Theoretical guarantees in KL for Diffusion Flow Matching

• URL: http://arxiv.org/abs/2409.08311v1
• Date: Thu, 12 Sep 2024 15:19:00 GMT
• Title: Theoretical guarantees in KL for Diffusion Flow Matching
• Authors: Marta Gentiloni Silveri, Giovanni Conforti, Alain Durmus,
• Abstract summary: Flow Matching (FM) aims to bridge in finite time the target distribution $nustar$ with an auxiliary distribution $mu$. We obtain non-asymptotics guarantees for Diffusion Flow Matching (DFM) models using as bridge the conditional distribution associated with the Brownian motion.
• Score: 9.618473763561418
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Flow Matching (FM) (also referred to as stochastic interpolants or rectified flows) stands out as a class of generative models that aims to bridge in finite time the target distribution $\nu^\star$ with an auxiliary distribution $\mu$, leveraging a fixed coupling $\pi$ and a bridge which can either be deterministic or stochastic. These two ingredients define a path measure which can then be approximated by learning the drift of its Markovian projection. The main contribution of this paper is to provide relatively mild assumptions on $\nu^\star$, $\mu$ and $\pi$ to obtain non-asymptotics guarantees for Diffusion Flow Matching (DFM) models using as bridge the conditional distribution associated with the Brownian motion. More precisely, we establish bounds on the Kullback-Leibler divergence between the target distribution and the one generated by such DFM models under moment conditions on the score of $\nu^\star$, $\mu$ and $\pi$, and a standard $L^2$-drift-approximation error assumption.

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