theorem 3 or Theorem 4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29577,"tokens_out":31111,"duration_ms":295086,"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":[{"comment":"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.","section":"Section IV-B, Lemma 2 (proof around Eqs. (28)-(29))"},{"comment":"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.","section":"Section V-A, Algorithm 1, line 13"}],"minor_comments":[{"comment":"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].","section":"Section IV-A, Theorem 3, proof Step 3 (Eq. (24))"},{"comment":"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.","section":"Section IV, Theorems 3 and 4"},{"comment":"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).","section":"Appendix A, Eqs. (48)-(50)"},{"comment":"The symbol Y_ηℓ appears to be a typo; it should be Y_ξℓ (or Y_{ξ_ℓ}) to match the surrounding text.","section":"Section V-A, Algorithm 1, line 9"},{"comment":"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.","section":"Section VI, Tables I and II"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper whose central deterministic uniqueness theorem appears sound. The main obstacle is the gap in the proof of Lemma 2, which is load-bearing for Theorem 4 but is repairable. The algorithm pseudocode issue in Algorithm 1 is also easily fixed. I do not see grounds for rejection; major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know: the central claims hold up. The personalized CTD model in (8)–(12) genuinely generalizes Kanatsoulis et al., Borsoi et al., and Sorensen/Sidiropoulos, and the uni-mode uniqueness conditions in Theorems 3 and 4 are new and interpretable. I checked the proofs carefully, and the stress-test is right: no gap invalidates them. A2 is genuinely necessary, and the block-diagonal permutation argument in Step 2 of Theorem 3 works. The paper does several things well. The conditions are stated in terms a practitioner can grasp: one fully unique tensor, mode-wise full-column-rank measurements, and a Kruskal-rank condition that prevents mixing common and distinct columns. The semi-algebraic algorithm is directly motivated by the proof, which is a nice feature. The paper is also honest: it acknowledges that the real cloud data do not exactly match the model, that the semi-algebraic method degrades at moderate SNR, and that rank selection remains open. The soft spots are real but not fatal. The experiments report means over 20 Monte Carlo runs with no error bars; for a signal processing journal I would want at least standard deviations. No code or data repository is provided, so exact reproduction is impossible. Lemma 2's proof is sketched loosely: the existence construction for a rank-T matrix of the block form is asserted with a matrix that does not obviously respect the block dimensions in the R≤M case. It is a standard genericity argument, so this is a minor gap, not a threat to the theorem. Also, the optimization algorithm with random initialization outperformed the semi-algebraic initialization at lower SNR, so the \"principled initialization\" claim is weaker than it first appears. These are presentation and reproducibility issues, not load-bearing flaws. The comparison to competing methods is fair, and the limitations are stated openly. This paper is for tensor decomposition researchers in signal processing and data fusion, especially those who need identifiability guarantees for shared/distinct component models. It deserves a serious referee. A careful referee should ask for error bars, code/data, and a tightened Lemma 2, but the core contribution is sound and worth engaging with.","headline":"Central uniqueness claims hold up; a solid, genuinely more general coupled tensor model with interpretable conditions, minor experimental and proof-presentation gaps.","tokens_in":30177,"tokens_out":1485,"would_cite":true,"duration_ms":14942,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A common tensor and its per-dataset distinct parts are uniquely recoverable under mild conditions.","keywords":["coupled tensor decomposition","multimodal data fusion","shared and distinct components","personalized learning","canonical polyadic decomposition","uni-mode uniqueness","identifiability","hyperspectral image fusion"],"falsifier":"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.","tokens_in":29116,"feed_emoji":"🧩","tokens_out":10730,"duration_ms":85974,"temperature":0.7,"pith_summary":"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.","feed_headline":"One fully unique view makes shared and private tensors recoverable","feed_subtitle":"New uniqueness conditions let data fusion separate shared from dataset-specific signals.