Recovering intrinsic low dimensional subspaces from data distributed on them
is a key preprocessing step to many applications. In recent years, there has
been a lot of work that models subspace recovery as low rank minimization
problems. We find that some representative models, such as Robust Principal
Component Analysis (R-PCA), Robust Low Rank Representation (R-LRR), and Robust
Latent Low Rank Representation (R-LatLRR), are actually deeply connected. More
specifically, we discover that once a solution to one of the models is
obtained, we can obtain the solutions to other models in closed-form
formulations. Since R-PCA is the simplest, our discovery makes it the center of
low rank subspace recovery models. Our work has two important implications.
First, R-PCA has a solid theoretical foundation. Under certain conditions, we
could find better solutions to these low rank models at overwhelming
probabilities, although these models are non-convex. Second, we can obtain
significantly faster algorithms for these models by solving R-PCA first. The
computation cost can be further cut by applying low complexity randomized
algorithms, e.g., our novel $\ell_{2,1}$ filtering algorithm, to R-PCA.
Experiments verify the advantages of our algorithms over other state-of-the-art
ones that are based on the alternating direction method.

本文发现一些代表性的子空间恢复模型（如 Robust Principal Component Analysis（R-PCA），Robust Low Rank Representation（R-LRR）和 Robust Latent Low Rank Representation（R-LatLRR））之间存在深层次联系，一旦获得一个模型的解决方案，可以通过封闭形式公式获得其他模型的解决方案，从而使得 R-PCA 成为低秩子空间恢复模型的核心。同时，通过先解决 R-PCA 可以获得显著更快的算法，并且可以通过应用低复杂度的随机算法进一步降低计算成本。