Reproduce “Learning the Infinitesimal Generator of Stochastic Diffusion Processes” on OpenResearch

About this paper

Vladimir R. Kostic, Karim Lounici, Helene Halconruy, Timothee Devergne, Massimiliano Pontil

The authors introduce an energy-weighted, reduced-rank regression (RRR) method for learning diffusion generator spectra. It fits the resolvent, a compact operator sharing the generator’s eigenfunctions, using known diffusion energy to estimate its leading modes. In a four-well example, one kernel-length-scale estimate tracks the ground-truth leading eigenfunction more closely than another. The learning bounds depend on stated assumptions.

Key insights

• Learning the bounded resolvent operator provides a theoretically tractable pathway to analyze the unbounded infinitesimal generator's spectral properties.
• An energy-based risk functional allows seamless integration of physical priors, improving the robustness and physical relevance of the learned system dynamics.
• The framework achieves dimension-free learning bounds and non-spurious spectral estimation, overcoming critical limitations of previous approaches.

Summary by alphaXiv.