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.

翻译：高斯去噪已成为构建生成建模中无模拟连续归一化流的一种强有力的原理。尽管取得了经验上的成功，但这些流的理论性质以及高斯去噪的正则化效应在很大程度上仍未得到探索。在本文中，我们旨在通过研究基于高斯去噪构建的无模拟连续归一化流的适定性来填补这一空白。通过一个称为高斯插值流的统一框架，我们针对几类丰富的目标分布，建立了流速度场的Lipschitz正则性、流的存在唯一性，以及流映射和时间反向流映射的Lipschitz连续性。这一分析还揭示了高斯插值流的自编码和循环一致性性质。此外，我们以二次Wasserstein距离为度量，深入研究了这些流在源分布和速度场扰动下的稳定性。我们的发现为生成建模中高斯插值流所采用的学习技术提供了宝贵的见解，为基于经验观测学习高斯插值流的端到端误差分析奠定了坚实的理论基础。