Pulling back symmetric Riemannian geometry for data analysis
Willem Diepeveen
Data sets tend to live in low-dimensional non-linear subspaces. Ideal data
analysis tools for such data sets should therefore account for such non-linear
geometry. The symmetric Riemannian geometry setting can be suitable for a
variety of reasons. First, it comes with a rich mathematical structure to
account for a wide range of non-linear geometries that has been shown to be
able to capture the data geometry through empirical evidence from classical
non-linear embedding. Second, many standard data analysis tools initially
developed for data in Euclidean space can also be generalised efficiently to
data on a symmetric Riemannian manifold. A conceptual challenge comes from the
lack of guidelines for constructing a symmetric Riemannian structure on the
data space itself and the lack of guidelines for modifying successful
algorithms on symmetric Riemannian manifolds for data analysis to this setting.
This work considers these challenges in the setting of pullback Riemannian
geometry through a diffeomorphism. The first part of the paper characterises
diffeomorphisms that result in proper, stable and efficient data analysis. The
second part then uses these best practices to guide construction of such
diffeomorphisms through deep learning. As a proof of concept, different types
of pullback geometries -- among which the proposed construction -- are tested
on several data analysis tasks and on several toy data sets. The numerical
experiments confirm the predictions from theory, i.e., that the diffeomorphisms
generating the pullback geometry need to map the data manifold into a geodesic
subspace of the pulled back Riemannian manifold while preserving local isometry
around the data manifold for proper, stable and efficient data analysis, and
that pulling back positive curvature can be problematic in terms of stability.