characterization of talagrand s transport entropy inequalities in metric spaces we give a characterization of transport entropy inequalities in metric spaces as an application we deduce that such inequalities are stable under bounded perturbation holley stroock perturbation lemma

from concentration to logarithmic sobolev and poincare inequalities abstract we give a new proof of the fact that gaussian concentration implies the logarithmic sobolev inequality when the curvature is bounded from below and also that exponential concentration implies poincare inequality under null curvature condition our proof holds on non smooth structures such as length spaces and provides a universal control of the constants we also give a new proof of the equivalence between dimension free gaussian concentration and talagrand s transport inequality

cut set bound is loose for gaussian relay networks the cut set bound developed by cover and el gamal in 1979 has since remained the best known upper bound on the capacity of the gaussian relay channel we develop a new upper bound on the capacity of the gaussian primitive relay channel which is tighter than the cut set bound our proof uses gaussian measure concentration to establish geometric relations satisfied with high probability between the n letter random variables associated with a reliable code for communicating over this channel we then translate these geometric relations into new information inequalities that cannot be obtained with classical methods combined with a tensorization argument proposed by courtade and ozgur in 2015 our result also implies that the current capacity approximations for gaussian relay networks which have linear gap to the cut set bound in the number of nodes are order optimal and lead to a lower bound on the pre constant

a characterization of dimension free concentration in terms of transportation inequalities the aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies talagrand s t 2 transportation cost inequality this theorem permits us to give a new and very short proof of a result of otto and villani generalizations to other types of concentration are also considered in particular one shows that the poincare inequality is equivalent to a certain form of dimension free exponential concentration the proofs of these results rely on simple large deviations techniques

measure concentration for euclidean distance in the case of dependent random variables let qn be a continuous density function in n dimensional euclidean space we think of qn as the density function of some random sequence xn with values in bbb r n for i 1 n let xi denote the collection of coordinates xi i i and let widebar x i denote the collection of coordinates xi i i we denote by q i x i bar x i the joint conditional density function of xi given widebar x i we prove measure concentration for qn in the case when for an appropriate class of sets i i the conditional densities q i x i bar x i as functions of xi uniformly satisfy a logarithmic sobolev inequality and ii these conditional densities also satisfy a contractivity condition related to dobrushin and shlosman s strong mixing condition

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

brief paper sensor control for multi object state space estimation using random finite sets the problem addressed in this paper is information theoretic sensor control for recursive bayesian multi object state space estimation using random finite sets the proposed algorithm is formulated in the framework of partially observed markov decision processes where the reward function associated with different sensor actions is computed via the renyi or alpha divergence between the multi object prior and the multi object posterior densities the proposed algorithm in implemented via the sequential monte carlo method the paper then presents a case study where the problem is to localise an unknown number of sources using a controllable moving sensor which provides range only detections four sensor control reward functions are compared in the study and the proposed scheme is found to perform the best

empirical distribution of good channel codes with nonvanishing error probability this paper studies several properties of channel codes that approach the fundamental limits of a given discrete or gaussian memoryless channel with a nonvanishing probability of error the output distribution induced by an capacity achieving code is shown to be close in a strong sense to the capacity achieving output distribution relying on the concentration of measure isoperimetry property enjoyed by the latter it is shown that regular lipschitz functions of channel outputs can be precisely estimated and turn out to be essentially nonrandom and independent of the actual code it is also shown that the output distribution of a good code and the capacity achieving one cannot be distinguished with exponential reliability the random process produced at the output of the channel is shown to satisfy the asymptotic equipartition property

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

strong data processing inequalities and phi sobolev inequalities for discrete channels the noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions for instance the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity by the data processing theorem however for a fixed reference input distribution this quantity may be strictly smaller than one giving the so called strong data processing inequalities sdpis the same considerations apply to an arbitrary phi divergence this paper presents a systematic study of optimal constants in the sdpis for discrete channels including their variational characterizations upper and lower bounds structural results for channels on product probability spaces and the relationship between the sdpis and the so called phi sobolev inequalities another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy like functionals of the input distribution by suitable measures of input output correlation several applications to information theory discrete probability and statistical physics are discussed

