Related papers: Universal sampling discretization
Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is…
We prove a sampling discretization theorem for the square norm of functions from a finite dimensional subspace satisfying Nikol'skii's inequality with an upper bound on the number of sampling points of the order of the dimension of the…
The paper addresses a problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under a standard assumption formulated in terms of…
Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set…
In the first part of the paper we study absolute error of sampling discretization of the integral $L_p$-norm for function classes of continuous functions. We use basic approaches from chaining technique to provide general upper bounds for…
There has been significant progress in the study of sampling discretization of integral norms for both a designated finite-dimensional function space and a finite collection of such function spaces (universal discretization). Sampling…
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun recently. In this paper we…
The paper addresses the problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under two standard kinds of assumptions --…
This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for…
Recently, it was discovered that for a given function class $\mathbf{F}$ the error of best linear recovery in the square norm can be bounded above by the Kolmogorov width of $\mathbf{F}$ in the uniform norm. That analysis is based on deep…
We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known…
The paper is devoted to discretization of integral norms of functions from a given collection of finite dimensional subspaces. For natural collections of subspaces of the multivariate trigonometric polynomials we construct sets of points,…
Given an $N$-dimensional subspace $X$ of $L_p([0,1])$, we consider the problem of choosing $M$-sampling points which may be used to discretely approximate the $L_p$ norm on the subspace. We are particularly interested in knowing when the…
It is known that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error measured in the square norm. In this paper we demonstrate how known results on universal sampling…
We prove optimal sampling bounds achieving $(1\pm\varepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms. This includes important functions such as logistic and…
The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(\mu)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms…
This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. We present here known results on…
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications but there is no systematic study of it. We present here a new technique, which…
Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,\mu)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame…
The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden…