High-dimensional sparse FFT based on sampling along multiple rank-1 lattices

TL;DR

This paper introduces an improved high-dimensional sparse FFT algorithm that combines multiple rank-1 lattices with a dimension-incremental approach, significantly reducing sample and computational complexities while maintaining reliable reconstruction.

Contribution

It couples the dimension-incremental reconstruction method with multiple rank-1 lattices, achieving lower complexities and improved performance in high-dimensional sparse FFTs.

Findings

Reduces sample complexity by almost linear factor compared to single rank-1 lattices.

Achieves lower arithmetic complexity, improving efficiency in high-dimensional settings.

Numerical tests confirm the theoretical advantages and high reliability of the method.

Abstract

The reconstruction of high-dimensional sparse signals is a challenging task in a wide range of applications. In order to deal with high-dimensional problems, efficient sparse fast Fourier transform algorithms are essential tools. The second and third authors have recently proposed a dimension-incremental approach, which only scales almost linear in the number of required sampling values and almost quadratic in the arithmetic complexity with respect to the spatial dimension $d$. Using reconstructing rank-1 lattices as sampling scheme, the method showed reliable reconstruction results in numerical tests but suffers from relatively large numbers of samples and arithmetic operations. Combining the preferable properties of reconstructing rank-1 lattices with small sample and arithmetic complexities, the first author developed the concept of multiple rank-1 lattices. In this paper, both concepts - dimension-incremental reconstruction and multiple rank-1 lattices - are coupled, which yields a distinctly improved high-dimensional sparse fast Fourier transform. Moreover, the resulting algorithm is analyzed in detail with respect to success probability, number of required samples, and arithmetic complexity. In comparison to single rank-1 lattices, the utilization of multiple rank-1 lattices results in a reduction in the complexities by an almost linear factor with respect to the sparsity. Various numerical tests confirm the theoretical results, the high performance, and the reliability of the proposed method.