TY - JOUR

T1 - Near Optimal Reconstruction of Spherical Harmonic Expansions

AU - Zandieh, Amir

AU - Han, Insu

AU - Avron, Haim

N1 - Publisher Copyright:
© 2023 Neural information processing systems foundation. All rights reserved.

PY - 2023

Y1 - 2023

N2 - We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the d-dimensional unit sphere Sd−1 using a near-optimal number of function evaluations. We show that for any f ∈ L2(Sd−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 Sd−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.

AB - We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the d-dimensional unit sphere Sd−1 using a near-optimal number of function evaluations. We show that for any f ∈ L2(Sd−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 Sd−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.

UR - http://www.scopus.com/inward/record.url?scp=85191158592&partnerID=8YFLogxK

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.conferencearticle???

AN - SCOPUS:85191158592

SN - 1049-5258

VL - 36

JO - Advances in Neural Information Processing Systems

JF - Advances in Neural Information Processing Systems

T2 - 37th Conference on Neural Information Processing Systems, NeurIPS 2023

Y2 - 10 December 2023 through 16 December 2023

ER -