{ "id": "ep46qrz8w3", "slug": "diffusion-map", "title": "Diffusion map", "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": "Diffusion map is a nonlinear dimensionality reduction method introduced by Ronald Coifman and Stéphane Lafon in 2006 that embeds high-dimensional data by running a random walk on a similarity graph.", "snippet": "Diffusion map is a nonlinear dimensionality reduction method introduced by Ronald Coifman and Stéphane Lafon in 2006 that embeds high-dimensional data by running a random walk on a similarity graph.", "node": "technology.computing.ai.supervised_unsupervised.dimensionality_reduction", "markdown": "# Diffusion map\n\nA diffusion map is a nonlinear dimensionality reduction method that embeds high-dimensional data into a low-dimensional [Euclidean space](https://www.edgepedia.org/euclidean-space) by running a random walk on a similarity graph built from the data, so that [Euclidean distance](https://www.edgepedia.org/euclidean-distance) in the embedding equals a diffusion-based measure of connectivity between points. The method was introduced by Ronald R. Coifman and Stéphane Lafon in a 2006 paper in Applied and Computational Harmonic Analysis.[1](https://doi.org/10.1016/j.acha.2006.04.006) Diffusion maps preserve local neighborhood structure and the connectivity of the data graph, and they produce a family of representations at different scales indexed by a diffusion time parameter.[2](https://dl.acm.org/doi/10.1109/TPAMI.2006.184) • [3](https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf) Typical uses include visualization, clustering, [manifold learning](https://www.edgepedia.org/manifold-learning), and trajectory analysis of single-cell gene expression data.\n\n| Key fact | Detail |\n|---|---|\n| Output | Coordinates \\( \\Psi_{t} \\) in which Euclidean distance equals the diffusion distance up to a preset relative accuracy \\( \\delta \\)[4](https://www.math.ucdavis.edu/~strohmer/courses/270/diffusion_maps.pdf) |\n| Introducing paper | Coifman and Lafon, Applied and Computational Harmonic Analysis, 2006 (received 29 October 2004)[1](https://doi.org/10.1016/j.acha.2006.04.006) |\n| Main parameters | Kernel bandwidth \\( \\epsilon \\) (neighborhood scale) and diffusion time \\( t \\)[5](https://proceedings.neurips.cc/paper_files/paper/2005/file/2a0f97f81755e2878b264adf39cba68e-Paper.pdf) • [6](https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf) |\n| Core algorithm | Four steps: kernel matrix, row normalization, eigendecomposition, mapping at time \\( t \\)[3](https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf) |\n| Cost | Reported as \\( O(n^{3}) \\) training and \\( O(n^{2}) \\) transformation in one comparative review; the single-cell application paper reports \\( n^{2} \\) time, prohibitive above roughly \\( 10^{4} \\) points[7](http://faculty.ist.psu.edu/vhonavar/Courses/dsmethods/dim1.pdf) • [8](https://mediatum.ub.tum.de/doc/1319938/document.pdf) |\n| Flagship application | Pseudotemporal ordering of differentiating single cells (Haghverdi, Buettner, and Theis, Bioinformatics, 2015)[9](https://doi.org/10.1093/bioinformatics/btv325) |\n\n## How it works\n\nThe construction starts from pairwise affinities, typically a Gaussian kernel \\( K(x_{i}, x_{j}) = \\exp(-\\|x_{i} - x_{j}\\|^{2} / (2\\epsilon)) \\).[10](https://arxiv.org/html/2503.03963v2) Row-normalizing this kernel produces a Markov transition matrix, so the entries describe a random walk on the data graph that is biased toward moves within dense regions.[11](https://stat.cmu.edu/~cshalizi/350/lectures/15/lecture-15.pdf) The entry \\( a^{(m)}(x, y) \\) of the \\( m \\)-th power of this matrix is the probability of transitioning from \\( x \\) to \\( y \\) in \\( m \\) steps.\n\nTwo points are connected by the diffusion distance, a weighted distance between their posterior distributions after \\( t \\) steps of the walk. Because it sums over all paths of length \\( t \\) through the graph, it is more robust to perturbations and noise than a geodesic distance, which depends on single shortest paths.[7](http://faculty.ist.psu.edu/vhonavar/Courses/dsmethods/dim1.pdf) Nadler, Lafon, Coifman, and Kevrekidis showed that this distance is exactly Euclidean distance in the eigenfunction coordinates:\n\n\\[ D_{t}^{2}(x_{0}, x_{1}) = \\sum_{j \\geq 1} \\lambda_{j}^{2t} \\left( \\psi_{j}(x_{0}) - \\psi_{j}(x_{1}) \\right)^{2} \\]\n\nwhere \\( \\lambda_{j} \\) and \\( \\psi_{j} \\) are the eigenvalues and eigenvectors of the Markov matrix.[5](https://proceedings.neurips.cc/paper_files/paper/2005/file/2a0f97f81755e2878b264adf39cba68e-Paper.pdf) The diffusion map \\( \\Psi_{t} \\) keeps the first coordinates \\( (\\lambda_{1}^{t}\\psi_{1}, \\ldots, \\lambda_{s}^{t}\\psi_{s}) \\), so Euclidean distance in the embedding reproduces the diffusion distance.[4](https://www.math.ucdavis.edu/~strohmer/courses/270/diffusion_maps.pdf) In the limit of infinitely many samples and vanishing bandwidth, the kernel approximates the heat kernel \\( e^{-t\\Delta} \\) of the [Laplace–Beltrami operator](https://www.edgepedia.org/laplace-beltrami-operator) on the underlying manifold, with normalization needed to correct the influence of sampling density.[10](https://arxiv.org/html/2503.03963v2)\n\n## How it is done\n\nThe basic algorithm has four steps[3](https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf):\n\n1. Compute the \\( N \\times N \\) pairwise distance matrix \\( d_{ij} = \\|x_{i} - x_{j}\\| \\) and form the kernel matrix \\( K \\) with \\( k_{ij} = \\exp(-\\Delta(i,j)/\\epsilon) \\). The kernel must be symmetric and nonnegative for the spectral analysis to apply.[12](https://doi.org/10.48550/arxiv.1706.09396) • [3](https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf)\n2. Normalize the rows of \\( K \\) to obtain the stochastic matrix \\( P \\), whose \\( t \\)-th power gives \\( t \\)-step transition probabilities.[6](https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf)\n3. Compute the eigenvectors and eigenvalues of \\( P \\).[3](https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf)\n4. Map each point at the chosen time \\( t \\).[3](https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf)\n\nChoosing \\( \\epsilon \\) and \\( t \\) is the main practical difficulty. The bandwidth \\( \\epsilon \\) has a dual interpretation as the squared radius of the neighborhood used to infer local geometry and as the discrete time step of the walk.[5](https://proceedings.neurips.cc/paper_files/paper/2005/file/2a0f97f81755e2878b264adf39cba68e-Paper.pdf) A common initial guess is \\( \\epsilon \\) equal to twice the mean of the row minima of the squared-distance matrix; a more systematic diagnostic, the Ksum test, plots the double sum \\( S(\\epsilon) \\approx \\int\\int k_{\\epsilon}\\, dx\\, dx' \\) on log-log axes and selects \\( \\epsilon \\) where the slope is approximately \\( d/2 \\), revealing the intrinsic dimension \\( d \\).[6](https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf) The bandwidth \\( \\varepsilon \\) controls direct kernel affinities, while the time \\( t \\) specifies the number of random-walk steps: for distinct data points, the one-step Markov matrix approaches the identity as \\( \\varepsilon \\) tends to zero, and for a connected, aperiodic walk, excessively large \\( t \\) makes transition distributions approach the same stationary distribution, reducing distinctions between points.[22](https://pmc.ncbi.nlm.nih.gov/articles/PMC7330274/) • [23](https://pmc.ncbi.nlm.nih.gov/articles/PMC4118417/) One practical rule is to pick \\( t \\) minimizing the Semi-Group Error, which is computationally easy and performs well in practice.[13](https://ar5iv.labs.arxiv.org/html/2203.02867)\n\nThe number of coordinates kept is set by an accuracy parameter \\( \\delta \\in (0,1) \\) through \\( s(\\delta,t) = \\max\\{m: |\\lambda_{m}|^{t} > \\delta|\\lambda_{1}|\\} \\); because eigenvalues decay below 1 in modulus, finitely many terms suffice for any preset accuracy.[4](https://www.math.ucdavis.edu/~strohmer/courses/270/diffusion_maps.pdf)\n\n## Origin\n\nThe method takes its name from the paper \"Diffusion maps\" by Ronald R. Coifman and Stéphane Lafon, published in Applied and Computational Harmonic Analysis in 2006.[1](https://doi.org/10.1016/j.acha.2006.04.006) • [4](https://www.math.ucdavis.edu/~strohmer/courses/270/diffusion_maps.pdf) The framework of eigenfunctions of Markov matrices and its multiscale companion methods were presented in a two-part 2005 PNAS paper, \"Geometric diffusions as a tool for harmonic analysis and structure definition of data\".[14](https://www.cs.yale.edu/homes/vision/zucker/papers/CLLMNWZ05_PNAS_2.pdf)\n\nThe method builds on earlier spectral embedding work, above all Laplacian eigenmaps by Mikhail Belkin and Partha Niyogi (2002)[15](https://doi.org/10.7551/mitpress/1120.003.0080); the two constructions are very similar, both starting from a graph on the data points.[6](https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf) The 2006 paper shows that kernel eigenmap methods, including local linear embedding, Laplacian eigenmaps, Hessian eigenmaps, and local tangent space alignment, fit into a general diffusion-process framework.[4](https://www.math.ucdavis.edu/~strohmer/courses/270/diffusion_maps.pdf)\n\n## Variants\n\n**Out-of-sample extension.** Geometric harmonics, by Coifman and Lafon (2006), extend empirical functions off the data set via the [Nyström method](https://www.edgepedia.org/nystrom-method), with the extension distance controlled by the eigenvalue.[16](https://doi.org/10.1016/j.acha.2005.07.005) • [14](https://www.cs.yale.edu/homes/vision/zucker/papers/CLLMNWZ05_PNAS_2.pdf) Landmark diffusion maps (L-dMaps), by Andrew W. Long and Andrew L. Ferguson (2017), accelerate manifold learning by computing the embedding on landmark points and extending it out-of-sample.[12](https://doi.org/10.48550/arxiv.1706.09396)\n\n**Multiscale and clustering variants.** The companion PNAS Part II paper introduced diffusion wavelets, a [multiresolution analysis](https://www.edgepedia.org/multiresolution-analysis) built from powers of the diffusion operator at logarithmically spaced times.[14](https://www.cs.yale.edu/homes/vision/zucker/papers/CLLMNWZ05_PNAS_2.pdf) Lafon and Lee showed that clustering in the diffusion embedding space is equivalent to compressing the diffusion operator, with quantization distortion bounding the compression error, which justifies k-means clustering in diffusion space.[2](https://dl.acm.org/doi/10.1109/TPAMI.2006.184)\n\n**Density correction and generative extensions.** When sampling density is nonuniform, right renormalization of the kernel, originally developed by Coifman and Lafon, modulates the density's effect.[6](https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf) A 2025 framework combines Double Diffusion Maps with [score-based generative models](https://www.edgepedia.org/score-based-generative-models) and Itô SDEs to sample densities on unknown low-dimensional manifolds: a first round of diffusion maps finds latent coordinates, and a second round builds Latent Harmonics that lift samples back to ambient space via the Nyström extension.[10](https://arxiv.org/html/2503.03963v2)\n\n## Applications\n\n**Single-cell genomics.** Haghverdi, Buettner, and Theis adapted diffusion maps to single-cell data by adequate choice of kernel width and inclusion of measurement uncertainties or missing values, enabling pseudotemporal ordering of differentiating cells in high-dimensional gene expression space.[9](https://doi.org/10.1093/bioinformatics/btv325) On two qPCR datasets (mouse hematopoietic stem cells and mouse embryonic stem cells) and an RNA-Seq dataset of human pre-implantation embryos, the adapted method performed considerably better than PCA and was advantageous over other nonlinear techniques such as t-SNE.[8](https://mediatum.ub.tum.de/doc/1319938/document.pdf)\n\n**Generative modeling.