Robust Recovery of Subspace Structures by Low-Rank Representation

摘要

In this work we address the subspace recovery problem. Given a set of datasamples (vectors) approximately drawn from a union of multiple subspaces, ourgoal is to segment the samples into their respective subspaces and correct thepossible errors as well. To this end, we propose a novel method termed Low-RankRepresentation (LRR), which seeks the lowest-rank representation among all thecandidates that can represent the data samples as linear combinations of thebases in a given dictionary. It is shown that LRR well solves the subspacerecovery problem: when the data is clean, we prove that LRR exactly capturesthe true subspace structures; for the data contaminated by outliers, we provethat under certain conditions LRR can exactly recover the row space of theoriginal data and detect the outlier as well; for the data corrupted byarbitrary errors, LRR can also approximately recover the row space withtheoretical guarantees. Since the subspace membership is provably determined bythe row space, these further imply that LRR can perform robust subspacesegmentation and error correction, in an efficient way.