Near Optimal Reconstruction of Spherical Harmonic Expansions

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