Improved Bounds on Fourier Entropy and Min-Entropy
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 f̂(S)². The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [E. Friedgut and G. Kalai, 1996] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that ℍ(f̂²)≤ C⋅ Inf(f), where ℍ(f̂²) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f?
In this paper we present three new contributions towards the FEI conjecture:
ii) Our first contribution shows that ℍ(f̂²) ≤ 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. [S. Chakraborty et al., 2016]. We further improve this bound for unambiguous DNFs.
iii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [R. O'Donnell et al., 2011; R. O'Donnell, 2014] which asks if ℍ_{∞}(f̂²) ≤ C⋅ Inf(f), where ℍ_{∞}(f̂²) is the min-entropy of the Fourier distribution. We show ℍ_{∞}(f̂²) ≤ 2⋅?_{min}^⊕(f), where ?_{min}^⊕(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have ℍ_{∞}(f̂²) ≤ 2log (‖f̂‖_{1,ε}/(1-ε)), where ‖f̂‖_{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).
iv) 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 2^ω(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.
Fourier analysis of Boolean functions
FEI conjecture
query complexity
polynomial approximation
approximate degree
certificate complexity
Computing methodologies~Representation of Boolean functions
Theory of computation~Models of computation
Mathematics of computing~Information theory
45:1-45:19
Regular Paper
A full version [S. Arunachalam et al., 2018] of the paper is available at https://arxiv.org/abs/1809.09819.
Part of this work was carried out when NS and SC visited CWI, Amsterdam and SA was part of MIT (supported by MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert Space) and University of Bristol (partially supported by EPSRC grant EP/L021005/1). SA thanks Ashley Montanaro for his hospitality. NS and SC would like to thank Satya Lokam for many helpful discussions on the Fourier entropy-Influence conjecture. SA and SC thank Jop Briët for pointing us to the literature on unbalancing lights and many useful discussions regarding Section 3.3. We also thank Penghui Yao and Avishay Tal for discussions during the course of this project, and Fernando Vieira Costa Júnior for pointing us to the reference [F. Bayart et al., 2014]. Finally, we thank the anonymous reviewers for many helpful comments.
Srinivasan
Arunachalam
Srinivasan Arunachalam
MIT, Cambridge, MA, USA
Work done when at QuSoft, CWI, Amsterdam, supported by ERC Consolidator Grant 615307 QPROGRESS and MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space.
Sourav
Chakraborty
Sourav Chakraborty
Indian Statistical Institute, Kolkata, India
Michal
Koucký
Michal Koucký
Computer Science Institute of Charles University, Prague, Czech Republic
Partially supported by ERC Consolidator Grant 616787 LBCAD, and GAČR grant 19-27871X.
Nitin
Saurabh
Nitin Saurabh
Max Planck Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany
Part of the work was done when the author was at IUUK, Prague and supported by the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 616787.
Ronald
de Wolf
Ronald de Wolf
QuSoft, CWI and University of Amsterdam, the Netherlands
Partially supported by ERC Consolidator Grant 615307 QPROGRESS (ended Feb 2019) and by NWO under QuantERA project QuantAlgo 680-91-034 and the Quantum Software Consortium.
10.4230/LIPIcs.STACS.2020.45
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Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, and Ronald de Wolf
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