mlsanityData to 7 Oct 2026

On UMAP's true loss function

arXiv 2103.14608 ↗First posted 26 Mar 2021cs.LG stat.MLcited by 32

Not on the map. Why?

Abstract (arXiv)

UMAP has supplanted t-SNE as state-of-the-art for visualizing high-dimensional datasets in many disciplines, but the reason for its success is not well understood. In this work, we investigate UMAP's sampling based optimization scheme in detail. We derive UMAP's effective loss function in closed form and find that it differs from the published one. As a consequence, we show that UMAP does not aim to reproduce its theoretically motivated high-dimensional UMAP similarities. Instead, it tries to reproduce similarities that only encode the shared k nearest neighbor graph, thereby challenging the previous understanding of UMAP's effectiveness. Instead, we claim that the key to UMAP's success is its implicit balancing of attraction and repulsion resulting from negative sampling. This balancing in turn facilitates optimization via gradient descent. We corroborate our theoretical findings on toy and single cell RNA sequencing data.

Key facts. Cited by 32 papers in the map (data through 7 Oct 2026). Builds on 8 of its 23 references that are in the map. First posted on arXiv 26 Mar 2021.

Builds on

8 of 23 references.

References · 8

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2018UMAP: Uniform Manifold Approximation and Projection for Dimension Reductioncited by 2,841
  3. 2020Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualizationcited by 170
  4. 2020Kernel Operations on the GPU, with Autodiff, without Memory Overflowscited by 101
  5. 2016Visualizing Large-scale and High-dimensional Datacited by 79
  6. 2019TriMap: Large-scale Dimensionality Reduction Using Tripletscited by 59
  7. 2017Clustering with t-SNE, provablycited by 51
  8. 2016Consistency of modularity clustering on random geometric graphscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 5; 2023: 6; 2024: 6; 2025: 6; 2026: 8.

32 papers.

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  1. 2023Learning Topology-Preserving Data Representationscited by 30
  2. 2022From t-SNE to UMAP with contrastive learningcited by 28
  3. 2023Geometric Autoencoders -- What You See is What You Decodecited by 23
  4. 2022Unsupervised visualization of image datasets using contrastive learningcited by 16
  5. 2021Uniform Manifold Approximation and Projection (UMAP) and its Variants: Tutorial and Surveycited by 10
  6. 2023ActUp: Analyzing and Consolidating tSNE and UMAPcited by 5
  7. 2025The Shape of Attraction in UMAP: Exploring the Embedding Forces in Dimensionality Reductioncited by 5
  8. 2022ParaDime: A Framework for Parametric Dimensionality Reductioncited by 4
  9. 2025RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Taskscited by 4
  10. 2023Relating tSNE and UMAP to Classical Dimensionality Reductioncited by 3
  11. 2023Persistent Homology for High-dimensional Data Based on Spectral Methodscited by 3
  12. 2024Towards One Model for Classical Dimensionality Reduction: A Probabilistic Perspective on UMAP and t-SNEcited by 3
20 more
  1. 2024IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrationscited by 3
  2. 2024Sailing in high-dimensional spaces: Low-dimensional embeddings through angle preservationcited by 2
  3. 2025EmbedOR: Provable Cluster-Preserving Visualizations with Curvature-Based Stochastic Neighbor Embeddingscited by 2
  4. 2024MS-IMAP -- A Multi-Scale Graph Embedding Approach for Interpretable Manifold Learningcited by 1
  5. 2024FedNE: Surrogate-Assisted Federated Neighbor Embedding for Dimensionality Reductioncited by 1
  6. 2024Attraction-Repulsion Swarming: A Generalized Framework of t-SNE via Force Normalization and Tunable Interactionscited by 1
  7. 2025Node Embeddings via Neighbor Embeddingscited by 1
  8. 2026MAPLE: Self-Supervised Learning-Enhanced Nonlinear Dimensionality Reduction for Visual Analysiscited by 1
  9. 2026On the continuum limit of t-SNE for data visualizationcited by 1
  10. 2022GiDR-DUN; Gradient Dimensionality Reduction -- Differences and Unificationcited by 0
  11. 2022On the Selection of Tuning Parameters for Patch-Stitching Embedding Methodscited by 0
  12. 2023Supervised Stochastic Neighbor Embedding Using Contrastive Learningcited by 0
  13. 2025Merging Hazy Sets with m-Schemes: A Geometric Approach to Data Visualizationcited by 0
  14. 2025Probabilistic Foundations of Fuzzy Simplicial Sets for Nonlinear Dimensionality Reductioncited by 0
  15. 2026UMAP Is Spectral Clustering on the Fuzzy Nearest-Neighbor Graphcited by 0
  16. 2026A Spectral Decomposition Framework for Multiscale Nonlinear Dimensionality Reductioncited by 0
  17. 2026On Out-of-sample Embedding in UMAPcited by 0
  18. 2026Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysiscited by 0
  19. 2026CosMAP: Contrastive Manifold Approximation and Projection for Dimensionality Reduction of Omics and Genealogical Datacited by 0
  20. 2026ibUMAP: Coherent and Scalable Field Evaluation for UMAP Optimizationcited by 0