Theoretical Analysis of Sparse Subspace Clustering with Missing Entries

TL;DR

This paper provides theoretical guarantees for Sparse Subspace Clustering (SSC) when applied to data with missing entries, showing that projecting zero-filled data improves clustering performance.

Contribution

It offers the first theoretical analysis of SSC with incomplete data and demonstrates the benefits of projecting onto observation patterns for better clustering accuracy.

Findings

Projection improves SSC performance with missing data

Projected data behave as complete points in projected subspaces

The analysis extends to self-expressive methods in general

Abstract

Sparse Subspace Clustering (SSC) is a popular unsupervised machine learning method for clustering data lying close to an unknown union of low-dimensional linear subspaces; a problem with numerous applications in pattern recognition and computer vision. Even though the behavior of SSC for complete data is by now well-understood, little is known about its theoretical properties when applied to data with missing entries. In this paper we give theoretical guarantees for SSC with incomplete data, and analytically establish that projecting the zero-filled data onto the observation pattern of the point being expressed leads to a substantial improvement in performance. The main insight that stems from our analysis is that even though the projection induces additional missing entries, this is counterbalanced by the fact that the projected and zero-filled data are in effect incomplete points associated with the union of the corresponding projected subspaces, with respect to which the point being expressed is complete. The significance of this phenomenon potentially extends to the entire class of self-expressive methods.