{ "id": "epy1zkxzzw", "slug": "umap", "title": "UMAP", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_computation" }, { "id": "technology.computing.ai.ml_methods", "label": "Machine learning methods", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_methods" }, { "id": "technology.computing.ai.supervised_unsupervised", "label": "Supervised, unsupervised, and semi-supervised learning", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.supervised_unsupervised" }, { "id": "technology.computing.ai.supervised_unsupervised.dimensionality_reduction", "label": "Dimensionality reduction and manifold learning", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.supervised_unsupervised.dimensionality_reduction" } ], "geo": [ { "id": "geo.nongeo.t2001.technology.computing.ai.supervised_unsupervised", "label": "Non-geographic · 2001 to 2020: Supervised, unsupervised, and semi-supervised learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.supervised_unsupervised", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t2001", "label": "Non-geographic · 2001 to 2020", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001" }, { "id": "geo.nongeo.t2001.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology" }, { "id": "geo.nongeo.t2001.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing" }, { "id": "geo.nongeo.t2001.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai" }, { "id": "geo.nongeo.t2001.technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.ml_computation" }, { "id": "geo.nongeo.t2001.technology.computing.ai.ml_methods", "label": "Machine learning methods", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.ml_methods" }, { "id": "geo.nongeo.t2001.technology.computing.ai.supervised_unsupervised", "label": "Supervised, unsupervised, and semi-supervised learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t2001.technology.computing.ai.supervised_unsupervised" } ] } ], "excerpt": "UMAP (Uniform Manifold Approximation and Projection) is a nonlinear dimension reduction method introduced in 2018, used for visualizing high-dimensional data and faster than most t-SNE implementations.", "snippet": "UMAP (Uniform Manifold Approximation and Projection) is a nonlinear dimension reduction method introduced in 2018, used for visualizing high-dimensional data and faster than most t-SNE implementations.", "node": "technology.computing.ai.supervised_unsupervised.dimensionality_reduction", "markdown": "# UMAP\n\nUMAP (Uniform Manifold Approximation and Projection) is a nonlinear dimension reduction method that embeds high-dimensional data into a low-dimensional space by representing the data as a fuzzy topological structure and optimizing a layout to match it. It is used for visualizing data and as pre-processing for machine-learning tasks such as clustering.[1](https://www.nature.com/articles/s43586-024-00363-x) Introduced by Leland McInnes, John Healy, and James Melville in 2018[2](https://doi.org/10.48550/arxiv.1802.03426), it is among the fastest manifold learning implementations available and significantly faster than most t-SNE implementations.[3](https://doi.org/10.21105/joss.00861) It has become a standard visualization tool in single-cell biology.[4](https://doi.org/10.1038/nbt.4314)\n\n| Key fact | Detail |\n|---|---|\n| Input / output | A high-dimensional dataset (sparse or dense, over a million dimensions in reported uses) and a low-dimensional embedding[5](https://github.com/lmcinnes/umap/) |\n| Principle | Fuzzy simplicial set representations of high- and low-dimensional data, matched by minimizing cross-entropy[2](https://doi.org/10.48550/arxiv.1802.03426) |\n| Default hyperparameters | n_neighbors = 15, min_dist = 0.1, negative_sample_rate = 5, metric = euclidean[6](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py) |\n| Reported speed | MNIST (70,000 samples, 784 dimensions) embedded in 42 seconds on a 3.1 GHz Intel Core i7[5](https://github.com/lmcinnes/umap/) |\n| GPU scaling | cuML 24.10 processes 50M points × 768 dimensions on one H100; multi-GPU cuML 25.06 embeds 106M vectors in 8 