  Several imaging algorithms including patch-based image denoising, image time
series recovery, and convolutional neural networks can be thought of as methods
that exploit the manifold structure of signals. While the empirical performance
of these algorithms is impressive, the understanding of recovery of the signals
and functions that live on manifold is less understood. In this paper, we focus
on the recovery of signals that live on a union of surfaces. In particular, we
consider signals living on a union of smooth band-limited surfaces in high
dimensions. We show that an exponential mapping transforms the data to a union
of low-dimensional subspaces. Using this relation, we introduce a sampling
theoretical framework for the recovery of smooth surfaces from few samples and
the learning of functions living on smooth surfaces. The low-rank property of
the features is used to determine the number of measurements needed to recover
the surface. Moreover, the low-rank property of the features also provides an
efficient approach, which resembles a neural network, for the local
representation of multidimensional functions on the surface.
