Tessellations of Semi-Discrete Flow Matching

2026-05-08

AI Summary

This work investigates the geometric structure induced by the terminal flow map in semi-discrete flow matching, focusing on the partitioning of space when transporting a Gaussian source distribution to a finite discrete target. By deriving closed-form solutions for the velocity field and combining techniques from differential equations and topological analysis, the study reveals—for the first time—that the resulting terminal assignment regions can be non-convex, bounded by curved edges, and topologically intricate, markedly differing from the Laguerre cells arising in semi-discrete optimal transport. Theoretical analysis establishes that these regions are open and simply connected, and under additional conditions, they are homeomorphic to the unit ball, thereby highlighting a fundamental distinction from classical optimal transport partitions.

Abstract

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.

Problem

• Flow Matching
• semi-discrete
• terminal assignment regions
• geometry
• generative modeling

Innovation

• Flow Matching
• semi-discrete transport
• terminal assignment regions
• geometric structure
• Laguerre cells