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Kernel ridge regression (KRR) is a widely used nonparametric method due to its strong theoretical guarantees and computational convenience. However, standard KRR does not distinguish between linear and nonlinear components in the signal,…
This paper studies the problem of how efficiently functions in the Sobolev spaces $\mathcal{W}^{s,q}([0,1]^d)$ and Besov spaces $\mathcal{B}^s_{q,r}([0,1]^d)$ can be approximated by deep ReLU neural networks with width $W$ and depth $L$,…
We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration…
In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed…
The central subspace of a pair of random variables $(y,x) \in \mathbb{R}^{p+1}$ is the minimal subspace $\mathcal{S}$ such that $y \perp \hspace{-2mm} \perp x\mid P_{\mathcal{S}}x$. In this paper, we consider the minimax rate of estimating…
In this article, we study the convergence behavior of the regularization-based algorithm for solving the polynomial regression model when both input data and responses are from infinite-dimensional Hilbert spaces. We derive convergence…
This paper establishes the first polynomial convergence rates for Gaussian kernel ridge regression (KRR) with a fixed hyperparameter in both the uniform and the $L^{2}$-norm. The uniform convergence result closes a gap in the theoretical…
Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these…
We investigate the approximation of weighted integrals over $\mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$…
Kernel interpolation in tensor product reproducing kernel Hilbert spaces allows for the use of sparse grids to mitigate the curse of the dimension. Typically, besides the generic constant, only a dimension dependent power of a logarithm…
Classical penalized likelihood regression problems deal with the case that the independent variables data are known exactly. In practice, however, it is common to observe data with incomplete covariate information. We are concerned with a…
We derive improved regression and classification rates for support vector machines using Gaussian kernels under the assumption that the data has some low-dimensional intrinsic structure that is described by the box-counting dimension. Under…
Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces (RKHS) usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian…
We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel…
Reinforcement learning (RL) has shown empirical success in various real world settings with complex models and large state-action spaces. The existing analytical results, however, typically focus on settings with a small number of…
In recent years, there has been a significant growth in research focusing on minimum $\ell_2$ norm (ridgeless) interpolation least squares estimators. However, the majority of these analyses have been limited to an unrealistic regression…
Under mild assumptions on the kernel, we obtain the best known error rates in a regularized learning scenario taking place in the corresponding reproducing kernel Hilbert space (RKHS). The main novelty in the analysis is a proof that one…
Greedy methods have recently been successfully applied to generalized kernel interpolation, or the recovery of a function from data stemming from the evaluation of linear functionals, including the approximation of solutions of linear PDEs…
We obtain robust and computationally efficient estimators for learning several linear models that achieve statistically optimal convergence rate under minimal distributional assumptions. Concretely, we assume our data is drawn from a…
Linear regression without correspondences concerns the recovery of a signal in the linear regression setting, where the correspondences between the observations and the linear functionals are unknown. The associated maximum likelihood…