Group-Invariant Subspace Clustering
This work addresses subspace clustering for data with group symmetries, which is incremental as it builds on existing methods like Sparse Subspace Clustering.
The paper tackles the problem of group-invariant subspace clustering, where data originates from subspaces invariant under group actions, by analyzing the Sparse Sub-module Clustering algorithm. It derives general conditions for identifying such subspaces, extending prior geometric analysis and linking it to a problem in geometric functional analysis.
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.