  Recent research indicates that the performance of machine learning models can
be improved by aligning the geometry of the latent space with the underlying
data structure. Rather than relying solely on Euclidean space, researchers have
proposed using hyperbolic and spherical spaces with constant curvature, or
combinations thereof, to better model the latent space and enhance model
performance. However, little attention has been given to the problem of
automatically identifying the optimal latent geometry for the downstream task.
We mathematically define this novel formulation and coin it as neural latent
geometry search (NLGS). More specifically, we introduce an initial attempt to
search for a latent geometry composed of a product of constant curvature model
spaces with a small number of query evaluations, under some simplifying
assumptions. To accomplish this, we propose a novel notion of distance between
candidate latent geometries based on the Gromov-Hausdorff distance from metric
geometry. In order to compute the Gromov-Hausdorff distance, we introduce a
mapping function that enables the comparison of different manifolds by
embedding them in a common high-dimensional ambient space. We then design a
graph search space based on the notion of smoothness between latent geometries
and employ the calculated distances as an additional inductive bias. Finally,
we use Bayesian optimization to search for the optimal latent geometry in a
query-efficient manner. This is a general method which can be applied to search
for the optimal latent geometry for a variety of models and downstream tasks.
We perform experiments on synthetic and real-world datasets to identify the
optimal latent geometry for multiple machine learning problems.
