Improved Bounds on Fourier Entropy and Min-Entropy

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10422306" target="_blank" >RIV/00216208:11320/20:10422306 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.4230/LIPIcs.STACS.2020.45" target="_blank" >https://doi.org/10.4230/LIPIcs.STACS.2020.45</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.STACS.2020.45" target="_blank" >10.4230/LIPIcs.STACS.2020.45</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Improved Bounds on Fourier Entropy and Min-Entropy

Popis výsledku v původním jazyce

Given a Boolean function f : { -1, l}(n) -&gt; { -1, 1}, define the Fourier distribution to be the distribution on subsets of [n], where each S subset of [n] is sampled with probability (f) over cap (S)(2). The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [24] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C &gt; 0 such that H((f) over cap (2)) &lt;= C . Inf(f), where H((f) over cap (2)) 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: (i) Our first contribution shows that H((f) over cap (2)) &lt;= 2 . aUC(circle plus) (f), where aUC(circle plus) (f) is the average unambiguous parity-certificate complexity of f . This improves upon several bounds shown by Chakraborty et al. [16]. We further improve this bound for unambiguous DNFs. (ii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O&apos;Donnell and others [43, 40] which asks if H-infinity ((f) over cap (2)) &lt;= C . lnf(f), where H-infinity ((f) over cap (2)) is the min-entropy of the Fourier distribution. We show H-infinity ((f) over cap (2)) &lt;= 2 . C-min(circle plus)(f), where C-min(circle plus )(f) is the minimum parity certificate complexity of f. We also show that for all epsilon &gt;= 0, we have H-infinity ((f) over cap (2)) &lt;= 2log(parallel to(f) over cap parallel to(1,epsilon) 1(1 - epsilon)) , where parallel to(f) over cap parallel to(1,epsilon )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). (iii) 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(omega(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.

Název v anglickém jazyce

Improved Bounds on Fourier Entropy and Min-Entropy

Popis výsledku anglicky

Given a Boolean function f : { -1, l}(n) -&gt; { -1, 1}, define the Fourier distribution to be the distribution on subsets of [n], where each S subset of [n] is sampled with probability (f) over cap (S)(2). The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [24] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C &gt; 0 such that H((f) over cap (2)) &lt;= C . Inf(f), where H((f) over cap (2)) 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: (i) Our first contribution shows that H((f) over cap (2)) &lt;= 2 . aUC(circle plus) (f), where aUC(circle plus) (f) is the average unambiguous parity-certificate complexity of f . This improves upon several bounds shown by Chakraborty et al. [16]. We further improve this bound for unambiguous DNFs. (ii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O&apos;Donnell and others [43, 40] which asks if H-infinity ((f) over cap (2)) &lt;= C . lnf(f), where H-infinity ((f) over cap (2)) is the min-entropy of the Fourier distribution. We show H-infinity ((f) over cap (2)) &lt;= 2 . C-min(circle plus)(f), where C-min(circle plus )(f) is the minimum parity certificate complexity of f. We also show that for all epsilon &gt;= 0, we have H-infinity ((f) over cap (2)) &lt;= 2log(parallel to(f) over cap parallel to(1,epsilon) 1(1 - epsilon)) , where parallel to(f) over cap parallel to(1,epsilon )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). (iii) 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(omega(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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GX19-27871X" target="_blank" >GX19-27871X: Efektivní aproximační algoritmy a obvodová složitost</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

37TH INTERNATIONAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS 2020)

ISBN

978-3-95977-140-5

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

19

Strana od-do

—

Název nakladatele

SCHLOSS DAGSTUHL, LEIBNIZ CENTER INFORMATICS

Místo vydání

WADEM

Místo konání akce

Montpellier

Datum konání akce

10. 3. 2020

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

000521377300045