  We propose a general framework to learn deep generative models via
\textbf{V}ariational \textbf{Gr}adient Fl\textbf{ow} (VGrow) on probability
spaces. The evolving distribution that asymptotically converges to the target
distribution is governed by a vector field, which is the negative gradient of
the first variation of the $f$-divergence between them. We prove that the
evolving distribution coincides with the pushforward distribution through the
infinitesimal time composition of residual maps that are perturbations of the
identity map along the vector field. The vector field depends on the density
ratio of the pushforward distribution and the target distribution, which can be
consistently learned from a binary classification problem. Connections of our
proposed VGrow method with other popular methods, such as VAE, GAN and
flow-based methods, have been established in this framework, gaining new
insights of deep generative learning. We also evaluated several commonly used
divergences, including Kullback-Leibler, Jensen-Shannon, Jeffrey divergences as
well as our newly discovered `logD' divergence which serves as the objective
function of the logD-trick GAN. Experimental results on benchmark datasets
demonstrate that VGrow can generate high-fidelity images in a stable and
efficient manner, achieving competitive performance with state-of-the-art GANs.
