# Multi-Scale Geometric Autoencoder

arXiv: https://arxiv.org/abs/2509.24168 · HTML: https://mlsanity.com/paper/2509.24168/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 21 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 6 of 6)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [LOCA: LOcal Conformal Autoencoder for standardized data coordinates](https://mlsanity.com/paper/2004.07234/) (2020)
- [Isometric Autoencoders](https://mlsanity.com/paper/2006.09289/) (2020)
- [Generalized Clustering and Multi-Manifold Learning with Geometric Structure Preservation](https://mlsanity.com/paper/2009.09590/) (2020)

## Built on by (most cited first, 1 of 1)

- [Bi-Lipschitz Autoencoder With Injectivity Guarantee](https://mlsanity.com/paper/2604.06701/) (2026)

## Abstract (arXiv)

Autoencoders have emerged as powerful models for visualization and dimensionality reduction based on the fundamental assumption that high-dimensional data is generated from a low-dimensional manifold. A critical challenge in autoencoder design is to preserve the geometric structure of data in the latent space, with existing approaches typically focusing on either global or local geometric properties separately. Global approaches often encounter errors in distance approximation that accumulate, while local methods frequently converge to suboptimal solutions that distort large-scale relationships. We propose Multi-Scale Geometric Autoencoder (MAE), which introduces an asymmetric architecture that simultaneously preserves both scales of the geometric structure by applying global distance constraints to the encoder and local geometric constraints to the decoder. Through theoretical analysis, we establish that this asymmetric design aligns naturally with the distinct roles of the encoder and decoder components. Our comprehensive experiments on both synthetic manifolds and real-world datasets demonstrate that MAE consistently outperforms existing methods across various evaluation metrics.