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Electronic Colloquium on Computational Complexity

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Reports tagged with hypercontractivity:
TR14-182 | 22nd December 2014
Dana Moshkovitz

Direct Product Testing With Nearly Identical Sets

Comments: 1

In this work we analyze a direct product test in which each of two provers receives a subset of size n of a ground set U,
and the two subsets intersect in about (1-\delta)n elements.
We show that if each of the provers provides labels to the n ... more >>>

TR15-084 | 21st May 2015
Or Ordentlich, Ofer Shayevitz, Omri Weinstein

Dictatorship is the Most Informative Balanced Function at the Extremes

Revisions: 2

Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that $I(f(X);Y)\leq 1-h(\alpha)$ for any Boolean function $f$. In this paper, we prove the conjecture for all ... more >>>

TR16-033 | 10th March 2016
Venkatesan Guruswami, Jaikumar Radhakrishnan

Tight bounds for communication assisted agreement distillation

Suppose Alice holds a uniformly random string $X \in \{0,1\}^N$ and Bob holds a noisy version $Y$ of $X$ where each bit of $X$ is flipped independently with probability $\epsilon \in [0,1/2]$. Alice and Bob would like to extract a common random string of min-entropy at least $k$. In this ... more >>>

TR18-016 | 25th January 2018
Naomi Kirshner, Alex Samorodnitsky

On $\ell_4$ : $\ell_2$ ratio of functions with restricted Fourier support

Revisions: 1

Given a subset $A\subseteq \{0,1\}^n$, let $\mu(A)$ be the maximal ratio between $\ell_4$ and $\ell_2$ norms of a function whose Fourier support is a subset of $A$. We make some simple observations about the connections between $\mu(A)$ and the additive properties of $A$ on one hand, and between $\mu(A)$ and ... more >>>

TR18-037 | 21st February 2018
Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

Inapproximability of Matrix $p \rightarrow q$ Norms

We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$, $q=1$), and has been ... more >>>

TR19-141 | 22nd October 2019
Mark Braverman, Subhash Khot, Dor Minzer

On Rich $2$-to-$1$ Games

We propose a variant of the $2$-to-$1$ Games Conjecture that we call the Rich $2$-to-$1$ Games Conjecture and show that it is equivalent to the Unique Games Conjecture. We are motivated by two considerations. Firstly, in light of the recent proof of the $2$-to-$1$ Games Conjecture, we hope to understand ... more >>>

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