arXiv subjects

Oliver Johnson

Publications and source records attributed to Oliver Johnson.

At least 37 records · Page 2Linked to original sources

arXiv2018

An extremal property of the normal distribution, with a discrete analog

We prove, using the Brascamp-Lieb inequality, that the Gaussian measure is the only strong log-concave measure having a strong log-concavity parameter equal to its covariance matrix. We also give a similar characterization of the Poisson measure in the discrete case, using "Chebyshev's other inequality". We briefly discuss how these results relate to Stein and Stein-Chen methods for Gaussian and Poisson approximation, and to the Bakry-Emery calculus.

math.PR↗

arXiv2018

Performance of group testing algorithms with near-constant tests-per-item

We consider the nonadaptive group testing with N items, of which $K = Θ(N^θ)$ are defective. We study a test design in which each item appears in nearly the same number of tests. For each item, we independently pick L tests uniformly at random with replacement, and place the item in those tests. We analyse the performance of these designs with simple and practical decoding algorithms in a range of sparsity regimes, and show that the performance is consistently improved in comparison with standard Bernoulli designs. We show that our new design requires 23% fewer tests than a Bernoulli design when paired with the simple decoding algorithms known as COMP and DD. This gives the best known nonadaptive group testing performance for $θ> 0.43$, and the best proven performance with a practical decoding algorithm for all $θ\in (0,1)$. We also give a converse result showing that the DD algorithm is optimal for these designs when $θ> 1/2$.

cs.IT↗

arXiv2018

A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation

In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically does not give a sharp converse result (error lower bound) for channel coding problems. Moreover, recent work has shown that an argument based on binary hypothesis testing gives tighter results. We adapt this technique to the statistical setting, and argue that Fano's inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case.

cs.IT↗

arXiv2017

Relaxation of monotone coupling conditions: Poisson approximation and beyond

It is well-known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also give explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.

math.PR↗

arXiv2017

A de Bruijn identity for discrete random variables

We discuss properties of the "beamsplitter addition" operation, which provides a non-standard scaled convolution of random variables supported on the non-negative integers. We give a simple expression for the action of beamsplitter addition using generating functions. We use this to give a self-contained and purely classical proof of a heat equation and de Bruijn identity, satisfied when one of the variables is geometric.

cs.IT↗

arXiv2016

A discrete log-Sobolev inequality under a Bakry-Emery type condition

We consider probability mass functions $V$ supported on the positive integers using arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry--Émery condition for a Markov birth and death operator with invariant measure $V$. Under this condition, we prove a modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.

math.PR↗

arXiv2016

Entropy and thinning of discrete random variables

We describe five types of results concerning information and concentration of discrete random variables, and relationships between them, motivated by their counterparts in the continuous case. The results we consider are information theoretic approaches to Poisson approximation, the maximum entropy property of the Poisson distribution, discrete concentration (Poincaré and logarithmic Sobolev) inequalities, monotonicity of entropy and concavity of entropy in the Shepp--Olkin regime.

math.PR↗

arXiv2016

Improved group testing rates with constant column weight designs

We consider nonadaptive group testing where each item is placed in a constant number of tests. The tests are chosen uniformly at random with replacement, so the testing matrix has (almost) constant column weights. We show that performance is improved compared to Bernoulli designs, where each item is placed in each test independently with a fixed probability. In particular, we show that the rate of the practical COMP detection algorithm is increased by 31% in all sparsity regimes. In dense cases, this beats the best possible algorithm with Bernoulli tests, and in sparse cases is the best proven performance of any practical algorithm. We also give an algorithm-independent upper bound for the constant column weight case; for dense cases this is again a 31% increase over the analogous Bernoulli result.

cs.IT↗

arXiv2016

Discrete versions of the transport equation and the Shepp-Olkin conjecture

We introduce a framework to consider transport problems for integer-valued random variables. We introduce weighting coefficients which allow us to characterize transport problems in a gradient flow setting, and form the basis of our introduction of a discrete version of the Benamou-Brenier formula. Further, we use these coefficients to state a new form of weighted log-concavity. These results are applied to prove the monotone case of the Shepp-Olkin entropy concavity conjecture.

math.PR↗

arXiv2016

Interference Management in Heterogeneous Networks with Blind Transmitters

Future multi-tier communication networks will require enhanced network capacity and reduced overhead. In the absence of Channel State Information (CSI) at the transmitters, Blind Interference Alignment (BIA) and Topological Interference Management (TIM) can achieve optimal Degrees of Freedom (DoF), minimising network's overhead. In addition, Non-Orthogonal Multiple Access (NOMA) can increase the sum rate of the network, compared to orthogonal radio access techniques currently adopted by 4G networks. Our contribution is two interference management schemes, BIA and a hybrid TIM-NOMA scheme, employed in heterogeneous networks by applying user-pairing and Kronecker Product representation. BIA manages inter- and intra-cell interference by antenna selection and appropriate message scheduling. The hybrid scheme manages intra-cell interference based on NOMA and inter-cell interference based on TIM. We show that both schemes achieve at least double the rate of TDMA. The hybrid scheme always outperforms TDMA and BIA in terms of Degrees of Freedom (DoF). Comparing the two proposed schemes, BIA achieves more DoF than TDMA under certain restrictions, and provides better Bit-Error-Rate (BER) and sum rate performance to macrocell users, whereas the hybrid scheme improves the performance of femtocell users.

