Gaussian denoising has emerged as a powerful principle for constructing
simulation-free continuous normalizing flows for generative modeling. Despite
their empirical successes, theoretical properties of these flows and the
regularizing effect of Gaussian denoising have remained largely unexplored. In
this work, we aim to address this gap by investigating the well-posedness of
simulation-free continuous normalizing flows built on Gaussian denoising.
Through a unified framework termed Gaussian interpolation flow, we establish
the Lipschitz regularity of the flow velocity field, the existence and
uniqueness of the flow, and the Lipschitz continuity of the flow map and the
time-reversed flow map for several rich classes of target distributions. This
analysis also sheds light on the auto-encoding and cycle-consistency properties
of Gaussian interpolation flows. Additionally, we delve into the stability of
these flows in source distributions and perturbations of the velocity field,
using the quadratic Wasserstein distance as a metric. Our findings offer
valuable insights into the learning techniques employed in Gaussian
interpolation flows for generative modeling, providing a solid theoretical
foundation for end-to-end error analyses of learning GIFs with empirical
observations.

我们研究了基于高斯降噪的模拟 - free 连续标准化流的良定义性、可微性和唯一性，通过建立高斯插值流的统一框架，我们证明了流速度场的利普希茨正则性、流的存在唯一性，以及流映射和时间反向流映射在多个丰富的目标分布类别中的利普希茨连续性，这也揭示了高斯插值流的自动编码和循环一致性特性。我们还研究了这些流在源分布和速度场扰动中的稳定性，利用二次 Wasserstein 距离作为度量。我们的发现为用于生成建模中的高斯插值流的学习技术提供了宝贵的见解，为使用经验观察进行端到端误差分析的学习 GIFs 提供了坚实的理论基础。