Near Optimal Reconstruction of Spherical Harmonic Expansions

Amir Zandieh, Insu Han, Haim Avron

Research output: Contribution to journalConference articlepeer-review


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


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