On Deterministic Conditions for Subspace Clustering under Missing Data

TL;DR

This paper establishes deterministic conditions under which sparse subspace clustering can succeed despite missing data, providing theoretical guarantees and empirical insights into clustering and data completion challenges.

Contribution

It introduces deterministic conditions for SSC success with missing data and compares two variants of SSC algorithms under the UoS model.

Findings

Deterministic conditions guarantee perfect clustering despite missing data

Accurate clustering does not necessarily lead to accurate subspace identification or data completion

Simulation results validate theoretical conditions and highlight differences between clustering and completion tasks

Abstract

In this paper we present deterministic conditions for success of sparse subspace clustering (SSC) under missing data, when data is assumed to come from a Union of Subspaces (UoS) model. We consider two algorithms, which are variants of SSC with entry-wise zero-filling that differ in terms of the optimization problems used to find affinity matrix for spectral clustering. For both the algorithms, we provide deterministic conditions for any pattern of missing data such that perfect clustering can be achieved. We provide extensive sets of simulation results for clustering as well as completion of data at 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.