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Pseudorandomness of Expander Walks via Fourier Analysis on Groups

Authors Fernando Granha Jeronimo , Tushant Mittal , Sourya Roy

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  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Expander graphs
  • pseudorandomness

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Abstract

A long line of work has studied the pseudorandomness properties of walks on expander graphs. A central goal is to measure how closely the distribution over n-length walks on an expander approximates the uniform distribution of n-independent elements. One approach to do so is to label the vertices of an expander with elements from an alphabet Σ, and study closeness of the mean of functions over Σⁿ, under these two distributions. We say expander walks ε-fool a function if the expander walk mean is ε-close to the true mean. There has been a sequence of works studying this question for various functions, such as the XOR function, the AND function, etc. We show that:  
- The class of symmetric functions is O(|Σ|λ)-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|) λ), by Golowich and Vadhan [CCC'22]. Moreover, if the expander is a Cayley graph over ℤ_|Σ|, we get a further improved bound of O(√{|Σ|} λ).  Morever, when Σ is a finite group G, we show the following for functions over Gⁿ:  
- The class of symmetric class functions is O({√|G|}/D λ}-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 ℤ₂ or ℤ. This enables us to get quantitatively better bounds even for unstructured sets.

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Fernando Granha Jeronimo, Tushant Mittal, and Sourya Roy. Pseudorandomness of Expander Walks via Fourier Analysis on Groups. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.49

Author Details

Fernando Granha Jeronimo
  • University of Illinois Urbana-Champaign, IL, USA
Tushant Mittal
  • Stanford University, CA, USA
Sourya Roy
  • University of Iowa, Iowa City, IA, USA

Funding

  • Mittal, Tushant: Research supported by NSF grants CCF-2143246 and CCF-2133154.
  • Roy, Sourya: Research supported by the Old Gold Summer Fellowship from The University of Iowa.

References

  1. M. Ajtai, J. Komlos, and E. Szemeredi. Deterministic simulation in LOGSPACE. In Proceedings of the 19th ACM Symposium on Theory of Computing, 1987. URL: https://doi.org/10.1145/28395.28410.
  2. David A. Mix Barrington. Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC¹. J. Comput. Syst. Sci., 38(1), 1989. URL: https://doi.org/10.1016/0022-0000(89)90037-8.
  3. Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, and Amnon Ta-Shma. Expander Random Walks: The General Case and Limitations. In Proceedings of the 49th International Colloquium on Automata, Languages and Programming, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.43.
  4. Gil Cohen, Noam Peri, and Amnon Ta-Shma. Expander Random Walks: A Fourier-Analytic Approach. In Proceedings of the 53nd ACM Symposium on Theory of Computing, 2021. URL: https://doi.org/10.1145/3406325.3451049.
  5. Anindya De. Pseudorandomness for Permutation and Regular Branching Programs. In Proceedings of the 26th IEEE Conference on Computational Complexity, 2011. URL: https://doi.org/10.1109/CCC.2011.23.
  6. Ankit Garg, Yin Tat Lee, Zhao Song, and Nikhil Srivastava. A matrix expander Chernoff bound. In Proceedings of the 50th ACM Symposium on Theory of Computing, 2018. URL: https://doi.org/10.1145/3188745.3188890.
  7. E.N. Gilbert. A Comparison of Signalling Alphabets. Bell System Technical Journal, 31:504-522, 1952. URL: https://doi.org/10.1002/j.1538-7305.1952.tb01393.x.
  8. D. Gillman. A Chernoff Bound for Random Walks on Expander Graphs. In focs93, pages 680-691, 1993. URL: https://doi.org/10.1109/SFCS.1993.366819.
  9. Louis Golowich. A New Berry-Esseen Theorem for Expander Walks. In Proceedings of the 55nd ACM Symposium on Theory of Computing, 2023. URL: https://doi.org/10.1145/3564246.3585141.
  10. Louis Golowich and Salil Vadhan. Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs. In Proceedings of the 37th IEEE Conference on Computational Complexity, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.27.
  11. W. T. Gowers. Quasirandom groups. Comb. Probab. Comput., 2008. URL: https://doi.org/10.1017/S0963548307008826.
  12. Pooya Hatami and William Hoza. Paradigms for Unconditional Pseudorandom Generators. Found. Trends Theor. Comput. Sci., 16(1-2), 2024. URL: https://doi.org/10.1561/0400000109.
  13. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander Graphs and Their Applications. Bull. Amer. Math. Soc., 43(04):439-562, August 2006. Google Scholar
  14. Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In Proceedings of the 26th ACM Symposium on Theory of Computing, 1994. URL: https://doi.org/10.1145/195058.195190.
  15. Fernando Granha Jeronimo, Tushant Mittal, Sourya Roy, and Avi Wigderson. Almost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification. In Proceedings of the 63st IEEE Symposium on Foundations of Computer Science, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00043.
  16. Fernando Granha Jeronimo, Tushant Mittal, Shashank Srivastava, and Madhur Tulsiani. Explicit Codes approaching Generalized Singleton Bound using Expanders. In Proceedings of the 57th ACM Symposium on Theory of Computing, 2025. URL: https://doi.org/10.48550/arXiv.2502.07308.
  17. Alexander Lubotzky. Discrete Groups, Expanding Graphs and Invariant Measures, volume 125 of Progress in mathematics. Birkhäuser, 1994. Google Scholar
  18. Raghu Meka and David Zuckerman. Small-Bias Spaces for Group Products. In APPROX-RANDOM 2009 Proceedings, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_49.
  19. Silas Richelson and Sourya Roy. Gilbert and Varshamov Meet Johnson: List-Decoding Explicit Nearly-Optimal Binary Codes. In Proceedings of the 64st IEEE Symposium on Foundations of Computer Science, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00021.
  20. Silas Richelson and Sourya Roy. Analyzing Ta-Shma’s Code via the Expander Mixing Lemma. IEEE Trans. Inf. Theory, 70(2):1040-1049, 2024. URL: https://doi.org/10.1109/TIT.2023.3304614.
  21. Ron M. Roth. Higher-Order MDS Codes. IEEE Transactions on Information Theory, 68(12):7798-7816, 2022. URL: https://doi.org/10.1109/TIT.2022.3194521.
  22. Jean-Pierre Serre. Linear Representations of Finite Groups, volume 42 of Graduate Texts in Mathematics. Springer New York, 1977. Google Scholar
  23. Chong Shangguan and Itzhak Tamo. Combinatorial list-decoding of Reed-Solomon codes beyond the Johnson radius. In Proceedings of the 52nd ACM Symposium on Theory of Computing, 2020. URL: https://doi.org/10.1145/3357713.3384295.
  24. Amnon Ta-Shma. Explicit, almost optimal, epsilon-balanced codes. In stoc17, pages 238-251. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055408.
  25. Amnon Ta-Shma and Ron Zadicario. The Expander Hitting Property When the Sets Are Arbitrarily Unbalanced. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024), 2024. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.31.
  26. Salil P. Vadhan. Pseudorandomness. Now Publishers Inc., 2012. URL: https://doi.org/10.1561/0400000010.
  27. R.R. Varshamov. Estimate of the number of signals in error correcting codes. Doklady Akademii Nauk SSSR, 117:739-741, 1957. Google Scholar
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