Quantitative Understanding of VAE as a Non-linearly Scaled Isometric Embedding
This work addresses the interpretability issue in VAEs for researchers and practitioners in machine learning, offering a theoretical framework to analyze model properties, but it is incremental as it builds on existing rate-distortion theory and PCA analogies.
The paper tackles the lack of transparency in Variational Autoencoders (VAEs) by providing a quantitative understanding through differential geometric and information-theoretic interpretations, showing that VAE can be mapped to an implicit isometric embedding with a scale factor derived from posterior parameters, enabling estimation of data probabilities and evaluation of latent variable importance similar to PCA eigenvalues.
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.