** A diffusion map particle system (DMPS) combines diffusion maps with Laplacian-adjusted Wasserstein gradient descent (LAWGD) for generative modeling; it requires no offline training, and the only tuned parameter is the kernel bandwidth.[17](https://www.aimsciences.org/article/doi/10.3934/fods.2024054)\n\n## Limitations and alternatives\n\n**Cost.** A comparative review lists \\( O(n^{3}) \\) training and \\( O(n^{2}) \\) transformation cost[7](http://faculty.ist.psu.edu/vhonavar/Courses/dsmethods/dim1.pdf), while the single-cell application paper reports \\( n^{2} \\) computation time as prohibitive for cell numbers above \\( 10^{4} \\) and proposes a k-nearest-neighbor variant as a remedy[8](https://mediatum.ub.tum.de/doc/1319938/document.pdf); published sources do not reconcile these figures.\n\n**Parameters and density.** The kernel choice is crucial for real datasets, although the Gaussian kernel is not the only possibility and does not change the asymptotic analysis qualitatively.[5](https://proceedings.neurips.cc/paper_files/paper/2005/file/2a0f97f81755e2878b264adf39cba68e-Paper.pdf) Nonuniform sampling density biases the operator unless right renormalization is applied.[6](https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf)\n\n**Failure modes.** For spectral clustering to succeed, the mean exit time from each cluster must significantly exceed the slowest relaxation time inside the clusters; for complex multiscale data this condition may fail.[18](https://www.weizmann.ac.il/math/Nadler/sites/math.Nadler/files/publications/diffusion_maps_-_a_probabilistic_interpretation_for_spectral_embedding_and_clustering_algorithms.pdf) A benchmark study found diffusion maps preserve local manifold structure robustly, even outperforming other spectral techniques in low-density sampling under a classification measure, but they struggle to recover global structure when data are scaled beyond \\( s \\in [0.5, 2] \\), and when intrinsic dimension \\( d > 4 \\) they show no significant advantage over PCA.[19](https://www2.compute.dtu.dk/~sohau/papers/icann2011/Mysling_ICANN_2011_Manifold_Learning.pdf)\n\n**Comparisons.** Against Isomap, the diffusion distance sums over many paths rather than relying on geodesic shortest paths, making it more robust to noise.[7](http://faculty.ist.psu.edu/vhonavar/Courses/dsmethods/dim1.pdf) UMAP, by Leland McInnes, John Healy, Nathaniel Saul, and Lukas Großberger (2018), is a popular heuristic method that minimizes mismatches between topological representations of the data.[20](https://doi.org/10.21105/joss.00861) • [21](https://arxiv.org/pdf/2311.03757v1) No quantitative head-to-head benchmark against t-SNE or UMAP appears in the published literature beyond the single-cell comparison favoring diffusion maps over PCA and t-SNE.[8](https://mediatum.ub.tum.de/doc/1319938/document.pdf)\n\n## References\n\n1. [Ronald R. Coifman, Stéphane Lafon (2006). Diffusion maps. Applied and Computational Harmonic Analysis.](https://doi.org/10.1016/j.acha.2006.04.006)\n2. [Diffusion Maps and Coarse-Graining: A Unified Framework for Dimensionality Reduction, Graph Partitioning, and Data Set Parameterization (Lafon & Lee, IEEE TPAMI vol. 28 no. 9)](https://dl.acm.org/doi/10.1109/TPAMI.2006.184)\n3. [An Introduction to Diffusion Maps (del Porte, Herbst, Hereman, van der Walt, PRASA 2008)](https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf)\n4. [Diffusion maps (Coifman & Lafon, Applied and Computational Harmonic Analysis, 2006)](https://www.math.ucdavis.edu/~strohmer/courses/270/diffusion_maps.pdf)\n5. [Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck Operators (Nadler, Lafon, Coifman, Kevrekidis, NeurIPS 2005)](https://proceedings.neurips.cc/paper_files/paper/2005/file/2a0f97f81755e2878b264adf39cba68e-Paper.pdf)\n6. [Dimensionality Reduction tutorial (UMD REU 2023)](https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf)\n7. [Dimensionality Reduction: A Comparative