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The process $(G_t)_{t\in[0,T]}$ is referred to as a fractional Gaussian process if the first-order partial derivative of the difference between its covariance function and that of the fractional Brownian motion $(B^H_t)_{t\in[0,T ]}$ is a…
Large-scale Gaussian process inference has long faced practical challenges due to time and space complexity that is superlinear in dataset size. While sparse variational Gaussian process models are capable of learning from large-scale data,…
In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure $\sigma$ on $\mathbb R^n$. The case when $\sigma$ is assumed absolutely continuous with respect to…
We use neural networks to represent the characteristic function of many-body Gaussian states in the quantum phase space. By a pullback mechanism, we model transformations due to unitary operators as linear layers that can be cascaded to…
This paper is devoted to filtering, smoothing, and prediction of polynomial processes that are partially observed. These problems are known to allow for an explicit solution in the simpler case of linear Gaussian state space models. The key…
Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the first and second moments, as well…
By departing from the previous attempt (Phys. Rev. {\bf E 51}, 4114, (1995)) we give a detailed construction of conditional and perturbed Markov processes, under the assumption that the Cauchy law of probability replaces the Gaussian law…
The sub-Gaussian stable distribution is a heavy-tailed elliptically contoured law which has interesting applications in signal processing and financial mathematics. This work addresses the problem of feasible estimation of distributions. We…
This tutorial aims to provide an intuitive introduction to Gaussian process regression (GPR). GPR models have been widely used in machine learning applications due to their representation flexibility and inherent capability to quantify…
The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic…
Gaussian process is a very promising novel technology that has been applied to both the regression problem and the classification problem. While for the regression problem it yields simple exact solutions, this is not the case for the…
We develop numerical algorithms for the efficient evaluation of quantities associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are based on…
We introduce a novel algorithm for controlling linear time invariant systems in a tracking problem. The controller is based on a Gaussian Process (GP) whose realizations satisfy a system of linear ordinary differential equations with…
The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic…
The Gaussian process is a powerful and flexible technique for interpolating spatiotemporal data, especially with its ability to capture complex trends and uncertainty from the input signal. This chapter describes Gaussian processes as an…
We study the filtering and smoothing problem for continuous-time linear Gaussian systems. While classical approaches such as the Kalman-Bucy filter and the Rauch-Tung-Striebel (RTS) smoother provide recursive formulas for the conditional…
In the framework of the supervised learning of a real function defined on a space X , the so called Kriging method stands on a real Gaussian field defined on X. The Euclidean case is well known and has been widely studied. In this paper, we…
We show that quantum circuits where the initial state and all the following quantum operations can be represented by positive Wigner functions can be classically efficiently simulated. This is true both for continuous-variable as well as…
It was shown in Mishura et al. (Stochastic Process. Appl. 123 (2013) 2353-2369), that any random variable can be represented as improper pathwise integral with respect to fractional Brownian motion. In this paper, we extend this result to…
This paper studies the application of the generalized method of moments (GMM) to multi-reference alignment (MRA): the problem of estimating a signal from its circularly-translated and noisy copies. We begin by proving that the GMM estimator…