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 Rd, 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 Rd of vertices of the corresponding hyperedge of H. Pinned subspaceincidence 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.