Department of Computer Science and Engineering, Shanghai Jiao Tong University, China.
IEEE Trans Pattern Anal Mach Intell. 2013 Jan;35(1):171-84. doi: 10.1109/TPAMI.2012.88.
In this paper, we address the subspace clustering problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to cluster the samples into their respective subspaces and remove possible outliers as well. To this end, we propose a novel objective function named Low-Rank Representation (LRR), which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that the convex program associated with LRR solves the subspace clustering problem in the following sense: When the data is clean, we prove that LRR exactly recovers the true subspace structures; when the data are contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for data corrupted by arbitrary sparse errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace clustering and error correction in an efficient and effective way.
在本文中，我们解决了子空间聚类问题。给定一组近似从多个子空间的并集中抽取的数据样本（向量），我们的目标是将样本聚类到它们各自的子空间中，并去除可能的异常值。为此，我们提出了一种名为低秩表示（LRR）的新目标函数，该函数在所有候选函数中寻找能够将数据样本表示为给定字典中基的线性组合的最低秩表示。结果表明，与 LRR 相关联的凸规划以以下方式解决子空间聚类问题：当数据是干净的时，我们证明 LRR 可以精确地恢复真实的子空间结构；当数据受到异常值的污染时，我们证明在某些条件下，LRR 可以精确地恢复原始数据的行空间，并检测到异常值；对于由任意稀疏误差损坏的数据，LRR 也可以在理论保证下进行近似恢复行空间。由于子空间成员身份可以通过行空间证明，这进一步表明 LRR 可以以高效有效的方式进行稳健的子空间聚类和错误校正。
