# A Universal Sampling Method for Reconstructing Signals with Simple Fourier Transforms

## TL;DR

This paper introduces a universal sampling method for reconstructing signals with simple Fourier structures, achieving near-optimal sample complexity and broad applicability across various signal classes.

## Contribution

It formalizes the relationship between Fourier spectral constraints and sample complexity, and presents a universal, efficient sampling strategy applicable to multiple signal types.

## Key findings

- Achieves near-optimal sample complexity based on statistical dimension.
- Provides a universal sampling strategy effective for all signal classes.
- Offers a simple, efficient algorithm matching state-of-the-art for bandlimited and sparse signals.

## Abstract

Reconstructing continuous signals from a small number of discrete samples is a fundamental problem across science and engineering. In practice, we are often interested in signals with 'simple' Fourier structure, such as bandlimited, multiband, and Fourier sparse signals. More broadly, any prior knowledge about a signal's Fourier power spectrum can constrain its complexity. Intuitively, signals with more highly constrained Fourier structure require fewer samples to reconstruct. We formalize this intuition by showing that, roughly, a continuous signal from a given class can be approximately reconstructed using a number of samples proportional to the *statistical dimension* of the allowed power spectrum of that class. Further, in nearly all settings, this natural measure tightly characterizes the sample complexity of signal reconstruction. Surprisingly, we also show that, up to logarithmic factors, a universal non-uniform sampling strategy can achieve this optimal complexity for *any class of signals*. We present a simple and efficient algorithm for recovering a signal from the samples taken. For bandlimited and sparse signals, our method matches the state-of-the-art. At the same time, it gives the first computationally and sample efficient solution to a broad range of problems, including multiband signal reconstruction and kriging and Gaussian process regression tasks in one dimension. Our work is based on a novel connection between randomized linear algebra and signal reconstruction with constrained Fourier structure. We extend tools based on statistical leverage score sampling and column-based matrix reconstruction to the approximation of continuous linear operators that arise in signal reconstruction. We believe that these extensions are of independent interest and serve as a foundation for tackling a broad range of continuous time problems using randomized methods.