cs.IT↗

arXiv2016

Thinning and Information Projections

In this paper we establish lower bounds on information divergence of a distribution on the integers from a Poisson distribution. These lower bounds are tight and in the cases where a rate of convergence in the Law of Thin Numbers can be computed the rate is determined by the lower bounds proved in this paper. General techniques for getting lower bounds in terms of moments are developed. The results about lower bound in the Law of Thin Numbers are used to derive similar results for the Central Limit Theorem.

math.PR↗

arXiv2015

Strong converses for group testing in the finite blocklength regime

We prove new strong converse results in a variety of group testing settings, generalizing a result of Baldassini, Johnson and Aldridge. These results are proved by two distinct approaches, corresponding to the non-adaptive and adaptive cases. In the non-adaptive case, we mimic the hypothesis testing argument introduced in the finite blocklength channel coding regime by Polyanskiy, Poor and Verdú. In the adaptive case, we combine a formulation based on directed information theory with ideas of Kemperman, Kesten and Wolfowitz from the problem of channel coding with feedback. In both cases, we prove results which are valid for finite sized problems, and imply capacity results in the asymptotic regime. These results are illustrated graphically for a range of models.

cs.IT↗

arXiv2015

Distributed Wideband Spectrum Sensing

We consider the problem of reconstructing wideband frequency spectra from distributed, compressive measurements. The measurements are made by a network of nodes, each independently mixing the ambient spectra with low frequency, random signals. The reconstruction takes place via local transmissions between nodes, each performing simple statistical operations such as ridge regression and shrinkage.

cs.IT↗

arXiv2015

A proof of the Shepp-Olkin entropy concavity conjecture

We prove the Shepp--Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof is a refinement of an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin's original conjecture, to consider Renyi and Tsallis entropies.

math.PR↗

arXiv2015

A hybrid TIM-NOMA scheme for the SISO Broadcast Channel

Future mobile communication networks will require enhanced network efficiency and reduced system overhead due to their user density and high data rate demanding applications of the mobile devices. Research on Blind Interference Alignment (BIA) and Topological Interference Management (TIM) has shown that optimal Degrees of Freedom (DoF) can be achieved, in the absence of Channel State Information (CSI) at the transmitters, reducing the network's overhead. Moreover, the recently emerged Non-Orthogonal Multiple Access (NOMA) scheme suggests a different multiple access approach, compared to the current orthogonal methods employed in 4G networks, resulting in high capacity gains. Our contribution is a hybrid TIM-NOMA scheme in Single-Input-Single-Output (SISO) K-user cells, in which users are divided into T groups, and 1/T DoF is achieved for each user. By superimposing users in the power domain, we introduce a two-stage decoding process, managing 'inter-group' interference based on the TIM principles, and 'intra-group' interference based on Successful Interference Cancellation (SIC), as proposed by NOMA. We show that for high SNR values the hybrid scheme can improve the sum rate by at least 100% when compared to Time Division Multiple Access (TDMA).

cs.IT↗

arXiv2014

The capacity of non-identical adaptive group testing

We consider the group testing problem, in the case where the items are defective independently but with non-constant probability. We introduce and analyse an algorithm to solve this problem by grouping items together appropriately. We give conditions under which the algorithm performs essentially optimally in the sense of information-theoretic capacity. We use concentration of measure results to bound the probability that this algorithm requires many more tests than the expected number. This has applications to the allocation of spectrum to cognitive radios, in the case where a database gives prior information that a particular band will be occupied.

cs.IT↗

arXiv2014

Blind Interference Alignment in General Heterogeneous Networks

Heterogeneous networks have a key role in the design of future mobile communication networks, since the employment of small cells around a macrocell enhances the network's efficiency and decreases complexity and power demand. Moreover, research on Blind Interference Alignment (BIA) has shown that optimal Degrees of Freedom (DoF) can be achieved in certain network architectures, with no requirement of Channel State Information (CSI) at the transmitters. Our contribution is a generalised model of BIA in a heterogeneous network with one macrocell with K users and K femtocells each with one user, by using Kronecker (Tensor) Product representation. We introduce a solution on how to vary beamforming vectors under power constraints to maximize the sum rate of the network and how optimal DoF can be achieved over K+1 time slots.

cs.IT↗

arXiv2013

Group testing algorithms: bounds and simulations

We consider the problem of non-adaptive noiseless group testing of $N$ items of which $K$ are defective. We describe four detection algorithms: the COMP algorithm of Chan et al.; two new algorithms, DD and SCOMP, which require stronger evidence to declare an item defective; and an essentially optimal but computationally difficult algorithm called SSS. By considering the asymptotic rate of these algorithms with Bernoulli designs we see that DD outperforms COMP, that DD is essentially optimal in regimes where $K \geq \sqrt N$, and that no algorithm with a nonadaptive Bernoulli design can perform as well as the best non-random adaptive designs when $K > N^{0.35}$. In simulations, we see that DD and SCOMP far outperform COMP, with SCOMP very close to the optimal SSS, especially in cases with larger $K$.

cs.IT↗