ReProCS：相关大噪声下递归鲁棒PCA与递归稀疏恢复之间缺失的一环
ReProCS: A Missing Link between Recursive Robust PCA and Recursive Sparse Recovery in Large but Correlated Noise
摘要：
本文研究递归鲁棒主元分析问题。这里，“鲁棒性”是指对独立和相关稀疏离群值的鲁棒性，尽管我们关注的是后者。出现此问题的一个关键应用是视频监控，其目标是将缓慢变化的背景与动态移动的前景对象分开。背景序列被很好地建模为位于一个低维子空间，可以随着时间的推移逐渐变化，而运动的前景对象构成相关的稀疏离群值。在这个和许多其他应用中，前景是PCA的离群值，但实际上是应用的“感兴趣的信号”;而背景是腐败或噪声。因此，我们的问题也可以解释为递归恢复的稀疏信号的时间序列中存在的大，但空间相关的噪声之一。这项工作有两个关键贡献。首先，我们提供了一种看待这个问题的新方法，并展示了我们的解决方案策略的关键部分是如何解决噪声压缩感知（CS）问题的。其次，我们展示了如何利用离群值的相关性来处理非常大的支持度离群值。主要思想如下。应用于先前支持估计的相关模型有助于预测当前支持。此预测作为“部分支持知识”，用于解决修改的CS问题，而不是CS。修改的CS重建的支持估计又用于使用卡尔曼滤波器（或任何自适应滤波器）来更新相关模型参数。我们称之为“支持预测修改CS”的方法。
英文摘要：
This work studies the recursive robust principal components' analysis (PCA) problem. Here, "robust" refers to robustness to both independent and correlated sparse outliers, although we focus on the latter. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background from moving foreground objects on-the-fly. The background sequence is well modeled as lying in a low dimensional subspace, that can gradually change over time, while the moving foreground objects constitute the correlated sparse outliers. In this and many other applications, the foreground is an outlier for PCA but is actually the "signal of interest" for the application; where as the background is the corruption or noise. Thus our problem can also be interpreted as one of recursively recovering a time sequence of sparse signals in the presence of large but spatially correlated noise. This work has two key contributions. First, we provide a new way of looking at this problem and show how a key part of our solution strategy involves solving a noisy compressive sensing (CS) problem. Second, we show how we can utilize the correlation of the outliers to our advantage in order to even deal with very large support sized outliers. The main idea is as follows. The correlation model applied to the previous support estimate helps predict the current support. This prediction serves as "partial support knowledge" for solving the modified-CS problem instead of CS. The support estimate of the modified-CS reconstruction is, in turn, used to update the correlation model parameters using a Kalman filter (or any adaptive filter). We call the resulting approach "support-predicted modified-CS".
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