相关论文: On optimal recovery in $L_2$

In this paper we consider the $L_q$-approximation of multivariate periodic functions $f$ with $L_p$-bounded mixed derivative (difference). The (possibly non-linear) reconstruction algorithm is supposed to recover the function from function…

We prove an analogue of the Korneichuk--Stechkin lemma for functions with values in $L$-spaces. As applications, we obtain sharp Ostrowski type inequalities and solve problems of optimal recovery of identity and convexifying operators, as…

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which…

In this paper, order estimates for the Kolmogorov $n$-widths of an intersection of an arbitrary family of balls $\nu_\alpha B^{\overline{k}}_{\overline{p}_\alpha}$ in $l_q^k$ are obtained for $1\le q\le 2$, $n\le \frac k2$. Here…

When attempting to recover functions from observational data, one naturally seeks to do so in an optimal manner with respect to some modeling assumption. With a focus put on the worst-case setting, this is the standard goal of Optimal…

A loss function measures the discrepancy between the true values (observations) and their estimated fits, for a given instance of data. A loss function is said to be proper (unbiased, Fisher consistent) if the fits are defined over a unit…

We consider the problem of the uniform (in $L_\infty$) recovery of ridge functions $f(x)=\varphi(\langle a,x\rangle)$, $x\in B_2^n$, using noisy evaluations $y_1\approx f(x^1),\ldots,y_N\approx f(x^N)$. It is known that for classes of…

In this note we show that sharp Kolmogorov-type inequalities that estimate the uniform norm $\|f^{(k)}\|$ of the $k$-th derivative of a function $f\colon \mathbb{R}\to\mathbb{R}$ by the values of the uniform norm of $f$ and uniform norms of…

Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in $L^2$. In particular, we show that in relevant cases such as…

Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere $S^m$ in the Euclidean space $\mathbb{R}^{m+1}$ are known to depend upon the Mercer's expansion of the compact and…

The reconstruction of unknown functions from a finite number of samples is a fundamental challenge in pure and applied mathematics. This survey provides a comprehensive overview of recent developments in sampling recovery, focusing on the…

Optimal lower bounds for discrepancy in Besov spaces with dominating mixed smoothness are known from the work of Triebel. Hinrichs proved upper bounds in the plane. In this work we systematically analyse the problem, starting with a survey…

We study $L_q$-approximation and integration for functions from the Sobolev space $W^s_p(\Omega)$ and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use iid sample points, uniformly distributed on the…

The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity for functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative…

We introduce the notion of kinetic maximal $L^2$-regularity with temporal weights for the (fractional) Kolmogorov equation. In particular, we determine the function spaces for the inhomogeneity and the initial value which characterize the…

A key problem in approximation theory is the recovery of high-dimensional functions from samples. In many cases, the functions of interest exhibit anisotropic smoothness, and, in many practical settings, the nature of this anisotropy may be…

Optimal recovery is a mathematical framework for learning functions from observational data by adopting a worst-case perspective tied to model assumptions on the functions to be learned. Working in a finite-dimensional Hilbert space, we…

We describe the set of parameters $(p_1,p_2,q_1,q_2)$ such that the balls $B_{q_1,q_2}^{s,b}$ are rigid in $\ell_{q_1,q_2}^{s,b}$ metric i.e. they are poorly approximated by linear subspaces of dimension $\le (1-\varepsilon)sb$, for large…

The paper concerns problems of the recovery of linear operators defined on sets of functions from information of these functions given with stochastic errors. The constructed optimal recovery methods, in general, do not use all the…

Models based on approximation capabilities have recently been studied in the context of Optimal Recovery. These models, however, are not compatible with overparametrization, since model- and data-consistent functions could then be…