{ "id": "epspt3pp5w", "slug": "uniform-manifold-approximation-and-projection", "title": "Uniform manifold approximation and projection", "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": "Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction method introduced in 2018 that embeds high-dimensional data into low dimensions while preserving local neighborhoods, and is a standard tool for visualizing single-cell datasets.", "snippet": "Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction method introduced in 2018 that embeds high-dimensional data into low dimensions while preserving local neighborhoods, and is a standard tool for visualizing single-cell datasets.", "node": "technology.computing.ai.supervised_unsupervised.dimensionality_reduction", "markdown": "# Uniform manifold approximation and projection\n\nUniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction method that embeds high-dimensional data into a low-dimensional space while preserving local neighborhood structure. It is used for visualizing data and as pre-processing for further machine-learning tasks such as clustering, and it has become a standard visualization tool for single-cell and other high-dimensional datasets.[1](https://doi.org/10.48550/arxiv.1802.03426) • [2](https://doi.org/10.21105/joss.00861) • [3](https://www.nature.com/articles/s43586-024-00363-x) Unlike t-SNE, it also scales well in embedding dimension, supports transforming new data into a fitted embedding, and accepts partial labels for semi-supervised reduction.[2](https://doi.org/10.21105/joss.00861)\n\n| Key fact | Detail |\n|---|---|\n| What it produces | A low-dimensional embedding (commonly 2D for plots, higher for clustering) of high-dimensional data[3](https://www.nature.com/articles/s43586-024-00363-x) |\n| Core assumptions | Data uniformly distributed on a locally connected Riemannian manifold with locally constant metric[4](https://github.com/lmcinnes/umap/) |\n| Objective | Cross-entropy between fuzzy simplicial set representations of high- and low-dimensional data[1](https://doi.org/10.48550/arxiv.1802.03426) |\n| Default hyperparameters | n_neighbors = 15, min_dist = 0.1, negative_sample_rate = 5[5](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py) |\n| Speed | 70,000-sample, 784-dimensional MNIST embedded in under a minute, versus about 45 minutes for scikit-learn's t-SNE[4](https://github.com/lmcinnes/umap/) |\n| Scale | Used directly on data with over a million dimensions; GPU versions handle tens to hundreds of millions of vectors[4](https://github.com/lmcinnes/umap/) • [6](https://developer.nvidia.com/blog/run-massive-scale-umap-in-minutes-using-multiple-gpus-without-losing-accuracy/) |\n| Main caveat | Strong on local structure, weak and parameter-sensitive on global structure[7](https://www.nature.com/articles/s42003-022-03628-x) |\n\n## How it works\n\nUMAP rests on three assumptions: the data are uniformly distributed on a [Riemannian manifold](https://www.edgepedia.org/riemannian-manifold), the [Riemannian metric](https://www.edgepedia.org/riemannian-metric) is locally constant or approximately so, and the manifold is locally connected.[4](https://github.com/lmcinnes/umap/) The uniformity assumption is handled by conformal changes of the geometry so the data distribution appears uniform in the representation.[8](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5) Practically, a unit ball about each point stretches to its k-th nearest neighbor, giving every point its own local distance scale, and a local connectivity constraint ensures no point is isolated.[9](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html)\n\nFrom these local approximations UMAP builds a fuzzy simplicial set, a weighted graph whose edges carry membership strengths. Disagreeing edge weights a and b are merged with the fuzzy union formula \\( a + b - a \\cdot b \\), interpreted as the probability that at least one edge exists, and low-dimensional membership strengths use curves of the form \\( \\frac{1}{1 + a \\cdot x^{2b}} \\).[9](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html) • [1](https://doi.org/10.48550/arxiv.1802.03426) The layout is then optimized to minimize the cross-entropy between the high- and low-dimensional representations,[1](https://doi.org/10.48550/arxiv.1802.03426) written over edges e as\n\n\\[ \\sum_{e} w_{h}(e)\\log\\frac{w_{h}(e)}{w_{l}(e)} + \\bigl(1 - w_{h}(e)\\bigr)\\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.