# Multiscale High-Dimensional Sparse Fourier Algorithms for Noisy Data

## TL;DR

This paper introduces a multiscale sparse Fourier algorithm that is both efficient and robust against noise, improving upon previous methods for high-dimensional signals that are sparse or nearly sparse in the Fourier domain.

## Contribution

It presents a novel multiscale algorithm capable of handling noisy high-dimensional Fourier data efficiently, extending prior work to practical noisy scenarios.

## Key findings

- Algorithm is robust against noise in high-dimensional Fourier data.
- Achieves efficient runtime comparable to prior sparse Fourier methods.
- Effectively approximates nearly sparse signals in noisy environments.

## Abstract

We develop an efficient and robust high-dimensional sparse Fourier algorithm for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm" (2016), an efficient sparse Fourier algorithm with $\Theta(ds \log s)$ average-case runtime and $\Theta(ds)$ sampling complexity under certain assumptions was developed for signals that are $s$-sparse and bandlimited in the $d$-dimensional Fourier domain, i.e. there are at most $s$ energetic frequencies and they are in $ \left[-N/2, N/2\right)^d\cap \mathbb{Z}^d$. However, in practice the measurements of signals often contain noise, and in some cases may only be nearly sparse in the sense that they are well approximated by the best $s$ Fourier modes. In this paper, we propose a multiscale sparse Fourier algorithm for noisy samples that proves to be both robust against noise and efficient.