Telegrapher's Generative Model via Kac Flows
This work addresses generative modeling for AI applications by proposing a novel method that improves upon diffusion models, though it appears incremental as it builds on existing flow-based frameworks.
The authors tackled the problem of generative modeling by introducing a new flow-based model based on the telegrapher's equation and Kac flows, which offers Lipschitz continuity and bounded velocity, and demonstrated its scalability and advantages over diffusion models in numerical experiments.
We break the mold in flow-based generative modeling by proposing a new model based on the damped wave equation, also known as telegrapher's equation. Similar to the diffusion equation and Brownian motion, there is a Feynman-Kac type relation between the telegrapher's equation and the stochastic Kac process in 1D. The Kac flow evolves stepwise linearly in time, so that the probability flow is Lipschitz continuous in the Wasserstein distance and, in contrast to diffusion flows, the norm of the velocity is globally bounded. Furthermore, the Kac model has the diffusion model as its asymptotic limit. We extend these considerations to a multi-dimensional stochastic process which consists of independent 1D Kac processes in each spatial component. We show that this process gives rise to an absolutely continuous curve in the Wasserstein space and compute the conditional velocity field starting in a Dirac point analytically. Using the framework of flow matching, we train a neural network that approximates the velocity field and use it for sample generation. Our numerical experiments demonstrate the scalability of our approach, and show its advantages over diffusion models.