minutes[7](https://developer.nvidia.com/blog/even-faster-and-more-scalable-umap-on-the-gpu-with-rapids-cuml/) • [8](https://developer.nvidia.com/blog/run-massive-scale-umap-in-minutes-using-multiple-gpus-without-losing-accuracy/) |\n| Main criticism | The negative-sampling optimization minimizes an effective loss that differs from the stated cross-entropy, under-weighting repulsion[9](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf) |\n\n## How it works\n\nUMAP assumes the data lie approximately uniformly distributed on a manifold. Under that assumption, a unit ball about each point stretches to the \\( k \\)-th nearest neighbor, giving every point its own locally varying distance function; the resulting local fuzzy simplicial sets are merged into a single topological representation of the data.[10](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html) Where two points disagree about an edge weight, the weights a and b are combined by the probabilistic t-conorm into \\( a + b - a \\cdot b \\), interpreted as the probability that at least one of the edges exists.[10](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html)\n\nGiven a low-dimensional layout, an equivalent fuzzy representation is built, and the layout is optimized to minimize the cross-entropy between the two representations over the 1-simplices E:\n\n\\[ C = \\sum_{e \\in E} w_{h}(e) \\log \\frac{w_{h}(e)}{w_{l}(e)} + (1 - w_{h}(e)) \\log \\frac{1 - w_{h}(e)}{1 - w_{l}(e)} \\]\n\nwhere \\( w_{h} \\) and \\( w_{l} \\) are the high- and low-dimensional edge weights.[10](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html) In principle this is the same loss other stochastic neighbor embedding methods use; the difference lies in how the probability distributions are constructed and their domain.[11](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5) The fuzzy-set machinery draws on Barr's fuzzy set theory[12](https://doi.org/10.4153/cmb-1986-079-9) and the fuzzy-set cross-entropy measure of Bhandari and Pal.[13](https://doi.org/10.1016/0020-0255%2893%2990073-u)\n\n## How it is done\n\nThe algorithm has two phases: construction of a weighted k-neighbor graph, then computation of a low-dimensional layout of that graph.[2](https://doi.org/10.48550/arxiv.1802.03426)\n\n1. **Nearest-neighbor graph.** Neighbors are found with Nearest-Neighbor-Descent[10](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html), and distances under the input metric to each point's nearest neighbors are locally rescaled to create a per-point fuzzy simplicial set, combined via fuzzy union.[6](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py)\n2. **Initialization.** The embedding is initialized with a spectral layout, Laplacian eigenmaps on the symmetrized fuzzy-weight matrix.[2](https://doi.org/10.48550/arxiv.1802.03426) • [11](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5)\n3. **Optimization.** Edges are sampled probabilistically, with negative sampling as used by word2vec and LargeVis, giving approximate stochastic gradient descent.[2](https://doi.org/10.48550/arxiv.1802.03426) The low-dimensional membership curve has the family \\( 1/(1 + a x^{2b}) \\).[10](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html)\n\nThe number of neighbors n sets the local scale of the manifold approximation: smaller values capture fine structure, larger values capture large-scale structure.[2](https://doi.org/10.48550/arxiv.1802.03426) The default is 15. min_dist controls how tightly points may be packed; the original authors call it an essentially aesthetic parameter[2](https://doi.org/10.48550/arxiv.1802.03426), though a later analysis argues it can strongly affect results by weighting the low-dimensional probability tails.[11](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5) The metric accepts many distances, including cosine and correlation.[5](https://github.com/lmcinnes/umap/) The optimization work scales with the number of edges in the fuzzy graph, giving complexity \\( O(k \\cdot N) \\) for fixed negative sampling; Nearest-Neighbor-Descent has an empirically reported complexity of approximately \\( O(N^{1.14}) \\).