Related papers: Infinite-Variate $L^2$-Approximation with Nested S…
We consider the problem of approximating a smooth function from finitely-many pointwise samples using $\ell^1$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three…
This paper addresses the problem of approximating an unknown function from point evaluations. When obtaining these point evaluations is costly, minimising the required sample size becomes crucial, and it is unreasonable to reserve a…
We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in $L^p$) by ReLU neural networks with an increasing number of coefficients, subject to…
In many linear inverse problems, we want to estimate an unknown vector belonging to a high-dimensional (or infinite-dimensional) space from few linear measurements. To overcome the ill-posed nature of such problems, we use a low-dimension…
This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…
We propose a nonlinear function-on-function regression model where both the covariate and the response are random functions. The nonlinear regression is carried out in two steps: we first construct Hilbert spaces to accommodate the…
Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be…
Many scientific problems involve data exhibiting both temporal and cross-sectional dependencies. While linear dependencies have been extensively studied, the theoretical analysis of regression estimators under nonlinear dependencies remains…
Many problems in science and engineering involve, as part of their solution process, the consideration of a separable function which is the sum of two convex functions, one of them possibly non-smooth. Recently a few works have discussed…
We study the consistency of minimum-norm interpolation in reproducing kernel Hilbert spaces corresponding to bounded kernels. Our main result give lower bounds for the generalization error of the kernel interpolation measured in a…
We study the Cowling approximation by analytical means as applied to a system of linear differential equations arising from models of non-radial stellar pulsation. We consider various asymptotic cases, including those of high harmonic…
We study the problem of approximating and estimating classification functions that have their decision boundary in the $RBV^2$ space. Functions of $RBV^2$ type arise naturally as solutions of regularized neural network learning problems and…
While the most recent Anderson acceleration (AA) convergence theory [Pollock et al, {\it IMA Num. An.}, 2021] requires that the AA optimization norm match the Hilbert space norm associated with the fixed point operator, in implementations…
We investigate the theoretical foundations of a recently introduced entropy-based formulation of weighted least squares for the approximation of overdetermined linear systems, motivated by robust data fitting in the presence of sparse gross…
In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important…
We propose a unified framework for likelihood-based regression modeling when the response variable has finite support. Our work is motivated by the fact that, in practice, observed data are discrete and bounded. The proposed methods assume…
In this paper we present results on asymptotic characteristics of multivariate function classes in the uniform norm. Our main interest is the approximation of functions with mixed smoothness parameter not larger than $1/2$. Our focus will…
Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the…
We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear (separable) Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field and prove its convergence…
We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d \in \mathbb{N} \cup \{\infty\}$ variables. The kernels $K$ are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic…