Hamiltonian dynamics for sampling on discrete spaces
arXiv:2608.17961
2026
Sampling
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
This paper provides a principled way to add continuous momentum to genuinely discrete state spaces without embedding discrete states into Euclidean space. The transferable asset is a lifted, non-reversible neighbor-jump process whose Gibbs target remains invariant while persistence produces ballistic rather than purely diffusive exploration. A practical neural-network use is sampling categorical latent variables, discrete diffusion states, or token substitutions with persistent momentum, followed by tau-leaping or splitting to batch many neighbor evaluations. The main engineering risk is the cost of evaluating all neighbors, so the first tests should use small categorical latents and compare effective samples per target-network evaluation against Gibbs, Metropolis, and ordinary discrete Langevin baselines.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Approximate the exact event-by-event lifted sampler by drawing independent Poisson jump counts over a short interval and applying compatible discrete moves in parallel. This converts sequential neighbor events into batched GPU-friendly updates while retaining the Hamiltonian rate structure; the step size controls the error-versus-throughput tradeoff.
Useful7/10
Difficulty5/10
Novelty8/10
✗ Failed on benchmark
2026
Replace independent categorical proposals or reversible Metropolis updates for discrete latent variables with a lifted sampler carrying persistent continuous edge momenta. Neighbor transitions are biased by the momentum and use a symmetric energy factor, so momentum reversal gives the required balance relation for the target Gibbs distribution while ordinary dynamics remain non-reversible. This should reduce random-walk behavior when sampling multimodal categorical latents or token sequences.
Useful7/10
Difficulty6/10
Novelty7/10