A generalisation of Mirsky's singular value inequalities

We prove an f-version of Mirsky's singular value inequalities for differences of matrices. This f-version consists in applying a positive concave function f, with f(0)=0, to every singular value in the original Mirsky inequalities.Comment: 8 page

Nonconvex third-order Tensor Recovery Based on Logarithmic Minimax Function

Recent researches have shown that low-rank tensor recovery based non-convex relaxation has gained extensive attention. In this context, we propose a new Logarithmic Minimax (LM) function. The comparative analysis between the LM function and the Logarithmic, Minimax concave penalty (MCP), and Minimax Logarithmic concave penalty (MLCP) functions reveals that the proposed function can protect large singular values while imposing stronger penalization on small singular values. Based on this, we define a weighted tensor LM norm as a non-convex relaxation for tensor tubal rank. Subsequently, we propose the TLM-based low-rank tensor completion (LRTC) model and the TLM-based tensor robust principal component analysis (TRPCA) model respectively. Furthermore, we provide theoretical convergence guarantees for the proposed methods. Comprehensive experiments were conducted on various real datasets, and a comparison analysis was made with the similar EMLCP method. The results demonstrate that the proposed method outperforms the state-of-the-art methods

Low-Rank Tensor Completion Based on Bivariate Equivalent Minimax-Concave Penalty

Low-rank tensor completion (LRTC) is an important problem in computer vision and machine learning. The minimax-concave penalty (MCP) function as a non-convex relaxation has achieved good results in the LRTC problem. To makes all the constant parameters of the MCP function as variables so that futherly improving the adaptability to the change of singular values in the LRTC problem, we propose the bivariate equivalent minimax-concave penalty (BEMCP) theorem. Applying the BEMCP theorem to tensor singular values leads to the bivariate equivalent weighted tensor $\Gamma$-norm (BEWTGN) theorem, and we analyze and discuss its corresponding properties. Besides, to facilitate the solution of the LRTC problem, we give the proximal operators of the BEMCP theorem and BEWTGN. Meanwhile, we propose a BEMCP model for the LRTC problem, which is optimally solved based on alternating direction multiplier (ADMM). Finally, the proposed method is applied to the data restorations of multispectral image (MSI), magnetic resonance imaging (MRI) and color video (CV) in real-world, and the experimental results demonstrate that it outperforms the state-of-arts methods.Comment: arXiv admin note: text overlap with arXiv:2109.1225

Spectral Properties of Structured Kronecker Products and Their Applications

We study certain spectral properties of some fundamental matrix functions of pairs of symmetric matrices. Our study includes eigenvalue inequalities and various interlacing properties of eigenvalues. We also discuss the role of interlacing in inverse eigenvalue problems for structured matrices. Interlacing is the main ingredient of many fundamental eigenvalue inequalities. This thesis also recounts a historical development of the eigenvalue inequalities relating the sum of two matrices to its summands with some recent findings motivated by problems arising in compressed sensing. One of the fundamental matrix functions on pairs of matrices is the Kronecker product. It arises in many fields such as image processing, signal processing, quantum information theory, differential equations and semidefinite optimization. Kronecker products enjoy useful algebraic properties that have proven to be useful in applications. The less-studied symmetric Kronecker product and skew-symmetric Kronecker product (a contribution of this thesis) arise in semidefinite optimization. This thesis focuses on certain interlacing and eigenvalue inequalities of structured Kronecker products in the context of semidefinite optimization. A popular method used in semidefinite optimization is the primal-dual interior point path following algorithms. In this framework, the Jordan-Kronecker products arise naturally in the computation of Newton search direction. This product also appears in many linear matrix equations, especially in control theory. We study the properties of this product and present some nice algebraic relations. Then, we revisit the symmetric Kronecker product and present its counterpart the skew-symmetric Kronecker product with its basic properties. We settle the conjectures posed by Tunçel and Wolkowicz, in 2003, on interlacing properties of eigenvalues of the Jordan-Kronecker product and inequalities relating the extreme eigenvalues of the Jordan-Kronecker product. We disprove these conjectures in general, but we also identify large classes of matrices for which the interlacing properties hold. Furthermore, we present techniques to generate classes of matrices for which these conjectures fail. In addition, we present a generalization of the Jordan-Kronecker product (by replacing the transpose operator with an arbitrary symmetric involution operator). We study its spectral structure in terms of eigenvalues and eigenvectors and show that the generalization enjoys similar properties of the Jordan-Kronecker product. Lastly, we propose a related structure, namely Lie-Kronecker products and characterize their eigenvectors