Tessellations of Semi-Discrete Flow Matching

TL;DR

This paper analyzes the geometric structure of semi-discrete flow matching in generative modeling, revealing properties of terminal assignment regions and their differences from classical optimal transport cells.

Contribution

It provides a theoretical analysis of the geometry of semi-discrete flow matching, including properties of terminal regions and their divergence from Laguerre cells.

Findings

Terminal regions are open, simply connected, and sometimes homeomorphic to a ball.

These regions can be non-convex with curved boundaries, unlike classical Laguerre cells.

The analysis clarifies the geometry before neural approximation effects.

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.