# Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders

arXiv: https://arxiv.org/abs/2208.10570 · HTML: https://mlsanity.com/paper/2208.10570/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 22 Aug 2022.

## Builds on (most cited first, 22 of 22)

- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Grammar Variational Autoencoder](https://mlsanity.com/paper/1703.01925/) (2017)
- [Unsupervised Anomaly Detection via Variational Auto-Encoder for Seasonal KPIs in Web Applications](https://mlsanity.com/paper/1802.03903/) (2018)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence](https://mlsanity.com/paper/2003.14286/) (2020)
- [Molecular generative model based on conditional variational autoencoder for de novo molecular design](https://mlsanity.com/paper/1806.05805/) (2018)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Discriminator Rejection Sampling](https://mlsanity.com/paper/1810.06758/) (2018)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Metropolis-Hastings Generative Adversarial Networks](https://mlsanity.com/paper/1811.11357/) (2018)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Learning disconnected manifolds: a no GANs land](https://mlsanity.com/paper/2006.04596/) (2020)
- [Diffusion Nets](https://mlsanity.com/paper/1506.07840/) (2015)
- [Representing Closed Transformation Paths in Encoded Network Latent Space](https://mlsanity.com/paper/1912.02644/) (2019)
- [Non-Parametric Estimation of Manifolds from Noisy Data](https://mlsanity.com/paper/2105.04754/) (2021)
- [Adaptive Geometric Multiscale Approximations for Intrinsically Low-dimensional Data](https://mlsanity.com/paper/1611.01179/) (2016)
- [Manifold Coordinates with Physical Meaning](https://mlsanity.com/paper/1811.11891/) (2018)
- [Disentanglement with Hyperspherical Latent Spaces using Diffusion Variational Autoencoders](https://mlsanity.com/paper/2003.08996/) (2020)
- [Turing approximations, toric isometric embeddings & manifold convolutions](https://mlsanity.com/paper/2110.02279/) (2021)

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

- [Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness](https://mlsanity.com/paper/2303.09863/) (2023)
- [Transformers for Learning on Noisy and Task-Level Manifolds: Approximation and Generalization Insights](https://mlsanity.com/paper/2505.03205/) (2025)
- [Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees](https://mlsanity.com/paper/2503.08642/) (2025)
- [Geometric regularization of autoencoders via observed stochastic dynamics](https://mlsanity.com/paper/2604.16282/) (2026)
- [Data-induced multiscale losses and efficient multirate gradient descent schemes](https://mlsanity.com/paper/2402.03021/) (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)
- [Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases](https://mlsanity.com/paper/2602.22873/) (2026)

## Abstract (arXiv)

Autoencoding is a popular method in representation learning. Conventional autoencoders employ symmetric encoding-decoding procedures and a simple Euclidean latent space to detect hidden low-dimensional structures in an unsupervised way. Some modern approaches to novel data generation such as generative adversarial networks askew this symmetry, but still employ a pair of massive networks--one to generate the image and another to judge the images quality based on priors learned from a training set. This work introduces a chart autoencoder with an asymmetric encoding-decoding process that can incorporate additional semi-supervised information such as class labels. Besides enhancing the capability for handling data with complicated topological and geometric structures, the proposed model can successfully differentiate nearby but disjoint manifolds and intersecting manifolds with only a small amount of supervision. Moreover, this model only requires a low-complexity encoding operation, such as a locally defined linear projection. We discuss the approximation power of such networks and derive a bound that essentially depends on the intrinsic dimension of the data manifold rather than the dimension of ambient space. Next we incorporate bounds for the sampling rate of training data need to faithfully represent a given data manifold. We present numerical experiments that verify that the proposed model can effectively manage data with multi-class nearby but disjoint manifolds of different classes, overlapping manifolds, and manifolds with non-trivial topology. Finally, we conclude with some experiments on computer vision and molecular dynamics problems which showcase the efficacy of our methods on real-world data.