HVQ-VAE: Variational auto-encoder with hyperbolic vector quantization

Vector quantized-variational autoencoder (VQ-VAE) and its variants have made significant progress in creating discrete latent space via learning a codebook. Previous works on VQ-VAE have focused on discrete latent spaces in Euclidean or in spherical spaces. This paper studies the geometric prior of hyperbolic spaces as a way to improve the learning capacity of VQ-VAE. That being said, working with the VQ-VAE in the hyperbolic space is not without difficulties, and the benefits of using hyperbolic space as the geometric prior for the latent space have never been studied in VQ-VAE. We bridge this gap by developing the VQ-VAE with hyperbolic vector quantization. To this end, we propose the hyperbolic VQ-VAE (HVQ-VAE), which learns the latent embedding of data and the codebook in the hyperbolic space. Specifically, we endow the discrete latent space in the Poincaré ball, such that the clustering algorithm can be formulated and optimized in the Poincaré ball. Thorough experiments against various baselines are conducted to evaluate the superiority of the proposed HVQ-VAE empirically. We show that HVQ-VAE enjoys better image reconstruction, effective codebook usage, and fast convergence than baselines. We also present evidence that HVQ-VAE outperforms VQ-VAE in low-dimensional latent space.