# Geometry of Deep Generative Models for Disentangled Representations

## TL;DR

This paper investigates the geometric properties of latent spaces in deep generative models designed for disentangled representations, revealing how curvature influences class separability and interpolation quality.

## Contribution

It provides a comparative analysis of the geometry of disentangled models versus VAEs, highlighting the role of Riemannian metrics in improving latent space interpretability.

## Key findings

- Disentangled models exhibit higher curvature in class-distinguishable features.
- Riemannian metrics enhance interpolation and distance measures in latent space.
- Latent space geometry impacts model robustness and interpretability.

## Abstract

Deep generative models like variational autoencoders approximate the intrinsic geometry of high dimensional data manifolds by learning low-dimensional latent-space variables and an embedding function. The geometric properties of these latent spaces has been studied under the lens of Riemannian geometry; via analysis of the non-linearity of the generator function. In new developments, deep generative models have been used for learning semantically meaningful `disentangled' representations; that capture task relevant attributes while being invariant to other attributes. In this work, we explore the geometry of popular generative models for disentangled representation learning. We use several metrics to compare the properties of latent spaces of disentangled representation models in terms of class separability and curvature of the latent-space. The results we obtain establish that the class distinguishable features in the disentangled latent space exhibits higher curvature as opposed to a variational autoencoder. We evaluate and compare the geometry of three such models with variational autoencoder on two different datasets. Further, our results show that distances and interpolation in the latent space are significantly improved with Riemannian metrics derived from the curvature of the space. We expect these results will have implications on understanding how deep-networks can be made more robust, generalizable, as well as interpretable.