mlsanityData to 7 Oct 2026

Learning Topology-Preserving Data Representations

arXiv 2302.00136 ↗First posted 31 Jan 2023cs.LG cs.AI math.ATcited by 30

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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.

Key facts. Cited by 30 papers in the map (data through 7 Oct 2026). Builds on 15 of its 43 references that are in the map. First posted on arXiv 31 Jan 2023.

Builds on

15 of 43 references.

References · 15

  1. 2017Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithmscited by 4,283
  2. 2018UMAP: Uniform Manifold Approximation and Projection for Dimension Reductioncited by 2,841
  3. 2017An introduction to Topological Data Analysis: fundamental and practical aspects for data scientistscited by 171
  4. 2020Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualizationcited by 170
  5. 2019Topological Autoencoderscited by 124
  6. 2020Parametric UMAP embeddings for representation and semi-supervised learningcited by 103
  7. 2021Representation Topology Divergence: A Method for Comparing Neural Network Representationscited by 66
  8. 2019Connectivity-Optimized Representation Learning via Persistent Homologycited by 43
  9. 2021On UMAP's true loss functioncited by 32
  10. 2021Manifold Topology Divergence: a Framework for Comparing Data Manifoldscited by 31
  11. 2019A Framework for Differential Calculus on Persistence Barcodescited by 29
  12. 2020GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodescited by 26
3 more
  1. 2021Topological Obstructions to Autoencodingcited by 20
  2. 2022Acceptability Judgements via Examining the Topology of Attention Mapscited by 19
  3. 2021Improving Metric Dimensionality Reduction with Distributed Topologycited by 10

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 4; 2024: 9; 2025: 12; 2026: 5.

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  1. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 65
  2. 2023Geometric Autoencoders -- What You See is What You Decodecited by 23
  3. 2023Differentiable Euler Characteristic Transforms for Shape Classificationcited by 18
  4. 2025Hypothesis-Driven Feature Manifold Analysis in LLMs via Supervised Multi-Dimensional Scalingcited by 6
  5. 2024Mapping the Multiverse of Latent Representationscited by 5
  6. 2025The Flood Complex: Large-Scale Persistent Homology on Millions of Pointscited by 5
  7. 2023Disentanglement Learning via Topologycited by 4
  8. 2024Deep Regression Representation Learning with Topologycited by 4
  9. 2024Relative Representations: Topological and Geometric Perspectivescited by 4
  10. 2025RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Taskscited by 4
  11. 2024MoE-Enhanced Explainable Deep Manifold Transformation for Complex Data Embedding and Visualizationcited by 3
  12. 2025Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reductioncited by 3
18 more
  1. 2024Topological Data Analysis Applications in Natural Language Processing: A Surveycited by 2
  2. 2025Topological Alignment of Shared Vision-Language Embedding Spacecited by 2
  3. 2023Data Topology-Dependent Upper Bounds of Neural Network Widthscited by 1
  4. 2024Dirichlet process mixture model based on topologically augmented signal representation for clustering infant vocalizationscited by 1
  5. 2025Structure-preserving contrastive learning for spatial time seriescited by 1
  6. 2025Topology-Aware CLIP Few-Shot Learningcited by 1
  7. 2025Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularizationcited by 1
  8. 2025Graph Persistence goes Spectralcited by 1
  9. 2026TopoPrune: Robust Data Pruning via Unified Latent Space Topologycited by 1
  10. 2026Topology-Aware Representation Alignment for Semi-Supervised Vision-Language Learningcited by 1
  11. 2024Scalar Function Topology Divergence: Comparing Topology of 3D Objectscited by 0
  12. 2024Normalized Space Alignment: A Versatile Metric for Representation Analysiscited by 0
  13. 2025Local-Order Auxiliary Losses Can Improve Autoencoder Reconstructioncited by 0
  14. 2025IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservationcited by 0
  15. 2025Torsion in Persistent Homology and Neural Networkscited by 0
  16. 2026Manifold-Matching Autoencoderscited by 0
  17. 2026Reparameterization through Coverings and Topological Weight Priorscited by 0
  18. 2026Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysiscited by 0