Hamiltonian Score Matching and Generative Flows
This work addresses the challenge of enhancing generative modeling in machine learning by extending Hamiltonian mechanics beyond classical applications, though it appears incremental as it builds on existing score matching and flow-based methods.
The paper tackles the problem of designing force fields for Hamiltonian ODEs to improve generative models, introducing Hamiltonian Score Matching (HSM) for score function estimation and Hamiltonian Generative Flows (HGFs) as a novel generative model that rivals leading techniques.
Classical Hamiltonian mechanics has been widely used in machine learning in the form of Hamiltonian Monte Carlo for applications with predetermined force fields. In this work, we explore the potential of deliberately designing force fields for Hamiltonian ODEs, introducing Hamiltonian velocity predictors (HVPs) as a tool for score matching and generative models. We present two innovations constructed with HVPs: Hamiltonian Score Matching (HSM), which estimates score functions by augmenting data via Hamiltonian trajectories, and Hamiltonian Generative Flows (HGFs), a novel generative model that encompasses diffusion models and flow matching as HGFs with zero force fields. We showcase the extended design space of force fields by introducing Oscillation HGFs, a generative model inspired by harmonic oscillators. Our experiments validate our theoretical insights about HSM as a novel score matching metric and demonstrate that HGFs rival leading generative modeling techniques.