Low-Rank Tensor Ring Alternating Least Squares With Tikhonov Regularization for Image Completion

Bayesian adaptive tensor ring decomposition with automatic model selection.

Robust tensor decomposition (RTD) is designed to distinguish low-rank and sparse tensors from noisy high-dimensional data, which holds fundamental significance in the fields of machine learning and computer vision. Nevertheless, current RTD-based methods fall short in addressing the issues of automatic noise adaptation and determination of model capacity. In response to these challenges, this paper introduces a robust non-parametric Bayesian method known as the Bayesian adaptive tensor ring decomposition (BATR) method. More precisely, BATR models unknown noise using a Dirichlet process Gaussian mixture model (DP-GMM), with automatic determination of the noise components. Besides, BATR accomplishes an adaptive low tensor ring (TR) rank model by incorporating a more advanced generalized hyperbolic (GH) prior into the probabilistic framework, thereby facilitating automatic TR rank determination. Furthermore, a variational Bayesian inference algorithm is employed to update the posteriors of the model. Extensive experiments on synthetic data, color images, face images, multispectral images, and hyperspectral images demonstrate the improved performance of BATR compared to other state-of-the-art methods.

Nonlocal and Nonconvex Robust Tensor Completion with Tensor Dictionary Learning for Multi-Dimensional Image Recovery

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.

Unified quadratic regularization for structured CP decomposition

Robust Tensor Recovery Using Second-Order Difference-Induced Adaptive Nuclear Norm

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.

STOD: Sparse Tensor Train Optimization via Orthogonal Decomposition for High-Dimensional Learning

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.

View-Aligned Nonlocal Low-Rank Tensor Reconstruction for Snapshot Compressive Multi-View Spectral Imaging System

Snapshot compressive multi-view spectral imaging (SC-MVSI) multiplexes view-spectral information into a single coded measurement, enabling compact acquisition with a two-dimensional detector. Because each reconstructed channel corresponds to both a selected spectral response and a view direction, direct cross-channel modeling at identical pixel coordinates can introduce structural mismatch caused by view-dependent displacement. This paper proposes a reference-guided view-aligned nonlocal low-rank tensor reconstruction method for SC-MVSI. The reconstruction is formulated as a coded inverse problem and solved using the alternating direction method of multipliers (ADMM) in a variable-splitting framework. In the prior update, a reference tensor guides block-level patch alignment before nonlocal tensor grouping, and the resulting fourth-order tensor groups are regularized by canonical polyadic (CP) low-rank approximation. Experiments on eight synthesized multispectral light-field scenes show that the proposed method achieves the highest average PSNR of 33.61 dB and the lowest average CAE of 5.69 degrees among the compared baselines, while obtaining the second-highest average SSIM of 0.8823. Real-system experiments further provide a qualitative demonstration of applying the proposed reconstruction framework to captured coded measurements.