Sample-Optimal Fourier Sampling in Any Constant Dimension -- Part I

TL;DR

This paper presents a new Fourier sampling algorithm for sparse recovery that is optimal in sample complexity, efficient in runtime, and extends to higher dimensions, with promising empirical performance.

Contribution

It introduces the first sample-optimal Fourier sampling algorithm for sparse recovery in any constant dimension, matching theoretical lower bounds and demonstrating practical effectiveness.

Findings

Achieves $O(k\,\log N)$ sample complexity matching lower bounds.

Runs in nearly linear time $\tilde O(N)$.

Empirical results show competitive sampling complexity and strong recovery guarantees.

Abstract

We give an algorithm for $\ell_2/\ell_2$ sparse recovery from Fourier measurements using $O(k\log N)$ samples, matching the lower bound of \cite{DIPW} for non-adaptive algorithms up to constant factors for any $k\leq N^{1-\delta}$. The algorithm runs in $\tilde O(N)$ time. Our algorithm extends to higher dimensions, leading to sample complexity of $O_d(k\log N)$, which is optimal up to constant factors for any $d=O(1)$. These are the first sample optimal algorithms for these problems. A preliminary experimental evaluation indicates that our algorithm has empirical sampling complexity comparable to that of other recovery methods known in the literature, while providing strong provable guarantees on the recovery quality.