Approximation of Functions over Manifolds: A Moving Least-Squares Approach

TL;DR

This paper introduces a manifold-based function approximation algorithm using Moving Least-Squares that works with noisy data, requires no prior knowledge of the manifold, and achieves high accuracy with linear complexity.

Contribution

The paper presents a novel Moving Least-Squares method for function approximation on manifolds that avoids dimension reduction and handles noisy samples effectively.

Findings

Achieves approximation order of O(h^{m+1}) in noiseless cases.

Has linear time complexity relative to ambient space dimension.

Performs favorably compared to statistical regression methods on manifolds.

Abstract

We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is $\mathcal{O}(h^{m+1})$, where $h$ is a local density of sample parameter (i.e., the fill distance) and $m$ is the degree of a local polynomial approximation, used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient-space's dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.