Near-optimal sampling strategies for multivariate function approximation on general domains

TL;DR

This paper introduces a novel sampling method for multivariate function approximation on general domains, achieving near-optimal sample complexity and improved accuracy and conditioning over standard methods.

Contribution

The paper proposes a new sampling strategy that ensures accurate, well-conditioned least-squares approximations with near-optimal sample complexity for functions on general domains.

Findings

Near-optimal sample complexity of O(N log N) for accurate approximation

The new sampling method improves conditioning over standard approaches

Numerical experiments confirm the effectiveness of the proposed method

Abstract

In this paper, we address the problem of approximating a multivariate function defined on a general domain in $d$ dimensions from sample points. We consider weighted least-squares approximation in an arbitrary finite-dimensional space $P$ from independent random samples taken according to a suitable measure. In general, least-squares approximations can be inaccurate and ill-conditioned when the number of sample points $M$ is close to $N = \dim(P)$. To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and well-conditioned approximations. The resulting sampling measure is discrete, and therefore straightforward to sample from. Our main result shows near-optimal sample complexity for this procedure; specifically, $M = \mathcal{O}(N \log(N))$ samples suffice for a well-conditioned and accurate approximation. Numerical experiments on polynomial approximation in general domains confirm the benefits of this method over standard sampling.