  Generative flows are attractive because they admit exact likelihood
optimization and efficient image synthesis. Recently, Kingma & Dhariwal (2018)
demonstrated with Glow that generative flows are capable of generating high
quality images. We generalize the 1 x 1 convolutions proposed in Glow to
invertible d x d convolutions, which are more flexible since they operate on
both channel and spatial axes. We propose two methods to produce invertible
convolutions that have receptive fields identical to standard convolutions:
Emerging convolutions are obtained by chaining specific autoregressive
convolutions, and periodic convolutions are decoupled in the frequency domain.
Our experiments show that the flexibility of d x d convolutions significantly
improves the performance of generative flow models on galaxy images, CIFAR10
and ImageNet.
