# From Points to Spheres: A Geometric Reinterpretation of Variational Autoencoders

arXiv: https://arxiv.org/abs/2507.17255 · HTML: https://mlsanity.com/paper/2507.17255/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 50 references that are in the map.
- First posted on arXiv 23 Jul 2025.

## Builds on (most cited first, 27 of 27)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [From Variational to Deterministic Autoencoders](https://mlsanity.com/paper/1903.12436/) (2019)
- [Don't Blame the ELBO! A Linear VAE Perspective on Posterior Collapse](https://mlsanity.com/paper/1911.02469/) (2019)
- [Understanding and Improving Interpolation in Autoencoders via an Adversarial Regularizer](https://mlsanity.com/paper/1807.07543/) (2018)
- [Generalized Multimodal ELBO](https://mlsanity.com/paper/2105.02470/) (2021)
- [Variational Autoencoders Pursue PCA Directions (by Accident)](https://mlsanity.com/paper/1812.06775/) (2018)
- [Learning Disentangled Joint Continuous and Discrete Representations](https://mlsanity.com/paper/1804.00104/) (2018)
- [Variational Autoencoder with Learned Latent Structure](https://mlsanity.com/paper/2006.10597/) (2020)
- [DVAE++: Discrete Variational Autoencoders with Overlapping Transformations](https://mlsanity.com/paper/1802.04920/) (2018)
- [Disentangling by Subspace Diffusion](https://mlsanity.com/paper/2006.12982/) (2020)
- [Vector Quantized Wasserstein Auto-Encoder](https://mlsanity.com/paper/2302.05917/) (2023)
- [Discovering Hidden Factors of Variation in Deep Networks](https://mlsanity.com/paper/1412.6583/) (2014)
- [Learning Manifold Dimensions with Conditional Variational Autoencoders](https://mlsanity.com/paper/2302.11756/) (2023)
- [Constraining Variational Inference with Geometric Jensen-Shannon Divergence](https://mlsanity.com/paper/2006.10599/) (2020)
- [Variational Clustering: Leveraging Variational Autoencoders for Image Clustering](https://mlsanity.com/paper/2005.04613/) (2020)
- [Generative adversarial interpolative autoencoding: adversarial training on latent space interpolations encourage convex latent distributions](https://mlsanity.com/paper/1807.06650/) (2018)
- [Joint Optimization of an Autoencoder for Clustering and Embedding](https://mlsanity.com/paper/2012.03740/) (2020)
- [Exploring the Latent Space of Autoencoders with Interventional Assays](https://mlsanity.com/paper/2106.16091/) (2021)
- [A Geometric Perspective on Autoencoders](https://mlsanity.com/paper/2309.08247/) (2023)
- [Semantic Interpolation in Implicit Models](https://mlsanity.com/paper/1710.11381/) (2017)
- [Deep Generative Clustering with VAEs and Expectation-Maximization](https://mlsanity.com/paper/2501.07358/) (2025)

## Abstract (arXiv)

Variational Autoencoder is typically understood from the perspective of probabilistic inference. In this work, we propose a new geometric reinterpretation which complements the probabilistic view and enhances its intuitiveness. We demonstrate that the proper construction of semantic manifolds arises primarily from the constraining effect of the KL divergence on the encoder. We view the latent representations as a Gaussian ball rather than deterministic points. Under the constraint of KL divergence, Gaussian ball regularizes the latent space, promoting a more uniform distribution of encodings. Furthermore, we show that reparameterization establishes a critical contractual mechanism between the encoder and decoder, enabling the decoder to learn how to reconstruct from these stochastic regions. We further connect this viewpoint with VQ-VAE, offering a unified perspective: VQ-VAE can be seen as an autoencoder where encodings are constrained to a set of cluster centers, with its generative capability arising from the compactness rather than its stochasticity. This geometric framework provides a new lens for understanding how VAE shapes the latent geometry to enable effective generation.