Near Optimal Reconstruction of Spherical Harmonic Expansions

Amir Zandieh, Insu Han, Haim Avron

Research output: Contribution to journalConference articlepeer-review

Abstract

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.

Original languageEnglish
JournalAdvances in Neural Information Processing Systems
Volume36
StatePublished - 2023
Event37th Conference on Neural Information Processing Systems, NeurIPS 2023 - New Orleans, United States
Duration: 10 Dec 202316 Dec 2023

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Information Systems
  • Signal Processing

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