Exact Subspace Segmentation and Outlier Detection by Low-Rank Representation

TL;DR

This paper introduces a method using Low-Rank Representation (LRR) to accurately segment data into subspaces and detect outliers without prior knowledge of outlier count or subspace properties, under mild conditions.

Contribution

The paper proves that LRR can exactly recover subspace structures and identify outliers in a unified, efficient framework, even with unknown number and rank of subspaces.

Findings

LRR exactly recovers the row space of data.

LRR detects outliers without prior knowledge.

The method is efficient and theoretically guaranteed.

Abstract

In this work, we address the following matrix recovery problem: suppose we are given a set of data points containing two parts, one part consists of samples drawn from a union of multiple subspaces and the other part consists of outliers. We do not know which data points are outliers, or how many outliers there are. The rank and number of the subspaces are unknown either. Can we detect the outliers and segment the samples into their right subspaces, efficiently and exactly? We utilize a so-called {\em Low-Rank Representation} (LRR) method to solve this problem, and prove that under mild technical conditions, any solution to LRR exactly recovers the row space of the samples and detect the outliers as well. Since the subspace membership is provably determined by the row space, this further implies that LRR can perform exact subspace segmentation and outlier detection, in an efficient way.