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the uni-mode uniqueness condition (Theorem 2) that each individual measured tensor must satisfy for its mode-j factor to be recoverable.","marker":"[45]"},{"why":"Provides the relaxed uni-mode uniqueness criteria (including the expression used for one-factor uniqueness) that underpin conditions A2 and A5.","marker":"[46]"},{"why":"Establishes uniqueness of coupled CPDs and the connection to LL1 block term decompositions, the basis for recovering a shared factor from coupled tensors.","marker":"[38]"},{"why":"States Kruskal's condition for full uniqueness of a CPD, which the paper uses as the definition of a fully unique measured tensor in A1.","marker":"[48]"},{"why":"Provides necessary conditions for CPD uniqueness, used in Step 3 of the proof to conclude that the common factor has no proportional columns.","marker":"[50]"},{"why":"Defines the coupled tensor factorization measurement model for hyperspectral/multispectral fusion that the proposed model generalizes by adding distinct components.","marker":"[9]"},{"why":"Introduces a coupled tensor decomposition with a distinct component in one dataset, the closest prior approach and a baseline in the real-data experiments.","marker":"[10]"},{"why":"Studies multi-set low-rank factorizations with shared and unshared components, providing the comparison point for uniqueness with a different low-rank model.","marker":"[34]"},{"why":"Reports the preliminary conference version of the model applied to multi-task fMRI data fusion, supporting the practical relevance of the framework.","marker":"[13]"}],"fun_headline_variants":["Shared vs private signals: unique separation in multimodal fusion","One fully unique view recovers shared and private tensors","Personalized CTD: uniqueness conditions for common and distinct parts","Telling common from dataset-specific in coupled tensor fusion","Multimodal fusion that separates shared and private factors uniquely"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Shared vs private signals: unique separation in multimodal fusion","One fully unique view recovers shared and private tensors","Personalized CTD: uniqueness conditions for common and distinct parts","Telling common from dataset-specific in coupled tensor fusion","Multimodal fusion that separates shared and private factors uniquely"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000568,"raw_usage":{"total_tokens":2721,"prompt_tokens":1009,"completion_tokens":1712,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":1632}},"tokens_in":625,"tokens_out":1712,"duration_ms":11747,"temperature":1.0,"reasoning_tokens":1632,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:42:07.983537+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Uni-mode and partial uniqueness conditions for CANDECOMP/PARAFAC of three-way ar- rays with linearly dependent loadings,","cited_arxiv_id":null,"evidence_quote":"Supplies the uni-mode uniqueness condition (Theorem 2) that each individual measured tensor must satisfy for its mode-j factor to be recoverable."},{"cited_title":"On the uniqueness of the canonical polyadic decomposition of third-order tensors–Part I: Basic results and uniqueness of one factor matrix,","cited_arxiv_id":null,"evidence_quote":"Provides the relaxed uni-mode uniqueness criteria (including the expression used for one-factor uniqueness) that underpin conditions A2 and A5."},{"cited_title":"Coupled canonical polyadic decompositions and (coupled) decompositions in multilinear rank- (Lr,n,Lr,n,1) terms–part I: Uniqueness,","cited_arxiv_id":null,"evidence_quote":"Establishes uniqueness of coupled CPDs and the connection to LL1 block term decompositions, the basis for recovering a shared factor from coupled tensors."},{"cited_title":"Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics,","cited_arxiv_id":null,"evidence_quote":"States Kruskal's condition for full uniqueness of a CPD, which the paper uses as the definition of a fully unique measured tensor in A1."},{"cited_title":"On the uniqueness of the canonical polyadic decomposition of third-order tensors–Part II: Uniqueness of the overall decomposition,","cited_arxiv_id":null,"evidence_quote":"Provides necessary conditions for CPD uniqueness, used in Step 3 of the proof to conclude that the common factor has no proportional columns."},{"cited_title":"Hyperspec- tral super-resolution: A coupled tensor factorization approach,","cited_arxiv_id":null,"evidence_quote":"Defines the coupled tensor factorization measurement model for hyperspectral/multispectral fusion that the proposed model generalizes by adding distinct components."},{"cited_title":"Coupled tensor decomposition for hyperspectral and multispectral image fusion with inter-image variability,","cited_arxiv_id":null,"evidence_quote":"Introduces a coupled tensor decomposition with a distinct component in one dataset, the closest prior approach and a baseline in the real-data experiments."},{"cited_title":"Multi-set low-rank factorizations with shared and unshared components,","cited_arxiv_id":null,"evidence_quote":"Studies multi-set low-rank factorizations with shared and unshared components, providing the comparison point for uniqueness with a different low-rank model."},{"cited_title":"Coupled CP tensor decomposition with shared and distinct components for multi-task fMRI data fusion,","cited_arxiv_id":null,"evidence_quote":"Reports the preliminary conference version of the model applied to multi-task fMRI data fusion, supporting the practical relevance of the framework."}],"review_version":1}