用波动方程构建生成模型，速度有界更稳定。
Telegrapher's Generative Model via Kac Flows
- 基于阻尼波动方程设计新型流模型，速度全局有界。
- 在Wasserstein距离下概率流为Lipschitz连续，可解析计算速度场。
- 适合追求稳定生成、避免扩散模型梯度爆炸的场景。
我们突破传统流模型范式，提出基于阻尼波动方程（即电报方程）的新生成模型。该方程与一维随机Kac过程存在费曼-卡克型对应关系。Kac流在时间上分步线性演化，使得概率流在Wasserstein距离下为Lipschitz连续，且速度范数全局有界，与扩散流相反。此外，该模型以扩散模型为其渐近极限。我们将此推广至多维情形，由各空间维度独立的一维Kac过程构成。我们证明该过程在Wasserstein空间中生成绝对连续路径，并在起始于狄拉克点时解析计算条件速度场。借助流匹配框架，训练神经网络逼近速度场并用于采样。数值实验表明该方法具备可扩展性，且优于扩散模型。
原文摘要 · Abstract (English)
We break the mold in flow-based generative modeling by proposing a new model based on the damped wave equation, also known as telegrapher's equation. Similar to the diffusion equation and Brownian motion, there is a Feynman-Kac type relation between the telegrapher's equation and the stochastic Kac process in 1D. The Kac flow evolves stepwise linearly in time, so that the probability flow is Lipschitz continuous in the Wasserstein distance and, in contrast to diffusion flows, the norm of the velocity remains globally bounded. Furthermore, the Kac model has the diffusion model as its asymptotic limit. We extend these considerations to a multi-dimensional stochastic process which consists of independent 1D Kac processes in each spatial component. We show that this process gives rise to an absolutely continuous curve in the Wasserstein space and analytically compute the conditional velocity field when starting in a Dirac point. Using the framework of flow matching, we train a neural network to approximate the velocity field and use it for sample generation. Our numerical experiments demonstrate the scalability of our approach, and show its advantages over diffusion models.
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