arXiv subjects

Emmanuel Abbe

Publications and source records attributed to Emmanuel Abbe.

94 records · Page 6Linked to original sources

arXiv2010

On the concentration of the number of solutions of random satisfiability formulas

Let $Z(F)$ be the number of solutions of a random $k$-satisfiability formula $F$ with $n$ variables and clause density $α$. Assume that the probability that $F$ is unsatisfiable is $O(1/\log(n)^{1+\e})$ for $\e>0$. We show that (possibly excluding a countable set of `exceptional' $α$'s) the number of solutions concentrate in the logarithmic scale, i.e., there exists a non-random function $ϕ(α)$ such that, for any $δ>0$, $(1/n)\log Z(F)\in [ϕ-δ,ϕ+δ]$ with high probability. In particular, the assumption holds for all $α<1$, which proves the above concentration claim in the whole satisfiability regime of random $2$-SAT. We also extend these results to a broad class of constraint satisfaction problems. The proof is based on an interpolation technique from spin-glass theory, and on an application of Friedgut's theorem on sharp thresholds for graph properties.

cs.DM↗

arXiv2010

Universal A Posteriori Metrics Game

Over binary input channels, uniform distribution is a universal prior, in the sense that it allows to maximize the worst case mutual information over all binary input channels, ensuring at least 94.2% of the capacity. In this paper, we address a similar question, but with respect to a universal generalized linear decoder. We look for the best collection of finitely many a posteriori metrics, to maximize the worst case mismatched mutual information achieved by decoding with these metrics (instead of an optimal decoder such as the Maximum Likelihood (ML) tuned to the true channel). It is shown that for binary input and output channels, two metrics suffice to actually achieve the same performance as an optimal decoder. In particular, this implies that there exist a decoder which is generalized linear and achieves at least 94.2% of the compound capacity on any compound set, without the knowledge of the underlying set.

cs.IT↗

arXiv2010

A Coordinate System for Gaussian Networks

This paper studies network information theory problems where the external noise is Gaussian distributed. In particular, the Gaussian broadcast channel with coherent fading and the Gaussian interference channel are investigated. It is shown that in these problems, non-Gaussian code ensembles can achieve higher rates than the Gaussian ones. It is also shown that the strong Shamai-Laroia conjecture on the Gaussian ISI channel does not hold. In order to analyze non-Gaussian code ensembles over Gaussian networks, a geometrical tool using the Hermite polynomials is proposed. This tool provides a coordinate system to analyze a class of non-Gaussian input distributions that are invariant over Gaussian networks.

cs.IT↗

arXiv2008

Linear Universal Decoding for Compound Channels: a Local to Global Geometric Approach

Over discrete memoryless channels (DMC), linear decoders (maximizing additive metrics) afford several nice properties. In particular, if suitable encoders are employed, the use of decoding algorithm with manageable complexities is permitted. Maximum likelihood is an example of linear decoder. For a compound DMC, decoders that perform well without the channel's knowledge are required in order to achieve capacity. Several such decoders have been studied in the literature. However, there is no such known decoder which is linear. Hence, the problem of finding linear decoders achieving capacity for compound DMC is addressed, and it is shown that under minor concessions, such decoders exist and can be constructed. This paper also develops a "local geometric analysis", which allows in particular, to solve the above problem. By considering very noisy channels, the original problem is reduced, in the limit, to an inner product space problem, for which insightful solutions can be found. The local setting can then provide counterexamples to disproof claims, but also, it is shown how in this problem, results proven locally can be "lifted" to results proven globally.

cs.IT↗