Review](http://faculty.ist.psu.edu/vhonavar/Courses/dsmethods/dim1.pdf)\n8. [Diffusion maps for high-dimensional single-cell analysis of differentiation data (Bioinformatics, 2015)](https://mediatum.ub.tum.de/doc/1319938/document.pdf)\n9. [Laleh Haghverdi, Florian Buettner, Fabian J. Theis (2015). Diffusion maps for high-dimensional single-cell analysis of differentiation data. Bioinformatics.](https://doi.org/10.1093/bioinformatics/btv325)\n10. [Generative Learning of Densities on Manifolds (arXiv, 2025)](https://arxiv.org/html/2503.03963v2)\n11. [Nonlinear Dimensionality Reduction II: Diffusion Maps (CMU lecture notes, Cosma Shalizi)](https://stat.cmu.edu/~cshalizi/350/lectures/15/lecture-15.pdf)\n12. [Long, Andrew W., Ferguson, Andrew L. (2017). Landmark Diffusion Maps (L-dMaps): Accelerated manifold learning out-of-sample extension. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.1706.09396)\n13. [Diffusion Maps: Using the Semigroup Property for Parameter Tuning (arXiv 2203.02867)](https://ar5iv.labs.arxiv.org/html/2203.02867)\n14. [Geometric diffusions as a tool for harmonic analysis and structure definition of data: Multiscale methods (PNAS 2005, Part II)](https://www.cs.yale.edu/homes/vision/zucker/papers/CLLMNWZ05_PNAS_2.pdf)\n15. [Mikhail Belkin, Partha Niyogi (2002). Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. The MIT Press eBooks.](https://doi.org/10.7551/mitpress/1120.003.0080)\n16. [Ronald R. Coifman, Stéphane Lafon (2006). Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions. Applied and Computational Harmonic Analysis.](https://doi.org/10.1016/j.acha.2005.07.005)\n17. [Diffusion map particle systems for generative modeling (Foundations of Data Science, 2024)](https://www.aimsciences.org/article/doi/10.3934/fods.2024054)\n18. [Diffusion Maps – a Probabilistic Interpretation for Spectral Embedding and Clustering Algorithms (Nadler et al.)](https://www.weizmann.ac.il/math/Nadler/sites/math.Nadler/files/publications/diffusion_maps_-_a_probabilistic_interpretation_for_spectral_embedding_and_clustering_algorithms.pdf)\n19. [An Empirical Study on the Performance of Spectral Manifold Learning Techniques (ICANN 2011)](https://www2.compute.dtu.dk/~sohau/papers/icann2011/Mysling_ICANN_2011_Manifold_Learning.pdf)\n20. [Leland McInnes and colleagues (2018). UMAP: Uniform Manifold Approximation and Projection. The Journal of Open Source Software.](https://doi.org/10.21105/joss.00861)\n21. [Manifold learning: what, how, and why (arXiv, November 2023)](https://arxiv.org/pdf/2311.03757v1)\n22. [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC7330274/)\n23. [pmc.ncbi.nlm.nih.gov](https://pmc.ncbi.nlm.nih.gov/articles/PMC4118417/)\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 · Reviewed: Sep 30, 2026 · Edited: Oct 15, 2026 · Last review: Sep 30, 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": [ "https://www.math.wustl.edu/~victor/classes/pmf/delaPorte-Herbst-Hereman-vanderWalt-PRASA-2008.pdf", "https://www.math.ucdavis.edu/~strohmer/courses/270/diffusion_maps.pdf", "https://www.math.umd.edu/~mariakc/REU2023/Tutorials/DimensionalityReduction.pdf", "https://stat.cmu.edu/~cshalizi/350/lectures/15/lecture-15.pdf" ], "url": "https://www.edgepedia.org/diffusion-map", "markdown_url": "https://www.edgepedia.org/diffusion-map.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": "\"Diffusion map\", Edgepedia (EdgeChat), https://www.edgepedia.org/diffusion-map. Edgepedia Community License 1.0.", "credit_md": "\"[Diffusion map](https://www.edgepedia.org/diffusion-map)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/diffusion-map](https://www.edgepedia.org/diffusion-map). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Diffusion map\", Edgepedia (EdgeChat), https://www.edgepedia.org/diffusion-map. Edgepedia Community License 1.0.", "speakable": "Diffusion map is a nonlinear dimensionality reduction method introduced by Ronald Coifman and Stéphane Lafon in 2006 that embeds high-dimensional data by running a random walk on a similarity graph." }