Subspace clustering of dimensionality-reduced data

TL;DR

This paper investigates how random dimensionality reduction affects the performance of subspace clustering algorithms, showing that reducing dimensions to the order of subspace dimensions preserves clustering accuracy.

Contribution

It provides theoretical analysis quantifying the impact of random projections on subspace affinities and clustering performance, guiding dimensionality reduction in practice.

Findings

Dimensionality reduction to subspace dimension order maintains clustering performance.

Random projections minimally alter subspace affinities under certain conditions.

Theoretical bounds relate subspace affinity changes to clustering accuracy.

Abstract

Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, assumed unknown. In practice one may have access to dimensionality-reduced observations of the data only, resulting, e.g., from "undersampling" due to complexity and speed constraints on the acquisition device. More pertinently, even if one has access to the high-dimensional data set it is often desirable to first project the data points into a lower-dimensional space and to perform the clustering task there; this reduces storage requirements and computational cost. The purpose of this paper is to quantify the impact of dimensionality-reduction through random projection on the performance of the sparse subspace clustering (SSC) and the thresholding based subspace clustering (TSC) algorithms. We find that for both algorithms dimensionality reduction down to the order of the subspace dimensions is possible without incurring significant performance degradation. The mathematical engine behind our theorems is a result quantifying how the affinities between subspaces change under random dimensionality reducing projections.