This work provides a theoretical analysis which establishes a non-asymptotic reconstruction error bound that characterizes the effects of sampling complexity, optimization convergence, and model mismatch in a structured tensor approximation problem.
Abstract
In this work, we consider a structured tensor approximation problem, where only a limited number of lateral slices are observed. The proposed algorithm , called Basis and Manifold prior Tensor Approximation (BMTA), exploits both global and local structures of the evolution of a global tensor. Specifically, BMTA integrates two signal models: (i) a quasi-basis model that captures smooth global variations along a physical trajectory, and (ii) a manifold-guided interpolation model that characterizes local relationships among tensor slices. A low-rank Tucker reconstruction framework is incorporated to efficiently capture the priors, resulting in coefficients for basis function estimation and a tensor optimization. In addition, we provide a theoretical analysis which establishes a non-asymptotic reconstruction error bound that characterizes the effects of sampling complexity, optimization convergence, and model mismatch. Numerical experiments are performed on both synthetic and real-world datasets, including quantum chemistry and spatiotemporal sensing applications.
An efficient numerical approach for compressing a high-dimensional discrete distribution function into a non-negative tensor train (NTT) format and observing that the proposed NTT fitting procedure exhibits drastically faster convergence than an alternative multiplicative update method that has been previously proposed is observed.
Xun Tang, R. Dwaraknath, Lexing Ying· SIAM Journal on Scientific C...· 0 citations
Robust tensor completion aims to recover a clean tensor from noisy and incomplete observations, where the observed tensor is corrupted by Gaussian noise and sparse noise simultaneously. Existing methods only exploit one or two priors out of global tensor low-rankness, local properties, and nonlocal self-similarity, leading to suboptimal recovery performance. In this paper, we propose a nonconvex model combined with nonlocal self-similarity and tensor dictionary learning for robust tensor completion. Specifically, by partitioning the tensor into several overlapping cubes, the similar cubes are grouped together. Then, we unfold the cubes into matrices and stack these matrices into a third-order tensor. Subsequently, the minimax concave penalty (MCP) is employed on the singular values of all frontal slices of the sub-tensors in the transformed domain to explore the low-rankness of the underlying sub-tensor. The tensor dictionary learning based on Tucker decomposition is used to explore the local patterns of the underlying sub-tensor. Moreover, the MCP is employed onto each entry of the sparse noise tensor to explore the sparsity. A proximal alternating linearized minimization algorithm is adopted to solve the resulting model. Extensive numerical experiments demonstrate that the proposed method outperforms the competing state-of-the-art methods in both visual quality and quantitative metrics.
Hongyue Sun, Duo Qiu, Jiahui Zhao· Mathematics· 0 citations
This paper proposes a novel Tensor Train (TT)-based tensor-on-tensor regression optimization framework for variable selection based on mode-1 hyperslice sparsity. The formulation incorporates an l2,0-regularized term on the first TT-core while imposing Stiefel manifold constraints on the remaining M−1 TT-cores. Leveraging the property that the group sparsity of the first core is equivalent to the hyperslice sparsity of the global structure, we establish theoretical guarantees for the uniform variable-selection consistency of the proposed model. To efficiently solve the proposed model, we design an alternating iterative algorithm equipped with a preconditioned metric and prove its convergence to a critical point. Extensive numerical experiments on both synthetic and real-world datasets demonstrate that the numerical solutions generated by our algorithm exhibit exact support recovery in practice, tightly aligning with our theoretical analysis.
Tensor data, such as hyperspectral images and videos, are often degraded by mixed noise, including Gaussian noise, sparse corruption, and outliers. In this paper, we propose a robust tensor recovery model based on second-order difference-induced adaptive tensor nuclear norm regularization. The underlying clean tensor is represented by a representative coefficient tensor and a learned orthogonal basis along the third mode, so that global low-rank correlations can be characterized in a compact and data-adaptive coefficient domain rather than in a fixed transform space. To incorporate local smoothness into the same representation, tensor nuclear norm penalties are imposed on the spatial second-order difference tensors of the representative coefficients. Compared with conventional first-order total variation, the proposed regularizer models local curvature variations and the correlations among second-order difference patterns, which helps reduce staircase artifacts while preserving structural details. A Hybrid Ordinary–Welsch fidelity term and an $\ell _{1}$ -norm sparse error term are further incorporated to improve robustness against mixed noise. The resulting optimization problem is solved by an ADMM-based algorithm. Experiments on hyperspectral image and video denoising demonstrate that the proposed method consistently improves PSNR and ERGAS under all tested noise settings while achieving competitive SSIM values.
This paper presents a convergence analysis for a newly developed nonlinear model reduction method: parametric probabilistic manifold decomposition (PPMD)~\cite{guo2026parametric}. In addition, existing analyzes of nonlinear reduced order models typically treat subspace reduction, manifold representation, regression, and nonlinear reconstruction as separate components and often remain at the level of discrete state vectors. To the best of our knowledge, no theory tracks the complete error propagation in a data-dependent model whose basis, residual geometry, spectral coordinates, parameter maps, and lifting operator are all learned from the same numerical solution data. We develop a coupled perturbation analysis for the entire PPMD procedure. A trajectory geometry induced by the spatial discretization and temporal quadrature connects discrete trajectory vectors isometrically with the corresponding PDE norm. Population spectral objects are introduced to align the empirical residual coordinates and derive a uniform coordinate error estimate, whose propagation through the Hilbert-valued kernel lifting estimator is then quantified. Combining these results with the full order discretization error, weighted low-rank approximation, parameter regression, and residual representation defect yields deterministic and high-probability trajectory error bounds and consistency in probability in the continuous PDE trajectory space. The theory identifies how the principal errors interact and which components limit the accuracy of the nonlinear reduced order model.