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Quantitative Understanding of VAE as a Non-linearly Scaled Isometric Embedding
Akira Nakagawa · Keizo Kato · Taiji Suzuki

Tue Jul 20 05:20 PM -- 05:25 PM (PDT) @ None

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.

Author Information

Akira Nakagawa (Fujitsu Limited)
Keizo Kato (Fujitsu Laboratories Ltd.)
Taiji Suzuki (The University of Tokyo / RIKEN)

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