  We consider the general problem of matching a subspace to a signal in R^N
that has been observed indirectly (compressed) through a random projection. We
are interested in the case where the collection of K-dimensional subspaces is
continuously parameterized, i.e. naturally indexed by an interval from the real
line, or more generally a region of R^D. Our main results show that if the
dimension of the random projection is on the order of K times a geometrical
constant that describes the complexity of the collection, then the match
obtained from the compressed observation is nearly as good as one obtained from
a full observation of the signal. We give multiple concrete examples of
collections of subspaces for which this geometrical constant can be estimated,
and discuss the relevance of the results to the general problems of template
matching and source localization.
