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Theoretical guarantees in KL for Diffusion Flow Matching

Marta Gentiloni Silveri, Giovanni Conforti, Alain Durmus

TL;DR

This paper establishes 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.

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 $ν^\star$ with an auxiliary distribution $μ$, leveraging a fixed coupling $π$ 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 $ν^\star$, $μ$ and $π$ 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 $ν^\star$, $μ$ and $π$, and a standard $L^2$-drift-approximation error assumption.

Theoretical guarantees in KL for Diffusion Flow Matching

TL;DR

This paper establishes 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 \mu\piL^2$-drift-approximation error assumption.

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 with an auxiliary distribution , leveraging a fixed coupling 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 , and 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 , and , and a standard -drift-approximation error assumption.
Paper Structure (19 sections, 8 theorems, 161 equations)

This paper contains 19 sections, 8 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Notation.
  4. Diffusion Flow Matching.
  5. Definition of the interpolated process
  6. Markovian projection and Diffusion Flow Matching
  7. Markovian projection.
  8. Diffusion Flow Matching.
  9. Main results
  10. Convergence Bounds.
  11. Related works and comparison with existing literature.
  12. The proposed methodology.
  13. Conclusion
  14. Postponed proofs
  15. The Markovian Projection
  16. ...and 4 more sections

Key Result

Theorem 1

Consider a $\pi \in \Pi(\mu,\nu^{\star})$ and $\mathbb{Q}^{\beta}$ associated with eq:sde_v0. Consider the drift field Under appropriate conditions (see section_markovian), the Markov process $(X^{\mathrm{M}}_t)_{t\in\left[0,1\right]}$ solution of mimics the one-dimensional time marginals of $\inter{\pi,\mathbb{Q}^{\beta}}$, i.e., for any $t \in [0,1)$, $X^{\mathrm{I}}_t \stackrel{\text{dist}}{=

Theorems & Definitions (26)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 2
  • Remark 6
  • Remark 7
  • Theorem 3
  • ...and 16 more