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

We aim to study the subspace structure of data approximately generated from multiple categories and remove errors (e.g., noise, corruptions, and outliers) in the data as well. Most previous methods for subspace analysis learn only one 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 when data contain errors. To tackle the problem, we propose a novel method, called Matrix Factorization with Column L₀-norm constraint (MFC₀), from the matrix factorization perspective. MFC₀ simultaneously discovers the multi-subspace structure of either clean or contaminated data, and learns the basis for each subspace. Specifically, the learnt basis with the orthonormal constraint shows high robustness to errors by adding a regularization term. Owing to the column l₀-norm constraint, the generated representation matrix can be (approximate) block-diagonal after reordering its columns, with each block characterizing one subspace. We develop an efficient first-order optimization scheme to stably solve the nonconvex and nonsmooth objective function of MFC₀. Experimental results on synthetic data and real-world face datasets demonstrate the superiority over traditional and state-of-the-art methods on both representation learning, subspace recovery and clustering.