A Discrete-Time Variational Objective for Degenerate Diffusion and Equivalence to Flow Matching

arXiv:2611.00800v1 Announce Type: new Abstract: Diffusion models and more recently Flow models have widespread application. In this work we consider degenerate Diffusion models in which the per-step generator is deterministic. We construct a modified variational framework and show a formal equivalence between Flow models and Degenerate Diffusion models. For our discrete-time variational objective we show that the optimal degenerate generator recovers the data distribution in the limit of an infinite number of generator steps. Our work therefore provides a unified perspective, showing that discrete-time implementations of flow matching and the discrete-time Degenerate Diffusion model can be considered within the same simple variational framework, removing the need for the continuous-time machinery normally used to justify flow matching models.