Fast Kernel Approximations for Latent Force Models and Convolved Multiple-Output Gaussian processes

A latent force model is a Gaussian process with a covariance function inspired by a differential operator. Such covariance function is obtained by performing convolution integrals between Green's functions associated to the differential operators, and covariance functions associated to latent functions. In the classical formulation of latent force models, the covariance functions are obtained analytically by solving a double integral, leading to expressions that involve numerical solutions of different types of error functions. In consequence, the covariance matrix calculation is considerably expensive, because it requires the evaluation of one or more of these error functions. In this paper, we use random Fourier features to approximate the solution of these double integrals obtaining simpler analytical expressions for such covariance functions. We show experimental results using ordinary differential operators and provide an extension to build general kernel functions for convolved multiple output Gaussian processes.Comment: 10 pages, 4 figures, accepted by UAI 201

Variational Inference of Joint Models using Multivariate Gaussian Convolution Processes

We present a non-parametric prognostic framework for individualized event prediction based on joint modeling of both longitudinal and time-to-event data. Our approach exploits a multivariate Gaussian convolution process (MGCP) to model the evolution of longitudinal signals and a Cox model to map time-to-event data with longitudinal data modeled through the MGCP. Taking advantage of the unique structure imposed by convolved processes, we provide a variational inference framework to simultaneously estimate parameters in the joint MGCP-Cox model. This significantly reduces computational complexity and safeguards against model overfitting. Experiments on synthetic and real world data show that the proposed framework outperforms state-of-the art approaches built on two-stage inference and strong parametric assumptions

Fast kernel approximations for latent force models and convolved multiple-output Gaussian processes

A latent force model is a Gaussian process with a covariance function inspired by a differential operator. Such covariance function is obtained by performing convolution integrals between Green's functions associated to the differential operators, and covariance functions associated to latent functions. In the classical formulation of latent force models, the covariance functions are obtained analytically by solving a double integral, leading to expressions that involve numerical solutions of different types of error functions. In consequence, the covariance matrix calculation is considerably expensive, because it requires the evaluation of one or more of these error functions. In this paper, we use random Fourier features to approximate the solution of these double integrals obtaining simpler analytical expressions for such covariance functions. We show experimental results using ordinary differential operators and provide an extension to build general kernel functions for convolved multiple output Gaussian processes