Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data
This provides stronger statistical support for robust PCA, which is used in various domains, but it is incremental as it builds on existing convex relaxation methods.
The paper tackles robust PCA by improving theoretical guarantees for convex programming in low-rank matrix estimation with noise, outliers, and missing data, showing it achieves near-optimal statistical accuracy in Euclidean and ℓ∞ loss even with a constant fraction of corrupted observations.
This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers, and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis vis-à-vis random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well-conditioned, incoherent, and of constant rank, we demonstrate that a principled convex program achieves near-optimal statistical accuracy, in terms of both the Euclidean loss and the $\ell_{\infty}$ loss. All of this happens even when nearly a constant fraction of observations are corrupted by outliers with arbitrary magnitudes. The key analysis idea lies in bridging the convex program in use and an auxiliary nonconvex optimization algorithm, and hence the title of this paper.