On deterministic conditions for subspace clustering under missing data

TL;DR

This paper provides deterministic conditions for sparse subspace clustering with missing data, analyzing two sampling scenarios and highlighting the challenges in subspace identification and data completion.

Contribution

It offers new dual conditions for perfect clustering under different missing data sampling cases, extending existing results and providing extensive simulations.

Findings

Accurate clustering does not guarantee subspace identification.

Results differ significantly between uniform and non-uniform sampling.

Simulation confirms the theoretical analysis of clustering and completion challenges.

Abstract

In this paper we present deterministic analysis of sufficient conditions for sparse subspace clustering under missing data, when data is assumed to come from a Union of Subspaces (UoS) model. In this context we consider two cases, namely Case I when all the points are sampled at the same co-ordinates, and Case II when points are sampled at different locations. We show that results for Case I directly follow from several existing results in the literature, while results for Case II are not as straightforward and we provide a set of dual conditions under which, perfect clustering holds true. We provide extensive set of simulation results for clustering as well as completion of data under missing entries, under the UoS model. Our experimental results indicate that in contrast to the full data case, accurate clustering does not imply accurate subspace identification and completion, indicating the natural order of relative hardness of these problems.