Deviation Inequalities Research Articles

A large deviation inequality for the rank of a random matrix

A large deviation inequality for the rank of a random matrix

Deviation inequalities for dependent sequences with applications to strong approximations

In this paper, we give precise rates of convergence in the strong invariance principle for stationary sequences of bounded real-valued random variables satisfying weak dependence conditions. One of the main ingredients is a new Fuk-Nagaev type inequality for a class of weakly dependent sequences. We describe also several classes of processes to which our results apply.

U-statistics of growing order and sub-Gaussian mean estimators with sharp constants

This paper addresses the following question: given a sample of i.i.d. random variables with finite variance, can one construct an estimator of the unknown mean that performs nearly as well as if the data were normally distributed? One of the most popular examples achieving this goal is the median of means estimator. However, it is inefficient in a sense that the constants in the resulting bounds are suboptimal. We show that a permutation-invariant modification of the median of means estimator admits deviation guarantees that are sharp up to 1+o(1) factor if the underlying distribution possesses more than \frac{3+\sqrt{5}}{2}\approx 2.62 moments and it is absolutely continuous with respect to the Lebesgue measure. This result yields potential improvements for a variety of algorithms that rely on the median of means estimator as a building block. At the core of our argument are the new deviation inequalities for the U-statistics of order that is allowed to grow with the sample size, a result that could be of independent interest.

Matrix deviation inequality for ℓp-norm

Motivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018), doi:10.1017/9781108231596 of Theorem 11.1.5], we show that this property holds for the [Formula: see text]-norm with [Formula: see text] and i.i.d. ensemble sub-Gaussian matrices, i.e. random matrices with i.i.d. mean-zero, unit variance, sub-Gaussian entries. As a consequence of our result, we establish the Johnson–Lindenstrauss lemma from [Formula: see text]-space to [Formula: see text]-space for all i.i.d. ensemble sub-Gaussian matrices.

Moderate deviations inequalities for Gaussian process regression

AbstractGaussian process regression is widely used to model an unknown function on a continuous domain by interpolating a discrete set of observed design points. We develop a theoretical framework for proving new moderate deviations inequalities on different types of error probabilities that arise in GP regression. Two specific examples of broad interest are the probability of falsely ordering pairs of points (incorrectly estimating one point as being better than another) and the tail probability of the estimation error at an arbitrary point. Our inequalities connect these probabilities to the mesh norm, which measures how well the design points fill the space.

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.

Deviation and concentration inequalities for dynamical systems with subexponential decay of correlations

We obtain large and moderate deviation estimates, as well as concentration inequalities, for a class of nonuniformly expanding maps with stretched exponential decay of correlations. In the large deviation regime, we also exhibit examples showing that the obtained upper bounds are essentially optimal.

Deviation inequalities and Cram&amp;eacute;r-type moderate deviations for non-ergodic bm&amp;alpha;-Brownian bridge process

考虑如下非遍历α-布朗桥过程 dX_t= -α/(T-t)X_t dt + d W_t，X₀=0， t∈[0,T), 其中0<α<1/2, T∈(0,∞)固定，W={W_t: t≥ 0}是标准的布朗运动。利用渐近分析的技巧及多重Wiener-Itô积分的偏差性质，本文研究了二次泛函∫^t₀1/(T-s) X_sd W_s和∫^t₀1/(T-s)² X_s²d s 的偏差不等式以及Cramér-型中偏差。作为应用，得到了对数似然率过程以及参数α极大似然估计的（自正则化）Cramér-型中偏差。

A deviation inequality for increment of a [formula omitted]-Brownian motion under [formula omitted]-expectation and applications

In this paper, we prove a deviation inequality for increments of a G-Brownian motion under G-expectation. As an application, we obtain a functional modulus of continuity for a G-Brownian motion.

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.

Sparse Hanson–Wright inequality for a bilinear form of sub‐Gaussian variables

In this paper, we derive a new version of Hanson–Wright inequality for a sparse bilinear form of sub‐Gaussian variables. Our results are a generalization of previous deviation inequalities that consider either sparse quadratic forms or dense bilinear forms. We apply the new concentration inequality to testing the cross‐covariance matrix when data are subject to missing. Using our results, we can find a threshold value of correlations that controls the family‐wise error rate. Furthermore, we discuss the multiplicative measurement error case for the bilinear form with a boundedness condition.

Deviation inequalities and Cramér-type moderate deviations for the explosive autoregressive process

This paper concerns the asymptotic properties of the quadratic functionals and associated ordinary least squares estimator in the explosive first-order Gaussian autoregressive process. By the deviation inequalities for multiple Wiener-Itô integrals and asymptotic analysis techniques, Cramér-type moderate deviations are achieved under the explosive and mildly explosive frameworks. As applications, the global and local powers for the unit root test are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results.

Adaptive invariant density estimation for continuous-time mixing Markov processes under sup-norm risk

Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes and in particular hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical statistical procedures beyond the beautiful, but limited framework available in the literature. To close this gap, we identify $\beta$-mixing of the process and heat kernel bounds on the transition density as a suitable combination to obtain $\sup$-norm and $L^2$ kernel invariant density estimation rates matching the case of reversible multidimenisonal diffusion processes and outperforming density estimation based on discrete i.i.d. or weakly dependent data. Moreover, we demonstrate how up to $\log$-terms, optimal $\sup$-norm adaptive invariant density estimation can be achieved within our general framework based on tight uniform moment bounds and deviation inequalities for empirical processes associated to additive functionals of Markov processes. The underlying assumptions are verifiable with classical tools from stability theory of continuous time Markov processes and PDE techniques, which opens the door to evaluate statistical performance for a vast amount of Markov models. We highlight this point by showing how multidimensional jump SDEs with L\'evy driven jump part under different coefficient assumptions can be seamlessly integrated into our framework, thus establishing novel adaptive $\sup$-norm estimation rates for this class of processes.

Large deviation inequalities of Bayesian estimator in nonlinear regression models

Large deviation inequalities of Bayesian estimator in nonlinear regression models

An elementary analysis of ridge regression with random design

In this note, we provide an elementary analysis of the prediction error of ridge regression with random design. The proof is short and self-contained. In particular, it bypasses the use of Rudelson's deviation inequality for covariance matrices, through a combination of exchangeability arguments, matrix perturbation and operator convexity.

Cramér-type moderate deviations for the log-likelihood ratio of inhomogeneous Ornstein–Uhlenbeck processes

We consider the Cramér-type moderate deviations for the log-likelihood ratio of the inhomogeneous Ornstein–Uhlenbeck processes in the stationary and explosive cases. The relative error of tail probability of the log-likelihood ratio is quantified by deviation inequalities for multiple Wiener–Itô integrals and mod-ϕ convergence approach. As the special cases, we get the Cramér-type moderate deviations of the Ornstein–Uhlenbeck process and α-Wiener bridge.

An embedding of the Morse boundary in the Martin boundary

We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic.

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.

Deviation properties for linear self-attracting diffusion process and applications

In this paper, we study explicitly the deviation properties, including the deviation inequalities and Cramér-type moderate deviations, for some quadratic functionals of linear self-attracting diffusion process. As applications, Cramér-type moderate deviations for the log-likelihood ratio process and drift parameter estimator are obtained. The main methods consist of the deviation inequalities and Cramér-type moderate deviations for multiple Wiener–Itô integrals, as well as the asymptotic analysis techniques.

Deviation inequalities for stochastic approximation by averaging

We introduce a class of Markov chains that includes models of stochastic approximation by averaging and non-averaging. Using a martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differences. Finally, we apply these inequalities to stochastic approximation by averaging and empirical risk minimization.