高斯插值流
Gaussian Interpolation Flows
AI总结：
该研究提出高斯插值流统一框架，探究基于高斯去噪的无仿真连续归一化流的适定性等理论性质，并分析其稳定性，为相关生成式建模学习技术提供理论支撑。
摘要：
高斯去噪已成为构建无仿真连续归一化流以用于生成式建模的一种强大方法。尽管这类方法在实验中取得了成功，但其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究。在本研究中，我们旨在通过研究基于高斯去噪构建的无仿真连续归一化流的适定性来填补这一空白。通过一个被称为高斯插值流（Gaussian interpolation flow）的统一框架，我们针对几类丰富的目标分布，证明了流速度场的利普希茨正则性、流的存在性与唯一性，以及流映射和时间反转流映射的利普希茨连续性。该分析还阐明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，研究了这些流在源分布及速度场扰动下的稳定性。我们的研究结果为高斯插值流在生成式建模中采用的学习技术提供了宝贵见解，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。
英文摘要：
Gaussian denoising has emerged as a powerful method 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 study 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 Gaussian interpolation flows with empirical observations.
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