# A Geometric Perspective on Autoencoders

arXiv: https://arxiv.org/abs/2309.08247 · HTML: https://mlsanity.com/paper/2309.08247/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 25 references that are in the map.
- First posted on arXiv 15 Sep 2023.

## Builds on (most cited first, 8 of 8)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Geometrically Enriched Latent Spaces](https://mlsanity.com/paper/2008.00565/) (2020)
- [Learning Flat Latent Manifolds with VAEs](https://mlsanity.com/paper/2002.04881/) (2020)
- [Extendable and invertible manifold learning with geometry regularized autoencoders](https://mlsanity.com/paper/2007.07142/) (2020)

## Built on by (most cited first, 8 of 8)

- [MMP++: Motion Manifold Primitives with Parametric Curve Models](https://mlsanity.com/paper/2310.17072/) (2023)
- [Motion Manifold Flow Primitives for Task-Conditioned Trajectory Generation under Complex Task-Motion Dependencies](https://mlsanity.com/paper/2407.19681/) (2024)
- [Differentiable Motion Manifold Primitives for Reactive Motion Generation under Kinodynamic Constraints](https://mlsanity.com/paper/2410.12193/) (2024)
- [Denoising Diffusion Generative Models Secretly Calculate Attentions](https://mlsanity.com/paper/2609.00885/) (2026)
- [From Points to Spheres: A Geometric Reinterpretation of Variational Autoencoders](https://mlsanity.com/paper/2507.17255/) (2025)
- [LEGO: Latent-space Exploration for Geometry-aware Optimization of Humanoid Kinematic Design](https://mlsanity.com/paper/2604.08636/) (2026)
- [Partially Linear Autoencoders for Manifold Learning and Dimensionality Reduction](https://mlsanity.com/paper/2608.29867/) (2026)
- [Train to Stitch: Topology-Preserving Regularization for Interoperable Representations](https://mlsanity.com/paper/2610.19401/) (2026)

## Abstract (arXiv)

This paper presents the geometric aspect of the autoencoder framework, which, despite its importance, has been relatively less recognized. Given a set of high-dimensional data points that approximately lie on some lower-dimensional manifold, an autoencoder learns the textitmanifold and its textitcoordinate chart, simultaneously. This geometric perspective naturally raises inquiries like "Does a finite set of data points correspond to a single manifold?" or "Is there only one coordinate chart that can represent the manifold?". The responses to these questions are negative, implying that there are multiple solution autoencoders given a dataset. Consequently, they sometimes produce incorrect manifolds with severely distorted latent space representations. In this paper, we introduce recent geometric approaches that address these issues.