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Spectral algorithms in higher-order Fourier analysis

Pablo Candela, Diego González-Sánchez, Balázs Szegedy

TL;DR

The paper develops a practical spectral framework for higher-order Fourier analysis on finite abelian groups by treating functions as $\mathop{Z}$-matrices and applying invariant operators to extract structured components. Central to the approach are the Fourier denoising operator $K_\varepsilon$ and the operator $\mathcal{K}_\varepsilon$, whose leading eigenvectors encode quadratic (order-2) structure via 2-step nilspace characters. A refined nilspace-analytic regularity lemma decomposes the quadratically structured part into nilspace characters, enabling inverse theorems and approximate diagonalization of Gowers norms, with quantitative bounds. The paper then proves a structure theorem for $\\mathcal{K}_\varepsilon(f\otimes \overline{f})$ in the quadratic case and derives algorithmic consequences: (i) recovering the quadratically structured part from the spectrum, and (ii) extracting dominant nilspace characters, including strategies to handle clustered eigenvalues through randomness. Collectively, these results bridge spectral theory, nilspace foundations, and higher-order Fourier analysis to produce direct, non-iterative algorithms for quadratic structure in bounded functions on finite abelian groups. The framework paves the way for practical quadratic denoising and nilspace-character extraction, with potential extensions to higher orders via the order-increment principle and ultralimit insights.

Abstract

Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the quadratic case. Our results reveal new spectral aspects of the theory underlying higher-order Fourier analysis. Along these lines, we prove new inverse and regularity theorems for the Gowers norms based on higher-order character decompositions. Using these results, we prove a spectral inverse theorem and a spectral regularity theorem in quadratic Fourier analysis.

Spectral algorithms in higher-order Fourier analysis

TL;DR

The paper develops a practical spectral framework for higher-order Fourier analysis on finite abelian groups by treating functions as -matrices and applying invariant operators to extract structured components. Central to the approach are the Fourier denoising operator and the operator , whose leading eigenvectors encode quadratic (order-2) structure via 2-step nilspace characters. A refined nilspace-analytic regularity lemma decomposes the quadratically structured part into nilspace characters, enabling inverse theorems and approximate diagonalization of Gowers norms, with quantitative bounds. The paper then proves a structure theorem for in the quadratic case and derives algorithmic consequences: (i) recovering the quadratically structured part from the spectrum, and (ii) extracting dominant nilspace characters, including strategies to handle clustered eigenvalues through randomness. Collectively, these results bridge spectral theory, nilspace foundations, and higher-order Fourier analysis to produce direct, non-iterative algorithms for quadratic structure in bounded functions on finite abelian groups. The framework paves the way for practical quadratic denoising and nilspace-character extraction, with potential extensions to higher orders via the order-increment principle and ultralimit insights.

Abstract

Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the quadratic case. Our results reveal new spectral aspects of the theory underlying higher-order Fourier analysis. Along these lines, we prove new inverse and regularity theorems for the Gowers norms based on higher-order character decompositions. Using these results, we prove a spectral inverse theorem and a spectral regularity theorem in quadratic Fourier analysis.
Paper Structure (27 sections, 82 theorems, 162 equations, 3 figures, 2 algorithms)

This paper contains 27 sections, 82 theorems, 162 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline of the spectral approach
  3. Main results with algorithmic consequences
  4. Nilspace theoretic foundations for the spectral method
  5. Connection with the ultralimit setting
  6. Basic notions
  7. A simple instance of the spectral approach: recovering classical Fourier analysis
  8. Structured functions of order $k$
  9. Characters of order $k$
  10. The Fourier denoising operator $K_\varepsilon$
  11. Elementary motivation of the spectral approach in quadratic Fourier analysis
  12. Spectrally isolated eigenvectors of $\mathcal{K}_\varepsilon(f\otimes\overline{f})$ are (weak) quadratic characters
  13. Further properties of the operator $K_\varepsilon$
  14. Nilspace characters
  15. Uniform approximation of Lipschitz functions by vertical Fourier sums
  16. ...and 12 more sections

Key Result

Theorem 1.1

For every $\rho_0\in[0,1)$ there exists $\varepsilon_0>0$ such that the following holds. Let $\mathop{\mathrm{Z}}\nolimits$ be a finite abelian group and let $f:\mathop{\mathrm{Z}}\nolimits\to \mathbb{C}$ be a 1-bounded function. Then there exists $\rho\in[\rho_0/2,\rho_0]$ and $\varepsilon\in[\vare

Figures (3)

  • Figure 1: Sketch of the spectral approach to higher-order Fourier analysis.
  • Figure 2: Removing added random noise from a quadratic structured function by a version of Algorithm 1 on the cyclic group $\mathbb{Z}_{500}$. A window of length $50$ is plotted for illustration. In this example, the function $f(i):=\sin(8i^2+3i+1)$, $i\in [500]$ (green graph) is perturbed by random noise, resulting in the function $g=f+r$ (red graph). The spectral algorithm is applied to $g$ and the reconstruction $f_2$ of $f$ (blue graph) is obtained by the projection of $g$ to the space spanned by the $6$ leading eigenvectors of the operator constructed from $g$. The plot highlights the reconstruction error $|f(i)-f_2(i)|$.
  • Figure 3: The nilspace approach to $k$-th order Fourier analysis

Theorems & Definitions (243)

  • Theorem 1.1: Spectral $U^3$-regularization
  • Remark 1.2
  • Remark 1.3: Choice of parameters
  • Remark 1.4: Higher-order versions
  • Remark 1.5: Continuous versions
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1: $\mathop{\mathrm{Z}}\nolimits$-matrices
  • Remark 2.2
  • ...and 233 more