Restructuring Vector Quantization with the Rotation Trick
Christopher Fifty Ronald Guenther Junkins Dennis Duan Aniketh Iyengar Jerry Weihong Liu Ehsan Amid Sebastian Thrun Christopher Re
Vector Quantized Variational AutoEncoders (VQ-VAEs) are designed to compress a continuous input to a discrete latent space and reconstruct it with minimal distortion. 
They operate by maintaining a set of vectors---often referred to as the codebook---and quantizing each encoder output to the nearest vector in the codebook. 
However, as vector quantization is non-differentiable, the gradient to the encoder flows _around_ the vector quantization layer rather than _through_ it in a straight-through approximation.
This approximation may be undesirable as all information from the vector quantization operation is lost. 
In this work, we propose a way to propagate gradients through the vector quantization layer of VQ-VAEs. 
We smoothly transform each encoder output into its corresponding codebook vector via a rotation and rescaling linear transformation that is treated as a constant during backpropagation. 
As a result, the relative magnitude and angle between encoder output and codebook vector becomes encoded into the gradient as it propagates through the vector quantization layer and back to the encoder.
Across 11 different VQ-VAE training paradigms, we find this restructuring improves reconstruction metrics, codebook utilization, and quantization error.
Vector Quantization, VQ-VAE