TL;DR
A structure-regularized CP (SR-CP) framework that uses a unified quadratic penalty to encode diverse structural priors through positive semidefinite matrices and a soft orthogonality-promoting penalty is further introduced to enhance component distinctiveness and numerical stability.
This work develops efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM) to enhance sparsity and identifiability of the learned factors in separable nonnegative matrix factorization.
This paper proposes a novel Tensor Train (TT)-based tensor-on-tensor regression optimization framework for variable selection based on mode-1 hyperslice sparsity, and designs an alternating iterative algorithm equipped with a preconditioned metric to efficiently solve the proposed model.
Multi-view clustering seeks a consensus partition from heterogeneous feature views while retaining view-specific information. We propose SGLog-γ∗-MSC, a nonconvex multi-view subspace clustering framework that combines graph total variation, an ℓ2,log penalty, and a tensor γ∗ spectral penalty. These terms promote locally consistent self-representations, model sample-wise corruption, and capture shared low-rank structure across views, respectively. We derive an alternating augmented-Lagrangian algorithm with candidate-selection rules that return global minimizers for both nonconvex proximal subproblems. We also state explicit conditions under which accumulation points satisfy the KKT system and establish conditional whole-sequence convergence through the Kurdyka–Łojasiewicz framework. Hyperparameters are selected by a label-free protocol specified before the audited rerun; ground-truth labels are never used during selection. Across 20 recorded k-means++ initializations of each fixed embedding, the method attains NMI 0.8206±0.0155 on Yale and 0.8962±0.0211 on Scene-15. Relative to nine literature-reported baselines, these are the highest reported NMI values on the two datasets. On UCI digits, BBCSport, and ORL, the method reaches NMI 0.9837, 0.9634, and 0.9892, respectively, ranking second only to HLR-M2VS in the descriptive cross-paper comparison. Component-wise ablation identifies the ℓ2,log term as the largest and most consistent contributor, while the effects of the γ∗ surrogate and sparse-gradient term depend on the dataset. Compared with the tensor nuclear norm, the γ∗ surrogate improves performance on BBCSport, ORL, and UCI digits; sensitivity analysis shows that an interior γ also improves performance on Scene-15. Together, these results support combining robust error modeling with local and shared structural regularization while emphasizing the dataset dependence of individual components.
Sampling a multidomain tensor from limited measurements is fundamental in structured linear inverse problems. Kronecker-structured sampling avoids the full high-dimensional sensing matrix, but design remains difficult: sequential discrete methods can commit the cross-mode budget too early, whereas standard continuous greedy avoids such early commitment at the cost of repeated state-dependent gradient evaluations. We propose spectral-residual continuous greedy (\alg) for frame-potential (FP) tensor sampling. \alg maintains a state-dependent fractional allocation before rounding. Mode-wise Gram matrices provide safe gradient intervals for direction certification, while Shapley values prioritize unresolved gradient queries. SR-CG then applies deterministic rounding followed by path-guided exchange (PGX), which reuses the final fractional state to restrict candidate swaps and accepts only exact FP-decreasing exchanges. We establish a finite-step approximation guarantee that approaches the classical $1-1/e$ factor as direction certification becomes exact and finite-step residual error vanishes. Experiments show fewer exact gradient evaluations and better FP designs, with clearer gains on instances where the cross-mode budget allocation is difficult to determine. \alg also achieves lower average normalized mean-squared error (NMSE) than Greedy-FP at all tested noise levels, although the reconstruction gain is smaller than the FP gain.
Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, unweighted spectral regularizers impose a common shrinkage profile across singular components, which may excessively attenuate dominant low-rank components, and factorized variants either lack component-specific weighting or require costly singular value decompositions (SVDs). This paper proposes two weighted Schatten-$p$ tensor factorization models, termed \WSpTFI{} and \WSpTFII{}, under the tensor-tensor product (t-product) framework to address these limitations. \WSpTFI{} is motivated by a factorized weighted tensor Schatten-$p$ norm identity and permits flexible, possibly asymmetric factor exponents. \WSpTFII{} constructs a regularizer from transform-domain column-pair energies, yielding SVD-free main factor updates and a column-pruning mechanism for reducing redundant rank components. This paper further develops an iteratively reweighted alternating direction method of multipliers (ADMM)-type scheme for \WSpTFI{} and an iteratively reweighted least squares (IRLS)--block successive upper-bound minimization (BSUM) scheme for \WSpTFII{}. Theoretical analysis establishes the weighted factorization relation and provides a conditional limiting Karush--Kuhn--Tucker (KKT) characterization for \WSpTFI{}. For \WSpTFII{}, the actual damped quadratic block updates yield a quantitative sufficient-decrease mechanism for the fixed-$\delta$ smoothed factor objective. This implies asymptotic regularity, and every accumulation point of the fixed-dimensional tail is stationary. Experiments on synthetic tensor completion, color-image restoration, hyperspectral inpainting, and printed-circuit-board defect detection demonstrate competitive reconstruction quality and robustness under various degradation conditions.
Multilinear principal component analysis (MPCA) reduces the dimension of tensor-valued data while preserving their mode-specific structure, but its quadratic scatter criterion can be unstable under heavy-tailed distributions and contamination. We propose spatial-sign-based multilinear principal component analysis (SMPCA), a robust dimension-reduction method that centers the observations by their spatial median, removes radial magnitude through spatial-sign normalization, and estimates the mode-wise loading spaces by alternating eigendecompositions. Under a separable tensor elliptical model, we show that the target mode-wise loading spaces uniquely maximize the population criterion and that one complete sweep of exact population block updates recovers them from any initialization. We also characterize exactly when their tensor-product subspace coincides with a leading unrestricted subspace of vectorized spatial-sign PCA and, when finite second moments exist, ordinary vectorized PCA. At the sample level, we derive explicit statistical rates for the mode-wise subspaces and the joint multilinear projector, obtain corresponding reconstruction guarantees, establish consistency of the cumulative-contribution dimension selector, and prove that the objective values generated by exact cyclic updates are nondecreasing and convergent. Simulations and an empirical application show that SMPCA is more accurate and stable than competitors under heavy-tailed distributions and outlier contamination, while retaining competitive performance under light-tailed settings.
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