Latent variable estimation with composite Hilbert space Gaussian processes

TL;DR

This paper introduces scalable composite Gaussian process models for latent variable estimation, leveraging spectral decomposition and Hilbert space approximations to handle large datasets efficiently, with applications in biology.

Contribution

It develops a novel scalable framework for latent variable estimation using composite Gaussian processes with spectral decomposition, especially for derivative GPs, applicable to large datasets.

Findings

Accurate latent variable estimation demonstrated in simulations

Models achieve well-calibrated uncertainty estimates

Significant speed improvements over exact Gaussian processes

Abstract

We develop a scalable class of models for latent variable estimation using composite Gaussian processes, with a focus on derivative Gaussian processes. We jointly model multiple data sources as outputs to improve the accuracy of latent variable inference under a single probabilistic framework. Similarly specified exact Gaussian processes scale poorly with large datasets. To overcome this, we extend the recently developed Hilbert space approximation methods for Gaussian processes to obtain a reduced-rank representation of the composite covariance function through its spectral decomposition. Specifically, we derive and analyze the spectral decomposition of derivative covariance functions and further study their properties theoretically. Using these spectral decompositions, our methods easily scale up to data scenarios involving thousands of samples. We validate our methods in terms of latent variable estimation accuracy, uncertainty calibration, and inference speed across diverse simulation scenarios. Finally, using a real world case study from single-cell biology, we demonstrate the potential of our models in estimating latent cellular ordering given gene expression levels, thus enhancing our understanding of the underlying biological process.