A Composition Theorem for the Fourier Entropy-Influence Conjecture

@article{ODonnell2013ACT,
  title={A Composition Theorem for the Fourier Entropy-Influence Conjecture},
  author={Ryan O'Donnell and Li-Yang Tan},
  journal={ArXiv},
  year={2013},
  volume={abs/1304.1347}
}
The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [1] seeks to relate two fundamental measures of Boolean function complexity: it states that H[f]≤C· Inf[f] holds for every Boolean function f, where H[f] denotes the spectral entropy of f, Inf[f] is its total influence, and C>0 is a universal constant. Despite significant interest in the conjecture it has only been shown to hold for a few classes of Boolean functions. Our main result is a composition theorem for the FEI… 

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  • G. Shalev
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  • 2018
TLDR
This work proves the conjecture for extremal cases, functions with small influence and functions with high entropy, and suggests a direction for proving FEI for read-k DNFs, and proves the Fourier Min-Entropy/Influence (FMEI) Conjecture for regular read-K DNF's.

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A natural interpretation of the Fourier Entropy-Influence conjecture, which states that there exists a communication protocol which, given subset of subsets of $[n]$ distributed as $\widehat{f}^2$, can communicate the value of $S$ using at most $C\cdot\operatorname{Inf}[f]$ bits in expectation.

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