  We develop a framework for derivative Gaussian process latent variable models
(DGP-LVM) that can handle multi-dimensional output data using modified
derivative covariance functions. The modifications account for complexities in
the underlying data generating process such as scaled derivatives, varying
information across multiple output dimensions as well as interactions between
outputs. Further, our framework provides uncertainty estimates for each latent
variable samples using Bayesian inference. Through extensive simulations, we
demonstrate that latent variable estimation accuracy can be drastically
increased by including derivative information due to our proposed covariance
function modifications. The developments are motivated by a concrete biological
research problem involving the estimation of the unobserved cellular ordering
from single-cell RNA (scRNA) sequencing data for gene expression and its
corresponding derivative information known as RNA velocity. Since the RNA
velocity is only an estimate of the exact derivative information, the
derivative covariance functions need to account for potential scale
differences. In a real-world case study, we illustrate the application of
DGP-LVMs to such scRNA sequencing data. While motivated by this biological
problem, our framework is generally applicable to all kinds of latent variable
estimation problems involving derivative information irrespective of the field
of study.
