Performance Analysis of Spectral Clustering on Compressed, Incomplete and Inaccurate Measurements

TL;DR

This paper analyzes how errors from compressed sensing and matrix completion impact spectral clustering, providing theoretical bounds and demonstrating robustness in multi-class clustering with real image data.

Contribution

It extends perturbation analysis of spectral clustering to multi-class scenarios incorporating compressed sensing and matrix completion techniques.

Findings

Small perturbations in affinity matrices affect spectral coordinates and clusterability.

Theoretical bounds depend on eigengaps between eigenvalues.

Numerical experiments confirm robustness in image data clustering.

Abstract

Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measurements of compressed sensing and matrix completion with the power of robust spectral clustering. Our analysis provides rigorous bounds on how small errors in the affinity matrix can affect the spectral coordinates and clusterability. This work generalizes the current perturbation results of two-class spectral clustering to incorporate multi-class clustering with k eigenvectors. We thoroughly track how small perturbation from using compressed sensing and matrix completion affect the affinity matrix and in succession the spectral coordinates. These perturbation results for multi-class clustering require an eigengap between the kth and (k+1)th eigenvalues of the affinity matrix, which naturally occurs in data with k well-defined clusters. Our theoretical guarantees are complemented with numerical results along with a number of examples of the unsupervised organization and clustering of image data.