Abstract
Given a Boolean function f:{ -1,1} ^{n}→ { -1,1, define the Fourier distribution to be the distribution on subsets of [n], where each S ⊆ [n] is sampled with probability f ˆ (S)2. The Fourier Entropy-influence (FEI) conjecture of Friedgut and Kalai [28] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C > 0 such that H(fˆ2) ≤ C ⋅ Inf (f), where H(fˆ2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f
In this article, we present three new contributions toward the FEI conjecture:
(1) | Our first contribution shows that H(fˆ2) ≤ 2 ⋅ aUC⊕(f), where aUC⊕(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour's conjecture for DNFs a natural next step toward resolution of the FEI conjecture. | ||||
(2) | We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O'Donnell and others [50, 53], which asks if H ∞ fˆ2) ≤ C ⋅ Inf(f), where H ∞ fˆ2) is the min-entropy of the Fourier distribution. We show H∞(fˆ2) ≤ 2⋅Cmin⊕(f), where Cmin⊕(f) is the minimum parity-certificate complexity of f. We also show that for all ε≥0, we have H∞(fˆ2)≤2 log(∥fˆ∥1,ε/(1−ε)), where ∥fˆ∥1,ε is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k). | ||||
(3) | Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial(whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2ω(d) can 1/3-approximate a Boolean function. This conjecture is known to be true assuming FEI, and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis. | ||||
- S. Aaronson and A. Ambainis. 2018. Forrelation: A problem that optimally separates quantum from classical computing. SIAM J. Comput. 47, 3 (2018), 982–1038.Google Scholar
Cross Ref
- A. Akavia, A. Bogdanov, S. Guo, A. Kamath, and A. Rosen. 2014. Candidate weak pseudorandom functions in ACMOD2. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science (ITCS'14). ACM, 251–260.Google Scholar
- N. Albuquerque, F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda. 2014. Sharp generalizations of the multilinear Bohnenblust-Hille inequality. J. Funct. Anal. 266, 6 (2014), 3276–3740.Google Scholar
Cross Ref
- N. Alon and J. Spencer. 2000. The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization.Google Scholar
- K. Amano. 2011. Tight bounds on the average sensitivity of k-CNF. Theory Comput. 7, 4 (2011), 45–48.Google Scholar
Cross Ref
- A. Ambainis, M. Kokainis, and R. Kothari. 2016. Nearly optimal separations between communication (or query) complexity and partitions. In Proceedings of the 31st Conference on Computational Complexity (CCC'16). 4:1–4:14.Google Scholar
- S. Arunachalam, S. Chakraborty, M. Koucký, N. Saurabh, and R. de Wolf. 2020. Improved bounds on fourier entropy and min-entropy. In Proceedings of the 37th International Symposium on Theoretical Aspects of Computer Science (STACS'20)(LIPIcs, Vol. 154). 45:1–45:19.Google Scholar
- F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda. 2014. The bohr radius of the n-dimensional polydisk is equivalent to . Adv. Math. 264 (2014), 726–746.Google Scholar
Cross Ref
- R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. 2001. Quantum lower bounds by polynomials. J. ACM 48, 4 (2001), 778–797.Google Scholar
Digital Library
- S. Ben-David, P. Hatami, and A. Tal. 2017. Low-sensitivity functions from unambiguous certificates. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS'17). 28:1–28:23.Google Scholar
- H. F. Bohnenblust and E. Hille. 1931. On the absolute convergence of dirichlet series. Ann. Math. 32, 3 (July 1931), 600–622.Google Scholar
Cross Ref
