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Tarik Kaced

Publications and source records attributed to Tarik Kaced.

10 recordsLinked to original sources
arXiv2018

Group Testing with Prior Statistics

We consider a new group testing model wherein each item is a binary random variable defined by an a priori probability of being defective. We assume that each probability is small and that items are independent, but not necessarily identically distributed. The goal of group testing algorithms is to identify with high probability the subset of defectives via non-linear (disjunctive) binary measurements. Our main contributions are two classes of algorithms: (1) adaptive algorithms with tests based either on a maximum entropy principle, or on a Shannon-Fano/Huffman code; (2) non-adaptive algorithms. Under loose assumptions and with high probability, our algorithms only need a number of measurements that is close to the information-theoretic lower bound, up to an explicitly-calculated universal constant factor. We provide simulations to support our results.

cs.IT↗
arXiv2017

Conditional Information Inequalities and Combinatorial Applications

We show that the inequality $H(A \mid B,X) + H(A \mid B,Y) \le H(A\mid B)$ for jointly distributed random variables $A,B,X,Y$, which does not hold in general case, holds under some natural condition on the support of the probability distribution of $A,B,X,Y$. This result generalizes a version of the conditional Ingleton inequality: if for some distribution $I(X: Y \mid A) = H(A\mid X,Y)=0$, then $I(A : B) \le I(A : B \mid X) + I(A: B \mid Y) + I(X : Y)$. We present two applications of our result. The first one is the following easy-to-formulate combinatorial theorem: assume that the edges of a bipartite graph are partitioned into $K$ matchings such that for each pair (left vertex $x$, right vertex $y$) there is at most one matching in the partition involving both $x$ and $y$; assume further that the degree of each left vertex is at least $L$ and the degree of each right vertex is at least $R$. Then $K\ge LR$. The second application is a new method to prove lower bounds for biclique coverings of bipartite graphs.

cs.IT↗
arXiv2016

Info-Clustering: A Mathematical Theory for Data Clustering

We formulate an info-clustering paradigm based on a multivariate information measure, called multivariate mutual information, that naturally extends Shannon's mutual information between two random variables to the multivariate case involving more than two random variables. With proper model reductions, we show that the paradigm can be applied to study the human genome and connectome in a more meaningful way than the conventional algorithmic approach. Not only can info-clustering provide justifications and refinements to some existing techniques, but it also inspires new computationally feasible solutions.

cs.IT↗
arXiv2016

The Entropy Region is not Closed Under Duality

We import a duality notion coming from polymatroids to define duality for information inequalities. We show that the entropy region for $n\ge 5$ is not closed under duality. Our result answers an open question of Matùš (1992).

cs.IT↗
arXiv2013

Conditional Information Inequalities for Entropic and Almost Entropic Points

We study conditional linear information inequalities, i.e., linear inequalities for Shannon entropy that hold for distributions whose entropies meet some linear constraints. We prove that some conditional information inequalities cannot be extended to any unconditional linear inequalities. Some of these conditional inequalities hold for almost entropic points, while others do not. We also discuss some counterparts of conditional information inequalities for Kolmogorov complexity.

cs.IT↗
arXiv2013

Equivalence of Two Proof Techniques for Non-Shannon-type Inequalities

We compare two different techniques for proving non-Shannon-type information inequalities. The first one is the original Zhang-Yeung's method, commonly referred to as the copy/pasting lemma/trick. The copy lemma was used to derive the first conditional and unconditional non-Shannon-type inequalities. The second technique first appeared in Makarychev et al paper [7] and is based on a coding lemma from Ahlswede and Körner works. We first emphasize the importance of balanced inequalities and provide a simpler proof of a theorem of Chan's for the case of Shannon-type inequalities. We compare the power of various proof systems based on a single technique.

cs.IT↗
arXiv2012

On the Non-robustness of Essentially Conditional Information Inequalities

We show that two essentially conditional linear inequalities for Shannon's entropies (including the Zhang-Yeung'97 conditional inequality) do not hold for asymptotically entropic points. This means that these inequalities are non-robust in a very strong sense. This result raises the question of the meaning of these inequalities and the validity of their use in practice-oriented applications.

cs.IT↗
arXiv2011

Almost-perfect secret sharing

Splitting a secret s between several participants, we generate (for each value of s) shares for all participants. The goal: authorized groups of participants should be able to reconstruct the secret but forbidden ones get no information about it. In this paper we introduce several notions of non- perfect secret sharing, where some small information leak is permitted. We study its relation to the Kolmogorov complexity version of secret sharing (establishing some connection in both directions) and the effects of changing the secret size (showing that we can decrease the size of the secret and the information leak at the same time).

cs.IT↗
arXiv2011

On essentially conditional information inequalities

In 1997, Z.Zhang and R.W.Yeung found the first example of a conditional information inequality in four variables that is not "Shannon-type". This linear inequality for entropies is called conditional (or constraint) since it holds only under condition that some linear equations are satisfied for the involved entropies. Later, the same authors and other researchers discovered several unconditional information inequalities that do not follow from Shannon's inequalities for entropy. In this paper we show that some non Shannon-type conditional inequalities are "essentially" conditional, i.e., they cannot be extended to any unconditional inequality. We prove one new essentially conditional information inequality for Shannon's entropy and discuss conditional information inequalities for Kolmogorov complexity.

cs.IT↗