Low-Rank Structure Learning via Log-Sum Heuristic Recovery

Recovering intrinsic data structure from corrupted observations plays an important role in various tasks in the communities of machine learning and signal processing. In this paper, we propose a novel model, named log-sum heuristic recovery (LHR), to learn the essential low-rank structure from corrupted data. Different from traditional approaches, which directly utilize $\ell_1$ norm to measure the sparseness, LHR introduces a more reasonable log-sum measurement to enhance the sparsity in both the intrinsic low-rank structure and in the sparse corruptions. Although the proposed LHR optimization is no longer convex, it still can be effectively solved by a majorization-minimization (MM) type algorithm, with which the non-convex objective function is iteratively replaced by its convex surrogate and LHR finally falls into the general framework of reweighed approaches. We prove that the MM-type algorithm can converge to a stationary point after successive iteration. We test the performance of our proposed model by applying it to solve two typical problems: robust principal component analysis (RPCA) and low-rank representation (LRR). For RPCA, we compare LHR with the benchmark Principal Component Pursuit (PCP) method from both the perspectives of simulations and practical applications. For LRR, we apply LHR to compute the low-rank representation matrix for motion segmentation and stock clustering. Experimental results on low rank structure learning demonstrate that the proposed Log-sum based model performs much better than the $\ell_1$-based method on for data with higher rank and with denser corruptions.