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

Will Burstein, Alex Iosevich, Hari Sarang Nathan

TL;DR

This paper introduces the Fourier ratio $FR(f)=\frac{\|c(f)\|_1}{\|c(f)\|_2}$ as a unifying, basis-invariant measure of effective dimension that governs recovery, localization, and learning-theoretic complexity. It proves that small FR guarantees stable recovery from randomly missing samples via $\ell^1$ minimization in a broad class of bounded orthonormal systems, and it shows a sharp localization obstruction—the global FR control fails under slice-by-slice localization with a degradation scaling like the square root of the number of slices. The authors further connect FR to algorithmic rate-distortion (via $K_U(f,\varepsilon)$) and to the statistical-query (SQ) dimension of the associated function class, establishing explicit bounds that unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter. Specializations to 2D Gabor bases demonstrate the framework's applicability to time-frequency structures, while discussions of constant-modulus bases, wavelets, and finite polynomials illustrate the broad reach of the FR-based approach. Overall, FR serves as a fundamental invariant bridging recovery guarantees, localization limitations, and dimension-like descriptions in information-theoretic signal processing.

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.

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

TL;DR

This paper introduces the Fourier ratio as a unifying, basis-invariant measure of effective dimension that governs recovery, localization, and learning-theoretic complexity. It proves that small FR guarantees stable recovery from randomly missing samples via minimization in a broad class of bounded orthonormal systems, and it shows a sharp localization obstruction—the global FR control fails under slice-by-slice localization with a degradation scaling like the square root of the number of slices. The authors further connect FR to algorithmic rate-distortion (via ) and to the statistical-query (SQ) dimension of the associated function class, establishing explicit bounds that unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter. Specializations to 2D Gabor bases demonstrate the framework's applicability to time-frequency structures, while discussions of constant-modulus bases, wavelets, and finite polynomials illustrate the broad reach of the FR-based approach. Overall, FR serves as a fundamental invariant bridging recovery guarantees, localization limitations, and dimension-like descriptions in information-theoretic signal processing.

Abstract

We introduce a generalized Fourier ratio, the 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 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.
Paper Structure (23 sections, 13 theorems, 116 equations, 1 figure)

This paper contains 23 sections, 13 theorems, 116 equations, 1 figure.

Table of Contents

  1. Background and motivation
  2. Road map
  3. Gabor recovery theorem with random frequency erasures
  4. Why the Fourier ratio can be more natural than approximate sparsity
  5. The global Fourier-ratio theorem and why it is genuinely global
  6. A basis recovery theorem
  7. Coefficient vectors and the Fourier ratio
  8. A soft sparsification lemma in Fourier-ratio language
  9. A basis version of the signal recovery theorem in FRdiscrete
  10. A quick coherence check for constant-modulus bases
  11. Application to two-dimensional Gabor bases
  12. Other natural orthonormal systems
  13. Why localization creates difficulties
  14. Measures of Complexity
  15. Connections with the theory of Kolmogorov complexity
  16. ...and 8 more sections

Key Result

Theorem 1

Suppose $f:{\mathbb Z}_N\times{\mathbb Z}_T\to{\mathbb C}$, and $T=T(N)$ satisfies $T(N)=o(\sqrt{N}e^N)$. Define Assume one transmits $Gf(m,a)$ for all $(m,a)\in{\mathbb Z}_N\times{\mathbb Z}_T$, and each transmitted frequency is lost independently with fixed probability $\theta$, where $0<\theta<\frac{1}{2E_{\max}}$. Let $M$ be the random set of missing frequencies and define Then, as $N\to\inf

Figures (1)

  • Figure 1: Fourier ratio as a unifying complexity parameter controlling recovery, localization, and dimension.

Theorems & Definitions (34)

  • Theorem 1: Row-wise Gabor transmission under binomial frequency losses GaborFrames
  • Remark 1
  • Theorem 2: Stable recovery for approximately sparse vectors CRT05
  • Example 1: A simple coefficient model where Fourier ratio is informative
  • Theorem 3: Global Fourier-ratio recovery FRdiscrete
  • Remark 2
  • Definition 1: Fourier ratio in an orthonormal basis
  • Lemma 1: Small Fourier ratio implies approximate sparsity
  • proof
  • Remark 3
  • ...and 24 more