Combinatorial Rigidity and Independence ofźGeneralized Pinned Subspace-Incidence Constraint Systems

Given a hypergraph H with m hyperedges and a set X of mpins, i.e. globally fixed subspaces in Euclidean space $$\mathbb {R}^{d}$$Rd, a pinned subspace-incidence system is the pair H,ï źX, with the constraint that each pin in X lies on the subspace spanned by the point realizations in $$\mathbb {R}^d$$Rd 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.