Optimal Data Scaling for Principal Component Pursuit: A Lyapunov Approach to Convergence

Abstract

In principle component pursuit (PCP), the essential idea is to replace the original non-convex optimization problem of the matrix rank and the count of non-zero entries by a convex optimization problem of the nuclear and I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norms. In the PCP literature, it is rigorously proved that the validity of this idea depends on the coherence of the uncontaminated data. Specifically, the lower the coherence is, the equivalence of the convex optimization problem to the original non-convex one will hold by a larger probability. Although the coherence index is fixed for a given data set, it is possible to adjust this index by introducing different scalings to the data. The target of this work is thus to find the optimal scaling of the data such that the coherence index is minimized. Based on the analysis of the PCP problem structure, a non-convex optimization problem with implicit dependence on the scaling parameters is firstly formulated. To solve this problem, a coordinate descent algorithm is proposed. Under mild conditions on the structure of the data matrix, the convergence of the algorithm to a global optimal point is rigorously proved by treating the algorithm as a discrete-time dynamic system and utilizing a Lyapunov-type approach. Monte Carlo simulation experiments are performed to verify the effectiveness of the developed results.