  Given a hypergraph $H$ with $m$ hyperedges and a set $X$ of $m$ \emph{pins},
i.e.\ globally fixed subspaces in Euclidean space $\mathbb{R}^d$, a
\emph{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
$H$. We are interested in 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. As is
customary, this is accompanied by a characterization of generic independence as
well as rigidity.
  In a previous paper \cite{sitharam2014incidence}, we used pinned
subspace-incidence systems towards solving the \emph{fitted dictionary
learning} problem, i.e.\ dictionary learning with specified underlying
hypergraph, and gave a combinatorial characterization of minimal rigidity for a
more restricted version of pinned subspace-incidence system, with $H$ being a
uniform hypergraph and pins in $X$ being 1-dimension subspaces. Moreover in a
recent paper \cite{Baker2015}, the special case of pinned line incidence
systems was used to model biomaterials such as cellulose and collagen fibrils
in cell walls. 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 fitted dictionary
learning problems.
