Hilbert space methods for approximating multi-output latent variable Gaussian processes

TL;DR

This paper extends Hilbert space Gaussian process methods to multi-output and latent input scenarios, demonstrating improved accuracy and calibration in simulations and real-world biological data, balancing speed and trustworthiness.

Contribution

It generalizes Hilbert space Gaussian process approximations to multi-output and latent input cases, enhancing their applicability and performance.

Findings

Faster computation with comparable or better accuracy.

Improved uncertainty calibration over exact methods.

Effective application to single cell biology data.

Abstract

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 developed class of Hilbert space Gaussian processes. They have been shown to significantly reduce computation time while retaining most of the favorable properties of exact Gaussian processes. However, Hilbert space approximations have so far only been developed for uni-dimensional outputs and manifest (known) inputs. Thus, we generalize Hilbert space methods to multi-output and latent input settings. Through extensive simulations, we show that the developed approximate Gaussian processes are indeed not only faster, but also provide similar or even better uncertainty calibration and accuracy of latent variable estimates compared to exact Gaussian processes. While not necessarily faster than alternative Gaussian process approximations, our new models provide better calibration and estimation accuracy, thus striking an excellent balance between trustworthiness and speed. We additionally illustrate our methods on a real-world case study from single cell biology.