ABSTRACT
In this work we ask the following basic question: assume the vertices of an expander graph are labelled by 0,1. What “test” functions f : { 0,1}t → {0,1} cannot distinguish t independent samples from those obtained by a random walk? The expander hitting property due to Ajtai, Komlos and Szemeredi (STOC 1987) is captured by the AND test function, whereas the fundamental expander Chernoff bound due to Gillman (SICOMP 1998), Heally (Computational Complexity 2008) is about test functions indicating whether the weight is close to the mean. In fact, it is known that all threshold functions are fooled by a random walk (Kipnis and Varadhan, Communications in Mathematical Physics 1986). Recently, it was shown that even the highly sensitive PARITY function is fooled by a random walk Ta-Shma (STOC 2017).
We focus on balanced labels. Our first main result is proving that all symmetric functions are fooled by a random walk. Put differently, we prove a central limit theorem (CLT) for expander random walks with respect to the total variation distance, significantly strengthening the classic CLT for Markov Chains that is established with respect to the Kolmogorov distance (Kipnis and Varadhan, Communications in Mathematical Physics 1986). Our approach significantly deviates from prior works. We first study how well a Fourier character χS is fooled by a random walk as a function of S. Then, given a test function f, we expand f in the Fourier basis and combine the above with known results on the Fourier spectrum of f.
We also proceed further and consider general test functions - not necessarily symmetric. As our approach is Fourier analytic, it is general enough to analyze such versatile test functions. For our second result, we prove that random walks on sufficiently good expander graphs fool tests functions computed by AC0 circuits, read-once branching programs, and functions with bounded query complexity.
- M. Ajtai. 1994. Recursive construction for $3$-regular expanders. Combinatorica 14, 4 (1994), 379–416. issn:0209-9683 Google ScholarCross Ref
- Miklós Ajtai, János Komlós, and Endre Szemerédi. 1987. Deterministic simulation in LOGSPACE. In Proceedings of the nineteenth annual ACM symposium on Theory of computing. 132–140.Google ScholarDigital Library
- Noga Alon. 1986. Eigenvalues and expanders. Combinatorica 6, 2 (1986), 83–96.Google ScholarDigital Library
- Noga Alon and Fan RK Chung. 1988. Explicit construction of linear sized tolerant networks. Discrete Mathematics 72, 1-3 (1988), 15–19.Google ScholarDigital Library
- Noga Alon, Jeff Edmonds, and Michael Luby. 1995. Linear time erasure codes with nearly optimal recovery. In Proceedings of IEEE 36th Annual Foundations of Computer Science. IEEE, 512–519.Google ScholarCross Ref
- N. Alon, Z. Galil, and V. D. Milman. 1987. Better expanders and superconcentrators. J. Algorithms 8, 3 (1987), 337–347. issn:0196-6774 Google ScholarDigital Library
- Avraham Ben-Aroya and Amnon Ta-Shma. 2011. A combinatorial construction of almost-Ramanujan graphs using the zig-zag product. SIAM J. Comput. 40, 2 (2011), 267–290. issn:0097-5397 Google ScholarDigital Library
- Yonatan Bilu and Nathan Linial. 2006. Lifts, discrepancy and nearly optimal spectral gap. Combinatorica 26, 5 (2006), 495–519. issn:0209-9683 Google ScholarDigital Library
- Mark Braverman. 2010. Polylogarithmic independence fools $\rm AC^0$ circuits. J. ACM 57, 5 (2010), Art. 28, 10. issn:0004-5411 Google ScholarDigital Library
- Mark Braverman, Gil Cohen, and Sumegha Garg. 2020. Pseudorandom Pseudo-distributions with Near-Optimal Error for Read-Once Branching Programs. SIAM J. Comput. 49, 5 (2020), STOC18–242–STOC18–299.Google ScholarDigital Library
- Harry Buhrman and Ronald De Wolf. 2002. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science 288, 1 (2002), 21–43.Google ScholarDigital Library
- Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, and Avishay Tal. 2018. Improved pseudorandomness for unordered branching programs through local monotonicity. In STOC'18–-Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. ACM, New York, 363–375. Google ScholarDigital Library
- Herman Chernoff. 1952. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics 23, 4 (1952), 493–507.Google ScholarCross Ref
- Michael B. Cohen. 2016. Ramanujan graphs in polynomial time. In 57th Annual IEEE Symposium on Foundations of Computer Science–-FOCS 2016. IEEE Computer Soc., Los Alamitos, CA, 276–281. Google ScholarCross Ref
- Irit Dinur. 2007. The PCP theorem by gap amplification. J. ACM 54, 3 (2007), Art. 12, 44. issn:0004-5411 Google ScholarDigital Library
- Ofer Gabber and Zvi Galil. 1981. Explicit constructions of linear-sized superconcentrators. J. Comput. System Sci. 22, 3 (1981), 407–420. issn:0022-0000 Special issued dedicated to Michael Machtey. Google ScholarCross Ref
- David Gillman. 1998. A Chernoff bound for random walks on expander graphs. SIAM J. Comput. 27, 4 (1998), 1203–1220.Google ScholarDigital Library
- Venkatesan Guruswami and Vinayak Kumar. 2020. Pseudobinomiality of the Sticky Random Walk. Technical Report. ECCC TR20-151.Google Scholar
- Alexander D Healy. 2008. Randomness-efficient sampling within nc. Computational Complexity 17, 1 (2008), 3–37.Google ScholarDigital Library
