Mathematics

We establish higher-order expansions for a difference between probability distributions of sums of i.i.d. random vectors in a Euclidean space. The derived bounds are uniform over two classes of sets: the set of all Euclidean balls and the set of all half-spaces. These results allow to account for an impact of higher-order moments or cumulants of the considered distributions; the obtained error terms depend on a sample size and a dimension explicitly. The new inequalities outperform accuracy of the normal approximation in existing Berry--Esseen inequalities under very general conditions. For symmetrically distributed random summands, the obtained results are optimal in terms of the ratio between the dimension and the sample size. Using the new higher-order inequalities, we study accuracy of the nonparametric bootstrap approximation and propose a bootstrap score test under possible model misspecification. The proposed results include also explicit error bounds for general elliptical confidence regions for an expected value of the random summands, and optimality of the Gaussian anti-concentration inequality over the set of all Euclidean balls.

In this paper, we derive a new dimension-free non-asymptotic upper bound for the quadratic k -means excess risk related to the quantization of an i.i.d sample in a separable Hilbert space. We improve the bound of order O(k/ n ??????) of Biau, Devroye and Lugosi, recovering the rate k/n ????????that has already been proved by Fefferman, Mitter, and Narayanan and by Klochkov, Kroshnin and Zhivotovskiy but with worse log factors and constants. More precisely, we bound the mean excess risk of an empirical minimizer by the explicit upper bound 16 B 2 log(n/k) klog(k)/n ??????????????????, in the bounded case when P(?�X?�≤B)=1 . This is essentially optimal up to logarithmic factors since a lower bound of order O( k 1??/d /n ????????????????) is known in dimension d . Our technique of proof is based on the linearization of the k -means criterion through a kernel trick and on PAC-Bayesian inequalities. To get a 1/ n ??????speed, we introduce a new PAC-Bayesian chaining method replacing the concept of δ -net with the perturbation of the parameter by an infinite dimensional Gaussian process. In the meantime, we embed the usual k -means criterion into a broader family built upon the Kullback divergence and its underlying properties. This results in a new algorithm that we named information k -means, well suited to the clustering of bags of words. Based on considerations from information theory, we also introduce a new bounded k -means criterion that uses a scale parameter but satisfies a generalization bound that does not require any boundedness or even integrability conditions on the sample. We describe the counterpart of Lloyd's algorithm and prove generalization bounds for these new k -means criteria.

It is well known that Hoeffding's inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding's inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz \cite{r1}. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding's inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding's inequality and its refinement one by Hertz only considered the first order moment of random variables. In this paper, we present a new type of Hoeffding's inequalities, where the high order moments of random variables are taken into account. It can get some considerable improvements in the tail bounds evaluation compared with the known results. It is expected that the developed new type Hoeffding's inequalities could get more interesting applications in some related fields that use Hoeffding's results.

In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We build the computational inference method using a finite-dimensional Galerkin method. We show that the proposed approximation has a convergence-in-probability property to the solution of the original multi-layered model. We then carry out Bayesian inference using the preconditioned Crank--Nicolson algorithm which is modified to work with multi-layered Gaussian fields. We show via numerical experiments in signal deconvolution and computerized X-ray tomography problems that the proposed method can offer both smoothing and edge preservation at the same time.

Under the framework of reproducing kernel Hilbert space (RKHS), we consider the penalized least-squares of the partially functional linear models (PFLM), whose predictor contains both functional and traditional multivariate part, and the multivariate part allows a divergent number of parameters. From the non-asymptotic point of view, we focus on the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form under a more general assumption known as the effective dimension to the model, by which we also show the prediction consistency when the number of multivariate covariates p slightly increases with the sample size n . Our new finding implies a trade-off between the number of non-functional predictors and the effective dimension of the kernel principal components to ensure the prediction consistency in the increasing-dimensional setting. The analysis in our proof hinges on the spectral condition of the sandwich operator of the covariance operator and the reproducing kernel, and on the concentration inequalities for the random elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower bound under the regularity assumption of Kullback-Leibler divergence of the models.

We study the effects of rounding on the moments of random variables. Specifically, given a random variable X and its rounded counterpart rd(X) , we study |E[ X k ]−E[rd(X ) k ]| for non-negative integer k . We consider the case that the rounding function rd:R→F corresponds either to (i) rounding to the nearest point in some discrete set F or (ii) rounding randomly to either the nearest larger or smaller point in this same set with probabilities proportional to the distances to these points. In both cases, we show, under reasonable assumptions on the density function of X , how to compute a constant C such that |E[ X k ]−E[rd(X ) k ]|<C ϵ 2 , provided |rd(x)−x|≤ϵE(x) , where E:R→ R ≥0 is some fixed positive piecewise linear function. Refined bounds for the absolute moments E[| X k −rd(X ) k |] are also given.

The main problem considered in this paper is construction and theoretical study of efficient n -point coverings of a d -dimensional cube [−1,1 ] d . Targeted values of d are between 5 and 50; n can be in hundreds or thousands and the designs (collections of points) are nested. This paper is a continuation of our paper \cite{us}, where we have theoretically investigated several simple schemes and numerically studied many more. In this paper, we extend the theoretical constructions of \cite{us} for studying the designs which were found to be superior to the ones theoretically investigated in \cite{us}. We also extend our constructions for new construction schemes which provide even better coverings (in the class of nested designs) than the ones numerically found in \cite{us}. In view of a close connection of the problem of quantization to the problem of covering, we extend our theoretical approximations and practical recommendations to the problem of construction of efficient quantization designs in a cube [−1,1 ] d . In the last section, we discuss the problems of covering and quantization in a d -dimensional simplex; practical significance of this problem has been communicated to the authors by Professor Michael Vrahatis, a co-editor of the present volume.

Importance sampling is a popular variance reduction method for Monte Carlo estimation, where a notorious question is how to design good proposal distributions. While in most cases optimal (zero-variance) estimators are theoretically possible, in practice only suboptimal proposal distributions are available and it can often be observed numerically that those can reduce statistical performance significantly, leading to large relative errors and therefore counteracting the original intention. In this article, we provide nonasymptotic lower and upper bounds on the relative error in importance sampling that depend on the deviation of the actual proposal from optimality, and we thus identify potential robustness issues that importance sampling may have, especially in high dimensions. We focus on path sampling problems for diffusion processes, for which generating good proposals comes with additional technical challenges, and we provide numerous numerical examples that support our findings.

In the linear regression model with possibly autoregressive errors, we propose a family of nonparametric tests for regression under a nuisance autoregression. The tests avoid the estimation of nuisance parameters, in contrast to the tests proposed in the literature.

Nonparametric estimation for semilinear SPDEs, namely stochastic reaction-diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, based on a precise analysis of the Hölder regularity of the solution process and its nonlinear component, we show that the asymptotic properties of diffusivity and volatility estimators derived from realized quadratic variations in the linear setup generalize to the semilinear SPDE. In particular, we obtain a rate-optimal joint estimator of the two parameters. Secondly, we derive a nonparametric estimator for the reaction function specifying the underlying equation. The estimate is chosen from a finite-dimensional function space based on a simple least squares criterion. Oracle inequalities with respect to both the empirical and usual L 2 -risk provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection.