Matrix Factorization with Column L0-Norm Constraint for Robust Multi-subspace Analysis

We aim to study the subspace structure of dataapproximately generated from multiple categories and removeerrors (e.g., noise, corruptions, and outliers) in the data aswell. Most previous methods for subspace analysis learn onlyone subspace, failing to discover the intrinsic complex structure, while state-of-the-art methods use data itself as the basis (self-expressiveness property), showing degraded performance whendata contain errors. To tackle the problem, we propose anovel method, called Matrix Factorization with Column L0-normconstraint (MFC0), from the matrix factorization perspective. MFC0 simultaneously discovers the multi-subspace structure ofeither clean or contaminated data, and learns the basis for eachsubspace. Speciflcally, the learnt basis with the orthonormal constraint shows high robustness to errors by adding a regularizationterm. Owing to the column l0-norm constraint, the generatedrepresentation matrix can be (approximate) block-diagonal afterreordering its columns, with each block characterizing onesubspace. We develop an efflcient flrst-order optimization schemeto stably solve the nonconvex and nonsmooth objective function ofMFC0. Experimental results on synthetic data and real-world facedatasets demonstrate the superiority over traditional and state-of-the-art methods on both representation learning, subspacerecovery and clustering.