Skeleton decomposition analysis for subspace clustering

Abstract

This paper provides a comprehensive analysis of skeleton decomposition used for segmentation of data W = [w 1 ···W N ] ⊂ Rd drawn from a union U = UM i=1 S i of linearly independent subspaces {S i }M i=1 of dimensions of {d i }M i=1 . Our previous work developed a generalized theoretical framework for computing similarity matrices by matrix factorization. Skeleton decomposition is a special case of this general theory. First, a square sub-matrix A ∊ Rr×r of W with the same rank r as W is found. Then, the corresponding row restriction R of W is constructed. This leads to P = A−1R and corresponding similarity matrix S W = (pTp)dmax, where d max is the maximum subspace dimension. Since most of the data matrices are low-rank in many subspace segmentation problems, this is computationally efficient compared to the other constructions of similarity matrices. It is also shown (with some limitations) that center-of-mass based sorting of data columns in S W can be used to quickly assess clustering performance while algorithm development in both noisy or noise-free cases.