Related papers: An Optimal Weighted Least-Squares Method for Opera…
We study the approximation numbers of weighted composition operators $f\mapsto w\cdot(f\circ\varphi)$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers…
This paper investigates system identification problems with Gaussian inputs and quantized observations under fixed thresholds. By reinterpreting the nonlinear effects induced by quantization as the product of the unknown parameter and an…
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…
We consider the fundamental learning problem of estimating properties of distributions over large domains. Using a novel piecewise-polynomial approximation technique, we derive the first unified methodology for constructing sample- and…
We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems in Hilbert scales under general source conditions. Our analysis does not require the regression function to be contained in the hypothesis…
Learning a robust classifier from a few samples remains a key challenge in machine learning. A major thrust of research has been focused on developing $k$-nearest neighbor ($k$-NN) based algorithms combined with metric learning that…
The paper presents analytic expressions of minimax (worst-case) estimates for solutions of linear abstract Neumann problems in Hilbert space with uncertain (not necessarily bounded!) inputs and boundary conditions given incomplete…
We consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best…
We consider the problem of heteroscedastic linear regression, where, given $n$ samples $(\mathbf{x}_i, y_i)$ from $y_i = \langle \mathbf{w}^{*}, \mathbf{x}_i \rangle + \epsilon_i \cdot \langle \mathbf{f}^{*}, \mathbf{x}_i \rangle$ with…
We consider an incremental approximation method for solving variational problems in infinite-dimensional Hilbert spaces, where in each step a randomly and independently selected subproblem from an infinite collection of subproblems is…
Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically…
This paper develops an interpretable, non-intrusive reduced-order modeling technique using regularized kernel interpolation. Existing non-intrusive approaches approximate the dynamics of a reduced-order model (ROM) by solving a data-driven…
Traditional machine learning models, particularly neural networks, are rooted in finite-dimensional parameter spaces and nonlinear function approximations. This report explores an alternative formulation where learning tasks are expressed…
We consider $\mathbb{L}_2$-approximation of elements of a Hermite space of analytic functions over $\mathbb{R}^s$. The Hermite space is a weighted reproducing kernel Hilbert space of real valued functions for which the Hermite coefficients…
This paper studies sparse covariance operator estimation for nonstationary processes with sharply varying marginal variance and small correlation lengthscale. We introduce a covariance operator estimator that adaptively thresholds the…
We present Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions based on the foundational ideas of function encoders. We decompose the task of learning operators into two parts:…
Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys…
We consider a high-probability non-asymptotic confidence estimation in the $\ell^2$-regularized non-linear least-squares setting with fixed design. In particular, we study confidence estimation for local minimizers of the regularized…
This short study reformulates the statistical Bayesian learning problem using a quantum mechanics framework. Density operators representing ensembles of pure states of sample wave functions are used in place probability densities. We show…
We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of…