Algorithm

Mathematical optimization

Mathematical analysis

Pattern recognition (psychology)

Artificial intelligence

Tensor (intrinsic definition)

Laplace operator

Pure mathematics

Subspace topology

Mathematics

Computer science

IEEE Transactions on Cybernetics

Recently, tensor sparsity modeling has achieved great success in the tensor completion (TC) problem. In real applications, the sparsity of a tensor can be rationally measured by low-rank tensor decomposition. However, existing methods either suffer from limited modeling power in estimating accurate rank or have difficulty in depicting hierarchical structure underlying such data ensembles. To address these issues, we propose a parametric tensor sparsity measure model, which encodes the sparsity for a general tensor by Laplacian scale mixture (LSM) modeling based on three-layer transform (TLT) for factor subspace prior with Tucker decomposition. Specifically, the sparsity of a tensor is first transformed into factor subspace, and then factor sparsity in the gradient domain is used to express the local similarity in within-mode. To further refine the sparsity, we adopt LSM by the transform learning scheme to self-adaptively depict deeper layer structured sparsity, in which the transformed sparse matrices in the sense of a statistical model can be modeled as the product of a Laplacian vector and a hidden positive scalar multiplier. We call the method as parametric tensor sparsity delivered by LSM-TLT. By a progressive transformation operator, we formulate the LSM-TLT model and use it to address the TC problem, and then the alternating direction method of multipliers-based optimization algorithm is designed to solve the problem. The experimental results on RGB images, hyperspectral images (HSIs), and videos demonstrate the proposed method outperforms state of the arts.

When Laplacian Scale Mixture Meets Three-Layer Transform: A Parametric Tensor Sparsity for Tensor Completion