  Gaussian processes (GPs) are ubiquitously used in sciences and engineering as
metamodels. Standard GPs, however, can only handle numerical or quantitative
variables. In this paper, we introduce latent map Gaussian processes (LMGPs)
that inherit the attractive properties of GPs and are also applicable to mixed
data which have both quantitative and qualitative inputs. The core idea behind
LMGPs is to learn a continuous, low-dimensional latent space or manifold which
encodes all qualitative inputs. To learn this manifold, we first assign a
unique prior vector representation to each combination of qualitative inputs.
We then use a low-rank linear map to project these priors on a manifold that
characterizes the posterior representations. As the posteriors are
quantitative, they can be directly used in any standard correlation function
such as the Gaussian or Matern. Hence, the optimal map and the corresponding
manifold, along with other hyperparameters of the correlation function, can be
systematically learned via maximum likelihood estimation. Through a wide range
of analytic and real-world examples, we demonstrate the advantages of LMGPs
over state-of-the-art methods in terms of accuracy and versatility. In
particular, we show that LMGPs can handle variable-length inputs, have an
explainable neural network interpretation, and provide insights into how
qualitative inputs affect the response or interact with each other. We also
employ LMGPs in Bayesian optimization and illustrate that they can discover
optimal compound compositions more efficiently than conventional methods that
convert compositions to qualitative variables via manual featurization.
