相关论文: A Lanczos-Stieltjes method for one-dimensional rid…
This work is devoted to the obtaining of a new numerical scheme based in quadrature formulas for the Lebesgue-Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically…
Approximating a manifold-valued function from samples of input-output pairs consists of modeling the relationship between an input from a vector space and an output on a Riemannian manifold. We propose a function approximation method that…
Gaussian processes provide a powerful probabilistic kernel learning framework, which allows learning high quality nonparametric regression models via methods such as Gaussian process regression. Nevertheless, the learning phase of Gaussian…
We study asymptotic minimax problems for estimating a $d$-dimensional regression parameter over spheres of growing dimension ($d\to \infty$). Assuming that the data follows a linear model with Gaussian predictors and errors, we show that…
This paper describes our experience developing polynomial approximations for trigonometric functions that produce correctly rounded results for multiple representations and rounding modes using the RLIBM approach. A key challenge with…
We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge…
Working with a toy model whose partition function consists of a discrete summation, we introduce the statistical field-theory methodology by transforming a partition function via a formal Gaussian integral relation (the Hubbard-Stratonovich…
We characterize the best $L_{2}$ approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions. In the special case when the weight functions are constants, we propose explicit…
This paper introduces a novel algorithmic solution for the approximation of a given multivariate function by a nomographic function that is composed of a one-dimensional continuous and monotone outer function and a sum of univariate…
Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…
Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for non-smooth functions with kinks. For example, such kinks can arise in the uncertainty…
We consider the approximation of a continuous function, defined on a compact set of the $d$-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best…
We address the challenge of correlated predictors in high-dimensional GLMs, where regression coefficients range from sparse to dense, by proposing a data-driven random projection method. This is particularly relevant for applications where…
We introduce and study the approximation properties of $g$-polynomials, defined as linear combinations of iterated Stieltjes integrals of a constant function. Focusing on the case where the derivator $g$ has finitely many discontinuities,…
Because of the advance in technologies, modern statistical studies often encounter linear models with the number of explanatory variables much larger than the sample size. Estimation and variable selection in these high-dimensional problems…
Random feature ridge regression is often analyzed in the high-dimensional regime under the homogeneous sampling model $x_i=\Sigma^{1/2}x_i'$, where the vectors $x_i'$ have iid entries and the same covariance matrix $\Sigma$ is shared by all…
A structure-preserving kernel ridge regression method is presented that allows the recovery of nonlinear Hamiltonian functions out of datasets made of noisy observations of Hamiltonian vector fields. The method proposes a closed-form…
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
A space-filling function is a bijection from the unit line segment to the unit square, cube, or hypercube. The function from the unit line segment is continuous. The inverse function, while well-defined, is not continuous. Space-filling…
High-dimensional linear regression has been thoroughly studied in the context of independent and identically distributed data. We propose to investigate high-dimensional regression models for independent but non-identically distributed…