arXiv subjects

Oliver Johnson

Publications and source records attributed to Oliver Johnson.

At least 55 records · Page 3Linked to original sources

arXiv2013

A de Bruijn identity for symmetric stable laws

We show how some attractive information--theoretic properties of Gaussians pass over to more general families of stable densities. We define a new score function for symmetric stable laws, and use it to give a stable version of the heat equation. Using this, we derive a version of the de Bruijn identity, allowing us to write the derivative of relative entropy as an inner product of score functions. We discuss maximum entropy properties of symmetric stable densities.

cs.IT↗

arXiv2013

The Capacity of Adaptive Group Testing

We define capacity for group testing problems and deduce bounds for the capacity of a variety of noisy models, based on the capacity of equivalent noisy communication channels. For noiseless adaptive group testing we prove an information-theoretic lower bound which tightens a bound of Chan et al. This can be combined with a performance analysis of a version of Hwang's adaptive group testing algorithm, in order to deduce the capacity of noiseless and erasure group testing models.

cs.IT↗

arXiv2012

Bounds on the Poincare constant under negative dependence

We give bounds on the Poincare (inverse spectral gap) constant of a non-negative, integer-valued random variable W, under negative dependence assumptions such as ultra log-concavity and total negative dependence. We show that the bounds obtained compare well to others in the literature. Examples treated include some occupancy and urn models, a random graph model and small spacings on the circumference of a circle. Applications to Poisson convergence theorems are considered.

math.PR↗

arXiv2012

Delay-rate tradeoff in ergodic interference alignment

Ergodic interference alignment, as introduced by Nazer et al (NGJV), is a technique that allows high-rate communication in n-user interference networks with fast fading. It works by splitting communication across a pair of fading matrices. However, it comes with the overhead of a long time delay until matchable matrices occur: the delay is q^n^2 for field size q. In this paper, we outline two new families of schemes, called JAP and JAP-B, that reduce the expected delay, sometimes at the cost of a reduction in rate from the NGJV scheme. In particular, we give examples of good schemes for networks with few users, and show that in large n-user networks, the delay scales like q^T, where T is quadratic in n for a constant per-user rate and T is constant for a constant sum-rate. We also show that half the single-user rate can be achieved while reducing NGJV's delay from q^n^2 to q^(n-1)(n-2). This extended version includes complete proofs and more details of good schemes for small n.

cs.IT↗

arXiv2011

Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures

Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, {\em Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-trivial extension of this semigroup approach that the natural analog of the Poisson maximum entropy property remains valid if the compound Poisson distributions under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. Sufficient conditions for compound distributions to be log-concave are discussed and applications to combinatorics are examined; new bounds are derived on the entropy of the cardinality of a random independent set in a claw-free graph, and a connection is drawn to Mason's conjecture for matroids. The present results are primarily motivated by the desire to provide an information-theoretic foundation for compound Poisson approximation and associated limit theorems, analogous to the corresponding developments for the central limit theorem and for Poisson approximation. Our results also demonstrate new links between some probabilistic methods and the combinatorial notions of log-concavity and ultra-log-concavity, and they add to the growing body of work exploring the applications of maximum entropy characterizations to problems in discrete mathematics.

math.CO↗

arXiv2011

Non-parametric change-point detection using string matching algorithms

Given the output of a data source taking values in a finite alphabet, we wish to detect change-points, that is times when the statistical properties of the source change. Motivated by ideas of match lengths in information theory, we introduce a novel non-parametric estimator which we call CRECHE (CRossings Enumeration CHange Estimator). We present simulation evidence that this estimator performs well, both for simulated sources and for real data formed by concatenating text sources. For example, we show that we can accurately detect the point at which a source changes from a Markov chain to an IID source with the same stationary distribution. Our estimator requires no assumptions about the form of the source distribution, and avoids the need to estimate its probabilities. Further, we establish consistency of the CRECHE estimator under a related toy model, by establishing a fluid limit and using martingale arguments.