[2](https://doi.org/10.48550/arxiv.1802.03426)\n\n## Origin\n\nUMAP was reported by McInnes, Healy, and Melville in a 2018 arXiv preprint[2](https://doi.org/10.48550/arxiv.1802.03426), with a companion software paper by McInnes, Healy, Nathaniel Saul, and Lukas Großberger in the Journal of Open Source Software the same year.[3](https://doi.org/10.21105/joss.00861) It builds on earlier work the Primer identifies as precursors: [Isomap](https://www.edgepedia.org/isomap) by Tenenbaum, de Silva, and Langford[14](https://doi.org/10.1126/science.290.5500.2319), t-SNE by van der Maaten and Hinton[1](https://www.nature.com/articles/s43586-024-00363-x), and Laplacian eigenmaps by Belkin and Niyogi, which also underlie the spectral initialization.[15](https://doi.org/10.1162/089976603321780317) A peer-reviewed Primer by Healy and McInnes appeared in Nature Reviews Methods Primers in 2024.[1](https://www.nature.com/articles/s43586-024-00363-x)\n\n## Variants\n\nThe reference implementation supports supervised and semi-supervised reduction via partial labels, and adding new points to a fitted embedding with the sklearn transform method.[3](https://doi.org/10.21105/joss.00861) • [5](https://github.com/lmcinnes/umap/) Parametric UMAP, by Sainburg, McInnes, and Gentner, uses the UMAP cost function as the loss of a neural network trained in mini-batches, enabling embedding of large datasets.[16](https://doi.org/10.1162/neco_a_01434) densMAP, by Narayan, Berger, and Cho, regularizes the cost function to preserve local density information that standard UMAP's uniformity assumption removes.[17](https://doi.org/10.1038/s41587-020-00801-7) Aligned-UMAP has been applied to longitudinal biomedical studies.[18](https://doi.org/10.1016/j.patter.2023.100741) On the GPU, Nolet and colleagues introduced a fully GPU-accelerated UMAP in cuML[19](https://doi.org/10.13016/m2sqzx-hybf), and cuML 25.06 added multi-GPU kNN graph construction.[8](https://developer.nvidia.com/blog/run-massive-scale-umap-in-minutes-using-multiple-gpus-without-losing-accuracy/)\n\n## Applications\n\nSingle-cell transcriptomics is a widely documented domain: Becht and colleagues applied UMAP to single-cell data visualization in 2018.[4](https://doi.org/10.1038/nbt.4314) In a benchmark of ten dimension reduction methods on 30 simulated and five real single-cell datasets, UMAP showed the highest stability, moderate accuracy, and the second-highest computing cost, with [Silhouette](https://www.edgepedia.org/silhouette) scores significantly higher than other methods in all simulation tests.[20](https://www.frontiersin.org/journals/genetics/articles/10.3389/fgene.2021.646936/full)\n\nOn the reference implementation's benchmarks, MNIST embeds in 42 seconds and Fashion MNIST in 49 seconds on a 3.1 GHz Intel Core i7, against roughly 45 minutes for scikit-learn's t-SNE on MNIST.[5](https://github.com/lmcinnes/umap/) UMAP scales better than t-SNE into embedding dimensions above 2 because it needs no global normalization and no quad-trees or oct-trees, which scale exponentially with dimension.[2](https://doi.org/10.48550/arxiv.1802.03426) GPU implementations have since changed the picture: cuML 24.10's batched NN-descent gave up to 311x speedup, cutting a 20M-point, 384-dimension run from 10 hours to 2 minutes[7](https://developer.nvidia.com/blog/even-faster-and-more-scalable-umap-on-the-gpu-with-rapids-cuml/), and cuML 25.06's multi-GPU graph construction embedded the 106M-vector MIRACL dataset in 8 minutes on eight H100 GPUs, a 74x end-to-end speedup over projected CPU runtime.[8](https://developer.nvidia.com/blog/run-massive-scale-umap-in-minutes-using-multiple-gpus-without-losing-accuracy/)\n\n## Limitations and alternatives\n\nDamrich and Hamprecht showed that UMAP's negative-sampling optimization minimizes an effective loss that differs significantly from its purported cross-entropy: the repulsive term's weight is drastically reduced, so UMAP approximates a binarized version of the high-dimensional similarities and most information beyond shared kNN graph connectivity is essentially ignored.