concentration for coulomb gases and coulomb transport inequalities we study the non asymptotic behavior of coulomb gases in dimension two and more such gases are modeled by an exchangeable boltzmann gibbs measure with a singular two body interaction we obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure with respect to bounded lipschitz and wasserstein distances this implies macroscopic as well as mesoscopic convergence in such distances in particular we improve the concentration inequalities known for the empirical spectral distribution of ginibre random matrices our approach is remarkably simple and bypasses the use of renormalized energy it crucially relies on new inequalities between probability metrics including coulomb transport inequalities which can be of independent interest our work is inspired by the one of maida and maurel segala itself inspired by large deviations techniques our approach allows to recover extend and simplify previous results by rougerie and serfaty

information inequalities and concentration of measure we derive inequalities of the form p q h p r h q r which hold for every choice of probability measures p q r where h p r denotes the relative entropy of p with respect to r and p q stands for a coupling type distance between p and q using the chain rule for relative entropies and then specializing to q with a given support we recover some of talagrand s concentration of measure inequalities for product spaces

sharp inequalities for f divergences f divergences are a general class of divergences between probability measures which include as special cases many commonly used divergences in probability mathematical statistics and information theory such as kullback leibler divergence chi squared divergence squared hellinger distance total variation distance and so on in this paper we study the problem of maximizing or minimizing an f divergence between two probability measures subject to a finite number of constraints on other f divergences we show that these infinite dimensional optimization problems can all be reduced to optimization problems over small finite dimensional spaces which are tractable our results lead to a comprehensive and unified treatment of the problem of obtaining sharp inequalities between f divergences we demonstrate that many of the existing results on inequalities between f divergences can be obtained as special cases of our results we also improve on some existing non sharp inequalities

the heat flow on metric random walk spaces abstract in this paper we study the heat flow on metric random walk spaces which unifies into a broad framework the heat flow on locally finite weighted connected graphs the heat flow determined by finite markov chains and some nonlocal evolution problems we give different characterizations of the ergodicity and prove that a metric random walk space with positive ollivier ricci curvature is ergodic furthermore we prove a cheeger inequality and as a consequence we show that a poincare inequality holds if and only if an isoperimetric inequality holds we also study the bakry emery curvature dimension condition and its relation with functional inequalities like the poincare inequality and the transport information inequalities

statistical inference under multiterminal data compression this paper presents a survey of the literature on the information theoretic problems of statistical inference under multiterminal data compression with rate constraints significant emphasis is put on problems 1 multiterminal hypothesis testing 2 multiterminal parameter estimation and 3 multiterminal pattern classification in either case of positive rates or zero rates in addition the paper includes three new results i e the converse theorems for all problems of multiterminal hypothesis testing multiterminal parameter estimation and multiterminal pattern classification at the zero rate

a simple proof of the blowing up lemma corresp the blowing up lemma says that if the probability with respect to a product measure of a set a subseteq cal x n cal x finite n large is not exponentially small then its l n neighborhood has probability almost one for some l n o n here an information theoretic proof of the blowing up lemma generalizing it to continuous alphabets is given

capacity upper bounds for the relay channel via reverse hypercontractivity we revisit the primitive relay channel introduced by cover in 1987 recent work derived upper bounds on the capacity of this channel that are tighter than the classical cutset bound using the concentration of measure in this paper we recover generalize and improve upon some of these upper bounds with simpler proofs using reverse hypercontractivity to our knowledge this is the first application of reverse hypercontractivity in proving first order converses in network information theory

poincare type inequalities for group measure spaces and related transportation cost inequalities let g be a countable discrete group with an orthogonal representation on a real hilbert space h we prove lp poincare inequalities for the group measure space l h g where both the group action and the gaussian measure space h are associated with the representation the idea of proof comes from pisier s method on the boundedness of riesz transform and lust piquard s work on spin systems then we deduce a transportation type inequality from the lp poincare inequalities in the general noncommutative setting this inequality is sharp up to a constant in the gaussian setting several applications are given including wiener rademacher chaos estimation and new examples of rieffel s compact quantum metric spaces

an on line universal lossy data compression algorithm via continuous codebook refinement i basic results a new on line universal lossy data compression algorithm is presented for finite memoryless sources with unknown statistics its performance asymptotically approaches the fundamental rate distortion limit the codebook is generated on the fly and continuously adapted by simple rules there is no separate codebook training or codebook transmission candidate codewords are randomly generated according to an arbitrary and possibly suboptimal distribution through a carefully designed gold washing or information theoretic sieve mechanism good codewords and only good codewords are promoted to permanent status with high probability we also determine the rate at which our algorithm approaches the fundamental limit

