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.

翻译：本文通过概率密度函数空间中的梯度下降方法处理无监督学习问题。主要结果表明，在由分布相关的常微分方程诱导的梯度流中，未知数据分布将在长时间极限下显现。即通过模拟该分布相关常微分方程可揭示数据分布。有趣的是，该常微分方程的模拟等价于生成对抗网络的训练过程。这一等价性为生成对抗网络提供了新的"合作"视角，更重要的是，揭示了生成对抗网络发散问题的新认知。具体而言，该等价性表明生成对抗网络算法隐式地最小化两组样本之间的均方误差，而这种均方误差拟合本身即可导致生成对抗网络发散。为构建分布相关常微分方程的解，我们首先通过巴拿赫空间中微分方程的Crandall-Liggett定理，证明关联的非线性福克-普朗克方程存在唯一弱解。基于该福克-普朗克方程的解，我们运用Trevisan叠加原理构建了常微分方程的唯一解。通过分析福克-普朗克方程，最终获得诱导梯度流向数据分布的收敛性结果。