Near Optimal Reconstruction of Spherical Harmonic Expansions

Published: 21 Sept 2023, Last Modified: 02 Nov 2023NeurIPS 2023 posterEveryoneRevisionsBibTeX
Keywords: spherical harmonic expansion, Gegenbauer polynomials, interpolation, leverage score sampling
TL;DR: We present an algorithm for efficiently recovering spherical harmonic expansions of functions on the $d$-dimensional unit sphere, requiring a near-optimal number of function evaluations.
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 kernel regression-based 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 built upon the connections between spherical harmonics and Gegenbauer polynomials. Unlike the prior results on fast spherical harmonic transform, our proposed algorithm works efficiently using a nearly optimal number of samples in any dimension $d$. Furthermore, we illustrate the empirical performance of our algorithm on numerical examples.
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Submission Number: 14713