A General Framework for Understanding Compressed Subspace Clustering Algorithms

Abstract

Subspace clustering (SC) refers to the problem of clustering unlabeled high-dimensional data into a union of low-dimensional linear subspaces. In many practical scenarios, one may have access to only the compressed data due to constraints of measurement or computation. In this paper, based on the recently proposed restricted isometric property of Gaussian random projection for low-dimensional subspaces, we propose a general framework for analyzing the performance of various subspace clustering algorithms when applied to the compressed data. Our framework captures the connection between the problems of compressed subspace clustering (CSC) and noisy subspace clustering. With the existing study on noisy SC, our framework makes it possible to easily extend the results in noisy SC to CSC. In this paper, we select the most commonly used subspace clustering algorithms, i.e. , sparse SC (SSC), SSC-orthogonal matching pursuit (SSC-OMP), and thresholding based SC (TSC), as representatives and analyze their performance using the proposed framework. Our results are consistent with related work obtained by previous researchers, where our methodology is much more direct and has more universal implications. Finally, the practicability and efficiency of CSC are verified by numerical experiments.