Combinatorial Rigidity and Independence of Generalized Pinned Subspace-Incidence Constraint Systems

Abstract

Given a hypergraph H with m hyperedges and a set X of m pins, i.e. globally fixed subspaces in Euclidean space $\mathbb {R}^{d}$, a pinned subspace-incidence system is the pair (H, X), with the constraint that each pin in X lies on the subspace spanned by the point realizations in $\mathbb {R}^d$ of vertices of the corresponding hyperedge of H. Pinned subspace-incidence systems arise in modeling dictionary learning problems as well as biomaterials such as cell wall microfibrils. We are interested in combinatorial characterization of pinned subspace-incidence systems that are minimally rigid, i.e. those systems that are guaranteed to generically yield a locally unique realization. As is customary, this is accompanied by a characterization of generic independence as well as rigidity. Previously, such a combinatorial rigidity characterization is only known for a more restricted version of pinned subpsace-incidence systems, with H being a uniform hypergraph and pins in X being 1-dimension subspaces. In this paper, we extend the combinatorial characterization to general pinned subspace-incidence systems, with H being a non-uniform hypergraph and pins in X being subspaces with arbitrary dimension. As there are generally many data points per subspace in a dictionary learning problem, which can only be modeled with pins of dimension larger than 1, such an extension enables application to a much larger class of dictionary learning problems.

This research was supported in part by the grant NSF CCF-1117695.