Subspace Clustering via Thresholding and Spectral Clustering

TL;DR

This paper introduces a low-complexity subspace clustering algorithm that uses correlation thresholding and spectral clustering, capable of handling intersecting subspaces, erasures, and outliers with high probability.

Contribution

The paper presents a novel, simple, and efficient subspace clustering method with rigorous probabilistic guarantees for intersecting subspaces, erasures, and outlier detection.

Findings

Algorithm succeeds with high probability for intersecting subspaces.

Handles data with erasures up to a linear scale in ambient dimension.

Provably detects outliers effectively.

Abstract

We consider the problem of clustering a set of high-dimensional data points into sets of low-dimensional linear subspaces. The number of subspaces, their dimensions, and their orientations are unknown. We propose a simple and low-complexity clustering algorithm based on thresholding the correlations between the data points followed by spectral clustering. A probabilistic performance analysis shows that this algorithm succeeds even when the subspaces intersect, and when the dimensions of the subspaces scale (up to a log-factor) linearly in the ambient dimension. Moreover, we prove that the algorithm also succeeds for data points that are subject to erasures with the number of erasures scaling (up to a log-factor) linearly in the ambient dimension. Finally, we propose a simple scheme that provably detects outliers.