- R. Boppana. 1997. The average sensitivity of bounded-depth circuits. Inform. Process. Lett. 63, 5 (1997), 257–261.Google Scholar
Digital Library
- J. Bourgain and G. Kalai. 1997. Influences of variables and threshold intervals under group symmetries. Geom. Funct. Anal. 7, 3 (1997), 438–461.Google Scholar
Cross Ref
- Y. Brandman, A. Orlitsky, and J. Hennessy. 1990. A spectral lower bound technique for the size of decision trees and two-level AND/OR circuits. IEEE Trans. Comput. 39, 2 (1990), 282–287.Google Scholar
Digital Library
- J. Briët, H. Buhrman, T. Lee, and T. Vidick. 2013. Multipartite entanglement in XOR games. Quant. Info. Comput. 13, 3-4 (2013), 334–360. Retrieved from https://arXiv:0911.4007.Google Scholar
- H. Buhrman and R. de Wolf. 2002. Complexity measures and decision tree complexity: A survey. Theor. Comput. Sci. 288, 1 (2002), 21–43.Google Scholar
Digital Library
- M. Bun and J. Thaler. 2013. Dual lower bounds for approximate degree and markov-bernstein inequalities. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP'13). 303–314.Google Scholar
- S. Chakraborty, A. Chattopadhyay, N. S. Mande, and M. Paraashar. 2020. Quantum query-to-communication simulation needs a logarithmic overhead. In Proceedings of the 35th Computational Complexity Conference (CCC'20)(LIPIcs, Vol. 169). 32:1–32:15.Google Scholar
- S. Chakraborty, S. Karmalkar, S. Kundu, S. V. Lokam, and N. Saurabh. 2018. Fourier entropy-influence conjecture for random linear threshold functions. In Proceedings of the 13th Latin American Theoretical Informatics Symposium (LATIN'18). 275–289.Google Scholar
- S. Chakraborty, R. Kulkarni, S.V. Lokam, and N. Saurabh. 2016. Upper bounds on fourier entropy. Theor. Comput. Sci. 654 (2016), 92–112.Google Scholar
Cross Ref
- M. Cheraghchi, E. Grigorescu, B. Juba, K. Wimmer, and N. Xie. 2018. AC MOD2 lower bounds for the Boolean inner product. J. Comput. System Sci. 97 (2018), 45–59.Google Scholar
Cross Ref
- G. Cohen and I. Shinkar. 2016. The complexity of DNF of parities. In Proceedings of the 7th Conference on Innovations in Theoretical Computer Science (ITCS'16). ACM, 47–58.Google Scholar
- T. M. Cover and J. A. Thomas. 1991. Elements of Information Theory. John Wiley & Sons.Google Scholar
- B. Das, M. Pal, and V. Visavaliya. 2011. The Entropy Influence Conjecture Revisited. Retrieved from https://arxiv:1110.4301.Google Scholar
- A. Defant, L. Frerick, J. Ortega-Cerdá, M. Ounaïes, and K. Seip. 2011. The bohnenblust-hille inequality for homogeneous polynomials is hypercontractive. Ann. Math. 174, 1 (2011), 485–497.Google Scholar
Cross Ref
- A. Defant, D. Popa, and U. Schwarting. 2010. Coordinatewise multiple summing operators in banach spaces. J. Funct. Anal. 259, 1 (2010), 220–242.Google Scholar
Cross Ref
- R. Eldan and R. Gross. 2020. Concentration on the Boolean hypercube via pathwise stochastic analysis. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC'20). 208–221.Google Scholar
- E. Friedgut and G. Kalai. 1996. Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc. 124, 10 (1996), 2993–3002.Google Scholar
Cross Ref
- M. Göös. 2015. Lower bounds for clique vs. independent set. In Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS'15). 1066–1076.Google Scholar
Digital Library
- P. Gopalan, A. Kalai, and A. R. Klivans. 2008. A query algorithm for agnostically learning DNF? In Proceedings of the 21st Annual Conference on Learning Theory (COLT'08). 515–516.Google Scholar
- P. Gopalan, A. T. Kalai, and A. Klivans. 2008. Agnostically learning decision trees. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC'08). 527–536.Google Scholar
