Robust Recovery of Subspace Structures by Low-Rank Representation

TLDR

Data samples are approximately drawn from a union of multiple subspaces. The study addresses the subspace recovery problem. We propose Low‑Rank Representation (LRR), which finds the lowest‑rank representation that expresses data samples as linear combinations of dictionary bases. LRR exactly recovers true subspace structures for clean data, recovers the row space and detects outliers under certain conditions for data with outliers, and approximately recovers the row space with guarantees for arbitrary errors, thereby enabling efficient robust subspace segmentation and error correction.

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.