Infinite Dimensional Stochastic Process Research Articles

A Survey on Bayesian Nonparametric Learning

Bayesian (machine) learning has been playing a significant role in machine learning for a long time due to its particular ability to embrace uncertainty, encode prior knowledge, and endow interpretability. On the back of Bayesian learning’s great success, Bayesian nonparametric learning (BNL) has emerged as a force for further advances in this field due to its greater modelling flexibility and representation power. Instead of playing with the fixed-dimensional probabilistic distributions of Bayesian learning, BNL creates a new “game” with infinite-dimensional stochastic processes. BNL has long been recognised as a research subject in statistics, and, to date, several state-of-the-art pilot studies have demonstrated that BNL has a great deal of potential to solve real-world machine-learning tasks. However, despite these promising results, BNL has not created a huge wave in the machine-learning community. Esotericism may account for this. The books and surveys on BNL written by statisticians are overcomplicated and filled with tedious theories and proofs. Each is certainly meaningful but may scare away new researchers, especially those with computer science backgrounds. Hence, the aim of this article is to provide a plain-spoken, yet comprehensive, theoretical survey of BNL in terms that researchers in the machine-learning community can understand. It is hoped this survey will serve as a starting point for understanding and exploiting the benefits of BNL in our current scholarly endeavours. To achieve this goal, we have collated the extant studies in this field and aligned them with the steps of a standard BNL procedure—from selecting the appropriate stochastic processes through manipulation to executing the model inference algorithms. At each step, past efforts have been thoroughly summarised and discussed. In addition, we have reviewed the common methods for implementing BNL in various machine-learning tasks along with its diverse applications in the real world as examples to motivate future studies.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval [0,1] with dependence on a single parameter, λ. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on λ and the behavior of the initial data around 1. The second scaling leads to a measure-valued Fleming–Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

Convergence of an infinite dimensional stochastic process to a spatially structured trait substitution sequence

We consider an individual-based spatially structured population for Darwinian evolution in an asexual population. The individuals move randomly on a bounded continuous space according to a reflected brownian motion. The dynamics involves also a birth rate, a density-dependent logistic death rate and a probability of mutation at each birth event. We study the convergence of the microscopic process in a long time scale when the population size grows to $$+\infty $$ and the mutation probability decreases to 0. We prove the convergence towards a jump process that jumps in the infinite dimensional space containing the monomorphic stable spatial distributions. The proof requires specific studies of the microscopic model. First, we study the extinction time of the branching diffusion processes that approximate small size populations. Then, we examine the upper bound of large deviation principle around the deterministic large population limit of the microscopic process. Finally, we find a lower bound on the exit time of a neighborhood of a stationary spatial distribution.

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

A stochastic pseudospectral and T-matrix algorithm for acoustic scattering by a class of multiple particle configurations

We consider absorption and scattering of acoustic waves from uncertain configurations comprising multiple two dimensional bodies with various material properties (sound-soft, sound-hard, absorbing and penetrable) and develop tools to address the problem of quantifying uncertainties in the acoustic cross sections of the configurations. The uncertainty arises because the locations and orientations of the particles in the configurations are described through random variables, and statistical moments of the far-fields induced by the stochastic configurations facilitate quantification of the uncertainty. We develop an efficient algorithm, based on a hybrid of the stochastic pseudospectral discretization (to truncate the infinite dimensional stochastic process) and an efficient stable truncated version of Waterman's T-matrix approach (for cost effective realization at each multiple particle configuration corresponding to the pseudospectral quadrature points) to simulate the statistical properties of the stochastic model. We demonstrate the efficiency of the algorithm for configurations with non-smooth and non-convex bodies with distinct material properties, and random locations and orientations with normal and log-normal distributions.

A stochastic process associated with the weighted white noise differentiation

In this paper, we give a relationship between the weighted white noise dierentiation and the Levy Laplacian introducing an operator changing the white noise ˙ B(t) by ˙ B(t) 2 . In addition, we give an infinite dimensional stochastic process generated by the weighted white noise dierentiation and a relationship between the stochastic process and the Levy Laplacian.

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

DIAGONALIZATION OF THE LÉVY LAPLACIAN AND RELATED STABLE PROCESSES

Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

Compensator design for stability enhancement with collocated controllers

A continuum model rather than a finite element model is used. The optimal compensator design is formulated as a stochastic regulator problem and is shown to be solvable by the general infinite-dimensional theory developed by the author despite the lack of exponential stabilizability. Infinite-dimensional steady-state Riccati equations characterizing the feedback control gain and the Kalman filter gain operators can be solved explicitly. The associated performance indexes including the mean-square control-effort are calculated in closed form. As a first approximation, the compensator transfer function can be realized as a bank of bandpass filters in parallel centered at the undamped mode frequencies. Numerical calculations for the gain and bandwidths for a typical configuration are presented. The performance of the compensator is evaluated when in fact in the true model there is no actuator noise. The theoretical problem involved is to show that the infinite-dimensional stochastic process is asymptotically stationary. It is possible to calculate the steady-state covariance in closed form and thereby calculate performance indexes of interest explicitly, facilitating the choice of optimal design parameters. >

Compensator Design for Stability Enhancement with Co-Located Controllers

In this paper we present a stochastic-optimization based approach to the design of compensators for stability enhancement applicable to flexible multibody systems with co-located rate sensros/actuators--for example a truss with offset antennas at one or both ends. A continuum model--rather than a finite ele-ment model--is used, and an essential step in the Compensator Design theory is the formulation of the problem as a wave equation in a Hilbert space. In particular using white noise theory as opposed to the Wiener Process theory the fact that steady state covariances are not nuclear poses no difficulty. The optimal Compensator Design problem is formulated as a stochastic regulator problem. In particular it turns out that we can solve explicitly the infinite dimensional steady state Riccati equations characterizing the feedback control gain and the Kaiman filter gain operators. We can also calculate in closed form the associated performance indices including the "mean square" control effort. We show that, as a first approximation, the Compensator Transfer Function can be realized as a bank of band-pass filters in parallel centered at the undamped mode frequencies. Numerical calculations for the gains and bandwidths for the COFS configuration are presented. We also evaluate the performance of the compensator when in fact in the truth model there is no acutator noise. The theoretical problem involved here is to show that the infinite dimensional stochastic process is asymptotically stationary. We are able to calculate the steady state covariance in closed form and thereby calculate performance indices of interest explicitly, facilitating the choice of optimal design paramters.