Low-rank Matrix Recovery from Local Coherence Perspective

We investigate the robust PCA problem of decomposing an observed matrix intothe sum of a low-rank and a sparse error matrices via convex programming PrincipalComponent Pursuit (PCP). In contrast to previous studies that assume the supportof the sparse error matrix is generated by uniform Bernoulli sampling, we allow non-uniform sampling, i.e., entries of the low-rank matrix are corrupted by errors withunequal probabilities. We characterize conditions on error corruption of each individualentry based on the local coherence of the low-rank matrix, under which correct matrixdecomposition by PCP is guaranteed. Such a refined analysis of robust PCA captureshow robust each entry of the low-rank matrix combats error corruption. Moreover,this result has several immediate implications on graph clustering problem, which havebeen partially studied in random graph clustering literatures. In order to deal withnon-uniform error corruption, our technical proof introduces a new weighted norm anddevelops/exploits the concentration properties that such a norm satisfies. We alsoinvestigate the partial observation setting and establish the general theory of matrixrecovery from both error corruption and partial observation based on local coherence.