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Learning kernels in operators from data lies at the intersection of inverse problems and statistical learning, providing a powerful framework for capturing non-local dependencies in function spaces and high-dimensional settings. In contrast…
We introduce a novel data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system…
We study feature learning in a compositional variant of kernel ridge regression in which the predictor is applied to a learnable linear transformation of the input. When the response depends on the input only through a low-dimensional…
We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data…
In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space, with noise affecting both in the data and values of the functions. Due to the curse of…
This paper introduces a local linear smoother for regression surfaces on the simplex. The estimator solves a least-squares regression problem weighted by a locally adaptive Dirichlet kernel, ensuring good boundary properties. Asymptotic…
The global dimensionality of a neural representation manifold provides rich insight into the computational process underlying both artificial and biological neural networks. However, all existing measures of global dimensionality are…
In this paper, we propose a dimension reduction model for spatially dependent variables. Namely, we investigate an extension of the \emph{inverse regression} method under strong mixing condition. This method is based on estimation of the…
Kernel ridge regression (KRR) is widely used for nonparametric regression over reproducing kernel Hilbert spaces. It offers powerful modeling capabilities at the cost of significant computational costs, which typically require $O(n^3)$…
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 propose a new method for functional nonparametric regression with a predictor that resides on a finite-dimensional manifold but is only observable in an infinite-dimensional space. Contamination of the predictor due to discrete/noisy…
This paper proves an abstract theorem addressing in a unified manner two important problems in function approximation: avoiding curse of dimensionality and estimating the degree of approximation for out-of-sample extension in manifold…
The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite dimensional Hilbert space. It allows us, for example, to define a distance…
It is of importance to develop statistical techniques to analyze high-dimensional data in the presence of both complex dependence and possible outliers in real-world applications such as imaging data analyses. We propose a new robust…
Precise asymptotics have revealed many surprises in high-dimensional regression. These advances, however, have not extended to perhaps the simplest estimator: direct Nadaraya-Watson (NW) kernel smoothing. Here, we describe how one can use…
Representing, comparing, and measuring the distance between probability distributions is a key task in computational statistics and machine learning. The choice of representation and the associated distance determine properties of the…
We present a new kernel-based algorithm for modeling evenly distributed multidimensional datasets that does not rely on input space sparsification. The presented method reorganizes the typical single-layer kernel-based model into a deep…
Generative networks have shown remarkable success in learning complex data distributions, particularly in generating high-dimensional data from lower-dimensional inputs. While this capability is well-documented empirically, its theoretical…
We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding…
Any applied mathematical model contains parameters. The paper proposes to use kernel learning for the parametric analysis of the model. The approach consists in setting a distribution on the parameter space, obtaining a finite training…