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

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.