  Flow-based generative models have become an important class of unsupervised
learning approaches. In this work, we incorporate the key ideas of
renormalization group (RG) and sparse prior distribution to design a
hierarchical flow-based generative model, RG-Flow, which can separate
information at different scales of images and extract disentangled
representations at each scale. We demonstrate our method on synthetic
multi-scale image datasets and the CelebA dataset, showing that the
disentangled representations enable semantic manipulation and style mixing of
the images at different scales. To visualize the latent representations, we
introduce receptive fields for flow-based models and show that the receptive
fields of RG-Flow are similar to those of convolutional neural networks. In
addition, we replace the widely adopted isotropic Gaussian prior distribution
by the sparse Laplacian distribution to further enhance the disentanglement of
representations. From a theoretical perspective, our proposed method has
$O(\log L)$ complexity for inpainting of an image with edge length $L$,
compared to previous generative models with $O(L^2)$ complexity.
