# Learning Topology-Preserving Data Representations

arXiv: https://arxiv.org/abs/2302.00136 · HTML: https://mlsanity.com/paper/2302.00136/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 43 references that are in the map.
- First posted on arXiv 31 Jan 2023.

## Builds on (most cited first, 15 of 15)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Parametric UMAP embeddings for representation and semi-supervised learning](https://mlsanity.com/paper/2009.12981/) (2020)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [Manifold Topology Divergence: a Framework for Comparing Data Manifolds](https://mlsanity.com/paper/2106.04024/) (2021)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes](https://mlsanity.com/paper/2003.07989/) (2020)
- [Topological Obstructions to Autoencoding](https://mlsanity.com/paper/2102.08380/) (2021)
- [Acceptability Judgements via Examining the Topology of Attention Maps](https://mlsanity.com/paper/2205.09630/) (2022)
- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)

## Built on by (most cited first, 30 of 31)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [Hypothesis-Driven Feature Manifold Analysis in LLMs via Supervised Multi-Dimensional Scaling](https://mlsanity.com/paper/2510.01025/) (2025)
- [The Flood Complex: Large-Scale Persistent Homology on Millions of Points](https://mlsanity.com/paper/2509.22432/) (2025)
- [Mapping the Multiverse of Latent Representations](https://mlsanity.com/paper/2402.01514/) (2024)
- [Disentanglement Learning via Topology](https://mlsanity.com/paper/2308.12696/) (2023)
- [Deep Regression Representation Learning with Topology](https://mlsanity.com/paper/2404.13904/) (2024)
- [Relative Representations: Topological and Geometric Perspectives](https://mlsanity.com/paper/2409.10967/) (2024)
- [Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction](https://mlsanity.com/paper/2502.20215/) (2025)
- [RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Tasks](https://mlsanity.com/paper/2503.11910/) (2025)
- [MoE-Enhanced Explainable Deep Manifold Transformation for Complex Data Embedding and Visualization](https://mlsanity.com/paper/2410.19504/) (2024)
- [Topological Data Analysis Applications in Natural Language Processing: A Survey](https://mlsanity.com/paper/2411.10298/) (2024)
- [Graph Persistence goes Spectral](https://mlsanity.com/paper/2506.06571/) (2025)
- [Topological Alignment of Shared Vision-Language Embedding Space](https://mlsanity.com/paper/2510.10889/) (2025)
- [Data Topology-Dependent Upper Bounds of Neural Network Widths](https://mlsanity.com/paper/2305.16375/) (2023)
- [Dirichlet process mixture model based on topologically augmented signal representation for clustering infant vocalizations](https://mlsanity.com/paper/2407.05760/) (2024)
- [Structure-preserving contrastive learning for spatial time series](https://mlsanity.com/paper/2502.06380/) (2025)
- [Topology-Aware CLIP Few-Shot Learning](https://mlsanity.com/paper/2505.01694/) (2025)
- [Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularization](https://mlsanity.com/paper/2505.04412/) (2025)
- [TopoPrune: Robust Data Pruning via Unified Latent Space Topology](https://mlsanity.com/paper/2602.02739/) (2026)
- [Topology-Aware Representation Alignment for Semi-Supervised Vision-Language Learning](https://mlsanity.com/paper/2604.26370/) (2026)
- [Scalar Function Topology Divergence: Comparing Topology of 3D Objects](https://mlsanity.com/paper/2407.08364/) (2024)
- [Normalized Space Alignment: A Versatile Metric for Representation Analysis](https://mlsanity.com/paper/2411.04512/) (2024)
- [Local-Order Auxiliary Losses Can Improve Autoencoder Reconstruction](https://mlsanity.com/paper/2504.04202/) (2025)
- [IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation](https://mlsanity.com/paper/2505.06288/) (2025)
- [Torsion in Persistent Homology and Neural Networks](https://mlsanity.com/paper/2506.03049/) (2025)
- [Manifold-Matching Autoencoders](https://mlsanity.com/paper/2603.16568/) (2026)
- [Reparameterization through Coverings and Topological Weight Priors](https://mlsanity.com/paper/2604.23804/) (2026)
- [Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysis](https://mlsanity.com/paper/2606.06342/) (2026)

## Abstract (arXiv)

We propose a method for learning topology-preserving data representations (dimensionality reduction). The method aims to provide topological similarity between the data manifold and its latent representation via enforcing the similarity in topological features (clusters, loops, 2D voids, etc.) and their localization. The core of the method is the minimization of the Representation Topology Divergence (RTD) between original high-dimensional data and low-dimensional representation in latent space. RTD minimization provides closeness in topological features with strong theoretical guarantees. We develop a scheme for RTD differentiation and apply it as a loss term for the autoencoder. The proposed method "RTD-AE" better preserves the global structure and topology of the data manifold than state-of-the-art competitors as measured by linear correlation, triplet distance ranking accuracy, and Wasserstein distance between persistence barcodes.