math.PR↗

arXiv2011

A natural derivative on [0,n] and a binomial Poincaré inequality

We consider probability measures supported on a finite discrete interval $[0,n]$. We introduce a new finitedifference operator $\nabla_n$, defined as a linear combination of left and right finite differences. We show that this operator $\nabla_n$ plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures.

math.PR↗

arXiv2010

Note on Noisy Group Testing: Asymptotic Bounds and Belief Propagation Reconstruction

An information theoretic perspective on group testing problems has recently been proposed by Atia and Saligrama, in order to characterise the optimal number of tests. Their results hold in the noiseless case, where only false positives occur, and where only false negatives occur. We extend their results to a model containing both false positives and false negatives, developing simple information theoretic bounds on the number of tests required. Based on these bounds, we obtain an improved order of convergence in the case of false negatives only. Since these results are based on (computationally infeasible) joint typicality decoding, we propose a belief propagation algorithm for the detection of defective items and compare its actual performance to the theoretical bounds.

cs.IT↗

arXiv2010

Monotonicity, thinning and discrete versions of the Entropy Power Inequality

We consider the entropy of sums of independent discrete random variables, in analogy with Shannon's Entropy Power Inequality, where equality holds for normals. In our case, infinite divisibility suggests that equality should hold for Poisson variables. We show that some natural analogues of the Entropy Power Inequality do not in fact hold, but propose an alternative formulation which does always hold. The key to many proofs of Shannon's Entropy Power Inequality is the behaviour of entropy on scaling of continuous random variables. We believe that Rényi's operation of thinning discrete random variables plays a similar role to scaling, and give a sharp bound on how the entropy of ultra log-concave random variables behaves on thinning. In the spirit of the monotonicity results established by Artstein, Ball, Barthe and Naor, we prove a stronger version of concavity of entropy, which implies a strengthened form of our discrete Entropy Power Inequality.

cs.IT↗

arXiv2010

Asymptotic Sum-Capacity of Random Gaussian Interference Networks Using Interference Alignment

We consider a dense n-user Gaussian interference network formed by paired transmitters and receivers placed independently at random in Euclidean space. Under natural conditions on the node position distributions and signal attenuation, we prove convergence in probability of the average per-user capacity C_Sigma/n to 1/2 E log(1 + 2SNR). The achievability result follows directly from results based on an interference alignment scheme presented in recent work of Nazer et al. Our main contribution comes through the converse result, motivated by ideas of `bottleneck links' developed in recent work of Jafar. An information theoretic argument gives a capacity bound on such bottleneck links, and probabilistic counting arguments show there are sufficiently many such links to tightly bound the sum-capacity of the whole network.

cs.IT↗

arXiv2010

Compound Poisson Approximation via Information Functionals

An information-theoretic development is given for the problem of compound Poisson approximation, which parallels earlier treatments for Gaussian and Poisson approximation. Let $P_{S_n}$ be the distribution of a sum $S_n=\Sumn Y_i$ of independent integer-valued random variables $Y_i$. Nonasymptotic bounds are derived for the distance between $P_{S_n}$ and an appropriately chosen compound Poisson law. In the case where all $Y_i$ have the same conditional distribution given $\{Y_i\neq 0\}$, a bound on the relative entropy distance between $P_{S_n}$ and the compound Poisson distribution is derived, based on the data-processing property of relative entropy and earlier Poisson approximation results. When the $Y_i$ have arbitrary distributions, corresponding bounds are derived in terms of the total variation distance. The main technical ingredient is the introduction of two "information functionals," and the analysis of their properties. These information functionals play a role analogous to that of the classical Fisher information in normal approximation. Detailed comparisons are made between the resulting inequalities and related bounds.

math.PR↗

arXiv2009

Interference alignment-based sum capacity bounds for random dense Gaussian interference networks

