Related papers: Infinite-dimensional optimization and Bayesian non…
The aim of the book is to present some recent results in the theory of stochastic It\^o equations with singular deterministic part (drift) and its applications to second-order elliptic and parabolic equations with singular first-order…
Driven by the need to solve increasingly complex optimization problems in signal processing and machine learning, there has been increasing interest in understanding the behavior of gradient-descent algorithms in non-convex environments.…
Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to…
Reconstruction of the tridimensional geometry of a visual scene using the binocular disparity information is an important issue in computer vision and mobile robotics, which can be formulated as a Bayesian inference problem. However,…
In this work, we present a novel approach for solving stochastic shape optimization problems. Our method is the extension of the classical stochastic gradient method to infinite-dimensional shape manifolds. We prove convergence of the…
We introduce a novel paradigm for learning non-parametric drift and diffusion functions for stochastic differential equation (SDE). The proposed model learns to simulate path distributions that match observations with non-uniform time…
Bayesian optimization is a sample-efficient method for solving expensive, black-box optimization problems. Stochastic programming concerns optimization under uncertainty where, typically, average performance is the quantity of interest. In…
We first describe a general class of optimization problems that describe many natural, economic, and statistical phenomena. After noting the existence of a conserved quantity in a transformed coordinate system, we outline several instances…
Due to their conjugate posteriors, Gaussian process priors are attractive for estimating the drift of stochastic differential equations with continuous time observations. However, their performance strongly depends on the choice of the…
It seems that in the current age, computers, computation, and data have an increasingly important role to play in scientific research and discovery. This is reflected in part by the rise of machine learning and artificial intelligence,…
We derive a stochastic gradient algorithm for semidefinite optimization using randomization techniques. The algorithm uses subsampling to reduce the computational cost of each iteration and the subsampling ratio explicitly controls…
A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence, uniqueness and path-continuity of infinite-time solutions is proved by an extension of the Ovsyannikov method. This…
Fine-scale simulation of complex systems governed by multiscale partial differential equations (PDEs) is computationally expensive and various multiscale methods have been developed for addressing such problems. In addition, it is…
Recent work on Bayesian optimization has shown its effectiveness in global optimization of difficult black-box objective functions. Many real-world optimization problems of interest also have constraints which are unknown a priori. In this…
We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed…
This paper introduces two variational inference approaches for infinite-dimensional inverse problems, developed through gradient descent with a constant learning rate. The proposed methods enable efficient approximate sampling from the…
Along the ideas of Curtain and Glover, we extend the balanced truncation method for infinite-dimensional linear systems to bilinear and stochastic systems. Specifically , we apply Hilbert space techniques used in many-body quantum mechanics…
We consider the problem of the Bayesian inference of drift and diffusion coefficient functions in a stochastic differential equation given discrete observations of a realisation of its solution. We give conditions for the well-posedness and…
Inverse problems in scientific computing often require optimization over infinite-dimensional Hilbert spaces. A commonly used solver in such settings is stochastic gradient descent (SGD), where gradients are approximated using randomly…
We study backward stochastic differential equations (BSDEs) in infinite horizon and design efficient numerical schemes for solving them. We establish a probabilistic representation of the solution of the BSDE using Malliavin derivative and…