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Workshop: The Symbiosis of Deep Learning and Differential Equations -- III

Enhanced Distribution Modelling via Augmented Architectures For Neural ODE Flows

Etrit Haxholli · Marco Lorenzi

Keywords: [ Neural ODEs ] [ Density Estimation ] [ Continuous Normalizing Flows ] [ Augmented Neural ODEs ] [ normalizing flows ]

Abstract: While the neural ODE formulation of normalizing flows such as in FFJORD enables us to calculate the determinants of free form Jacobians in $\mathcal{O}(D)$ 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 $\textit{cable rule}$, 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 ($32\times32$), show that AFFJORD outperforms the baseline FFJORD through the improved flexibility of the underlying vector field.

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