The Fourier Ratio and complexity of signals

TL;DR

This paper introduces the Fourier ratio as a measure of signal complexity, showing that signals with small Fourier ratio can be efficiently approximated by low-degree polynomials and have low algorithmic complexity.

Contribution

It establishes a quantitative link between Fourier ratio, signal structure, and learnability, using advanced probabilistic and approximation theory.

Findings

Large Fourier ratio signals are concentrated on sparse sets.

Small Fourier ratio signals are well-approximated by low-degree polynomials.

Small Fourier ratio implies low algorithmic rate--distortion complexity.

Abstract

We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, \[ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(\mu)}}{\|\widehat f\|_{L^2(\mu)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, \] as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small $\mathrm{FR}(f)$ forces $f$ to be well-approximated in both $L^2$ and $L^\infty$ by low-degree trigonometric polynomials. Quantitatively, the class $\{f:\mathrm{FR}(f)\le r\}$ admits degree $O(r^2)$ $L^2$-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.