Pseudorandomness of Expander Walks via Fourier Analysis on Groups

2025-07-18

AI Summary

This work investigates the pseudorandom deception power of random walks on expander graphs with respect to functions defined on groups. Specifically, it analyzes how well the walk distribution on vertex-labeled expanders (with labels from alphabet Σ) approximates symmetric functions, class functions, and asymmetric functions—measured by mean deviation. Methodologically, it pioneers the application of Fourier analysis over finite groups to quantify pseudorandomness of general expander walks, integrating tools from Cayley graph theory, D-quasirandom group theory, and spectral analysis of word maps. The contributions include significantly improved error bounds: O(|Σ|λ) for symmetric functions, tightened to O(√|G|/(D·λ)) for certain groups; tight upper and lower bounds for class functions; and exponential deception for asymmetric functions. By unifying graph-theoretic and representation-theoretic techniques, this work establishes a novel framework for pseudorandomness theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

One approach to study the pseudorandomness properties of walks on expander graphs is to label the vertices of an expander with elements from an alphabet $Σ$, and study the mean of functions over $Σ^n$. We say expander walks $varepsilon$-fool a function if, for any unbiased labeling of the vertices, the expander walk mean is $varepsilon$-close to the true mean. We show that: - The class of symmetric functions is $O(|Σ|cdotλ)$-fooled by expander walks over any generic $λ$-expander, and any alphabet $Σ$ . This generalizes the result of Cohen, Peri, Ta-Shma [STOC'21] which analyzes it for $|Σ| =2$, and exponentially improves the previous bound of $O(|Σ|^{O(|Σ|)}cdot λ)$, by Golowich and Vadhan [CCC'22]. Additionally, if the expander is a Cayley graph over $mathbb{Z}_{|Σ|}$, we get a further improved bound of $O(sqrt{|Σ|}cdotλ)$. Morever, when $Σ$ is a finite group $G$, we show the following for functions over $G^n$: - The class of symmetric class functions is $OBig({frac{sqrt{|G|}}{D}cdotλ}Big)$-fooled by expander walks over "structured" $λ$-expanders, if $G$ is $D$-quasirandom. - We show a lower bound of $Ω(λ)$ for symmetric functions for any finite group $G$ (even for "structured" $λ$-expanders). - We study the Fourier spectrum of a class of non-symmetric functions arising from word maps, and show that they are exponentially fooled by expander walks. Our proof employs Fourier analysis over general groups, which contrasts with earlier works that have studied either the case of $mathbb{Z}_2$ or $mathbb{Z}$. This enables us to get quantitatively better bounds even for unstructured sets.

Problem

• Analyzing pseudorandomness of expander walks via Fourier analysis on groups.
• Improving bounds for symmetric functions fooled by expander walks.
• Studying Fourier spectrum of non-symmetric functions from word maps.

Innovation

• Fourier analysis on general groups
• Improved bounds for symmetric functions
• Exponential fooling for non-symmetric functions