Hypocoercivity of Tempered Bouncy Particle Samplers for Heavy-Tailed Targets

What the math gives to ML

The paper's transferable asset is a state-dependent kinetic temperature that leaves a prescribed heavy-tailed position law invariant, rather than merely tuning a global sampler temperature. The joint law uses velocities with covariance \(\sigma^2(x)I\), allowing exploration to accelerate in remote or low-density regions while retaining the target \(\mu(x)\propto e^{-U(x)}\). For neural energy models, latent-variable models, or posterior sampling, this can be converted into an exactly invariant state-dependent Langevin sampler with the required divergence correction. The hypocoercive viewpoint also suggests testing mixing on heavy-tailed targets through weighted-Poincare or effective-sample-size measurements rather than only short-run loss curves.