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We introduce an approach to quickly and accurately approximate the cumulative distribution function of multivariate Gaussian distributions arising from spatial Gaussian processes. This approximation is trivially parallelizable and simple to…
From a given nuclear triplet we construct a nuclear triplet of sequence spaces and introduce a correlated Gaussian measure via the Bochner-Minlos theorem. Considering special types of correlation operators on such sequence spaces, certain…
Spectral approximation and variational inducing learning for the Gaussian process are two popular methods to reduce computational complexity. However, in previous research, those methods always tend to adopt the orthonormal basis functions,…
In this paper, we analyze Gaussian processes using statistical mechanics. Although the input is originally multidimensional, we simplify our model by considering the input as one-dimensional for statistical mechanical analysis. Furthermore,…
Proximal gradient methods are a popular tool for the solution of structured, nonsmooth minimization problems. In this work, we investigate an extension of the former to general Banach spaces and provide worst-case convergence rates for,…
Sparse variational Gaussian process (GP) approximations based on inducing points have become the de facto standard for scaling GPs to large datasets, owing to their theoretical elegance, computational efficiency, and ease of implementation.…
We propose and analyze a generalized splitting method to sample approximately from a distribution conditional on the occurrence of a rare event. This has important applications in a variety of contexts in operations research, engineering,…
Multivariate Gaussian distributions enjoy Gaussian conditional distributions that makes conditioning easy: conditioning boils down to implementing analytical formulae for conditional means and covariances. For more general distributions,…
Predictive models for binary data are fundamental in various fields, and the growing complexity of modern applications has motivated several flexible specifications for modeling the relationship between the observed predictors and the…
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of…
We introduce a variational Bayesian neural network where the parameters are governed via a probability distribution on random matrices. Specifically, we employ a matrix variate Gaussian \cite{gupta1999matrix} parameter posterior…
This article proposes a co-variance operator for Banach valued random elements using the concept of $U$-statistic. We then study the asymptotic distribution of the proposed co-variance operator along with related large sample properties.…
Gaussian processes are a powerful class of non-linear models, but have limited applicability for larger datasets due to their high computational complexity. In such cases, approximate methods are required, for example, the recently…
Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rate of…
In this paper we extend existing Bayesian methods for variable selection in Gaussian process regression, to select both the regression terms and the active covariates in the spatial correlation structure. We then use the estimated posterior…
Gaussian processes are probabilistic models that are commonly used as functional priors in machine learning. Due to their probabilistic nature, they can be used to capture the prior information on the statistics of noise, smoothness of the…
Gaussian comparison inequalities provide a way of bounding probabilities relating to multivariate Gaussian random vectors in terms of probabilities of random variables with simpler correlation structures. In this paper, we establish the…
A recent novel extension of multi-output Gaussian processes handles heterogeneous outputs assuming that each output has its own likelihood function. It uses a vector-valued Gaussian process prior to jointly model all likelihoods' parameters…
We propose a safe approximation to joint chance-constrained programming where the constraint functions are additively dependent on a normally-distributed random vector. The approximation is analytical, meaning that it requires neither…
In this paper, we consider a stochastic system described by a differential equation admitting a spatially varying random coefficient. The differential equation has been employed to model various static physics systems such as elastic…