[9](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html)\n\nWhat the optimizer actually minimizes differs from this stated objective. Böhm, Berens, and colleagues showed that negative-sampling-based SGD yields an effective loss in which the repulsive term's weight is drastically reduced, so UMAP approximates a binarized version of the high-dimensional similarities, using little information beyond shared kNN graph connectivity.[10](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf) This binarization explains UMAP's over-contraction into crisp substructures, and the same authors note that negative sampling distorts the gradient and under-represents repulsive forces.[10](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf) • [8](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5)\n\n## How it is done\n\nA standard run has two phases: graph construction in the high-dimensional space, then optimization of the low-dimensional layout.[9](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html) • [11](https://jmlr.org/papers/volume22/20-1061/20-1061.pdf)\n\n1. **Build the kNN graph.** For each point, find its k nearest neighbors under the chosen metric using the nearest-neighbor-descent algorithm; compute \\( \\rho_{i} \\), the minimal positive distance to a neighbor, which guarantees local connectivity, and solve for the scaling parameter \\( \\sigma_{i} \\) by binary search, analogous to t-SNE's perplexity search.[1](https://doi.org/10.48550/arxiv.1802.03426) • [11](https://jmlr.org/papers/volume22/20-1061/20-1061.pdf) • [12](https://ar5iv.labs.arxiv.org/html/2109.02508)\n2. **Initialize the embedding** with the spectral (Laplacian eigenmap) embedding of the symmetrized fuzzy-weight graph.[12](https://ar5iv.labs.arxiv.org/html/2109.02508)\n3. **Optimize by SGD** with probabilistic edge sampling and negative sampling, drawing m negative samples per positive edge (default negative_sample_rate = 5).[1](https://doi.org/10.48550/arxiv.1802.03426) • [5](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py) n_epochs defaults to 500 for small datasets and 200 for large ones.[5](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py)\n4. **Optionally transform new data** into the pretrained embedding space.[2](https://doi.org/10.21105/joss.00861)\n\nThe main hyperparameters are n_neighbors (default 15; recommended range 5–50), which sets the local/global tradeoff: low values concentrate on very local structure, large values such as 200 capture broader structure at the cost of fine detail.[5](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py) • [13](https://umap-learn.readthedocs.io/en/latest/parameters.html) • [4](https://github.com/lmcinnes/umap/) min_dist (default 0.1; sensible range 0.001–0.5) is the minimum distance allowed between embedded points; low values give clumpier embeddings that follow the manifold more closely, higher values spread points out, and the original authors call it an essentially aesthetic parameter.[13](https://umap-learn.readthedocs.io/en/latest/parameters.html) • [1](https://doi.org/10.48550/arxiv.1802.03426) The metric is selectable (euclidean, cosine, jaccard for sparse data, hellinger, and custom compiled metrics), and because UMAP scales well in embedding dimension, n_components of 10 or 50 is practical for downstream clustering.[13](https://umap-learn.readthedocs.io/en/latest/parameters.html) • [14](https://docs.nvidia.com/cuml/26.08/api/generated/cuml.manifold.UMAP/)\n\n## Origin\n\nUMAP was presented by Leland McInnes, John Healy, and James Melville in the arXiv preprint 1802.03426 in 2018,[1](https://doi.org/10.48550/arxiv.1802.03426) with a companion software paper by Leland McInnes and colleagues in the Journal of Open Source Software 3(29):861, published in 2018.[2](https://doi.org/10.21105/joss.00861) The method builds on earlier work on Laplacian eigenmaps and on David Spivak's category-theoretic fuzzy simplicial sets.[1](https://doi.org/10.48550/arxiv.1802.03426) It belongs to the neighbor-embedding family alongside t-SNE, introduced by [Laurens van der Maaten](https://www.edgepedia.org/laurens-van-der-maaten) and Geoffrey E. Hinton in 2008, and LargeVis; unlike t-SNE, UMAP and LargeVis do not normalize embedding-space probabilities over all pairs.[12](https://ar5iv.labs.arxiv.org/html/2109.02508)\n\n## Variants\n\nThe reference implementation supports supervised and semi-supervised reduction through labels or partial labels, transformation of new points, inverse transform, and non-Euclidean embeddings such as hyperbolic ones.