  Longitudinal datasets measured repeatedly over time from individual subjects,
arise in many biomedical, psychological, social, and other studies. A common
approach to analyse high-dimensional data that contains missing values is to
learn a low-dimensional representation using variational autoencoders (VAEs).
However, standard VAEs assume that the learnt representations are i.i.d., and
fail to capture the correlations between the data samples. We propose the
Longitudinal VAE (L-VAE), that uses a multi-output additive Gaussian process
(GP) prior to extend the VAE's capability to learn structured low-dimensional
representations imposed by auxiliary covariate information, and derive a new KL
divergence upper bound for such GPs. Our approach can simultaneously
accommodate both time-varying shared and random effects, produce structured
low-dimensional representations, disentangle effects of individual covariates
or their interactions, and achieve highly accurate predictive performance. We
compare our model against previous methods on synthetic as well as clinical
datasets, and demonstrate the state-of-the-art performance in data imputation,
reconstruction, and long-term prediction tasks.
