# Variational Autoencoders Pursue PCA Directions (by Accident)

## TL;DR

This paper explains why Variational Autoencoders naturally align with PCA directions, attributing it to the encoder's diagonal approximation and stochasticity promoting local orthogonality, supported by theory and experiments.

## Contribution

It provides a theoretical and experimental explanation for VAE's alignment with PCA directions, revealing an unintended but fundamental property.

## Key findings

- VAE's encoder induces local orthogonality in the decoder.
- The alignment with PCA directions arises from the encoder's diagonal approximation and stochasticity.
- Theoretical analysis confirms the connection between VAE behavior and PCA principles.

## Abstract

The Variational Autoencoder (VAE) is a powerful architecture capable of representation learning and generative modeling. When it comes to learning interpretable (disentangled) representations, VAE and its variants show unparalleled performance. However, the reasons for this are unclear, since a very particular alignment of the latent embedding is needed but the design of the VAE does not encourage it in any explicit way. We address this matter and offer the following explanation: the diagonal approximation in the encoder together with the inherent stochasticity force local orthogonality of the decoder. The local behavior of promoting both reconstruction and orthogonality matches closely how the PCA embedding is chosen. Alongside providing an intuitive understanding, we justify the statement with full theoretical analysis as well as with experiments.