  This paper describes an entropy regularization term for vector quantization
(VQ) based on the analysis of persistent homology of the VQ embeddings. Higher
embedding entropy positively correlates with higher codebook utilization,
mitigating overfit towards the identity and codebook collapse in VQ-based
autoencoders [1]. We show that homology-constrained regularization is an
effective way to increase entropy of the VQ process (approximated to input
entropy) while preserving the approximated topology in the quantized latent
space, averaged over mini batches. This work further explores some patterns of
persistent homology diagrams of latents formed by vector quantization. We
implement and test the proposed algorithm as a module integrated into a sample
VQ-VAE. Linked code repository provides a functioning implementation of the
proposed architecture, referred to as homology-constrained vector quantization
(HC-VQ) further in this work.
