# Extendable and invertible manifold learning with geometry regularized autoencoders

arXiv: https://arxiv.org/abs/2007.07142 · HTML: https://mlsanity.com/paper/2007.07142/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2020.

## Built on by (most cited first, 20 of 20)

- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [Geometry- and Accuracy-Preserving Random Forest Proximities](https://mlsanity.com/paper/2201.12682/) (2022)
- [DLME: Deep Local-flatness Manifold Embedding](https://mlsanity.com/paper/2207.03160/) (2022)
- [Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds](https://mlsanity.com/paper/2410.12779/) (2024)
- [Learning shared neural manifolds from multi-subject FMRI data](https://mlsanity.com/paper/2201.00622/) (2021)
- [EVNet: An Explainable Deep Network for Dimension Reduction](https://mlsanity.com/paper/2211.15478/) (2022)
- [A Geometric Perspective on Autoencoders](https://mlsanity.com/paper/2309.08247/) (2023)
- [GenURL: A General Framework for Unsupervised Representation Learning](https://mlsanity.com/paper/2110.14553/) (2021)
- [UDRN: Unified Dimensional Reduction Neural Network for Feature Selection and Feature Projection](https://mlsanity.com/paper/2207.03809/) (2022)
- [Unveiling Multiple Descents in Unsupervised Autoencoders](https://mlsanity.com/paper/2406.11703/) (2024)
- [Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation](https://mlsanity.com/paper/2507.11789/) (2025)
- [Random Forest Autoencoders for Guided Representation Learning](https://mlsanity.com/paper/2502.13257/) (2025)
- [Inductive Global and Local Manifold Approximation and Projection](https://mlsanity.com/paper/2406.08097/) (2024)
- [Are We Using Autoencoders in a Wrong Way?](https://mlsanity.com/paper/2309.01532/) (2023)
- [Guided Manifold Alignment with Geometry-Regularized Twin Autoencoders](https://mlsanity.com/paper/2509.22913/) (2025)
- [MIOFlow 2.0: A unified framework for inferring cellular stochastic dynamics from single cell and spatial transcriptomics data](https://mlsanity.com/paper/2603.22564/) (2026)
- [Pseudo-Nonlinear Data Augmentation: A Constrained Energy Minimization Viewpoint](https://mlsanity.com/paper/2410.00718/) (2024)
- [Freeze, Diffuse, Decode: Task-Aware Adaptation of Transformer Embeddings for Antimicrobial Peptide Design](https://mlsanity.com/paper/2511.23120/) (2025)
- [On Out-of-sample Embedding in UMAP](https://mlsanity.com/paper/2606.04451/) (2026)
- [FADEx: Feature Attribution and Distortion-based Explanation of Dimensionality Reduction](https://mlsanity.com/paper/2607.27463/) (2026)

## Abstract (arXiv)

A fundamental task in data exploration is to extract simplified low dimensional representations that capture intrinsic geometry in data, especially for faithfully visualizing data in two or three dimensions. Common approaches to this task use kernel methods for manifold learning. However, these methods typically only provide an embedding of fixed input data and cannot extend to new data points. Autoencoders have also recently become popular for representation learning. But while they naturally compute feature extractors that are both extendable to new data and invertible (i.e., reconstructing original features from latent representation), they have limited capabilities to follow global intrinsic geometry compared to kernel-based manifold learning. We present a new method for integrating both approaches by incorporating a geometric regularization term in the bottleneck of the autoencoder. Our regularization, based on the diffusion potential distances from the recently-proposed PHATE visualization method, encourages the learned latent representation to follow intrinsic data geometry, similar to manifold learning algorithms, while still enabling faithful extension to new data and reconstruction of data in the original feature space from latent coordinates. We compare our approach with leading kernel methods and autoencoder models for manifold learning to provide qualitative and quantitative evidence of our advantages in preserving intrinsic structure, out of sample extension, and reconstruction. Our method is easily implemented for big-data applications, whereas other methods are limited in this regard.