multiple description source coding with no excess marginal rate multiple description source coding concerns situations in which the transmission of the source information is distributed over two data streams at rates r sub 1 and r sub 2 respectively when both data streams are received the decoder uses the combined data at rate r sub 1 r sub 2 to reconstruct the source information with average distortion d sub 0 if a communication breakdown prevents one of the data streams from reaching the receiver the decoder has to base its reconstruction solely on the available data at rate either r sub 1 or r sub 2 this results in a higher distortion of either d sub 1 or d sub 2 respectively the region spl rscr of all achievable quintuples r sub 1 r sub 2 d sub 0 d sub 1 d sub 2 has been determined in the so called no excess rate sum case defined by imposing the requirement r sub 1 r sub 2 r d sub 0 where r spl middot is the rate distortion function of the source the case with excess rate sum characterized by r sub 1 r sub 2 r d sub 0 is challenging we study in this paper a special case of it in which the requirements r sub t r d sub t t 1 2 are imposed we refer to this as the no excess marginal rate case the lower and upper bounds on d sub 0 we obtain are separated by only a tiny gap when evaluated for a binary equiprobable source and the hamming distortion measure

random regularization of brown spectral measure abstract we generalize a recent result of haagerup namely we show that a convolution with a standard gaussian random matrix regularizes the behavior of fuglede kadison determinant and brown spectral distribution measure in this way it is possible to establish a connection between the limit eigenvalues distributions of a wide class of random matrices and the brown measure of the corresponding limits

strong converse using change of measure arguments the strong converse for a coding theorem shows that the optimal asymptotic rate possible with vanishing error cannot be improved by allowing a fixed error building on a method introduced by gu and effros for centralized coding problems we develop a general and simple recipe for proving strong converse that is applicable for distributed problems as well heuristically our proof of strong converse mimics the standard steps for proving a weak converse except that we apply those steps to a modified distribution obtained by conditioning the original distribution on the event that no error occurs a key component of our recipe is the replacement of the hard markov constraints implied by the distributed nature of the problem with a soft information cost using a variational formula introduced by oohama we illustrate our method by providing a short proof of the strong converse for the wyner ziv problem and strong converse theorems for interactive function computation common randomness and secret key agreement and the wiretap channel the latter three strong converse problems were open prior to this work

a large deviation approach to some transportation cost inequalities new transportation cost inequalities are derived by means of elementary large deviation reasonings their dual characterization is proved this provides an extension of a well known result of s bobkov and f gotze their tensorization properties are investigated sufficient conditions and necessary conditions too for these inequalities are stated in terms of the integrability of the reference measure applying these results leads to new deviation results concentration of measure and deviations of empirical processes

the redundancy of source coding with a fidelity criterion 1 known statistics the problem of redundancy of source coding with respect to a fidelity criterion is considered for any fixed rate r 0 and any memoryless source with finite source and reproduction alphabets and a common distribution p the nth order distortion redundancy d sub n r of fixed rate coding is defined as the minimum of the difference between the expected distortion per symbol of any block code with length n and rate r and the distortion rate function d p r of the source p it is demonstrated that for sufficiently large n d sub n r is equal to spl part spl part r d p r ln n 2n o ln n n where spl part spl part r d p r is the partial derivative of d p r evaluated at r and assumed to exist for any fixed distortion level d 0 and any memoryless source p the nth order rate redundancy r sub n d of coding at fixed distortion level d or by using d semifaithful codes is defined as the minimum of the difference between the expected rate per symbol of any d semifaithful code of length n and the rate distortion function r p d of p evaluated at d it is proved that for sufficiently large n r sub n d is upper bounded by ln n n o ln n n and lower bounded by in n 2n o in n n as a by product the lower bound of r sub n d derived in this paper gives a positive answer to a conjecture proposed by yu and speed 1993

wasserstein continuity of entropy and outer bounds for interference channels it is shown that under suitable regularity conditions differential entropy is o sqrt n lipschitz as a function of probability distributions on mathbb r n with respect to the quadratic wasserstein distance under similar conditions discrete shannon entropy is shown to be o n lipschitz in distributions over the product space with respect to ornstein s bar d distance wasserstein distance corresponding to the hamming distance these results together with talagrand s and marton s transportation information inequalities allow one to replace the unknown multi user interference with its independent identically distributed approximations as an application a new outer bound for the two user gaussian interference channel is proved which in particular settles the missing corner point problem of costa 1985