Large Deviation Inequality Research Articles

A large deviation inequality for the rank of a random matrix

A large deviation inequality for the rank of a random matrix

Exponential Decay of Intersection Volume With Applications on List-Decodability and Gilbert-Varshamov Type Bound

We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) wellspread, and (iii) a certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of balls in Hamming, Johnson spaces and symmetric groups decay exponentially as their centers drift apart. To verify condition (iii), we prove some large deviation inequalities ‘on a slice’ for functions with Lipschitz conditions. We then use these estimates on intersection volumes to ∙ obtain a sharp lower bound on list-decodability of random <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary codes, confirming a conjecture of Li and Wootters; and ∙ improve the classical bound of Levenshtein from 1971 on constant weight codes by a factor linear in dimension, resolving a problem raised by Jiang and Vardy. Our probabilistic point of view also offers a unified framework to obtain improvements on other Gilbert-Varshamov type bounds, giving conceptually simple and calculation-free proofs for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary codes, permutation codes, and spherical codes. Another consequence is a counting result on the number of codes, showing ampleness of large codes.

Random sampling in reproducing kernel spaces with mixed norm

In this work we consider random sampling of signals in (in)finite-dimensional reproducing kernel spaces with mixed norm. Here the random sampling refers to randomly taken sampling positions according to some probability measure. We study the stability of random sampling procedure by establishing sampling inequality that holds with high probability when the sampling size is large. We establish the probabilistic sampling inequality though a combination of mathematical analysis and probabilistic analysis. The main tools we use are covering number of signal (function) space and (uniform) large deviation inequality for a sequence of random variables. We provide a concise proof and our proof leads to explicit and transparent estimates involved in the probability with which the sampling inequality holds.

Large deviation inequalities of Bayesian estimator in nonlinear regression models

Large deviation inequalities of Bayesian estimator in nonlinear regression models

Outage Constrained Robust Beamforming Optimization for Multiuser IRS-Assisted Anti-Jamming Communications With Incomplete Information

Malicious jamming attacks have been regarded as a serious threat to Internet of Things (IoT) networks, which can significantly degrade the Quality of Service (QoS) of users. This article utilizes an intelligent reflecting surface (IRS) to enhance anti-jamming performance due to its capability in reconfiguring the wireless propagation environment via dynamically adjusting each IRS reflecting elements. To enhance the communication performance against jamming attacks, a robust beamforming optimization problem is formulated in a multiuser IRS-assisted anti-jamming communications scenario with or without imperfect jammer’s channel state information (CSI). In addition, we further consider the fact that the jammer’s transmit beamforming can not be known at BS. Specifically, with no knowledge of jammers transmit beamforming, the total transmit power minimization problems are formulated subject to the outage probability requirements of legitimate users with the jammer’s statistical CSI, and signal-to-interference-plus-noise ratio requirements of legitimate users without the jammer’s CSI, respectively. By applying the decomposition-based large deviation inequality, Bernstein-type inequality, Cauchy–Schwarz inequality, and penalty nonsmooth optimization method, we efficiently solve the initial intractable and nonconvex problems. Numerical simulations demonstrate that the proposed anti-jamming approaches achieve superior anti-jamming performance and lower power-consumption compared to the non-IRS scheme and reveal the impact of key parameters on the achievable system performance.

Quasi-likelihood analysis and its applications

The Ibragimov–Khasminskii theory established a scheme that gives asymptotic properties of the likelihood estimators through the convergence of the likelihood ratio random field. This scheme is extending to various nonlinear stochastic processes, combined with a polynomial type large deviation inequality proved for a general locally asymptotically quadratic quasi-likelihood random field. We give an overview of the quasi-likelihood analysis and its applications to ergodic/non-ergodic statistics for stochastic processes.

Outage Constraint Robust Cooperative Beamforming and Jamming Design for Secure AF Relay Networks

This paper investigates a joint robust scheme in a secrecy relay network where distributed relays perform cooperative beamforming and a friendly jammer transmits jamming signal to enhance the information security. Specifically, we consider the outage constraint secrecy rate maximization design with imperfect channel state information. Through semidefinite relaxation and one-dimensional search, we propose a two-layer optimization method to solve the nonconvex problem. In addition, the Bernstein-type inequality and large deviation inequality are utilized to convert the probabilistic constraint. Simulation results demonstrate the performance of the proposed design.

