Demystifying Transition Matching: When and Why It Can Beat Flow Matching

TL;DR

This paper analyzes when and why Transition Matching (TM) surpasses Flow Matching (FM) in generative modeling, demonstrating TM's advantages in specific distribution settings through theoretical proofs and experiments.

Contribution

It provides the first theoretical analysis of TM's superiority over FM, especially for unimodal and well-separated Gaussian mixtures, supported by empirical validation.

Findings

TM achieves lower KL divergence than FM for finite steps in unimodal Gaussian distributions.

TM converges faster than FM under a fixed compute budget in the unimodal Gaussian setting.

TM outperforms FM when the target distribution has well-separated modes and non-zero variance.

Abstract

Flow Matching (FM) underpins many state-of-the-art generative models, yet recent results indicate that Transition Matching (TM) can achieve higher quality with fewer sampling steps. This work answers the question of when and why TM outperforms FM. First, when the target is a unimodal Gaussian distribution, we prove that TM attains strictly lower KL divergence than FM for finite number of steps. The improvement arises from stochastic difference latent updates in TM, which preserve target covariance that deterministic FM underestimates. We then characterize convergence rates, showing that TM achieves faster convergence than FM under a fixed compute budget, establishing its advantage in the unimodal Gaussian setting. Second, we extend the analysis to Gaussian mixtures and identify local-unimodality regimes in which the sampling dynamics approximate the unimodal case, where TM can outperform FM. The approximation error decreases as the minimal distance between component means increases, highlighting that TM is favored when the modes are well separated. However, when the target variance approaches zero, each TM update converges to the FM update, and the performance advantage of TM diminishes. In summary, we show that TM outperforms FM when the target distribution has well-separated modes and non-negligible variances. We validate our theoretical results with controlled experiments on Gaussian distributions, and extend the comparison to real-world applications in image and video generation.