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%2F21%3A10438299" target="_blank" >RIV/00216208:11320/21:10438299 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=VC5Rh0kxzB" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=VC5Rh0kxzB</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1145/3470860" target="_blank" >10.1145/3470860</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 : {-1.1}(n) {-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 [28] 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 article, we present three new contributions toward the FEI conjecture: (1) 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 off. This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour&apos;s conjecture for DNFs a natural next step toward resolution of the FEI conjecture. (2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O&apos;Donnell and others [50, 53], which asks if H-infinity((f) over cap (2)) &lt;= C . Inf(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;= 2 log(parallel to(f) over cap parallel to(1,epsilon)/(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 DN Fs (for constant k). (3) 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 : {-1.1}(n) {-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 [28] 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 article, we present three new contributions toward the FEI conjecture: (1) 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 off. This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs. We also discuss how our work makes Mansour&apos;s conjecture for DNFs a natural next step toward resolution of the FEI conjecture. (2) We next consider the weaker Fourier Min-entropy-influence (FMEI) conjecture posed by O&apos;Donnell and others [50, 53], which asks if H-infinity((f) over cap (2)) &lt;= C . Inf(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;= 2 log(parallel to(f) over cap parallel to(1,epsilon)/(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 DN Fs (for constant k). (3) 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

J_imp - Článek v periodiku v databázi Web of Science

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

2021

Kód důvěrnosti údajů

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

Název periodika

ACM Transactions on Computation Theory

ISSN

1942-3454

e-ISSN

—

Svazek periodika

13

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

40

Strana od-do

22

Kód UT WoS článku

000692902500002

EID výsledku v databázi Scopus

2-s2.0-85114356171