(Maybe) in-consistency between VQ-VAE paper and its implementation.

FIrst of all, maybe it is my misunderstanding of the paper, so hope somebody could explain it for me, thanks! :

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in the paper, the loss is defined as

where e is the codebook defined at the beginning of the Section:

So, in the paper, the codebook loss and commitment loss are MSE between z_e(x) and e.

However, in the implementation, they are implemented as MSE between z_e(x)(inputs) and z_q(x)(quantized), where variable quantized means quantized encoding of the image, namely z_q:

Are they actually the same thing? why?

• If the paper stated is right. how the dimension matches between z_e(x)(H' * W' * D) and e(K * D)?
• if the implementation is right. how z_q(x)(quantized) backprop since its calculation contains argmin?