{Improved Bounds on Fourier Entropy and Min-entropy}

Given a Boolean function $f:\{-1,1\}^n\to \{-1,1\}$, the Fourier distribution assigns probability $\widehat{f}(S)^2$ to $S\subseteq [n]$. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai asks if there exist a universal constant C>0 such that $H(\hat{f}^2)\leq C Inf(f)$, where $H(\hat{f}^2)$ is the Shannon entropy of the Fourier distribution of $f$ and $Inf(f)$ is the total influence of $f$. 1) We consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture. This asks if $H_{\infty}(\hat{f}^2)\leq C Inf(f)$, where $H_{\infty}(\hat{f}^2)$ is the min-entropy of the Fourier distribution. We show $H_{\infty}(\hat{f}^2)\leq 2C_{\min}^\oplus(f)$, where $C_{\min}^\oplus(f)$ is the minimum parity certificate complexity of $f$. We also show that for every $\epsilon\geq 0$, we have $H_{\infty}(\hat{f}^2)\leq 2\log (\|\hat{f}\|_{1,\epsilon}/(1-\epsilon))$, where $\|\hat{f}\|_{1,\epsilon}$ is the approximate spectral norm of $f$. As a corollary, we verify the FMEI conjecture for the class of read-$k$ $DNF$s (for constant $k$). 2) We show that $H(\hat{f}^2)\leq 2 aUC^\oplus(f)$, where $aUC^\oplus(f)$ is the average unambiguous parity certificate complexity of $f$. This improves upon Chakraborty et al. An important consequence of the FEI conjecture is the long-standing Mansour's conjecture. We show that a weaker version of FEI already implies Mansour's conjecture: is $H(\hat{f}^2)\leq C \min\{C^0(f),C^1(f)\}$?, where $C^0(f), C^1(f)$ are the 0- and 1-certificate complexities of $f$, respectively. 3) We study what FEI implies about the structure of polynomials that 1/3-approximate a Boolean function. We pose a conjecture (which is implied by FEI): no "flat" degree-$d$ polynomial of sparsity $2^{\omega(d)}$ can 1/3-approximate a Boolean function. We prove this conjecture unconditionally for a particular class of polynomials

Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function $f : \{-1, 1\}^n \to \{-1, 1\}$ and $\bullet : \{-1, 1\}^2 \to \{-1, 1\}$ the two-party bounded-error quantum communication complexity of $(f \circ \bullet)$ is $O(Q(f) \log n)$, where $Q(f)$ is the bounded-error quantum query complexity of $f$. Note that the bounded-error randomized communication complexity of $(f \circ \bullet)$ is bounded by $O(R(f))$, where $R(f)$ denotes the bounded-error randomized query complexity of $f$. Thus, the BCW simulation has an extra $O(\log n)$ factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. H{\o}yer and de Wolf (STACS'02) showed that for the Set-Disjointness function, this can be reduced to $c^{\log^* n}$ for some constant $c$, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is $\mathsf{NOR}_n \circ \wedge$) is $O(Q(\mathsf{NOR}_n))$. Perhaps somewhat surprisingly, we show that when $\bullet = \oplus$, then the extra $\log n$ factor in the BCW simulation is unavoidable. In other words, we exhibit a total function $F : \{-1, 1\}^n \to \{-1, 1\}$ such that $Q^{cc}(F \circ \oplus) = \Theta(Q(F) \log n)$. To the best of our knowledge, it was not even known prior to this work whether there existed a total function $F$ and 2-bit function $\bullet$, such that $Q^{cc}(F \circ \bullet) = \omega(Q(F))$

Fractional Pseudorandom Generators from Any Fourier Level

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay {et al.} [CHHL19,CHLT19] that exploit $L_1$ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the $k$-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with $k$. This interpolates previous works, which either require Fourier bounds on all levels [CHHL19], or have polynomial dependence on the error parameter in the seed length [CHLT10], and thus answers an open question in [CHLT19]. As an example, we show that for polynomial error, Fourier bounds on the first $O(\log n)$ levels is sufficient to recover the seed length in [CHHL19], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor's theorem and bounding the degree-$k$ Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the \emph{level-k unsigned Fourier sum}, which is potentially a much smaller quantity than the $L_1$ notion in previous works. By generalizing a connection established in [CHH+20], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for $\mathbb{F}_2$ polynomials with seed length close to the state-of-the-art construction due to Viola [Vio09], which was not known to be possible using this framework

Improved bounds on Fourier entropy and min-entropy

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 ⊆ [n] is sampled with probability fb(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 > 0 such that H(f-2) ≤ C · Inf(f), where H(f-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-2) ≤ 2 · aUC (f), where aUC (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'Donnell and others [43, 40] which asks if H∞(f-2) ≤ C · Inf(f), where H∞(f-2) is the min-entropy of the Fourier distribution. We show H∞(f-2) ≤ 2 · C min(f), where C min(f) is the minimum parity certificate complexity of f. We also show that for all ε ≥ 0, we have H∞(f-2) ≤ 2 log(kfbk1,ε/(1 − ε)), where kfbk1,ε 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 th