Quantitative Understanding of VAE as a Non-linearly Scaled Isometric Embedding

Akira Nakagawa, Keizo Kato

Keywords: unsupervised representation learning, deep image compression

Abstract: 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 by interpreting VAE as a non-linearly scaled isometric embedding. According to the Rate-distortion theory, the optimal transform coding is achieved by using a PCA-like orthonormal transform where the transform space is isometric to the input. From this analogy, we show 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. In addition, the quantitative importance of each latent variable can be evaluated like the eigenvalue of PCA.

One-sentence Summary: VAE can provide a practical quantitative analysis of input data by mapping to implicit isometric embedding.

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