[9](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf) With defaults \\( k = 15 \\) and \\( m = 5 \\), input similarities above 0.2 map to target similarities above 0.83 for \\( n = 500 \\) points, explaining the crisp, over-contracted substructures UMAP produces.[9](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf)\n\nThe original paper claims UMAP \"arguably preserves more of the global structure\" than t-SNE[2](https://doi.org/10.48550/arxiv.1802.03426), but later peer-reviewed evaluations find that t-SNE and UMAP preserve local structure well while struggling on global structure, and that neither can be adjusted smoothly from local to global preservation.[21](https://jmlr.org/papers/volume22/20-1061/20-1061.pdf) A systematic evaluation of eight methods on transcriptomic data found t-SNE and UMAP highly sensitive to parameter and pre-processing choices and weak on global-structure metrics, while PCA, TriMap, PaCMAP, and ForceAtlas2 were robust.[22](https://www.nature.com/articles/s42003-022-03628-x) UMAP results also changed dramatically across hyperparameter settings in a grid search on single-cell data[20](https://www.frontiersin.org/journals/genetics/articles/10.3389/fgene.2021.646936/full), and are not generally robust to the number of principal components chosen in pre-processing.[22](https://www.nature.com/articles/s42003-022-03628-x) The scDEED method, by Lucy Xia, Christy Lee, and Jingyi Jessica Li, detects dubious 2D single-cell embeddings and optimizes t-SNE and UMAP hyperparameters.[23](https://doi.org/10.1038/s41467-024-45891-y)\n\nFalse clusters are the practical risk: methods that preserve local but not global structure can present \"false\" clusters as real, generating false hypotheses. On a 59,286-cell PBMC benchmark, UMAP separated the two dendritic cell subsets into spatially distant groups where t-SNE, TriMap, and PaCMAP mapped them close together.[22](https://www.nature.com/articles/s42003-022-03628-x) Chari and Pachter argued in 2023 that such embeddings are specious; a 2024 rebuttal found kNN accuracy above 90% for UMAP and t-SNE versus below 62% for PCA on three scRNA-seq datasets, though kNN recall was below 40% for all methods, and endorsed 2D embeddings for exploratory hypothesis generation while agreeing they should not be used for quantitative downstream analysis.[24](https://doi.org/10.1371/journal.pcbi.1011288) • [25](https://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1012403)\n\nFailure modes include the connected-manifold assumption: the reference implementation offers a disconnection_distance parameter to disconnect vertices at distances above a threshold when points are maximally different from all others.[6](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py) Alternatives include PaCMAP, which adds mid-near and further-point loss terms to preserve both local and global structure[21](https://jmlr.org/papers/volume22/20-1061/20-1061.pdf), TriMap[26](https://doi.org/10.48550/arxiv.1910.00204), and densMAP for density preservation.[17](https://doi.org/10.1038/s41587-020-00801-7)\n\n## References\n\n1. [Uniform manifold approximation and projection | Nature Reviews Methods Primers (Healy & McInnes, 2024)](https://www.nature.com/articles/s43586-024-00363-x)\n2. [McInnes, Leland, Healy, John, Melville, James (2018). UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1802.03426)\n3. [Leland McInnes and colleagues (2018). UMAP: Uniform Manifold Approximation and Projection. The Journal of Open Source Software.](https://doi.org/10.21105/joss.00861)\n4. [Etienne Becht and colleagues (2018). Dimensionality reduction for visualizing single-cell data using UMAP. Nature Biotechnology.](https://doi.org/10.1038/nbt.4314)\n5. [lmcinnes/umap (official repository README)](https://github.com/lmcinnes/umap/)\n6. [umap/umap_.py reference implementation source (docstrings)](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py)\n7. [Even Faster and More Scalable UMAP on the GPU with NVIDIA cuML](https://developer.nvidia.com/blog/even-faster-and-more-scalable-umap-on-the-gpu-with-rapids-cuml/)\n8. [Run Massive-Scale UMAP in Minutes