# Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness

arXiv: https://arxiv.org/abs/2303.09863 · HTML: https://mlsanity.com/paper/2303.09863/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 66 references that are in the map.
- First posted on arXiv 17 Mar 2023.

## Builds on (most cited first, 19 of 19)

- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem](https://mlsanity.com/paper/1906.11945/) (2019)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [Deep nets for local manifold learning](https://mlsanity.com/paper/1607.07110/) (2016)
- [Adaptive Geometric Multiscale Approximations for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/1611.01179/) (2016)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks](https://mlsanity.com/paper/2011.01797/) (2020)
- [Structure-adaptive manifold estimation](https://mlsanity.com/paper/1906.05014/) (2019)
- [Side Effects of Learning from Low-dimensional Data Embedded in a Euclidean Space](https://mlsanity.com/paper/2203.00614/) (2022)
- [Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint](https://mlsanity.com/paper/2206.04569/) (2022)

## Built on by (most cited first, 10 of 10)

- [Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study](https://mlsanity.com/paper/2410.00357/) (2024)
- [Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning](https://mlsanity.com/paper/2401.10490/) (2024)
- [Transformers for Learning on Noisy and Task-Level Manifolds: Approximation and Generalization Insights](https://mlsanity.com/paper/2505.03205/) (2025)
- [A Statistical Analysis of Wasserstein Autoencoders for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2402.15710/) (2024)
- [Connections between Operator-splitting Methods and Deep Neural Networks with Applications in Image Segmentation](https://mlsanity.com/paper/2307.09052/) (2023)
- [Distributional Autoencoders Know the Score](https://mlsanity.com/paper/2502.11583/) (2025)
- [Latent Autoencoder Ensemble Kalman Filter for Nonlinear Data assimilation](https://mlsanity.com/paper/2603.06752/) (2026)
- [Geometric regularization of autoencoders via observed stochastic dynamics](https://mlsanity.com/paper/2604.16282/) (2026)
- [Learning Theory of Transformers: Local-to-Global Approximation via Softmax Partition of Unity](https://mlsanity.com/paper/2605.08811/) (2026)
- [Data-Driven Model Reduction using WeldNet: Windowed Encoders for Learning Dynamics](https://mlsanity.com/paper/2512.11090/) (2025)

## Abstract (arXiv)

Autoencoders have demonstrated remarkable success in learning low-dimensional latent features of high-dimensional data across various applications. Assuming that data are sampled near a low-dimensional manifold, we employ chart autoencoders, which encode data into low-dimensional latent features on a collection of charts, preserving the topology and geometry of the data manifold. Our paper establishes statistical guarantees on the generalization error of chart autoencoders, and we demonstrate their denoising capabilities by considering n noisy training samples, along with their noise-free counterparts, on a d-dimensional manifold. By training autoencoders, we show that chart autoencoders can effectively denoise the input data with normal noise. We prove that, under proper network architectures, chart autoencoders achieve a squared generalization error in the order of displaystyle n^-frac2d+2log4 n, which depends on the intrinsic dimension of the manifold and only weakly depends on the ambient dimension and noise level. We further extend our theory on data with noise containing both normal and tangential components, where chart autoencoders still exhibit a denoising effect for the normal component. As a special case, our theory also applies to classical autoencoders, as long as the data manifold has a global parametrization. Our results provide a solid theoretical foundation for the effectiveness of autoencoders, which is further validated through several numerical experiments.