# Chart Auto-Encoders for Manifold Structured Data

arXiv: https://arxiv.org/abs/1912.10094 · HTML: https://mlsanity.com/paper/1912.10094/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2019.

## Built on by (most cited first, 30 of 33)

- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Partition of unity networks: deep hp-approximation](https://mlsanity.com/paper/2101.11256/) (2021)
- [Convergence Analysis of Flow Matching in Latent Space with Transformers](https://mlsanity.com/paper/2404.02538/) (2024)
- [Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning](https://mlsanity.com/paper/2401.10490/) (2024)
- [Density estimation on smooth manifolds with normalizing flows](https://mlsanity.com/paper/2106.03500/) (2021)
- [Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness](https://mlsanity.com/paper/2303.09863/) (2023)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Decoder ensembling for learned latent geometries](https://mlsanity.com/paper/2408.07507/) (2024)
- [Atlas Generative Models and Geodesic Interpolation](https://mlsanity.com/paper/2102.00264/) (2021)
- [Deep ReLU Networks Have Surprisingly Simple Polytopes](https://mlsanity.com/paper/2305.09145/) (2023)
- [Real-time Animation Generation and Control on Rigged Models via Large Language Models](https://mlsanity.com/paper/2310.17838/) (2023)
- [Transformers for Learning on Noisy and Task-Level Manifolds: Approximation and Generalization Insights](https://mlsanity.com/paper/2505.03205/) (2025)
- [Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations](https://mlsanity.com/paper/2401.15482/) (2024)
- [Distributional Autoencoders Know the Score](https://mlsanity.com/paper/2502.11583/) (2025)
- [Learning Geometry and Topology via Multi-Chart Flows](https://mlsanity.com/paper/2505.24665/) (2025)
- [Manifoldron: Direct Space Partition via Manifold Discovery](https://mlsanity.com/paper/2201.05279/) (2022)
- [Probabilistic partition of unity networks for high-dimensional regression problems](https://mlsanity.com/paper/2210.02694/) (2022)
- [Are We Using Autoencoders in a Wrong Way?](https://mlsanity.com/paper/2309.01532/) (2023)
- [Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flow](https://mlsanity.com/paper/2312.10235/) (2023)
- [A Theoretical Study of Neural Network Expressive Power via Manifold Topology](https://mlsanity.com/paper/2410.16542/) (2024)
- [Geometric regularization of autoencoders via observed stochastic dynamics](https://mlsanity.com/paper/2604.16282/) (2026)
- [Is an encoder within reach?](https://mlsanity.com/paper/2206.01552/) (2022)
- [SMIXAE: Towards Unsupervised Manifold Discovery in Language Models](https://mlsanity.com/paper/2605.09224/) (2026)
- [Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders](https://mlsanity.com/paper/2408.16138/) (2024)
- [Atlas Gaussian processes on restricted domains and point clouds](https://mlsanity.com/paper/2511.15822/) (2025)
- [Data-Driven Model Reduction using WeldNet: Windowed Encoders for Learning Dynamics](https://mlsanity.com/paper/2512.11090/) (2025)

## Abstract (arXiv)

Deep generative models have made tremendous advances in image and signal representation learning and generation. These models employ the full Euclidean space or a bounded subset as the latent space, whose flat geometry, however, is often too simplistic to meaningfully reflect the manifold structure of the data. In this work, we advocate the use of a multi-chart latent space for better data representation. Inspired by differential geometry, we propose a Chart Auto-Encoder (CAE) and prove a universal approximation theorem on its representation capability. We show that the training data size and the network size scale exponentially in approximation error with an exponent depending on the intrinsic dimension of the data manifold. CAE admits desirable manifold properties that auto-encoders with a flat latent space fail to obey, predominantly proximity of data. We conduct extensive experimentation with synthetic and real-life examples to demonstrate that CAE provides reconstruction with high fidelity, preserves proximity in the latent space, and generates new data remaining near the manifold. These experiments show that CAE is advantageous over existing auto-encoders and variants by preserving the topology of the data manifold as well as its geometry.