- Shlomo Hoory, Nathan Linial, and Avi Wigderson. 2006. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43, 4 (2006), 439–561. issn:0273-0979 Google ScholarCross Ref
- Hao Huang. 2019. Induced subgraphs of hypercubes and a proof of the sensitivity conjecture. Ann. of Math. (2) 190, 3 (2019), 949–955. issn:0003-486X Google ScholarCross Ref
- Russell Impagliazzo, Noam Nisan, and Avi Wigderson. 1994. Pseudorandomness for network algorithms. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing. 356–364.Google ScholarDigital Library
- Nabil Kahale. 1995. Eigenvalues and expansion of regular graphs. J. Assoc. Comput. Mach. 42, 5 (1995), 1091–1106. issn:0004-5411 Google ScholarDigital Library
- Claude Kipnis and SR Srinivasa Varadhan. 1986. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Communications in Mathematical Physics 104, 1 (1986), 1–19.Google ScholarCross Ref
- Beno\^\it Kloeckner. 2017. Effective limit theorems for Markov chains with a spectral gap. arXiv preprint arXiv:1703.09623 (2017).Google Scholar
- Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf. 2017. High-rate locally correctable and locally testable codes with sub-polynomial query complexity. J. ACM 64, 2 (2017), Art. 11, 42. issn:0004-5411 Google ScholarDigital Library
- Pascal Lezaud. 2001. Chernoff and Berry-Esséen inequalities for Markov processes. ESAIM: Probability and Statistics 5 (2001), 183–201.Google ScholarCross Ref
- Nathan Linial, Yishay Mansour, and Noam Nisan. 1993. Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach. 40, 3 (1993), 607–620. issn:0004-5411 Google ScholarDigital Library
- Alexander Lubotzky. 2012. Expander graphs in pure and applied mathematics. Bull. Amer. Math. Soc. (N.S.) 49, 1 (2012), 113–162. issn:0273-0979 Google ScholarCross Ref
- Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. 1988. Ramanujan graphs. Combinatorica 8, 3 (1988), 261–277.Google ScholarCross Ref
- Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava. 2015. Interlacing families I: Bipartite Ramanujan graphs of all degrees. Ann. of Math. (2) 182, 1 (2015), 307–325. issn:0003-486X Google ScholarCross Ref
- Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava. 2018. Interlacing families IV: Bipartite Ramanujan graphs of all sizes. SIAM J. Comput. 47, 6 (2018), 2488–2509. issn:0097-5397 Google ScholarDigital Library
- Grigorii Aleksandrovich Margulis. 1988. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy peredachi informatsii 24, 1 (1988), 51–60.Google Scholar
- Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. 2020. Explicit near-Ramanujan graphs of every degree. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing. 510–523.Google ScholarDigital Library
- A. Nilli. 1991. On the second eigenvalue of a graph. Discrete Math. 91, 2 (1991), 207–210. issn:0012-365X Google ScholarDigital Library
- Ryan O'Donnell. 2014. Analysis of Boolean Functions. Cambridge University Press. Google ScholarCross Ref
- Ran Raz, Omer Reingold, and Salil Vadhan. 1999. Error reduction for extractors. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039). IEEE, 191–201.Google ScholarDigital Library
- Omer Reingold. 2005. Undirected ST-connectivity in log-space. In STOC'05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing. ACM, New York, 376–385. Google ScholarDigital Library
- Omer Reingold, Thomas Steinke, and Salil Vadhan. 2013. Pseudorandomness for regular branching programs via Fourier analysis. In Approximation, randomization, and combinatorial optimization. Lecture Notes in Comput. Sci., Vol. 8096. Springer, Heidelberg, 655–670. Google ScholarCross Ref
- Omer Reingold, Salil Vadhan, and Avi Wigderson. 2000. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Proceedings 41st Annual Symposium on Foundations of Computer Science. IEEE, 3–13.Google ScholarDigital Library
- Eyal Rozenman and Salil Vadhan. 2005. Derandomized squaring of graphs. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. Springer, 436–447.Google Scholar
- Michael Sipser and Daniel A Spielman. 1996. Expander codes. IEEE transactions on Information Theory 42, 6 (1996), 1710–1722.Google ScholarDigital Library
- Amnon Ta-Shma. 2017. Explicit, almost optimal, epsilon-balanced codes. In STOC'17–-Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. ACM, New York, 238–251. Google ScholarDigital Library
- Avishay Tal. 2017. Tight bounds on the Fourier spectrum of AC0. In 32nd Computational Complexity Conference (CCC 2017). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.Google ScholarDigital Library
- Terence Tao. 2015. What's new: 275A, Notes 5. Variants of the central limit theorem. https://terrytao.wordpress.com/2015/11/19/275a-notes-5-variants-of-the-central-limit-theorem/\#more-8566.Google Scholar
- Luca Trevisan. 2017. Lecture Notes on Graph Partitioning, Expanders and Spectral Methods. University of California, Berkeley, https://people. eecs. berkeley. edu/ luca/books/expanders-2016. pdf (2017).Google Scholar
- Salil P Vadhan. 2012. Pseudorandomness. Vol. 7. Now.Google Scholar
- Leslie G. Valiant. 1976. Graph-theoretic properties in computational complexity. J. Comput. System Sci. 13, 3 (1976), 278–285. issn:0022-0000 Google ScholarDigital Library
Index Terms
- Expander random walks: a Fourier-analytic approach
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