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Given a hypergraph $H$ with $m$ hyperedges and a set $Q$ of $m$ \emph{pinningsubspaces}, i.e.\ globally fixed subspaces in Euclidean space $\mathbb{R}^d$, a\emph{pinned subspace-incidence system} is the pair $(H, Q)$, with theconstraint that each pinning subspace in $Q$ is contained in the subspacespanned by the point realizations in $\mathbb{R}^d$ of vertices of thecorresponding hyperedge of $H$. This paper provides a combinatorialcharacterization of pinned subspace-incidence systems that are \emph{minimallyrigid}, i.e.\ those systems that are guaranteed to generically yield a locallyunique realization.  Pinned subspace-incidence systems have applications in the \emph{DictionaryLearning (aka sparse coding)} problem, i.e.\ the problem of obtaining a sparserepresentation of a given set of data vectors by learning \emph{dictionaryvectors} upon which the data vectors can be written as sparse linearcombinations. Viewing the dictionary vectors from a geometry perspective as thespanning set of a subspace arrangement, the result gives a tight bound on thenumber of dictionary vectors for sufficiently randomly chosen data vectors, andgives a way of constructing a dictionary that meets the bound. For lessstringent restrictions on data, but a natural modification of the dictionarylearning problem, a further dictionary learning algorithm is provided. Althoughthere are recent rigidity based approaches for low rank matrix completion, weare unaware of prior application of combinatorial rigidity techniques in thesetting of Dictionary Learning. We also provide a systematic classification ofproblems related to dictionary learning together with various algorithms, theirassumptions and performance.

Combinatorial rigidity of Incidence systems and Application to Dictionary learning