A Unified Framework for Low-Rank plus Sparse Matrix Recovery
This work provides a general solution for matrix recovery problems, which is incremental as it builds on existing methods but offers a unified approach with proven convergence rates.
The authors tackled the problem of recovering low-rank plus sparse matrices by proposing a unified framework with a generic algorithm that converges at a locally linear rate and matches the best-known robustness guarantees for sparsity tolerance. They demonstrated its application in robust matrix sensing and robust PCA, with experiments on synthetic and real datasets supporting their theoretical results.
We propose a unified framework to solve general low-rank plus sparse matrix recovery problems based on matrix factorization, which covers a broad family of objective functions satisfying the restricted strong convexity and smoothness conditions. Based on projected gradient descent and the double thresholding operator, our proposed generic algorithm is guaranteed to converge to the unknown low-rank and sparse matrices at a locally linear rate, while matching the best-known robustness guarantee (i.e., tolerance for sparsity). At the core of our theory is a novel structural Lipschitz gradient condition for low-rank plus sparse matrices, which is essential for proving the linear convergence rate of our algorithm, and we believe is of independent interest to prove fast rates for general superposition-structured models. We illustrate the application of our framework through two concrete examples: robust matrix sensing and robust PCA. Experiments on both synthetic and real datasets corroborate our theory.