Speaker
Description
In score-based diffusion models, the underlying probability path is governed by an Ornstein-Uhlenbeck (OU) process,
connecting a target data variable with a Gaussian.
This variance-preserving OU path is driven by the Wiener process,
and it is known that due to its unbounded variation,
the OU path may admit certain singularities concerning its score at times close to the data.
To overcome this shortcoming,
we study the Ornstein-Uhlenbeck-Kac (OU-K) process, where instead of the Wiener process
a finite-speed Kac process defines the driving process.
The OU-K process can be analyzed via a time-inhomogeneous Kac process.
By extending the latter to a piecewise-deterministic Markov process,
we derive its Fokker-Planck equation in form of a specifically damped wave equation,
the Euler-Poisson-Darboux (EPD) equation.
For Kac switching rates being multiples of the OU convergence rate,
we can explicitly describe the law of the inhomogeneous Kac process via the sum of two Legendre polynomials.
This allows us to apply the OU-K process in a generative flow model. We demonstrate its advantages over diffusion
and flow matching models on singular data sets and on few-step CIFAR generation.
This is joint work with J. Auffenberg abd R. Duong (TU Berlin).