Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective
Neta Shaul Itai Gat Marton Havasi Daniel Severo Anuroop Sriram Peter Holderrieth Brian Karrer Yaron Lipman Ricky T. Q. Chen
The design space of discrete-space diffusion or flow generative models are significantly less well-understood than their continuous-space counterparts, with many works focusing only on a simple masked construction.
In this work, we aim to take a holistic approach to the construction of discrete generative models based on continuous-time Markov chains, and for the first time, allow the use of arbitrary discrete probability paths, or colloquially, corruption processes. 
Through the lens of optimizing the symmetric kinetic energy, we propose velocity formulas that can be applied to any given probability path, completely decoupling the probability and velocity, and giving the user the freedom to specify any desirable probability path based on expert knowledge specific to the data domain. 
Furthermore, we find that a special construction of mixture probability paths optimizes the symmetric kinetic energy for the discrete case.
We empirically validate the usefulness of this new design space across multiple modalities: text generation, inorganic material generation, and image generation. We find that we can outperform the mask construction even in text with kinetic-optimal mixture paths, while we can make use of domain-specific constructions of the probability path over the visual domain.
flow matching, discrete generative modeling
Through the lens of kinetic optimality, we expand the design space of Discrete Flow Matching, allowing the use of any probability path and simultaneously justifying existing mixture paths.