Using Multiple GPUs, Without Losing Accuracy](https://developer.nvidia.com/blog/run-massive-scale-umap-in-minutes-using-multiple-gpus-without-losing-accuracy/)\n9. [On UMAP's True Loss Function (NeurIPS 2021)](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf)\n10. [How UMAP Works (author documentation)](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html)\n11. [UMAP (Springer chapter critically assessing the algorithm's strategy)](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5)\n12. [Michael Barr (1986). Fuzzy Set Theory and Topos Theory. Canadian Mathematical Bulletin.](https://doi.org/10.4153/cmb-1986-079-9)\n13. [Some new information measures for fuzzy sets (Information Sciences, 1993)](https://doi.org/10.1016/0020-0255%2893%2990073-u)\n14. [Joshua B. Tenenbaum, Vin de Silva, John C. Langford (2000). A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science.](https://doi.org/10.1126/science.290.5500.2319)\n15. [Mikhail Belkin, Partha Niyogi (2003). Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computation.](https://doi.org/10.1162/089976603321780317)\n16. [Tim Sainburg, Leland McInnes, Timothy Q. Gentner (2021). Parametric UMAP Embeddings for Representation and Semisupervised Learning. Neural Computation.](https://doi.org/10.1162/neco_a_01434)\n17. [Ashwin Narayan, Bonnie Berger, Hyunghoon Cho (2021). Assessing single-cell transcriptomic variability through density-preserving data visualization. Nature Biotechnology.](https://doi.org/10.1038/s41587-020-00801-7)\n18. [Anant Dadu and colleagues (2023). Application of Aligned-UMAP to longitudinal biomedical studies. Patterns.](https://doi.org/10.1016/j.patter.2023.100741)\n19. [Nolet, Corey J. and colleagues (2020). Bringing UMAP Closer to the Speed of Light with GPU Acceleration. arXiv (Cornell University).](https://doi.org/10.13016/m2sqzx-hybf)\n20. [A Comparison for Dimensionality Reduction Methods of Single-Cell RNA-seq Data (Frontiers in Genetics 2021)](https://www.frontiersin.org/journals/genetics/articles/10.3389/fgene.2021.646936/full)\n21. [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMap, and PaCMAP (Wang et al., JMLR)](https://jmlr.org/papers/volume22/20-1061/20-1061.pdf)\n22. [Towards a comprehensive evaluation of dimension reduction methods for transcriptomic data visualization (Communications Biology 2022)](https://www.nature.com/articles/s42003-022-03628-x)\n23. [Lucy Xia, Christy Lee, Jingyi Jessica Li (2024). Statistical method scDEED for detecting dubious 2D single-cell embeddings and optimizing t-SNE and UMAP hyperparameters. Nature Communications.](https://doi.org/10.1038/s41467-024-45891-y)\n24. [Tara Chari, Lior Pachter (2023). The specious art of single-cell genomics. PLoS Computational Biology.](https://doi.org/10.1371/journal.pcbi.1011288)\n25. [The art of seeing the elephant in the room: 2D embeddings of single-cell data do make sense (PLOS Computational Biology, 2024)](https://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1012403)\n26. [Amid, Ehsan, Warmuth, Manfred K. (2019). TriMap: Large-scale Dimensionality Reduction Using Triplets. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1910.00204)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Machine learning and neural computation › Machine learning methods › Supervised, unsupervised, and semi-supervised learning › Dimensionality reduction and manifold learning*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/umap", "markdown_url": "https://www.edgepedia.org/umap.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"UMAP\", Edgepedia (EdgeChat), https://www.edgepedia.org/umap. Edgepedia Community License 1.0.", "credit_md": "\"[UMAP](https://www.edgepedia.org/umap)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/umap](https://www.edgepedia.org/umap). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"UMAP\", Edgepedia (EdgeChat), https://www.edgepedia.org/umap. Edgepedia Community License 1.0.", "speakable": "UMAP is a nonlinear dimension reduction method introduced in 2018, used for visualizing high-dimensional data and faster than most t-SNE implementations." }