  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.
