On Geometric Analysis of Affine Sparse Subspace Clustering

TL;DR

This paper introduces a geometric analysis of affine sparse subspace clustering (ASSC), demonstrating conditions under which ASSC guarantees subspace-preserving affinity and exploring its density properties, validated through synthetic and real data.

Contribution

It develops a new geometric framework for ASSC, including affine independence, and shows weaker conditions for subspace-preserving recovery and the potential for dense affinities.

Findings

ASSC guarantees subspace-preserving affinity under affine independence.

Weaker conditions suffice for subspace-preserving recovery for most data points.

ASSC can produce subspace-dense affinity, aiding correct clustering.

Abstract

Sparse subspace clustering (SSC) is a state-of-the-art method for segmenting a set of data points drawn from a union of subspaces into their respective subspaces. It is now well understood that SSC produces subspace-preserving data affinity under broad geometric conditions but suffers from a connectivity issue. In this paper, we develop a novel geometric analysis for a variant of SSC, named affine SSC (ASSC), for the problem of clustering data from a union of affine subspaces. Our contributions include a new concept called affine independence for capturing the arrangement of a collection of affine subspaces. Under the affine independence assumption, we show that ASSC is guaranteed to produce subspace-preserving affinity. Moreover, inspired by the phenomenon that the $\ell_1$ regularization no longer induces sparsity when the solution is nonnegative, we further show that subspace-preserving recovery can be achieved under much weaker conditions for all data points other than the extreme points of samples from each subspace. In addition, we confirm a curious observation that the affinity produced by ASSC may be subspace-dense---which could guarantee the subspace-preserving affinity of ASSC to produce correct clustering under rather weak conditions. We validate the theoretical findings on carefully designed synthetic data and evaluate the performance of ASSC on several real data sets.