Efficient Bayesian inference on Grassmann manifolds leveraging Lagrangian Hamiltonian Monte Carlo

We introduce a Bayesian framework for geodesic regression on Grassmann manifolds. Since a geodesic on the manifold is fully determined by its initial position and velocity, we reformulate the regression from function estimation to inferring these two initial conditions. We perform this inference via a Grassmann-specific Lagrangian Hamiltonian Monte Carlo method that exploits the manifold’s Riemannian geometry. Solving the associated Lagrangian dynamics reduces to following the geodesic flow, yielding an efficient inference procedure. We analyze the asymptotic properties of the resulting posterior, providing theoretical guarantees, and confirm the computational efficiency and accuracy of the approach experimentally on complex Bayesian modelling tasks.