  Clifford Group Equivariant Neural Networks (CGENNs) leverage Clifford
algebras and multivectors as an alternative approach to incorporating group
equivariance to ensure symmetry constraints in neural representations. In
principle, this formulation generalizes to orthogonal groups and preserves
equivariance regardless of the metric signature. However, previous works have
restricted internal network representations to Euclidean or Minkowski
(pseudo-)metrics, handpicked depending on the problem at hand. In this work, we
propose an alternative method that enables the metric to be learned in a
data-driven fashion, allowing the CGENN network to learn more flexible
representations. Specifically, we populate metric matrices fully, ensuring they
are symmetric by construction, and leverage eigenvalue decomposition to
integrate this additional learnable component into the original CGENN
formulation in a principled manner. Additionally, we motivate our method using
insights from category theory, which enables us to explain Clifford algebras as
a categorical construction and guarantee the mathematical soundness of our
approach. We validate our method in various tasks and showcase the advantages
of learning more flexible latent metric representations. The code and data are
available at https://github.com/rick-ali/Metric-Learning-for-CGENNs
