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Register a new userA short note on the joint entropy of n/2-wise independence
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Added by
Aditya Potukuchi
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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In this note, we prove a tight lower bound on the joint entropy of $n$ unbiased Bernoulli random variables which are $n/2$-wise independent. For general $k$-wise independence, we give new lower bounds by adapting Navon and Samorodnitskys Fourier proof of the `LP bound on error correcting codes. This counts as partial progress on a problem asked by Gavinsky and Pudlak.
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How low can the joint entropy of $n$ $d$-wise independent (for $dge2$) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$, for $p<1$)? This question has been posed and partially answered in a recent work of Babai. In this paper we improve some of his bounds, prove new bounds in a wider range of parameters and show matching upper bounds in some special cases. In particular, we prove tight lower bounds for the min-entropy (as well as the entropy) of pairwise and three-wise independent balanced binary variables for infinitely many values of $n$.
We consider an agent trying to bring a system to an acceptable state by repeated probabilistic action. Several recent works on algorithmizations of the Lovasz Local Lemma (LLL) can be seen as establishing sufficient conditions for the agent to succeed. Here we study whether such stochastic control is also possible in a noisy environment, where both the process of state-observation and the process of state-evolution are subject to adversarial perturbation (noise). The introduction of noise causes the tools developed for LLL algorithmization to break down since the key LLL ingredient, the sparsity of the causality (dependence) relationship, no longer holds. To overcome this challenge we develop a new analysis where entropy plays a central role, both to measure the rate at which progress towards an acceptable state is made and the rate at which noise undoes this progress. The end result is a sufficient condition that allows a smooth tradeoff between the intensity of the noise and the amenability of the system, recovering an asymmetric LLL condition in the noiseless case.
This note resolves an open problem asked by Bezrukov in the open problem session of IWOCA 2014. It shows an equivalence between regular graphs and graphs for which a sequence of invariants presents some symmetric property. We extend this result to a few other sequences.
The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is $o(log n)$. However, both results become useless when the total influence of the function is $omega(log n)$. The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of $S_n$. In this paper, we build and improve on the techniques of the Bourgain-Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: 1. A quantitative improvement of the Bourgain--Kalai result regarding the total influence of functions that are transitively symmetric. 2. A slightly weaker version of the Fourier--Entropy Conjecture of Friedgut and Kalai. This weaker version implies in particular that the Fourier spectrum of a constant variance, Boolean function $f$ is concentrated on $2^{O(I[f]log I[f])}$ characters, improving an earlier result of Friedgut. Removing the $log I[f]$ factor would essentially resolve the Fourier--Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas. Our concentration result has new implications in learning theory: it implies that the class of functions whose total influence is at most $K$ is agnostically learnable in time $2^{O(Klog K)}$, using membership queries.
In this short note, we show two NP-completeness results regarding the emph{simultaneous representation problem}, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some $k$ graphs can be represented so that every vertex is represented by the same interval in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when $k$ is a part of the input and graphs are not in a sunflower position.
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