We consider the problem of Robust PCA in the the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many corruptions can we tolerate) via polynomial-time algorithms is by now understood. This paper presents and analyzes a non-convex optimization approach that greatly reduces the computational complexity of the above problems, compared to the best available algorithms. In particular, in the fully observed case, with $r$ denoting rank and $d$ dimension, we reduce the complexity from $\mathcal{O}(r^2d^2\log(1/\varepsilon))$ to $\mathcal{O}(rd^2\log(1/\varepsilon))$ -- a big savings when the rank is big. For the partially observed case, we show the complexity of our algorithm is no more than $\mathcal{O}(r^4d \log d \log(1/\varepsilon))$. Not only is this the best-known run-time for a provable algorithm under partial observation, but in the setting where $r$ is small compared to $d$, it also allows for near-linear-in-$d$ run-time that can be exploited in the fully-observed case as well, by simply running our algorithm on a subset of the observations.

本文介绍了一种非凸优化方法，用于解决全观测和部分观测情况下的鲁棒主成分分析问题，该方法与现有最佳算法相比，显著降低了计算复杂度，并且在部分观测情况下，我们的算法在有可证明的情况下也是已知的运行时间最短的算法。

通过梯度下降的快速鲁棒PCA算法