The Fourier Ratio: A Unifying Measure of Complexity for Recovery, Localization, and Learning

TL;DR

This paper introduces the Fourier ratio as a basis-invariant measure of effective dimension that unifies limits in signal recovery, localization, and learning, with broad implications across analytic, algorithmic, and learning domains.

Contribution

It defines the Fourier ratio as a unifying complexity measure and demonstrates its fundamental role in recovery guarantees, localization limits, and complexity bounds.

Findings

Functions with small Fourier ratio can be stably recovered from missing samples.

Localization attempts increase the Fourier ratio, indicating a global-local complexity trade-off.

The Fourier ratio bounds Kolmogorov and SQ complexity measures.

Abstract

We introduce a generalized Fourier ratio, the \(\ell^1/\ell^2\) norm ratio of coefficients in an \emph{arbitrary} orthonormal system, as a single, basis-invariant measure of \emph{effective dimension} that governs fundamental limits across signal recovery, localization, and learning. First, we prove that functions with small Fourier ratio can be stably recovered from random missing samples via \(\ell^1\) minimization, extending and clarifying compressed sensing guarantees for general bounded orthonormal systems. Second, we establish a sharp \emph{localization obstruction}: any attempt to localize recovery to subslices of a product space necessarily inflates the Fourier ratio by a factor scaling with the square root of the slice count, demonstrating that global complexity cannot be distributed locally. Finally, we show that the same parameter controls key complexity-theoretic measures: it provides explicit upper bounds on Kolmogorov rate-distortion description length and on the statistical query (SQ) dimension of the associated function class. These results unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter, revealing the Fourier ratio as a fundamental invariant in information-theoretic signal processing.