论文标题
稳健的PCA中的桥接凸和非凸优化：噪声，异常值和缺少数据
Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data
论文作者
论文摘要
本文在（1）随机噪声，（（2）总稀疏异常值和（3）缺少数据的情况下，在低级矩阵估计中提供了凸编程方法的改进理论保证。这个问题通常被称为可靠的主成分分析（鲁棒PCA），在各个域中找到了应用程序。尽管凸松弛的广泛适用性，但可用的统计支持（尤其是相对于随机噪声的稳定性分析）仍然高于最佳速度，我们在本文中得到了加强。当未知的矩阵具有良好的条件，不连贯的且恒定的等级时，我们证明了原则上的凸面程序可以在欧几里得损失和$ \ ell _ {\ el _ {\ fly iftty} $损失方面达到近乎最佳的统计精度。所有这些都会发生，即使几乎持续的观察结果被任意幅度的异常值所破坏。关键分析思想在于桥接使用的凸程序和辅助非凸优化算法，因此是本文的标题。
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.
