Fast First-Order Methods for Stable Principal Component Pursuit

TL;DR

This paper introduces and analyzes several fast first-order optimization methods for the stable principal component pursuit problem, demonstrating their efficiency and convergence properties, and providing the first algorithm with O(1/eps) complexity and SVD-level per-iteration cost.

Contribution

The paper presents new fast first-order algorithms for SPCP, including the first with O(1/eps) iteration complexity and SVD-level per-iteration cost.

Findings

The proposed methods efficiently solve SPCP with theoretical convergence guarantees.

One method directly handles the non-smooth objective and converges under mild conditions.

Computational tests show the new method outperforms existing approaches.

Abstract

The stable principal component pursuit (SPCP) problem is a non-smooth convex optimization problem, the solution of which has been shown both in theory and in practice to enable one to recover the low rank and sparse components of a matrix whose elements have been corrupted by Gaussian noise. In this paper, we show how several fast first-order methods can be applied to this problem very efficiently. Specifically, we show that the subproblems that arise when applying optimal gradient methods of Nesterov, alternating linearization methods and alternating direction augmented Lagrangian methods to the SPCP problem either have closed-form solutions or have solutions that can be obtained with very modest effort. All but one of the methods analyzed require at least one of the non-smooth terms in the objective function to be smoothed and obtain an eps-optimal solution to the SPCP problem in O(1/eps) iterations. The method that works directly with the fully non-smooth objective function, is proved to be convergent under mild conditions on the sequence of parameters it uses. Our preliminary computational tests show that the latter method, although its complexity is not known, is fastest and substantially outperforms existing methods for the SPCP problem. To best of our knowledge, an algorithm for the SPCP problem that has O(1/eps) iteration complexity and has a per iteration complexity equal to that of a singular value decomposition is given for the first time.