Gaussian Interpolation Flows
Yuan Gao Jian Huang Yuling Jiao
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.