Combinatorial rigidity of Incidence systems and Application to Dictionary learning

Abstract: Given a hypergraph $H$ with $m$ hyperedges and a set $Q$ of $m$ \emph{pinning subspaces}, i.e.\ globally fixed subspaces in Euclidean space $\mathbb{R}^d$, a \emph{pinned subspace-incidence system} is the pair $(H, Q)$, with the constraint that each pinning subspace in $Q$ is contained in the subspace spanned by the point realizations in $\mathbb{R}^d$ of vertices of the corresponding hyperedge of $H$. This paper provides a combinatorial characterization of pinned subspace-incidence systems that are \emph{minimally rigid}, i.e.\ those systems that are guaranteed to generically yield a locally unique realization. Pinned subspace-incidence systems have applications in the \emph{Dictionary Learning (aka sparse coding)} problem, i.e.\ the problem of obtaining a sparse representation of a given set of data vectors by learning \emph{dictionary vectors} upon which the data vectors can be written as sparse linear combinations. Viewing the dictionary vectors from a geometry perspective as the spanning set of a subspace arrangement, the result gives a tight bound on the number of dictionary vectors for sufficiently randomly chosen data vectors, and gives a way of constructing a dictionary that meets the bound. For less stringent restrictions on data, but a natural modification of the dictionary learning problem, a further dictionary learning algorithm is provided. Although there are recent rigidity based approaches for low rank matrix completion, we are unaware of prior application of combinatorial rigidity techniques in the setting of Dictionary Learning. We also provide a systematic classification of problems related to dictionary learning together with various algorithms, their assumptions and performance.