Necessary and sufficient conditions for sketched subspace clustering

Abstract

This paper is about an interesting phenomenon: two r-dimensional subspaces, even if they are orthogonal to one an other, can appear identical if they are only observed on a subset of coordinates. Understanding this phenomenon is of particular importance for many modern applications of subspace clustering where one would like to subsample in order to improve computational efficiency. Examples include real-time video surveillance and datasets so large that cannot even be stored in memory. In this paper we introduce a new metric between subspaces, which we call partial coordinate discrepancy. This metric captures a notion of similarity between subsampled subspaces that is not captured by other distance measures between subspaces. With this, we are able to show that subspace clustering is theoretically possible in lieu of coherence assumptions using only r + 1 rows of the dataset at hand. This gives precise information-theoretic necessary and sufficient conditions for sketched subspace clustering. This can greatly improve computational efficiency without compromising performance. We complement our theoretical analysis with synthetic and real data experiments.