Outage-Constrained Robust Multigroup Multicast Beamforming for Satellite-Based Internet of Things Coexisting With Terrestrial Networks

Satellite-based Internet of Things (IoT) is recognized as a cost-effective approach for global access. In this article, we aim at improving the spectrum efficiency of satellite systems to serve a huge number of IoT devices. To this end, we present a cognitive satellite-terrestrial framework, where a multibeam satellite system with full frequency reuse shares the spectrum with terrestrial networks based on the underlay paradigm. Considering DVB-S2X recommendations, geometric configurations, and channel characteristics, we investigate a robust multigroup multicast beamforming design for the satellite-based IoT coexisting with terrestrial networks in the presence of a phase error on channel state information, and characterize the achievable rate region under the outage probability constraint for the terminal and the power consumption constraint for the satellite. Based on the concept of rate profile, an associated optimization problem is formulated to design robust beamformers and determine the Pareto boundary of the region. To solve the intractable problem, we propose a two-level iterative algorithm on the basis of joint bisection search and penalty function enabled nonsmooth optimization. In particular, we develop a Bernstein-type inequality aided method and a large deviation inequality aided method to obtain a tractable and conservative approximation for the probabilistic constraint, respectively. Numerical results are provided to confirm the validity and superiority of our proposed scheme over the existing approaches and reveal the impact of key parameters on the achievable system performance.

Group Fairness in Committee Selection

In this article, we study fairness in committee selection problems. We consider a general notion of fairness via stability: A committee is stable if no coalition of voters can deviate and choose a committee of proportional size, so that all these voters strictly prefer the new committee to the existing one. Our main contribution is to extend this definition to stability of a distribution (or lottery) over committees. We consider two canonical voter preference models: the A pproval S et setting where each voter approves a set of candidates and prefers committees with larger intersection with this set; and the R anking R epresentative setting where each voter ranks committees based on how much she likes her favorite candidate in a committee. Our main result is to show that stable lotteries always exist for these canonical preference models. Interestingly, given preferences of voters over committees, the procedure for computing an approximately stable lottery is the same for both models and therefore extends to the setting where some voters have the former preference structure and others have the latter. Our existence proof uses the probabilistic method and a new large deviation inequality that may be of independent interest.

Strict Inequality for the Chemical Distance Exponent in Two‐Dimensional Critical Percolation

AbstractWe provide the first nontrivial upper bound for the chemical distance exponent in two‐dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length n in critical percolation on ℤ2 is bounded by Cn2 − δπ3(n) for some δ &gt; 0, where π3(n) is the “three‐arm probability to distance n.” This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of n with high probability. In the case of site percolation on the triangular lattice, we obtain a strict upper bound for the exponent of 4/3.The proof builds on the strategy developed in our previous paper, but with a new iterative scheme, and a new large deviation inequality for events in annuli conditional on arm events, which may be of independent interest. © 2020 Wiley Periodicals LLC

Large deviation inequalities of LS estimator in nonlinear regression models

In the paper, the large deviation inequalities of the LS estimator for the nonlinear regression model with martingale differences errors are established. The assumptions for the errors are (conditional) exponential integrability which weaken the bounded condition in Hu (1993). As an application, we give the large deviation inequalities of LS estimator for the simple Michaelis–Menten model.

Quasi-likelihood analysis and Bayes-type estimators of an ergodic diffusion plus noise

We consider adaptive maximum-likelihood-type estimators and adaptive Bayes-type ones for discretely observed ergodic diffusion processes with observation noise whose variance is constant. The quasi-likelihood functions for the diffusion and drift parameters are introduced and the polynomial-type large deviation inequalities for those quasi-likelihoods are shown to see the asymptotic properties of the adaptive Bayes-type estimators and the convergence of moments for both adaptive maximum-likelihood-type estimators and adaptive Bayes-type ones.

On general maximum likelihood empirical Bayes estimation of heteroscedastic IID normal means

We propose a general maximum likelihood empirical Bayes (GMLEB) method for the heteroscedastic normal means estimation with known variances. The idea is to plug the generalized maximum likelihood estimator in the oracle Bayes rule. From the point of view of restricted empirical Bayes, the general empirical Bayes aims at a benchmark risk smaller than the linear empirical Bayes methods when the unknown means are i.i.d. variables. We prove an oracle inequality which states that under mild conditions, the regret of the GMLEB is of smaller order than $(\log n)^{5}/n$. The proof is based on a large deviation inequality for the generalized maximum likelihood estimator. The oracle inequality leads to the property that the GMLEB is adaptive minimax in $L_{p}$ balls when the order of the norm of the ball is larger than $((\log n)^{5/2}/\sqrt{n})^{1/(p\wedge 2)}$. We demonstrate the superb risk performance of the GMLEB through simulation experiments.

