# Ensuring Topological Data-Structure Preservation under Autoencoder Compression due to Latent Space Regularization in Gauss--Legendre nodes

arXiv: https://arxiv.org/abs/2309.08228 · HTML: https://mlsanity.com/paper/2309.08228/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 81 references that are in the map.
- First posted on arXiv 15 Sep 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Fairness Testing: Testing Software for Discrimination](https://mlsanity.com/paper/1709.03221/) (2017)
- [Variational Autoencoders Pursue PCA Directions (by Accident)](https://mlsanity.com/paper/1812.06775/) (2018)
- [On Implicit Regularization in β-VAEs](https://mlsanity.com/paper/2002.00041/) (2020)
- [Robustness Verification of Tree-based Models](https://mlsanity.com/paper/1906.03849/) (2019)
- [Local Disentanglement in Variational Auto-Encoders Using Jacobian L1 Regularization](https://mlsanity.com/paper/2106.02923/) (2021)
- [Towards Understanding the Invertibility of Convolutional Neural Networks](https://mlsanity.com/paper/1705.08664/) (2017)
- [Replacing Automatic Differentiation by Sobolev Cubatures fastens Physics Informed Neural Nets and strengthens their Approximation Power](https://mlsanity.com/paper/2211.15443/) (2022)
- [Data-Driven Shadowgraph Simulation of a 3D Object](https://mlsanity.com/paper/2106.00317/) (2021)
- [Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces](https://mlsanity.com/paper/2301.04887/) (2023)

## Built on by (most cited first, 2 of 2)

- [Complex variational autoencoders admit K"ahler structure](https://mlsanity.com/paper/2511.15172/) (2025)
- [On Adversarial Vulnerability of Vision-Language Models through the Lens of Intermediate Spectral Subspaces](https://mlsanity.com/paper/2607.07375/) (2026)

## Abstract (arXiv)

We formulate a data independent latent space regularisation constraint for general unsupervised autoencoders. The regularisation rests on sampling the autoencoder Jacobian in Legendre nodes, being the centre of the Gauss-Legendre quadrature. Revisiting this classic enables to prove that regularised autoencoders ensure a one-to-one re-embedding of the initial data manifold to its latent representation. Demonstrations show that prior proposed regularisation strategies, such as contractive autoencoding, cause topological defects already for simple examples, and so do convolutional based (variational) autoencoders. In contrast, topological preservation is ensured already by standard multilayer perceptron neural networks when being regularised due to our contribution. This observation extends through the classic FashionMNIST dataset up to real world encoding problems for MRI brain scans, suggesting that, across disciplines, reliable low dimensional representations of complex high-dimensional datasets can be delivered due to this regularisation technique.