Robust recovery of subspace structures by low-rank representation

Guangcan Liu, Zhouchen Lin, Shuicheng Yan, Ju Sun, Yong Yu, Yi Ma

Research output: Contribution to journalArticlepeer-review

1756 Scopus citations

Abstract

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.

Original languageEnglish (US)
Article number6180173
Pages (from-to)171-184
Number of pages14
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Volume35
Issue number1
DOIs
StatePublished - Jan 1 2013
Externally publishedYes

Keywords

  • Low-rank representation
  • outlier detection
  • segmentation
  • subspace clustering

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