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Convergence Analysis of Flow Matching in Latent Space with Transformers

Yuling Jiao, Yanming Lai, Yang Wang, Bokai Yan

TL;DR

This work provides an end-to-end statistical learning theory for flow matching in latent spaces using transformers. By embedding high-dimensional data into a latent space via a pre-trained autoencoder and modeling the velocity field with Lipschitz transformers, the authors establish convergence in Wasserstein-2 distance between the generated and target distributions, accounting for pre-training domain shift and reconstruction error. They prove universal approximation results for Lipschitz transformers, derive generalization and discretization bounds for velocity-field estimation, and quantify the end-to-end error introduced by autoencoder pre-training, yielding precise conditions under which convergence is guaranteed. The results offer theoretical justification for latent-space ODE-based generative models with transformers and guide practical deployment by clarifying the roles of pre-training, discretization, and Lipschitz regularity in achieving reliable sample generation.

Abstract

We present theoretical convergence guarantees for ODE-based generative models, specifically flow matching. We use a pre-trained autoencoder network to map high-dimensional original inputs to a low-dimensional latent space, where a transformer network is trained to predict the velocity field of the transformation from a standard normal distribution to the target latent distribution. Our error analysis demonstrates the effectiveness of this approach, showing that the distribution of samples generated via estimated ODE flow converges to the target distribution in the Wasserstein-2 distance under mild and practical assumptions. Furthermore, we show that arbitrary smooth functions can be effectively approximated by transformer networks with Lipschitz continuity, which may be of independent interest.

Convergence Analysis of Flow Matching in Latent Space with Transformers

TL;DR

This work provides an end-to-end statistical learning theory for flow matching in latent spaces using transformers. By embedding high-dimensional data into a latent space via a pre-trained autoencoder and modeling the velocity field with Lipschitz transformers, the authors establish convergence in Wasserstein-2 distance between the generated and target distributions, accounting for pre-training domain shift and reconstruction error. They prove universal approximation results for Lipschitz transformers, derive generalization and discretization bounds for velocity-field estimation, and quantify the end-to-end error introduced by autoencoder pre-training, yielding precise conditions under which convergence is guaranteed. The results offer theoretical justification for latent-space ODE-based generative models with transformers and guide practical deployment by clarifying the roles of pre-training, discretization, and Lipschitz regularity in achieving reliable sample generation.

Abstract

We present theoretical convergence guarantees for ODE-based generative models, specifically flow matching. We use a pre-trained autoencoder network to map high-dimensional original inputs to a low-dimensional latent space, where a transformer network is trained to predict the velocity field of the transformation from a standard normal distribution to the target latent distribution. Our error analysis demonstrates the effectiveness of this approach, showing that the distribution of samples generated via estimated ODE flow converges to the target distribution in the Wasserstein-2 distance under mild and practical assumptions. Furthermore, we show that arbitrary smooth functions can be effectively approximated by transformer networks with Lipschitz continuity, which may be of independent interest.
Paper Structure (32 sections, 33 theorems, 267 equations, 1 figure)

This paper contains 32 sections, 33 theorems, 267 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our main contributions
  3. Organization
  4. Preliminaries
  5. Transformer networks
  6. Pre-training
  7. Flow matching
  8. Approximation
  9. Generalization and Sampling
  10. End-to-end Error Analysis
  11. Conclusion and Future Work
  12. Approximation Error
  13. Input embedding
  14. Approximation of polynomials with single-head attention
  15. Approximation of polynomials with multi-head attention
  16. ...and 17 more sections

Key Result

Theorem 8

Let $0<\varepsilon<1$ and $\beta>0$. For any function $\boldsymbol{f} \in \mathcal{H}_{d,d^\prime}^\beta([0, 1]^{d}, K)$, there exists a transformer network $\boldsymbol{\phi} \in \mathcal{T}_{d, d^{\prime}}\left(N, h, d_k, d_v, d_{ff}, B, J, \gamma\right)$, where such that Furthermore, if $\beta>1$, we may choose

Figures (1)

  • Figure 1: An illustration of our framework. Pre-training: Based on $m$ samples $\mathcal{Y} = \{\boldsymbol{y}_i\}_{i=1}^m$ drawn i.i.d. from pre-trained data distribution $\widetilde{\gamma}_1$, we minimize the empirical reconstruction loss to obtain an encoder $\widehat{\boldsymbol{E}}: [0,1]^D \rightarrow [0,1]^d$ and the corresponding decoder $\widehat{\boldsymbol{D}}: [0,1]^d \rightarrow \mathbb{R}^D$. These will serve as the bridge linking the high-dimensional input space and the low-dimensional latent space. Flow matching: For the target distribution $\gamma_1$ and $n$ samples $\mathcal{X} = \{\boldsymbol{x}_i\}_{i=1}^n$ drawn from it, the encoder $\widehat{\boldsymbol{E}}$ maps them to the latent space with $\pi_1 = \widehat{\boldsymbol{E}}_{\#} \gamma_1$ and $\widehat{\boldsymbol{E}}(\mathcal{X}) = \{\widehat{\boldsymbol{E}}(\boldsymbol{x}_i)\}_{i=1}^n$. Flow matching is then applied within the latent space, where a transformer network is trained to predict the velocity field of the transformation from a standard normal distribution $\pi_0 = \mathcal{N}(0, I_d)$ to the target latent distribution $\pi_1$. Sampling: Given the estimated velocity field, we can generate samples from an approximation of the continuous flow ODE starting from the prior distribution $\pi_0$. The generated latent data distribution $\widehat{\pi}_T$ will be mapped back to the high-dimensional space by the decoder $\widehat{\boldsymbol{D}}$, resulting in the generated data distribution $\widehat{\gamma}_T = \widehat{\boldsymbol{D}}_{\#} \widehat{\pi}_T$.

Theorems & Definitions (46)

  • Remark 1
  • Definition 2: Pseudo-dimension
  • Definition 3: Covering number
  • Definition 4: Lipschitz functions
  • Definition 5: Hölder classes
  • Definition 6: Differentiability classes
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Remark 10
  • ...and 36 more