- P. Gopalan, R. Meka, and O. Reingold. 2013. DNF sparsification and a faster deterministic counting algorithm. Comput. Complex. 22, 2 (June 2013), 275–310.Google Scholar
Cross Ref
- P. Gopalan, R. A. Servedio, A. Tal, and A. Wigderson. 2016. Degree and sensitivity: Tails of two distributions. In Proceedings of the 31st Conference on Computational Complexity (CCC'16). 13:1–13:23.Google Scholar
- L. Gross. 1975. Logarithmic sobolev inequalities. Amer. J. Math. 97, 4 (1975), 1061–1083.Google Scholar
Cross Ref
- R. Hod. 2017. Improved Lower Bounds for the Fourier Entropy/Influence Conjecture via Lexicographic Functions. Retrieved from https://arxiv:1711.00762.Google Scholar
- J. Kahn, G. Kalai, and Nathan Linial. 1988. The influence of variables on Boolean functions. In Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science (FOCS'88). 68–80.Google Scholar
Digital Library
- G. Kalai. 2007. The Entropy/Influence Conjecture. Retrieved from https://terrytao.wordpress.com/2007/08/16/gil-kalai-the-entropyinfluence-conjecture/.Google Scholar
- N. Keller, E. Mossel, and T. Schlank. 2012. A note on the entropy/influence conjecture. Discrete Math. 312, 22 (2012), 3364–3372.Google Scholar
Cross Ref
- E. Kelman, G. Kindler, N. Lifshitz, D. Minzer, and M. Safra. 2020. Towards a proof of the fourier-entropy conjecture?. In Proceedings of the 61st IEEE Annual Symposium on Foundations of Computer Science (FOCS'20). IEEE, 247–258.Google Scholar
- A. Klivans, H. Lee, and A. Wan. 2010. Mansour's conjecture is true for random DNF formulas. In Proceedings of the 23rd Conference on Learning Theory (COLT'10). 368–380.Google Scholar
- T. Lee and A. Shraibman. 2009. Lower bounds in communication complexity. Found. Trends Theor. Comput. Sci. 3, 4 (2009), 263–398.Google Scholar
Cross Ref
- N. Linial, Y. Mansour, and N. Nisan. 1993. Constant depth circuits, fourier transform, and learnability. J. ACM 40, 3 (July 1993), 607–620.Google Scholar
Digital Library
- J. E. Littlewood. 1930. On bounded bilinear forms in an infinite number of variables. Quart. J. Math. 1 (1930), 164–174.Google Scholar
Cross Ref
- S. Lovett, K. Wu, and J. Zhang. 2020. Decision list compression by mild random restrictions. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC'20). ACM, 247–254.Google Scholar
- S. Lovett and J. Zhang. 2019. DNF sparsification beyond sunflowers. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC'19). ACM, 454–460.Google Scholar
- Y. Mansour. 1994. Learning Boolean functions via the fourier transform. In Theoretical Advances in Neural Computation and Learning, V. Roychowdhury, K.-Y. Siu, and A. Orlitsky (Eds.). Springer U.S., 391–424.Google Scholar
- Y. Mansour. 1995. An learning algorithm for DNF under the uniform distribution. J. Comput. System Sci. 50, 3 (1995), 543–550.Google Scholar
Digital Library
- A. Montanaro. 2012. Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53, 12 (2012), 122206.Google Scholar
Cross Ref
- A. Montanaro and T. Osborne. 2009. On the communication complexity of XOR functions. Retrieved from https://arXiv:0909.3392.Google Scholar
- R. O'Donnell. 2014. Analysis of Boolean Functions. Cambridge University Press.Google Scholar
- R. O'Donnell and L.-Y. Tan. 2013. A composition theorem for the fourier entropy-influence conjecture. In Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP'13). 780–791.Google Scholar
Digital Library
- R. O'Donnell, J. Wright, Y. Zhao, X. Sun, and L.-Y. Tan. 2014. A composition theorem for parity kill number. In Proceedings of the 29th Conference on Computational Complexity (CCC'14). 144–154.Google Scholar
