Robust principal component analysis with rank and cardinality regularization under matrix factorization

TL;DR

This paper introduces a novel nonconvex relaxation framework for robust PCA involving rank and sparsity regularization, establishing equivalences and proposing a proximal gradient algorithm with convergence guarantees.

Contribution

It develops an equivalent nonconvex relaxation framework for rank and $ ext{ extlbrack}0 ext{ extbrack}$-norm regularized PCA and proposes a convergent proximal gradient algorithm for the resulting factorization problem.

Findings

Established equivalence between regularized PCA and matrix factorization solutions.

Designed a proximal gradient algorithm with convergence to stationary points.

Demonstrated the algorithm's ability to attain strong stationary points.

Abstract

Robust principal component analysis is an important representative method in data analysis. It is usually viewed as an optimization problem involving the rank and $\ell_0$-norm of matrices. In this paper, we study the rank and $\ell_0$ regularized optimization problem and its matrix factorization problem. We establish their equivalences on global minimizers and stationary points, respectively. Furthermore, we construct a broadly applicable equivalent nonconvex relaxation framework for the constrained factorization model in the sense of global minimizers and stationary points with strong optimality conditions (called strong stationary points). For the general factorization problem with lower semicontinuous regularizers and a loss function whose gradient is locally Lipschitz, we propose a novel proximal gradient-based algorithm based on joint and alternating calculation with convergence to its limiting-critical points. The algorithm can attain the stationary points of the original problem and its adaptive counterpart can attain the strong stationary points of the factorization problem.