Categorical Flow Matching on Statistical Manifolds

Understand

We introduce Statistical Flow Matching (SFM), a novel and mathematically rigorous flow-matching framework on the manifold of parameterized probability measures inspired by the results from information geometry.

• We demonstrate the effectiveness of our method on the discrete generation problem by instantiating SFM on the manifold of categorical distributions whose geometric properties remain unexplored in previous discrete generative models.
• Utilizing the Fisher information metric, we equip the manifold with a Riemannian structure whose intrinsic geometries are effectively leveraged by following the shortest paths of geodesics.
• We develop an efficient training and sampling algorithm that overcomes numerical stability issues with a diffeomorphism between manifolds.