  Traditional deep generative models of images and other spatial modalities can
only generate fixed sized outputs. The generated images have exactly the same
resolution as the training images, which is dictated by the number of layers in
the underlying neural network. Recent work has shown, however, that feeding
spatial noise vectors into a fully convolutional neural network enables both
generation of arbitrary resolution output images as well as training on
arbitrary resolution training images. While this work has provided impressive
empirical results, little theoretical interpretation was provided to explain
the underlying generative process. In this paper we provide a firm theoretical
interpretation for infinite spatial generation, by drawing connections to
spatial stochastic processes. We use the resulting intuition to improve upon
existing spatially infinite generative models to enable more efficient training
through a model that we call an infinite generative adversarial network, or
$\infty$-GAN. Experiments on world map generation, panoramic images and texture
synthesis verify the ability of $\infty$-GAN to efficiently generate images of
arbitrary size.
