# Sampling of surfaces and functions in high dimensional spaces

## TL;DR

This paper presents a novel sampling framework for recovering smooth surfaces and functions in high-dimensional spaces using a nonlinear approach that leverages low-rank features in a lifted space, enabling efficient learning from limited data.

## Contribution

It introduces a nonlinear sampling method based on exponential lifting and low-rank features, generalizing union of subspace models for surface and function recovery.

## Key findings

- Effective surface recovery from few samples.
- Low-rank feature properties enable efficient computation.
- Resembles neural networks with fewer parameters.

## Abstract

We introduce a sampling theoretic framework for the recovery of smooth surfaces and functions living on smooth surfaces from few samples. The proposed approach can be thought of as a nonlinear generalization of union of subspace models widely used in signal processing. This scheme relies on an exponential lifting of the original data points to feature space, where the features live on union of subspaces. The low-rank property of the features are used to recover the surfaces as well as to determine the number of measurements needed to recover the surface. 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 significantly reduced number of parameters make the computational structure attractive for learning inference from limited labeled training data.