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Analysing and computing with Gaussian processes arising from infinitely wide neural networks has recently seen a resurgence in popularity. Despite this, many explicit covariance functions of networks with activation functions used in modern…
We study a birth and death model for the adapatation of a sexual population to an environment. The population is structured by a phenotypical trait, and, possibly, an age variable. Recombination is modeled by Fisher's infinitesimal…
We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a…
In this paper we investigate the problem of estimating the regression function in models with correlated observations. The data is obtained from several experimental units each of them forms a time series. We propose a new estimator based…
Historically, matrix projection models (MPMs) have been employed to study population dynamics with regard to size, age or structure. To work with continuous traits, in the past decade, integral projection models (IPMs) have been proposed.…
Understanding how the brain encodes stimuli has been a fundamental problem in computational neuroscience. Insights into this problem have led to the design and development of artificial neural networks that learn representations by…
Naively, the analytic index of a family of self-adjoint Fredholm operators ought to be (an equivalence class of) the family of the kernels of these operators. The present paper is devoted to a rigorous version of this idea based on ideas of…
Estimating the innovation probability density is an important issue in any regression analysis. This paper focuses on functional autoregressive models. A residual-based kernel estimator is proposed for the innovation density. Asymptotic…
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert…
Time-series classification is essential across diverse domains, including medical diagnosis, industrial monitoring, financial forecasting, and human activity recognition. The Rocket algorithm has emerged as a simple yet powerful method,…
We study the long-time asymptotic behaviour of semigroups generated by non-local Schr\"odinger operators of the form $H = -L+V$; the free operator $L$ is the generator of a symmetric L\'evy process in $\mathbb R^d$, $d > 1$ (with…
Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The conventional description, known as the Ulam-Galerkin method, involves projecting onto basis functions represented as…
The distributional statistical framework replaces classical probability densities by distribution-kernel pairs $(T, \varphi)$, where $T$ is a tempered distribution and $\varphi$ is a rapidly decaying kernel. We develop the thesis that the…
Probabilistic solutions of the so called Schr\"{o}dinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input-output statistics data for the…
Functional linear and single-index models are core regression methods in functional data analysis and are widely used for performing regression in a wide range of applications when the covariates are random functions coupled with scalar…
We propose a tensor network framework for approximating the evolution of observables of measure-preserving ergodic systems. Our approach is based on a spectrally-convergent approximation of the skew-adjoint Koopman generator by a…
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on…
Continuous convolution has recently gained prominence due to its ability to handle irregularly sampled data and model long-term dependency. Also, the promising experimental results of using large convolutional kernels have catalyzed the…
We consider an operator-based latent Markov representation of a stochastic nonlinear dynamical system, where the stochastic evolution of the latent state embedded in a reproducing kernel Hilbert space is described with the corresponding…
In this paper we show how specific families of positive definite kernels serve as powerful tools in analyses of iteration algorithms for multiple layer feedforward Neural Network models. Our focus is on particular kernels that adapt well to…