Related papers: Function values are enough for $L_2$-approximation
We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a…
In applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements. Series method offers that by approximating the unknown function based on $k$ basis…
It is known that for a $\rho$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to…
Least-squares approximation is one of the most important methods for recovering an unknown function from data. While in many applications the data is fixed, in many others there is substantial freedom to choose where to sample. In this…
In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the…
Most commonly used \emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy 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 this paper, we prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., for approximation by functions of the form $\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n \varrho_k(A_k…
We study the functional linear regression model with a scalar response and a Hilbert space-valued predictor, a canonical example of an ill-posed inverse problem. We show that the functional partial least squares (PLS) estimator attains…
In this article, we construct semiparametrically efficient estimators of linear functionals of a probability measure in the presence of side information using an easy empirical likelihood approach. We use estimated constraint functions and…
In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both…
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…
We study the approximation of expectations $\E(f(X))$ for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms,…
A common technique for compressing a neural network is to compute the $k$-rank $\ell_2$ approximation $A_{k,2}$ of the matrix $A\in\mathbb{R}^{n\times d}$ that corresponds to a fully connected layer (or embedding layer). Here, $d$ is the…
We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - \theta)$ over the $n$-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every…
Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into…
We obtain the optimal value of the constant K(n,s) in the Sobolev-Nirenberg-Gagliardo inequality $ \|\,u\,\|_{L^{\infty}(\mathbb{R}^{n})} \leq K(n,s) \,\|\, u \,\|_{L^{2}(\mathbb{R}^{n})}^{1 - n/(2s)} \|\, u…
Lasserre [La] proved that for every compact set $K\subset\mathbb R^n$ and every even number $d$ there exists a unique homogeneous polynomial $g_0$ of degree $d$ with $K\subset G_1(g_0)=\{x\in\mathbb R^n:g_0(x)\leq 1\}$ minimizing $|G_1(g)|$…
In this paper we study the sampling recovery problem for certain relevant multivariate function classes which are not compactly embedded into $L_\infty$. Recent tools relating the sampling numbers to the Kolmogorov widths in the uniform…
Let $\xi = \{x^j\}_{j=1}^n$ be a grid of $n$ points in the $d$-cube ${\II}^d:=[0,1]^d$, and $\Phi = \{\phi_j\}_{j =1}^n$ a family of $n$ functions on ${\II}^d$. We define the linear sampling algorithm $L_n(\Phi,\xi,\cdot)$ for an…