在这项工作中，我们研究了在线鲁棒主成分分析（RPCA）问题。在最近的工作中，RPCA已被定义为一个问题的分离低秩矩阵（真实数据），$L$，和稀疏矩阵（离群值），$S$，从他们的总和，$M:=L + S$。该问题的更一般的版本是从$M:=L + S + W$恢复$L$和$S$，其中$W$是非结构化小噪声/破坏的矩阵。出现该问题的重要应用是在视频分析中尝试分离稀疏前景（例如，移动物体）从缓慢变化的背景中。虽然有大量的解决方案和保证批量RPCA问题的最新工作，在线问题在很大程度上是开放的。“在线”RPCA是在运行中进行上述操作的问题，其中额外假设初始子空间是准确已知的，并且生成$l_t$的子空间随时间缓慢变化。我们开发和研究一种新的“在线”RPCA算法的基础上最近推出的递归投影压缩感知（ReProCS）框架。我们的算法改进了原始的ReProCS算法，它还返回更准确的离线估计。这项工作的关键贡献是一个正确的结果（完整的性能保证），该算法在合理的温和的假设。通过使用额外的假设-准确的初始子空间知识，缓慢的子空间变化和聚类特征值-我们能够消除批量RPCA结果的一个重要限制和ReProCS在线RPCA最近结果的两个关键限制。据我们所知，这项工作是在线RPCA的前几个正确性结果之一。大多数早期的结果只是部分结果，即，它们需要对中间算法估计进行假设。
英文摘要：
In this work, we study the online robust principal components' analysis (RPCA) problem. In recent work, RPCA has been defined as a problem of separating a low-rank matrix (true data), $L$, and a sparse matrix (outliers), $S$, from their sum, $M:=L + S$. A more general version of this problem is to recover $L$ and $S$ from $M:=L + S + W$ where $W$ is the matrix of unstructured small noise/corruptions. An important application where this problem occurs is in video analytics in trying to separate sparse foregrounds (e.g., moving objects) from slowly changing backgrounds. While there has been a large amount of recent work on solutions and guarantees for the batch RPCA problem, the online problem is largely open."Online" RPCA is the problem of doing the above on-the-fly with the extra assumptions that the initial subspace is accurately known and that the subspace from which $l_t$ is generated changes slowly over time. We develop and study a novel "online" RPCA algorithm based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our algorithm improves upon the original ReProCS algorithm and it also returns even more accurate offline estimates. The key contribution of this work is a correctness result (complete performance guarantee) for this algorithm under reasonably mild assumptions. By using extra assumptions -- accurate initial subspace knowledge, slow subspace change, and clustered eigenvalues -- we are able to remove one important limitation of batch RPCA results and two key limitations of a recent result for ReProCS for online RPCA. To our knowledge, this work is among the first few correctness results for online RPCA. Most earlier results were only partial results, i.e., they required an assumption on intermediate algorithm estimates.
Provable Inductive Robust PCA via Iterative Hard Thresholding
Microsoft Corporation（微软公司）；Conduent Labs India（Conduent印度实验室）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；University of California Irvine（加利福尼亚大学尔湾分校）；Xerox Research Centre India（施乐印度研究中心）
