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arXiv · 2112.09039

Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Renyi entropy

Niv Levhari·Alex Samorodnitsky
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Abstract

Let $T_{\epsilon}$, $0 \le \epsilon \le 1/2$, be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a distribution on $\{0,1\}^n$ and let $q > 1$. We prove tight Mrs. Gerber-type results for the second Renyi entropy of $T_{\epsilon} f$ which take into account the value of the $q^{th}$ Renyi entropy of $f$. For a general function $f$ on $\{0,1\}^n$ we prove tight hypercontractive inequalities for the $\ell_2$ norm of $T_{\epsilon} f$ which take into account the ratio between $\ell_q$ and $\ell_1$ norms of $f$.

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Niv Levhari, Alex Samorodnitsky. 2021-12-16. Hypercontractive inequalities for the second norm of highly concentrated functions, and Mrs. Gerber's-type inequalities for the second Renyi entropy. https://doi.org/10.3390/e24101376

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