A sample efficient sparse FFT for arbitrary frequency candidate sets in high dimensions

TL;DR

This paper introduces a highly efficient, dimension-incremental sparse Fourier transform algorithm capable of reconstructing high-dimensional functions with arbitrary candidate sets, significantly reducing sample complexity and computational effort.

Contribution

The proposed method is the first to handle arbitrary candidate sets in high dimensions without structural assumptions, improving sample complexity and efficiency over existing approaches.

Findings

Achieves accurate reconstruction of s-sparse Fourier functions with high probability.

Demonstrates high efficiency in numerical tests for both sparse and compressible cases.

Scales mildly with dimension, enabling practical high-dimensional Fourier analysis.

Abstract

In this paper a sublinear time algorithm is presented for the reconstruction of functions that can be represented by just few out of a potentially large candidate set of Fourier basis functions in high spatial dimensions, a so-called high-dimensional sparse fast Fourier transform. In contrast to many other such algorithms, our method works for arbitrary candidate sets and does not make additional structural assumptions on the candidate set. Our transform significantly improves upon the other approaches available for such a general framework in terms of the scaling of the sample complexity. Our algorithm is based on sampling the function along multiple rank-1 lattices with random generators. Combined with a dimension-incremental approach, our method yields a sparse Fourier transform whose computational complexity only grows mildly in the dimension and can hence be efficiently computed even in high dimensions. Our theoretical analysis establishes that any Fourier $s$-sparse function can be accurately reconstructed with high probability. This guarantee is complemented by several numerical tests demonstrating the high efficiency and versatile applicability for the exactly sparse case and also for the compressible case.