Related papers: Universal sampling discretization

We study the problem of sampling from a distribution $\mu$ with density $\propto e^{-V}$ for some potential function $V:\mathbb R^d\to \mathbb R$ with query access to $V$ and $\nabla V$. We start with the following standard assumptions: (1)…

We prove $L_q(\R^m)$--discretization inequalities for entire functions $f$ of exponential type in the form \ba C_2\|f\|_{L_q(\R^m)} \le \left(\sum_{\nu=1}^\iy \left\vert f\left(X_\nu\right) \right\vert^q\right)^{1/q} \le…

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and possibly conditioned) continuous time…

We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on…

We provide new sufficient conditions for Chebyshev estimates for Beurling generalized primes. It is shown that if the counting function $N$ of a generalized number system satisfies the $L^{1}$-condition $$…

Assume that $V_h$ is a space of piecewise polynomials of degree less than $r\geq 1$ on a family of quasi-uniform triangulation of size $h$. Then the following well-known upper bound holds for a sufficiently smooth function $u$ and $p\in [1,…

This paper concerns the construction of tests for universal hypothesis testing problems, in which the alternate hypothesis is poorly modeled and the observation space is large. The mismatched universal test is a feature-based technique for…

The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector m and scatter matrix S of an elliptically symmetric t…

Let $X$ be a random variable with distribution function $F,$ and $X_{1},X_{2},...,X_{n}$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\leq x\}.$ Using majorization…

Let $\Lambda$ be a uniformly discrete set and $S$ be a compact set in $R$. We prove that if there exists a bounded sequence of functions in Paley--Wiener space $PW_S$, which approximates $\delta-$functions on $\Lambda$ with $l^2-$error $d$,…

Let $(\Omega,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$ and $u_{\lambda}:= \phi_{\lambda} |_{\partial \Omega}$ the associated…

A loss function measures the discrepancy between the true values and their estimated fits, for a given instance of data. In classification problems, a loss function is said to be proper if a minimizer of the expected loss is the true…

A classical regularity result is that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We…

We study the problem of finding global minimizers of $V(x):\mathbb{R}^d\rightarrow\mathbb{R}$ approximately via sampling from a probability distribution $\mu_{\sigma}$ with density $p_{\sigma}(x)=\dfrac{\exp(-V(x)/\sigma)}{\int_{\mathbb…

We study subspaces of Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,\dots$, of a function $f\in L_M$, $\int_0^1 f(t)\,dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of…

We consider the random lasing from a weakly scattering medium and demonstrate that the distribution of the threshold gain over the ensemble of statistically independent finite-size samples is universal. Universality stems from the facts…

We find sufficient conditions for a discrete sequence to be interpolating or sampling for certain generalized Bergman spaces on open Riemann surfaces. As in previous work of Bendtsson, Ortega-Cerda, Seip, Wallsten and others, our conditions…

Let $M$ be a complete Riemannian manifold, $N\in \NN$ and $p\ge 1$. We prove that almost everywhere on $x=(x_1,...,x_N)\in M^N$ for Lebesgue measure in $M^N$, the measure $\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$ has a unique $p$-mean $e_p(x)$.…

We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure mu on [-1,1]. Assume that mu is a regular measure, and is absolutely continuous in an open…

A set $U\subseteq \reals^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \subseteq \reals^2$ consists of points chosen randomly and uniformly from the unit square then…