[4](https://github.com/lmcinnes/umap/) • [2](https://doi.org/10.21105/joss.00861) **Parametric UMAP** extends the optimization step to a neural network that learns a parametric mapping from data to embedding, performing comparably to non-parametric UMAP while enabling fast online embeddings of new data.[15](https://arxiv.org/abs/2009.12981v4) • [16](https://doi.org/10.1162/neco_a_01434) **densMAP**, by Ashwin Narayan, Bonnie Berger, and Hyunghoon Cho ([Nature Biotechnology](https://www.edgepedia.org/nature-biotechnology), 2021), regularizes the cost function to preserve local density information that standard UMAP discards.[17](https://doi.org/10.1038/s41587-020-00801-7) • [12](https://ar5iv.labs.arxiv.org/html/2109.02508) A Progressive UMAP variant supports streaming and out-of-sample embedding, while original UMAP handles out-of-sample embedding through its transform operation.[12](https://ar5iv.labs.arxiv.org/html/2109.02508) • [23](https://umap-learn.readthedocs.io/en/latest/api.html) **Aligned-UMAP**, applied to longitudinal biomedical studies by Anant Dadu and colleagues (Patterns, 2023), embeds data from multiple time points into consistent spaces.[18](https://doi.org/10.1016/j.patter.2023.100741) On the implementation side, Corey J. Nolet and colleagues introduced a fully GPU-accelerated UMAP (GPUMAP/cuML) in 2020, using FAISS for kNN search and a fused CUDA kernel for gradient updates,[19](https://doi.org/10.13016/m2sqzx-hybf) and the torchdr package accelerates every pipeline step (kNN, affinities, optimization) on GPU.[4](https://github.com/lmcinnes/umap/)\n\n## Applications\n\nUMAP is used for visualizing high-dimensional data and as pre-processing for clustering and other machine-learning tasks, and it has become a standard tool for single-cell datasets.[1](https://doi.org/10.48550/arxiv.1802.03426) • [3](https://www.nature.com/articles/s43586-024-00363-x) Its speed comes largely from skipping the global normalization t-SNE performs, which also makes it suitable for mini-batch optimization; its optimization cost is \\( O(k \\cdot N) \\) in the number of edges, with overall empirical complexity around \\( O(N^{1.14}) \\).[12](https://ar5iv.labs.arxiv.org/html/2109.02508) • [1](https://doi.org/10.48550/arxiv.1802.03426) On 784-dimensional MNIST with 70,000 samples, UMAP completes in under a minute versus about 45 minutes for scikit-learn's t-SNE.[4](https://github.com/lmcinnes/umap/) GPU implementations extend the scale sharply: cuML UMAP processed 3 million Google-News datapoints in 9.5 minutes,[19](https://doi.org/10.13016/m2sqzx-hybf) and cuML 25.06 embedded a 106-million-vector MIRACL dataset with 2048 dimensions end-to-end in 8 minutes on eight H100 GPUs, a 74x speedup over projected CPU runtime.[6](https://developer.nvidia.com/blog/run-massive-scale-umap-in-minutes-using-multiple-gpus-without-losing-accuracy/)\n\n## Limitations and alternatives\n\n**Global structure.** The original authors state UMAP \"arguably preserves more of the global structure\" than t-SNE, while cautioning that if global structure is of primary interest UMAP may not be the best choice and recommending multidimensional scaling instead.[1](https://doi.org/10.48550/arxiv.1802.03426) Independent evaluations disagree with the authors' claim: a systematic benchmark of eight methods on transcriptomic data found UMAP strong on local structure but weak on global-structure metrics, where PCA, TriMap, PaCMAP, and ForceAtlas2 performed well and were robust to parameter choices.[7](https://www.nature.com/articles/s42003-022-03628-x) Chari and colleagues argue in PLOS Computational Biology that t-SNE and UMAP distort global structure in single-cell data and that such embeddings should not be used for inference beyond local neighborhoods.[20](https://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1011288) A concrete failure example: on a 59,286-cell PBMC dataset, UMAP separated the two dendritic cell subsets (mDCs and pDCs) into spatially distant groups, whereas t-SNE, TriMap, and PaCMAP placed them close together.[7](https://www.nature.com/articles/s42003-022-03628-x)\n\n**Other failure modes.** UMAP results vary substantially under different parameter choices within the reasonable range, and it is highly sensitive to parameter and pre-processing choices.