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

arXiv: https://arxiv.org/abs/1805.07460 · HTML: https://mlsanity.com/paper/1805.07460/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2018.

## Built on by (most cited first, 5 of 5)

- [Compositional Modeling of Nonlinear Dynamical Systems with ODE-based Random Features](https://mlsanity.com/paper/2106.05960/) (2021)
- [Learning Nonparametric Volterra Kernels with Gaussian Processes](https://mlsanity.com/paper/2106.05582/) (2021)
- [Adjoint-aided inference of Gaussian process driven differential equations](https://mlsanity.com/paper/2202.04589/) (2022)
- [Deep Latent Force Models: ODE-based Process Convolutions for Bayesian Deep Learning](https://mlsanity.com/paper/2311.14828/) (2023)
- [Adaptive RKHS Fourier Features for Compositional Gaussian Process Models](https://mlsanity.com/paper/2407.01856/) (2024)

## Abstract (arXiv)

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.