Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery

By Qi Xie, Qian Zhao, Deyu Meng, Zongben Xu. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2018

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As a promising way for analyzing data, sparse modeling has achieved great success throughout science and engineering. It is well known that the sparsity/low-rank of a vector/matrix can be rationally measured by nonzero-entries-number ($l_0$ norm)/nonzero- singular-values-number (rank), respectively. However, data from real applications are often generated by the interaction of multiple factors, which obviously cannot be sufficiently represented by a vector/matrix, while a high order tensor is expected to provide more faithful representation to deliver the intrinsic structure underlying such data ensembles. Unlike the vector/matrix case, constructing a rational high order sparsity measure for tensor is a relatively harder task. To this aim, in this paper we propose a measure for tensor sparsity, called Kronecker-basis-representation based tensor sparsity measure (KBR briefly), which encodes both sparsity insights delivered by Tucker and CANDECOMP/PARAFAC (CP) low-rank decompositions for a general tensor. Then we study the KBR regularization minimization (KBRM) problem, and design an effective ADMM algorithm for solving it, where each involved parameter can be updated with closed-form equations. Such an efficient solver makes it possible to extend KBR to various tasks like tensor completion and tensor robust principal component analysis. A series of experiments, including multispectral image (MSI) denoising, MSI completion and background subtraction, substantiate the superiority of the proposed methods beyond state-of-the-arts.

Subfields: Computational Mathematics, Geophysics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/6407ac1f051d30ca621ce16cf4ca67beb05930c6