CUR Decompositions, Similarity Matrices, and Subspace Clustering

TL;DR

This paper introduces a CUR decomposition-based framework for subspace clustering that constructs similarity matrices capable of accurate clustering in both noise-free and noisy conditions, outperforming existing methods.

Contribution

It presents a novel CUR decomposition approach for subspace clustering, deriving multiple similarity matrices and demonstrating superior performance on motion segmentation data.

Findings

Exact clustering in noise-free scenarios

Flexible similarity matrices for noisy data

State-of-the-art results on Hopkins155 dataset

Abstract

A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces $\mathscr{U}=\underset{i=1}{\overset{M}\bigcup}S_i$. The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method. Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.