Fast Low-Rank Subspace Segmentation

Abstract

Subspace segmentation is the problem of segmenting (or grouping) a set of $n$ data points into a number of clusters, with each cluster being a (linear) subspace. The recently established algorithms such as Sparse Subspace Clustering (SSC), Low-Rank Representation (LRR) and Low-Rank Subspace Segmentation (LRSS) are effective in terms of segmentation accuracy, but computationally inefficient as they possess a complexity of $O(n^{3})$ , which is too high to afford for the case where $n$ is very large. In this paper we devise a fast subspace segmentation algorithm with complexity of $O(n\log (n))$ . This is achieved by firstly using partial Singular Value Decomposition (SVD) to approximate the solution of LRSS, secondly utilizing Locality Sensitive Hashing (LSH) to build a sparse affinity graph that encodes the subspace memberships, and finally adopting a fast Normalized Cut (NCut) algorithm to produce the final segmentation results. Besides of high efficiency, our algorithm also has comparable effectiveness as the original LRSS method.