Near Optimal Reconstruction of Spherical Harmonic Expansions

TL;DR

This paper introduces a nearly optimal algorithm for reconstructing spherical harmonic expansions of functions on high-dimensional spheres, using a minimal number of function evaluations and leveraging advanced sampling techniques.

Contribution

The paper presents a novel, efficient algorithm for recovering spherical harmonic expansions with near-optimal sample complexity in any dimension.

Findings

Algorithm achieves near-optimal sample complexity.

Works efficiently in high dimensions.

Empirical results demonstrate effectiveness.

Abstract

We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the d-dimensional unit sphere $\mathbb{S}^{d-1}$ using a near-optimal number of function evaluations. We show that for any $f \in L^2(\mathbb{S}^{d-1})$, the number of evaluations of $f$ needed to recover its degree-$q$ spherical harmonic expansion equals the dimension of the space of spherical harmonics of degree at most $q$ up to a logarithmic factor. Moreover, we develop a simple yet efficient algorithm to recover degree-$q$ expansion of $f$ by only evaluating the function on uniformly sampled points on $\mathbb{S}^{d-1}$. Our algorithm is based on the connections between spherical harmonics and Gegenbauer polynomials and leverage score sampling methods. Unlike the prior results on fast spherical harmonic transform, our proposed algorithm works efficiently using a nearly optimal number of samples in any dimension d. We further illustrate the empirical performance of our algorithm on numerical examples.