BRIDGING CONVEX AND NONCONVEX OPTIMIZATION IN ROBUST PCA: NOISE, OUTLIERS AND MISSING DATA

• 成果类型: Article
• 署名作者: Chen, Yuxin; Fan, Jianqing; Ma, Cong; Yan, Yuling
• 署名单位: Princeton University; Princeton University; University of Chicago
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2066
• 发表日期: 2021
• 页码: 2948-2971
• 关键词: rank matrix recovery; Optimal Rates; Gradient descent; completion; SPARSE; relaxation; DECOMPOSITION; Incoherence; algorithms; guarantees

摘要

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 in the presence of 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 l(infinity) 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.