Related papers: An error bound in the Sudakov-Fernique inequality

We introduce a Bernstein-type inequality which serves to uniformly control quadratic forms of gaussian variables. The latter can for example be used to derive sharp model selection criteria for linear estimation in linear regression and…

We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds…

Gaussian processes have become a promising tool for various safety-critical settings, since the posterior variance can be used to directly estimate the model error and quantify risk. However, state-of-the-art techniques for safety-critical…

For absolutely convergent series we state explicitly a one-sided summation estimate that can be viewed as the discrete analogue of the change of variable formula on the half line. This estimate is implicit in Pascal Lef\`evre's recent…

We develop novel empirical Bernstein inequalities for the variance of bounded random variables. Our inequalities hold under constant conditional variance and mean, without further assumptions like independence or identical distribution of…

The classical Cram\'er-Rao inequality gives a lower bound for the variance of a unbiased estimator of an unknown parameter, in some statistical model of a random process. In this note we rewrite the statment and proof of the bound using…

Let G be a finite abelian group of order n. For a complex valued function f on G, let \fht denote the Fourier transform of f. The uncertainty inequality asserts that if f \neq 0 then |supp(f)| |supp(\fht)| \geq n. Answering a question of…

Error bounds have been studied for more than seventy years, beginning with the seminal result of Hoffman (1952) [{\it J. Res. Natl. Bur. Standards}, 49 (1952), 263--265], which establishes an upper bound for the distance from an arbitrary…

Explicit classical states achieving maximal $f$-divergence are given, allowing for a simple proof of Matsumoto's Theorem, and the systematic extension of any inequality between classical $f$-divergences to quantum $f$-divergences. Our…

We prove a new inequality for Gaussian processes, this inequality implies the Gordon-Chevet inequality. Some remarks on Gaussian proofs of Dvoretzky's theorem are given.

We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…

In this expository note, we give a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables. We deduce a useful concentration inequality for sub-gaussian random vectors. Two examples are given to…

We derive an (almost) guaranteed upper bound on the error of deep neural networks under distribution shift using unlabeled test data. Prior methods either give bounds that are vacuous in practice or give estimates that are accurate on…

Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian…

We give here a simple proof of weighted logarithmic Sobolev inequality, for example for Cauchy type measures, with optimal weight, sharpening results of Bobkov-Ledoux. Some consequences are also discussed.

Tight bounds for several symmetric divergence measures are introduced, given in terms of the total variation distance. Each of these bounds is attained by a pair of 2 or 3-element probability distributions. An application of these bounds…

We consider the estimation of parametric fractional time series models in which not only is the memory parameter unknown, but one may not know whether it lies in the stationary/invertible region or the nonstationary or noninvertible…

We describe a general technique that yields the first {\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning…

We obtain non asymptotic concentration bounds for two kinds of stochastic approximations. We first consider the deviations between the expectation of a given function of the Euler scheme of some diffusion process at a fixed deterministic…

We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to…