As data is a predominant resource in applications, Riemannian geometry is a natural
framework to model and unify complex nonlinear sources of data. However, the …

Invariance and equivariance to geometrical transformations have proven to be very useful
inductive biases when training (convolutional) neural network models, especially in the low …

At the dawn of the Internet era graph analytics play an important role in high-and low-level
network policymaking across a wide array of fields so diverse as transportation network …

When moving an object endowed with continuous symmetry, an ambiguity arises in its
underlying rigid body transformation, induced by the arbitrariness of the portion of motion …

Lie-group variational integrators of arbitrary order are developed using the Galerkin method,
based on unit quaternion interpolation. To our knowledge, quaternions have not been used …

Geometric finite elements (GFE) generalize the idea of Galerkin methods to variational
problems for unknowns that map into nonlinear spaces. In particular, GFE methods …

We derive a family of high-order, structure-preserving approximations of the Riemannian
exponential map on several matrix manifolds, including the group of unitary matrices, the …

We describe a systematic mathematical approach to the geometric discretization of gauge
field theories that is based on Dirac and multi-Dirac mechanics and geometry, which provide …

A novel structure-preserving algorithm for general relativity in vacuum is derived from a
lattice gauge theoretic discretization of the tetradic Palatini action. The resulting model of …

Time-varying polar decomposition becomes important and has many applications in
potential fields. This paper first proposes a continuous-time model and a discrete-time …