Outage Constrained Robust Beamforming for Sum Rate Maximization in Multi-Beam Satellite Systems

This letter investigates robust beamforming for multi-beam satellite communications in face of a phase error of channel state information. From the perspective of a practical system, we first formulate a sum rate maximization problem under the outage probability constraint of each user and the power consumption constraint of each individual feed. To handle the challenging problem, we characterize the Pareto frontier of the outage constrained achievable rate region based on a rate-profile method. In particular, a conservative approximation of the probabilistic constraint is achieved by means of the Taylor series expansion and the large deviation inequality. Numerical results validate the effectiveness and superiority of our proposed scheme compared to the existing approaches and give an insight into the impact of on-board payloads on robust beamforming.

Robust secure transmission for wireless information and power transfer in heterogeneous networks

Here, the authors investigate robust secure transmission for wireless information and power transfer in heterogeneous networks (HeNets) under the deterministic and stochastic CSI errors, respectively, where one macrocell base station (MBS) and one femtocell base station (FBS) respectively serve to macrocell users (MUs) and femtocell users (FUs). Meanwhile, FUs acquire energy from the FBS and multiple eavesdroppers attempt to wiretap its downlink information. To secure the wireless information and power transfer, the authors propose an artificial noise (AN)-aided transmission design. Under the deterministic CSI error, the authors jointly design the beamforming and AN vectors of the MBS and FBS to minimise the total transmit power, while satisfying the constraints on the signal-to-interference plus noise (SINR) requirement at each MU, energy harvesting (EH) and secrecy outage probability requirement at each FU. This formulated problem is non-convex. Utilising S-procedure and successive convex approximation technique, the authors successfully transfer it into a series of convex problems. Under the stochastic CSI error, the joint design problem is formulated with a range of outage probabilistic constraints. Further, the authors, respectively, resort to large deviation and Bernstein-type inequalities, S-procedure to convert it into solvable forms. Simulation results demonstrate the effectiveness of the authors' proposed design.

Robust Beamforming for Multibeam Satellite Communication in the Face of Phase Perturbations

A precoding scheme is proposed for the downlink of multibeam satellite communication in the face of phase perturbations. In order to alleviate the signal-to-interference and noise ratio degradation induced by phase uncertainty and to minimize the transmission power consumption, we formulated the beamforming problem as a chance-constrained optimization. Moreover, we provided a large deviation inequality aided conservative approximation for the chance-constraints, followed by conceiving a tractable solution. Our simulation results showed the accuracy and the efficiency of our proposed approximation algorithm compared to the benchmarks.

Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋) (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a&lt;⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0&lt;β&lt;2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

Optimal rate for covariance operator estimators of functional autoregressive processes with random coefficients

We improve a result of Allam and Mourid (2014) by deriving the optimal n rate for the empirical covariance operators of a Hilbert-valued autoregressive process with random coefficients. Our approach is based on a suitable autoregressive representation of a sequence of covariance operators related to the model, which leads to a decomposition with Hilbert-valued martingale differences. Using large deviation inequalities for Hilbert-valued martingale differences, we then establish exponential bounds and derive the almost sure convergence of the empirical covariance operators in the Hilbert–Schmidt norm, achieving the parametric rate n up to a ln(n) factor in the bounded process case.

On the asymptotic properties of Bayes-type estimators with general loss functions

We study the asymptotic behavior of Bayes-type estimators and give sufficient conditions to obtain the asymptotic limit distribution of the estimation error. We assume so called polynomial-type large deviation inequalities and prove the asymptotic equivalence of the estimation errors of Bayes-type and M-estimators by virtue of Ibragimov–Has’minskiĭ theory. The results can be applied to several cases such as diffusion processes and jump diffusion processes. In this paper, we focus on the application to ergodic diffusion processes and ergodic jump diffusion processes, demonstrating asymptotic normality and convergence of moments for Bayes-type estimators with general loss functions.

Partial quasi-likelihood analysis

The quasi-likelihood analysis is generalized to the partial quasi-likelihood analysis. Limit theorems for the quasi-likelihood estimators, especially the quasi-Bayesian estimator, are derived in the situation where existence of a slow mixing component prohibits the Rosenthal type inequality from applying to the derivation of the polynomial type large deviation inequality for the statistical random field. We give two illustrative examples.