  Variational autoencoder (VAE) estimates the posterior parameters (mean and
variance) of latent variables corresponding to each input data. While it is
used for many tasks, the transparency of the model is still an underlying
issue. This paper provides a quantitative understanding of VAE property through
the differential geometric and information-theoretic interpretations of VAE.
According to the Rate-distortion theory, the optimal transform coding is
achieved by using an orthonormal transform with PCA basis where the transform
space is isometric to the input. Considering the analogy of transform coding to
VAE, we clarify theoretically and experimentally that VAE can be mapped to an
implicit isometric embedding with a scale factor derived from the posterior
parameter. As a result, we can estimate the data probabilities in the input
space from the prior, loss metrics, and corresponding posterior parameters, and
further, the quantitative importance of each latent variable can be evaluated
like the eigenvalue of PCA.
