Robust PCA with compressed data
The robust principal component analysis (RPCA) problem seeks to separate lowrank trends from sparse outliers within a data matrix, that is, to approximate a n⇥d matrix D as the sum of a low-rank matrix L and a sparse matrix S. We examine the robust principal component analysis (RPCA) problem under data compression, where the data Y is approximately given by (L+S)·C, that is, a low-rank + sparse data matrix that has been compressed to size n ⇥ m (with m substantially smaller than the original dimension d) via multiplication with a compression matrix C. We give a convex program for recovering the sparse component S along with the compressed low-rank component L · C, along with upper bounds on the error of this reconstruction that scales naturally with the compression dimension m and coincides with existing results for the uncompressed setting m = d. Our results can also handle error introduced through additive noise or through missing data. The scaling of dimension, compression, and signal complexity in our theoretical results is verified empirically through simulations, and we also apply our method to a data set measuring chlorine concentration across a network of sensors to test its performance in practice.
This paper proposes a convex robust PCA method that differs from previous work in that it is designed to be applied to compressed data (i.e. to a lower-dimensional linear transformation of the data).The contributions include the problem statement, its formulation as a convex program, a theoretical approximation bound, specializations for specific assumptions on the noise and compression matrix, empirical results on small synthetic data illustrating the rates of convergence are in accordance with theory, and one small synthetic result providing circumstantial evidence that the approach allowing for sparse perturbations appears beneficial to standard low-rank matrix factorization on the data at hand.The work presented includes non-trivial results that require a high degree of sophistication. The method itself is new and possibly important in a number of practical applications.On the other hand, none of the contributions are breaking new ground. E.g. the problem statement and formulation are straightforward given the literature on compressed sensing and convex approaches multi-task learning and multi-variate regression. The theory, while by no means trivial, contains results of a similar kind and relies on existing proof techniques presented in earlier work (e.g. [1]).The experiments raise questions regarding scalability (surely it is possible to solve this optimization problem for larger data sets?). While they do illustrate the rate of convergence is similar to what the bounds predict, it would have been useful to also see to what extent the constants in the bound are tight (e.g. by plotting the actual bounds on top of the same figures).