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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…
We propose a natural intrinsic extension of ridge regression from Euclidean spaces to general Riemannian manifolds for time-series prediction. Our approach combines Riemannian least-squares fitting via B\'ezier curves, empirical covariance…
When the regressors of a econometric linear model are nonorthogonal, it is well known that their estimation by ordinary least squares can present various problems that discourage the use of this model. The ridge regression is the most…
Kernel ridge regression (KRR) is a well-known and popular nonparametric regression approach with many desirable properties, including minimax rate-optimality in estimating functions that belong to common reproducing kernel Hilbert spaces…
In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are…
In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the…
This article provides, through theoretical analysis, an in-depth understanding of the classification performance of the empirical risk minimization framework, in both ridge-regularized and unregularized cases, when high dimensional data are…
This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear…
The problem of learning functions over spaces of probabilities - or distribution regression - is gaining significant interest in the machine learning community. A key challenge behind this problem is to identify a suitable representation…
We consider the problem of reconstructing a function from a finite set of noise-corrupted samples. Two kernel algorithms are analyzed, namely kernel ridge regression and $\varepsilon$-support vector regression. By assuming the ground-truth…
This study introduces a debiasing method for regression estimators, including high-dimensional and nonparametric regression estimators. For example, nonparametric regression methods allow for the estimation of regression functions in a…
In this work, we develop and study an empirical projection operator scheme for solving nonparametric regression problems. This scheme is based on an approximate projection of the regression function over a suitable reproducing kernel…
We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between $L_2$ and the hypothesis space, which we consider as a vector-valued reproducing kernel…
In this work we solve a degenerate parabolic equation for the half line with Dirichlet boundary data, and use some results from the theory of Reproducing Kernel Hilbert Spaces to show that the null reachable space of this degenerate…
Fr\'echet regression generalizes linear regression to metric-space-valued responses by defining fitted values as minimizers of weighted Fr\'echet functionals. Since these weights may have mixed signs, the resulting objective is a signed…
This paper examines the performance of ridge regression in reproducing kernel Hilbert spaces in the presence of noise that exhibits a finite number of higher moments. We establish excess risk bounds consisting of subgaussian and polynomial…
Assume we are given a set of items from a general metric space, but we neither have access to the representation of the data nor to the distances between data points. Instead, suppose that we can actively choose a triplet of items (A,B,C)…
Typical IRT rating-scale models assume that the rating category threshold parameters are the same over examinees. However, it can be argued that many rating data sets violate this assumption. To address this practical psychometric problem,…
In this paper, we study the statistical limits in terms of Sobolev norms of gradient descent for solving inverse problem from randomly sampled noisy observations using a general class of objective functions. Our class of objective functions…
Over-parameterized models like deep nets and random forests have become very popular in machine learning. However, the natural goals of continuity and differentiability, common in regression models, are now often ignored in modern…