Recovery of Surfaces and Functions in High Dimensions: Sampling Theory and Links to Neural Networks

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 the 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. The direct representation of such a function in high dimensions often suffers from the curse of dimensionality; the large number of parameters would translate to the need for extensive training data. The low-rank property of the features can significantly reduce the number of parameters, which makes the computational structure attractive for learning and inference from limited labeled training data.