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Towards a Proof of the Fourier-Entropy Conjecture?
Geometric and Functional Analysis ( IF 2.2 ) Pub Date : 2020-09-05 , DOI: 10.1007/s00039-020-00544-2
Esty Kelman , Guy Kindler , Noam Lifshitz , Dor Minzer , Muli Safra

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is \(o(\log n)\). However, both results become useless when the total influence of the function is \(\omega (\log n)\). The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of \(S_n\). In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include:

  1. 1.

    A quantitative improvement of the Bourgain–Kalai result regarding the total influence of functions that are transitively symmetric.

  2. 2.

    A slightly weaker version of the Fourier-Entropy Conjecture of Friedgut and Kalai. Our result establishes new bounds on the Fourier entropy of a Boolean function f, as well as stronger bounds on the Fourier entropy of low-degree parts of f. In particular, it implies that the Fourier spectrum of a constant variance, Boolean function f is concentrated on \(2^{O(I[f]\log I[f])}\) characters, improving an earlier result of Friedgut. Removing the \(\log I[f]\) factor would essentially resolve the Fourier-Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas.

Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time \(2^{O(K\log K)}\) using membership queries. Thus, the class of functions with total influence \(O(\log n/\log \log n)\) is agnostically learnable in \(\mathsf{poly}(n)\) time.



中文翻译:

证明傅里叶熵猜想?

函数的总影响力是布尔函数分析中的核心概念,而表征具有较小总影响力的函数是与此相关的最基本问题之一。只要总影响的界线是\(o(\ log n)\),KKL定理和Friedgut junta定理就可以很好地描述此类函数。但是,当函数的总影响为\(\ omega(\ log n)\)时,两个结果都将变得无用。Bourgain和Kalai证明了唯一的打破对数障碍的有趣类函数的情况,他们专注于\(S_n \)足够大的子组下对称的函数。在本文中,我们对Bourgain-Kalai论文的技术进行了改进和完善,并在布尔函数的傅立叶谱上建立了新的集中结果,而总影响较小。我们的结果包括:

  1. 1。

    关于传递对称函数的总影响,布尔加因-凯莱结果的定量改进。

  2. 2。

    弗里德古特和卡拉的傅里叶熵猜想的弱化版本。我们的结果建立在布尔函数的傅立叶熵新界˚F上的低度部分傅立叶熵,以及较强的边界˚F。特别是,这意味着恒定方差布尔函数f的傅立叶谱集中在\(2 ^ {O(I [f] \ log I [f])} \)字符上,从而改善了Friedgut的较早结果。去除\(\ log I [f] \)因子将基本上解决傅立叶熵猜想,并解决关于多项式大小DNF公式的傅立叶谱的曼苏猜想。

我们对傅立叶函数谱的集中结果具有很小的总影响力,这对学习理论也有新的启示。更具体地说,我们得出结论,使用成员资格查询可以在时间\(2 ^ {O(K \ log K)} \)上不可知地学习总影响力最多为K的函数类别。因此,具有总影响力\(O(\ log n / \ log \ log n)\)的函数类别在\(\ mathsf {poly}(n)\)时间内是不可知的。

更新日期:2020-09-06
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