  In this paper, we present a method of embedding physics data manifolds with
metric structure into lower dimensional spaces with simpler metrics, such as
Euclidean and Hyperbolic spaces. We then demonstrate that it can be a powerful
step in the data analysis pipeline for many applications. Using progressively
more realistic simulated collisions at the Large Hadron Collider, we show that
this embedding approach learns the underlying latent structure. With the notion
of volume in Euclidean spaces, we provide for the first time a viable solution
to quantifying the true search capability of model agnostic search algorithms
in collider physics (i.e. anomaly detection). Finally, we discuss how the ideas
presented in this paper can be employed to solve many practical challenges that
require the extraction of physically meaningful representations from
information in complex high dimensional datasets.
