Analysis of Robust PCA via Local Incoherence
We investigate the robust PCA problem of decomposing an observed matrix into the sum of a low-rank and a sparse error matrices via convex programming Principal Component Pursuit (PCP). In contrast to previous studies that assume the support of the 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 with unequal probabilities. We characterize conditions on error corruption of each individual entry based on the local incoherence of the low-rank matrix, under which correct matrix decomposition by PCP is guaranteed. Such a refined analysis of robust PCA captures how robust each entry of the low rank matrix combats error corruption. In order to deal with non-uniform error corruption, our technical proof introduces a new weighted norm and develops/exploits the concentration properties that such a norm satisfies.
Summary of paper:Until recently, the related but different problems of matrix completion and robust PCA - via nuclear norm minimization - had similar assumptions under which recovery was known: incoherence of the low-rank matrix, and uniform randomness in the samples (for completion) or errors (for robust PCA). Recently however it was shown that for matrix completion recovery is provably possible, via same convex methods, even when the sampling is non-uniform, provided this non-uniformity is adapted to LOCAL coherence. This paper establishes a similar result for robust PCA, showing that if the errors are non-uniromly random (again in a way that is adapted to the same local coherences) then exact recovery is again possible.Quality:Overall the result is likely to be important and of interest. The techniques are new, but along lines somewhat similar to those employed in the (recent) matrix completion analysis using local coherence.The experiments are not very convincing because the low-rank matrices are constructed to have very similar local coherences. It would have been better to see matrices with very different local coherences. This can be done by pre and post multiplying the low-rank matrices generated in the paper with diagonal matrices that have large dynamic range on the diagonal elements. Clarity:The paper is clearly written, though by the very nature of what it is trying to do is a bit dense to read.Originality:The paper takes the new recent understanding of how local coherence can allow for non-uniformity in sampling for matrix completion, and extends it to the case of errors in robust PCA. The results are original.Significance:Robust PCA has a huge number of applications, and by this token an improvement in its understanding is significant. However the paper does not provide any new method; but rather a better analysis of the already popular method. So its empirical significance may be lower.It would have been nice to see specific examples where this kind of non-uniformity both naturally arises, and leads to appreciably better recovery. For example, does this imply that for the graph clustering problem with unequal size clusters, larger clusters can be sparser but still recover-able ?