Bayesian optimisation (BO) using a Gaussian process (GP)-based surrogate model is a powerful tool for solving black-box optimisation problems but does not scale well to high-dimensional data. Previous works have proposed to use variational autoencoders (VAEs) to project high-dimensional data onto a low-dimensional latent space and to implement BO in the inferred latent space. In this work, we propose a conditional generative model for efficient high-dimensional BO that uses a GP surrogate model together with GP prior VAEs. A GP prior VAE extends the standard VAE by conditioning the generative and inference model on auxiliary covariates, capturing complex correlations across samples with a GP. Our model incorporates the observed target quantity values as auxiliary covariates learning a structured latent space that is better suited for the GP-based BO surrogate model. It handles partially observed auxiliary covariates using a unifying probabilistic framework and can also incorporate additional auxiliary covariates that may be available in real-world applications. We demonstrate that our method improves upon existing latent space BO methods on simulated datasets as well as on commonly used benchmarks.

Conditional variational autoencoders (CVAEs) are versatile deep latent variable models that extend the standard VAE framework by conditioning the generative model with auxiliary covariates. The original CVAE model assumes that the data samples are independent, whereas more recent conditional VAE models, such as the Gaussian process (GP) prior VAEs, can account for complex correlation structures across all data samples. While several methods have been proposed to learn standard VAEs from partially observed datasets, these methods fall short for conditional VAEs. In this work, we propose a method to learn conditional VAEs from datasets in which auxiliary covariates can contain missing values as well. The proposed method augments the conditional VAEs with a prior distribution for the missing covariates and estimates their posterior using amortised variational inference. At training time, our method accounts for the uncertainty associated with the missing covariates while simultaneously maximising the evidence lower bound. We develop computationally efficient methods to learn CVAEs and GP prior VAEs that are compatible with mini-batching. Our experiments on simulated datasets as well as on real-world biomedical datasets show that the proposed method outperforms previous methods in learning conditional VAEs from non-temporal, temporal, and longitudinal datasets.

The variational autoencoder (VAE) is a popular deep latent variable model used to analyse high-dimensional datasets by learning a low-dimensional latent representation of the data. It simultaneously learns a generative model and an inference network to perform approximate posterior inference. Recently proposed extensions to VAEs that can handle temporal and longitudinal data have applications in healthcare, behavioural modelling, and predictive maintenance. However, these extensions do not account for heterogeneous data, i.e., data comprising of continuous and discrete attributes, which is common in many real-life applications. In this work, we propose the heterogeneous longitudinal VAE (HL-VAE) that extends the existing temporal and longitudinal VAEs to heterogeneous data for imputing missing values and unseen time point prediction. HL-VAE provides efficient inference for high-dimensional datasets and includes likelihood models for continuous, count, categorical, and ordinal data while accounting for missing observations. We demonstrate our model’s efficacy through simulated as well as clinical datasets, and show that our proposed model achieves competitive performance in missing value imputation and predictive accuracy.

Clinical patient records are an example of high-dimensional data that is typically collected from disparate sources and comprises of multiple likelihoods with noisy as well as missing values. In this work, we propose an unsupervised generative model that can learn a low-dimensional representation among the observations in a latent space, while making use of all available data in a heterogeneous data setting with missing values. We improve upon the existing Gaussian process latent variable model (GPLVM) by incorporating multiple likelihoods and deep neural network parameterised back-constraints to create a non-linear dimensionality reduction technique for heterogeneous data. In addition, we develop a variational inference method for our model that uses numerical quadrature. We establish the effectiveness of our model and compare against existing GPLVM methods on a standard benchmark dataset as well as on clinical data of Parkinson’s disease patients treated at the HUS Helsinki University Hospital.

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.