We consider a dense $K$ user Gaussian interference network formed by paired transmitters and receivers placed independently at random in a fixed spatial region. Under natural conditions on the node position distributions and signal attenuation, we prove convergence in probability of the average per-user capacity $\csum/K$ to $\half \ep \log(1 + 2 \SNR)$. The achievability result follows directly from results based on an interference alignment scheme presented in recent work of Nazer et al. Our main contribution comes through an upper bound, motivated by ideas of `bottleneck capacity' developed in recent work of Jafar. By controlling the physical location of transmitter--receiver pairs, we can match a large proportion of these pairs to form so-called $ε$-bottleneck links, with consequent control of the sum capacity.

cs.IT↗

arXiv2009

Thinning, Entropy and the Law of Thin Numbers

Renyi's "thinning" operation on a discrete random variable is a natural discrete analog of the scaling operation for continuous random variables. The properties of thinning are investigated in an information-theoretic context, especially in connection with information-theoretic inequalities related to Poisson approximation results. The classical Binomial-to-Poisson convergence (sometimes referred to as the "law of small numbers" is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is provided for this limit, and nonasymptotic bounds are also established. This development parallels, in part, the development of Gaussian inequalities leading to the information-theoretic version of the central limit theorem. In particular, a "thinning Markov chain" is introduced, and it is shown to play a role analogous to that of the Ornstein-Uhlenbeck process in connection to the entropy power inequality.

cs.IT↗

arXiv2009

Concavity of entropy under thinning

Building on the recent work of Johnson (2007) and Yu (2008), we prove that entropy is a concave function with respect to the thinning operation T_a. That is, if X and Y are independent random variables on Z_+ with ultra-log-concave probability mass functions, then H(T_a X+T_{1-a} Y)>= a H(X)+(1-a)H(Y), 0 <= a <= 1, where H denotes the discrete entropy. This is a discrete analogue of the inequality (h denotes the differential entropy) h(sqrt(a) X + sqrt{1-a} Y)>= a h(X)+(1-a) h(Y), 0 <= a <= 1, which holds for continuous X and Y with finite variances and is equivalent to Shannon's entropy power inequality. As a consequence we establish a special case of a conjecture of Shepp and Olkin (1981).

cs.IT↗

arXiv2008

Information inequalities and a dependent Central Limit Theorem

We adapt arguments concerning information-theoretic convergence in the Central Limit Theorem to the case of dependent random variables under Rosenblatt mixing conditions. The key is to work with random variables perturbed by the addition of a normal random variable, giving us good control of the joint density and the mixing coefficient. We strengthen results of Takano and of Carlen and Soffer to provide entropy-theoretic, not weak convergence.

math.PR↗

arXiv2008

Theoretical properties of Cook's PFC dimension reduction algorithm for linear regression

We analyse the properties of the Principal Fitted Components (PFC) algorithm proposed by Cook. We derive theoretical properties of the resulting estimators, including sufficient conditions under which they are $\sqrt{n}$-consistent, and explain some of the simulation results given in Cook's paper. We use techniques from random matrix theory and perturbation theory. We argue that, under Cook's model at least, the PFC algorithm should outperform the Principal Components algorithm.

math.ST↗

arXiv2008

On the entropy and log-concavity of compound Poisson measures

Motivated, in part, by the desire to develop an information-theoretic foundation for compound Poisson approximation limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson approximation), this work examines sufficient conditions under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. We show that the natural analog of the Poisson maximum entropy property remains valid if the measures under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. The proofs are largely based on ideas related to the semigroup approach introduced in recent work by Johnson for the Poisson family. Sufficient conditions are given for compound distributions to be log-concave, and specific examples are presented illustrating all the above results.

cs.IT↗

arXiv2006

Log-concavity and the maximum entropy property of the Poisson distribution

We prove that the Poisson distribution maximises entropy in the class of ultra-log-concave distributions, extending a result of Harremoës. The proof uses ideas concerning log-concavity, and a semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is a concave function along this semigroup.

math.PR↗