  This paper approaches the unsupervised learning problem by gradient descent
in the space of probability density functions. A main result shows that along
the gradient flow induced by a distribution-dependent ordinary differential
equation (ODE), the unknown data distribution emerges as the long-time limit.
That is, one can uncover the data distribution by simulating the
distribution-dependent ODE. Intriguingly, the simulation of the ODE is shown
equivalent to the training of generative adversarial networks (GANs). This
equivalence provides a new "cooperative" view of GANs and, more importantly,
sheds new light on the divergence of GANs. In particular, it reveals that the
GAN algorithm implicitly minimizes the mean squared error (MSE) between two
sets of samples, and this MSE fitting alone can cause GANs to diverge. To
construct a solution to the distribution-dependent ODE, we first show that the
associated nonlinear Fokker-Planck equation has a unique weak solution, by the
Crandall-Liggett theorem for differential equations in Banach spaces. Based on
this solution to the Fokker-Planck equation, we construct a unique solution to
the ODE, using Trevisan's superposition principle. The convergence of the
induced gradient flow to the data distribution is obtained by analyzing the
Fokker-Planck equation.
