Learning GPLVM with arbitrary kernels using the unscented transformation

TL;DR

This paper introduces a novel approach for GPLVM using the unscented transformation, enabling efficient and deterministic handling of arbitrary kernels with linear complexity in dimensions, improving over traditional quadrature methods.

Contribution

The authors propose replacing quadrature with the unscented transformation in GPLVM, allowing arbitrary kernels and linear complexity, enhancing scalability and performance.

Findings

Comparable or better performance than existing solutions

Linear computational complexity in the number of dimensions

Effective in dimensionality reduction and uncertainty propagation tasks

Abstract

Gaussian Process Latent Variable Model (GPLVM) is a flexible framework to handle uncertain inputs in Gaussian Processes (GPs) and incorporate GPs as components of larger graphical models. Nonetheless, the standard GPLVM variational inference approach is tractable only for a narrow family of kernel functions. The most popular implementations of GPLVM circumvent this limitation using quadrature methods, which may become a computational bottleneck even for relatively low dimensions. For instance, the widely employed Gauss-Hermite quadrature has exponential complexity on the number of dimensions. In this work, we propose using the unscented transformation instead. Overall, this method presents comparable, if not better, performance than offthe-shelf solutions to GPLVM and its computational complexity scales only linearly on dimension. In contrast to Monte Carlo methods, our approach is deterministic and works well with quasi-Newton methods, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. We illustrate the applicability of our method with experiments on dimensionality reduction and multistep-ahead prediction with uncertainty propagation.