Poster
Near Optimal Reconstruction of Spherical Harmonic Expansions
Amir Zandieh · Insu Han · Haim Avron
Great Hall & Hall B1+B2 (level 1) #1906
Abstract:
We propose an algorithm for robust recovery of the spherical harmonic expansion of functions defined on the -dimensional unit sphere using a near-optimal number of function evaluations. We show that for any , the number of evaluations of needed to recover its degree- spherical harmonic expansion equals the dimension of the space of spherical harmonics of degree at most , up to a logarithmic factor. Moreover, we develop a simple yet efficient kernel regression-based algorithm to recover degree- expansion of by only evaluating the function on uniformly sampled points on . 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 . Furthermore, we illustrate the empirical performance of our algorithm on numerical examples.
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