Digital Library
- R. O'Donnell, J. Wright, and Y. Zhou. 2011. The fourier entropy-influence conjecture for certain classes of Boolean functions. In Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP'11). 330–341.Google Scholar
- R. O'Donnell and Y. Zhao. 2016. Polynomial bounds for decoupling, with applications. In Proceedings of the 31st Conference on Computational Complexity (CCC'16). 24:1–24:18.Google Scholar
- D. Pellegrino and J. Seoane-Sepúlveda. 2012. New upper bounds for the constants in the bohnenblust-hille inequality. J. Math. Anal. Appl. 386, 1 (2012), 300–307.Google Scholar
Cross Ref
- D. Pellegrino and E. V. Teixeira. 2018. Towards sharp Bohnenblust-Hille constants. Commun. Contemp. Math. 20, 3 (2018), 1750029.Google Scholar
Cross Ref
- R. A. Servedio and E. Viola. 2012. On a special case of rigidity. Retrieved from http://eccc.hpi-web.de/report/2012/144.Google Scholar
- G. Shalev. 2018. On the Fourier Entropy Influence Conjecture for Extremal Classes. Retrieved from https://arxiv:1806.03646.Google Scholar
- R. Shaltiel and E. Viola. 2010. Hardness amplification proofs require majority. SIAM J. Comput. 39, 7 (2010), 3122–3154.Google Scholar
Digital Library
- C. E. Shannon. 1948. A mathematical theory of communication. Bell Syst. Techn. J. 27, 3 (1948), 379–423.Google Scholar
Cross Ref
- A. Sherstov. 2018. Algorithmic polynomials. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC'18). 311–324.Google Scholar
Digital Library
- A. A. Sherstov. 2011. The pattern matrix method. SIAM J. Comput. 40, 6 (2011), 1969–2000.Google Scholar
Digital Library
- Y. Shi. 2000. Lower bounds of quantum black-box complexity and degree of approximating polynomials by influence of Boolean variables. Inform. Process. Lett. 75, 1–2 (2000), 79–83.Google Scholar
Digital Library
- A. Tal. 2014. Shrinkage of de morgan formulae by spectral techniques. In Proceedings of the 55th IEEE Annual Symposium on Foundations of Computer Science (FOCS'14). 551–560.Google Scholar
Digital Library
- A. Tal. 2017. Tight bounds on the fourier spectrum of AC. In Proceedings of the 32nd Computational Complexity Conference (CCC'17). 15:1–15:31.Google Scholar
Digital Library
- P. Traxler. 2009. Variable influences in conjunctive normal forms. In Proceedings of 12th International Conference on Theory and Applications of Satisfiability Testing (SAT'09). Springer, Berlin, 101–113.Google Scholar
Digital Library
- H. Y. Tsang, C. H. Wong, N. Xie, and S. Zhang. 2013. Fourier sparsity, spectral norm, and the log-rank conjecture. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS'13). 658–667.Google Scholar
- A. Wan, J. Wright, and C. Wu. 2014. Decision trees, protocols and the entropy-influence conjecture. In Proceedings of the 5th Innovations in Theoretical Computer Science (ITCS'14). 67–80.Google Scholar
- R. de Wolf. 2008. A brief introduction to fourier analysis on the Boolean cube. Theory Comput. Library Grad. Surveys 1 (2008), 1–20.Google Scholar
- S. Zhang. 2014. Efficient quantum protocols for XOR functions. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'14). 1878–1885.Google Scholar
Cross Ref
- Z. Zhang and Y. Shi. 2009. Communication complexities of symmetric XOR functions. Quant. Info. Comput. 9, 3 (2009), 255–263.Google Scholar
Digital Library
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Improved Bounds on Fourier Entropy and Min-entropy
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