Covariance-Adaptive Bouncy Particle Samplers via Split Lagrangian Dynamics

Piecewise Deterministic Markov Processes (PDMPs) provide a powerful framework for continuous-time Monte Carlo, with the Bouncy Particle Sampler (BPS) as a prominent example. Recent advances through the Metropolised PDMP framework allow local adaptivity in step size and effective path length, the latter acting as a refreshment rate. However, current PDMP samplers cannot adapt to local changes in the covariance structure of the target distribution. We extend BPS by introducing a position-dependent velocity distribution that varies with the local covariance structure of the target. Building on ideas from Riemannian Manifold Hamiltonian Monte Carlo and its velocity-based variant, Lagrangian Dynamical Monte Carlo, we construct a PDMP for which changes in the metric trigger additional velocity update events. Using a metric derived from the target Hessian, the resulting algorithm adapts to the local covariance structure. Through a series of controlled experiments, we provide practical guidance on when the proposed covariance-adaptive BPS should be preferred over standard PDMP algorithms.