While the neural ODE formulation of normalizing flows such as in FFJORD enables us to calculate the determinants of free form Jacobians in O(D) time, the flexibility of the transformation underlying neural ODEs has been shown to be suboptimal. In this paper, we present AFFJORD, a neural ODE-based normalizing flow which enhances the representation power of FFJORD by defining the neural ODE through special augmented transformation dynamics which preserve the topology of the space. Furthermore, we derive the Jacobian determinant of the general augmented form by generalizing the chain rule in the continuous sense into the Cable Rule, which expresses the forward sensitivity of ODEs with respect to their initial conditions. The cable rule gives an explicit expression for the Jacobian of a neural ODE transformation, and provides an elegant proof of the instantaneous change of variable. Our experimental results on density estimation in synthetic and high dimensional data, such as MNIST, CIFAR-10 and CelebA 32x32, show that AFFJORD outperforms the baseline FFJORD through the improved flexibility of the underlying vector field.

本文提出了AFFJORD，一种基于神经ODE的归一化流，通过定义保留空间拓扑特殊增广变换动力学，增强了FFJORD的表示能力，并通过广义连锁法将增广形式的Jacobian行列式推广到连续情况下，给出了一个神经ODE变换Jacobian的显式表达，并提供了一个优雅的瞬时变量变化的证明。实验结果表明，AFFJORD在密度估计方面优于基线FFJORD。

神经ODE流的增强结构改进分布建模