Group-Invariant Subspace Clustering

Abstract: In this paper we consider the problem of group invariant subspace clustering where the data is assumed to come from a union of group-invariant subspaces of a vector space, i.e. subspaces which are invariant with respect to action of a given group. Algebraically, such group-invariant subspaces are also referred to as submodules. Similar to the well known Sparse Subspace Clustering approach where the data is assumed to come from a union of subspaces, we analyze an algorithm which, following a recent work [1], we refer to as Sparse Sub-module Clustering (SSmC). The method is based on finding group-sparse self-representation of data points. In this paper we primarily derive general conditions under which such a group-invariant subspace identification is possible. In particular we extend the geometric analysis in [2] and in the process we identify a related problem in geometric functional analysis.