Sampling of Surfaces and Learning Functions in High Dimensions

The efficient representation of data in high-dimensional spaces is a key problem in several machine learning tasks. To capture the non-linear structure of the data, we model the data as points living on a smooth surface. We model the surface as the zero level-set of a bandlimited function. We show that this representation allows a non-linear lifting of the surface model, which will map the points to a low-dimensional subspace. This mapping between surfaces and the well-understood subspace model allows us to introduce novel algorithms (a) to recover the surface from few of its samples and (b) to learn a multidimensional bandlimited function from training data. The utility of these algorithms is introduced in practical applications including image denoising.