[7](https://www.nature.com/articles/s42003-022-03628-x) It can find spurious manifold structure in noise, its approximations can produce suboptimal embeddings for datasets smaller than 500 samples, and negative sampling under-represents repulsive forces, contributing to over-contraction.[1](https://doi.org/10.48550/arxiv.1802.03426) • [10](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf) • [8](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5) A broad evaluation of 11 dimensionality reduction methods found that input data distribution and parameter settings largely determine how much structure is preserved.[21](https://pmc.ncbi.nlm.nih.gov/articles/PMC7305633/) The statistical method scdeed, by Lucy Xia, Christy Lee, and Jingyi Jessica Li ([Nature Communications](https://www.edgepedia.org/nature-communications), 2024), detects dubious 2D single-cell embeddings and optimizes t-SNE and UMAP hyperparameters, addressing these reliability concerns.[22](https://doi.org/10.1038/s41467-024-45891-y)\n\n**Alternatives.** PaCMAP adds neighbor, mid-near, and further-point loss terms to preserve local and global structure simultaneously, addressing the tradeoff that t-SNE and UMAP resolve in favor of one or the other.[11](https://jmlr.org/papers/volume22/20-1061/20-1061.pdf) PCA, TriMap, and ForceAtlas2 are stronger where global structure or parameter robustness matters, MDS when preserving the full distance matrix is the goal, and t-SNE remains a comparable choice for purely local visualization.[7](https://www.nature.com/articles/s42003-022-03628-x) • [1](https://doi.org/10.48550/arxiv.1802.03426)\n\n## References\n\n1. [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)\n2. [Leland McInnes and colleagues (2018). UMAP: Uniform Manifold Approximation and Projection. The Journal of Open Source Software.](https://doi.org/10.21105/joss.00861)\n3. [Uniform manifold approximation and projection (Nature Reviews Methods Primers)](https://www.nature.com/articles/s43586-024-00363-x)\n4. [lmcinnes/umap GitHub repository (README)](https://github.com/lmcinnes/umap/)\n5. [umap/umap_.py (source code docstrings)](https://github.com/lmcinnes/umap/blob/master/umap/umap_.py)\n6. [Run Massive-Scale UMAP in Minutes Using Multiple GPUs, Without Losing Accuracy (cuML/cuVS 25.06)](https://developer.nvidia.com/blog/run-massive-scale-umap-in-minutes-using-multiple-gpus-without-losing-accuracy/)\n7. [Towards a comprehensive evaluation of dimension reduction methods for transcriptomic data visualization](https://www.nature.com/articles/s42003-022-03628-x)\n8. [UMAP (book chapter, Springer)](https://link.springer.com/chapter/10.1007/978-3-031-97973-6_5)\n9. [How UMAP Works (official documentation)](https://umap-learn.readthedocs.io/en/latest/how_umap_works.html)\n10. [On UMAP's True Loss Function (NeurIPS 2021)](https://proceedings.neurips.cc/paper_files/paper/2021/file/2de5d16682c3c35007e4e92982f1a2ba-Paper.pdf)\n11. [Understanding How Dimension Reduction Tools Work (PaCMAP paper, JMLR)](https://jmlr.org/papers/volume22/20-1061/20-1061.pdf)\n12. [UMAP and its Variants: Tutorial and Survey](https://ar5iv.labs.arxiv.org/html/2109.02508)\n13. [UMAP parameters documentation](https://umap-learn.readthedocs.io/en/latest/parameters.html)\n14. [UMAP, NVIDIA cuML API documentation](https://docs.nvidia.com/cuml/26.08/api/generated/cuml.manifold.UMAP/)\n15. [Parametric UMAP embeddings for representation and semi-supervised learning](https://arxiv.org/abs/2009.12981v4)\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. [The specious art of single-cell genomics](https://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1011288)\n21. [A Quantitative Framework for Evaluating Single-Cell Data Structure Preservation by Dimensionality Reduction Techniques](https://pmc.ncbi.nlm.nih.gov/articles/PMC7305633/)\n22. [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)\n23. [UMAP API Guide, umap 0.5.8 documentation](https://umap-learn.readthedocs.io/en/latest/api.html)\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/uniform-manifold-approximation-and-projection", "markdown_url": "https://www.edgepedia.org/uniform-manifold-approximation-and-projection.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. 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Edgepedia Community License 1.0.", "speakable": "Uniform manifold approximation and projection is a nonlinear dimensionality reduction method introduced in 2018 that embeds high-dimensional data into low dimensions while preserving local neighborhoods, and is a standard tool for visualizing single-cell datasets." }