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Normalizing flows model probability distributions by learning invertibletransformations that transfer a simple distribution into complex distributions.Since the architecture of ResNet-based normalizing flows is more flexible thanthat of coupling-based models, ResNet-based normalizing flows have been widelystudied in recent years. Despite their architectural flexibility, it iswell-known that the current ResNet-based models suffer from constrainedLipschitz constants. In this paper, we propose the monotone formulation toovercome the issue of the Lipschitz constants using monotone operators andprovide an in-depth theoretical analysis. Furthermore, we construct anactivation function called Concatenated Pila (CPila) to improve gradient flow.The resulting model, Monotone Flows, exhibits an excellent performance onmultiple density estimation benchmarks (MNIST, CIFAR-10, ImageNet32,ImageNet64). Code is available at https://github.com/mlvlab/MonotoneFlows.

Invertible Monotone Operators for Normalizing Flows