Efficient and Effective Nonconvex Low-Rank Subspace Clustering via SVT-Free Operators

Abstract

With the growing interest in convex and nonconvex low-rank matrix learning problems, the widely used singular value thresholding (SVT) operators associated with rank relaxation functions often face higher computational complexity, particularly for large-scale data matrices. To improve the efficacy of low-rank subspace clustering and overcome the issue of high computational complexity, this work proposes an efficient and effective method that avoids the need for singular value decomposition (SVD) computations in the iteration scheme. This can be achieved through the use of a computationally efficient and compact formulation, as well as automatic removal of the optimal mean, which reduces time consumption and enhances evaluation performance. A unified clustering framework based on Schatten- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> norm regularized by ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,<i>q</i></sub> -norm can be formulated using this processing way, where inner element suppression can be achieved by choosing appropriate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p, q</i> ∈ (0,1). Additionally, calculating the optimal mean enhances the robustness of the proposed method in the presence of outliers. Unlike the general iteration scheme of the alternating direction method of multiplier (ADMM) algorithms that introduce auxiliary splitting variables, the proposed alternating re-weighted least square (ARwLS) algorithm uses matrix inverse and multiplication computations to obtain analytic solutions, resulting in faster processing speeds for each sub-problem. To further investigate, we provide the computational complexity of each iteration and the theoretical analysis of the convergence property, where the derived solution is a stationary point. Experimental results on synthetic data and several benchmark datasets demonstrate the promising efficiency and efficacy of the proposed clustering method compared to classical and competing algorithms.