  The ability of deep learning methods to perform classification and regression
tasks relies heavily on their capacity to uncover manifolds in high-dimensional
data spaces and project them into low-dimensional representation spaces. In
this study, we investigate the structure and character of the manifolds
generated by classical variational autoencoder (VAE) approaches and deep kernel
learning (DKL). In the former case, the structure of the latent space is
determined by the properties of the input data alone, while in the latter, the
latent manifold forms as a result of an active learning process that balances
the data distribution and target functionalities. We show that DKL with active
learning can produce a more compact and smooth latent space which is more
conducive to optimization compared to previously reported methods, such as the
VAE. We demonstrate this behavior using a simple cards data set and extend it
to the optimization of domain-generated trajectories in physical systems. Our
findings suggest that latent manifolds constructed through active learning have
a more beneficial structure for optimization problems, especially in
feature-rich target-poor scenarios that are common in domain sciences, such as
materials synthesis, energy storage, and molecular discovery. The jupyter
notebooks that encapsulate the complete analysis accompany the article.
