Similarity matrix framework for data from union of subspaces

Abstract

This paper presents a framework for finding similarity matrices for the segmentation of data W = [ w 1 ⋯ w N ] ⊂ R D drawn from a union U = ⋃ i = 1 M S i of independent subspaces { S i } i = 1 M of dimensions { d i } i = 1 M . It is shown that any factorization of W = B P , where columns of B form a basis for data W and they also come from U , can be used to produce a similarity matrix Ξ W . In other words, Ξ W ( i , j ) ≠ 0 , when the columns w i and w j of W come from the same subspace, and Ξ W ( i , j ) = 0 , when the columns w i and w j of W come from different subspaces. Furthermore, Ξ W = Q d m a x , where d m a x = max ⁡ { d i } i = 1 M and Q ∈ R N × N with Q ( i , j ) = | P T P ( i , j ) | . It is shown that a similarity matrix obtained from the reduced row echelon form of W is a special case of the theory. It is also proven that the Shape Interaction Matrix defined as V V T , where W = U Σ V T is the skinny singular value decomposition of W , is not necessarily a similarity matrix. But, taking powers of its absolute value always generates a similarity matrix. An interesting finding of this research is that a similarity matrix can be obtained using a skeleton decomposition of W . First, a square sub-matrix A ∈ R r × r of W with the same rank r as W is found. Then, the matrix R corresponding to the rows of W that contain A is constructed. Finally, a power of the matrix P T P where P = A − 1 R provides a similarity matrix Ξ W . Since most of the data matrices are low-rank in many subspace segmentation problems, this is computationally efficient compared to other constructions of similarity matrices.