Group-Invariant Subspace Clustering

TL;DR

This paper extends subspace clustering to group-invariant subspaces, proposing a method called Sparse Sub-module Clustering (SSmC) that finds group-sparse representations, with theoretical conditions for successful identification.

Contribution

It introduces a novel clustering approach for group-invariant subspaces and derives general conditions for accurate subspace identification.

Findings

Extended geometric analysis of group-invariant subspace clustering

Identified conditions for successful subspace recovery

Connected the problem to geometric functional analysis

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.