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Several imaging algorithms including patch-based image denoising, image timeseries recovery, and convolutional neural networks can be thought of as methodsthat exploit the manifold structure of signals. While the empirical performanceof these algorithms is impressive, the understanding of recovery of the signalsand functions that live on manifold is less understood. In this paper, we focuson the recovery of signals that live on a union of surfaces. In particular, weconsider signals living on a union of smooth band-limited surfaces in highdimensions. We show that an exponential mapping transforms the data to a unionof low-dimensional subspaces. Using this relation, we introduce a samplingtheoretical framework for the recovery of smooth surfaces from few samples andthe learning of functions living on smooth surfaces. The low-rank property ofthe features is used to determine the number of measurements needed to recoverthe surface. Moreover, the low-rank property of the features also provides anefficient approach, which resembles a neural network, for the localrepresentation of multidimensional functions on the surface.

Recovery of surfaces and functions in high dimensions: sampling theory and links to neural networks