Robust Recovery of Subspace Structures by Low-Rank Representation

TL;DR

This paper introduces Low-Rank Representation (LRR), a novel method for robustly recovering and segmenting subspace structures from data, even with errors or outliers, with proven theoretical guarantees.

Contribution

The paper proposes LRR, a new approach that effectively recovers subspace structures and segments data robustly in the presence of various types of errors, with rigorous theoretical analysis.

Findings

LRR exactly captures true subspace structures for clean data.

LRR can recover the row space and detect outliers under certain conditions.

LRR approximately recovers the row space with guarantees for corrupted data.

Abstract

In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed 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 LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data 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 the data corrupted by arbitrary 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 segmentation and error correction, in an efficient way.