Efficient Image Classification via Structured Low-Rank Matrix Factorization Regression
Hengmin Zhang, Jian Yang, Jianjun Qian, Guangwei Gao, Xiangyuan Lan, Zhiyuan Zha, Bihan Wen
Abstract
In real-world applications involving sparse coding and low-rank matrix recovery problems, linear regression methods usually struggle to effectively capture the structured correlations present in data matrices. This limitation arises from representation approaches that treat images as vectors and handle testing samples individually, overlooking these correlations. To address these challenges, we propose a novel approach that leverages the low-rank property to capture the global and intrinsic structure of residual and coefficient matrices, departing from the assumption of independent and identically distributed (I.I.D) data. Our method introduces nonconvex and nonsmooth low-rank matrix regression models guided by the extended matrix variate power exponential distribution (M.P.E.D). By incorporating factorization strategies into the regression coefficient matrix and utilizing the Schatten- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> norm with three distinct values of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> , we enhance computational efficiency. Our formulation enables efficient subproblem solving through the introduction of auxiliary variables and the use of singular value threshold operators. We achieve closed-form solutions using the proposed multi-variable alternating direction method of multipliers (ADMM). Theoretical analysis establishes the local convergence properties and computational complexity of our optimization algorithm. Furthermore, we conduct numerical experiments on various image datasets, including face, object, and digital, to demonstrate the superior performance and computational efficiency of our methods compared to several related regression approaches. The source codes for our method are available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/ZhangHengMin/TIFS_SLRMFR</uri> .