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{{Short description|Geometric algorithm}}
[[File:Diffusion_map_of_a_torodial_helix.jpg|thumb|right|Given non-uniformly sampled data points on a toroidal helix (top), the first two Diffusion Map coordinates with Laplace–Beltrami normalization are plotted (bottom). The Diffusion Map unravels the toroidal helix recovering the underlying intrinsic circular geometry of the data.]]
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</math>
where <math>\{\lambda_l \}</math> is the sequence of eigenvalues of <math>M</math> and <math>\{\psi_l \}</math> and <math>\{\phi_l \}</math> are the biorthogonal
Due to the spectrum decay of the eigenvalues, only a few terms are necessary to achieve a given relative accuracy in this sum.
====Parameter α and the diffusion operator====
The reason to introduce the normalization step involving <math>\alpha</math> is to tune the influence of the data point density on the infinitesimal transition of the diffusion. In some applications, the sampling of the data is generally not related to the geometry of the manifold we are interested in describing. In this case, we can set <math>\alpha=1</math> and the diffusion operator approximates the [[Laplace–Beltrami operator]]. We then recover the Riemannian geometry of the data set regardless of the distribution of the points. To describe the long-term behavior of the point distribution of a system of stochastic differential equations, we can use <math>\alpha=0.5</math> and the resulting Markov chain approximates the [[Fokker–Planck equation|Fokker–Planck diffusion]]. With <math>\alpha=0</math>, it reduces to the classical graph Laplacian normalization.
===Diffusion distance===
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: <math>
D_t(x_i,x_j)^2
</math>
so the Euclidean distance in the diffusion coordinates approximates the diffusion distance.
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| journal = Proceedings of the IEEE International Conference on Computer Vision 2013
| pages = 1960–1967
}}</ref> [[spectral clustering]], low dimensional representation of images, image segmentation,<ref name="Farbman" /> 3D model segmentation,<ref name="sidi11" /> speaker verification<ref name="spk">{{cite
| last1 = Barkan
| first1 = Oren
| last2 = Aronowitz
| first2 = Hagai
|
|
▲ | journal = Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
| date = 2013
| pages = 7639–7643
| doi = 10.1109/ICASSP.2013.6639149
}}</ref> and identification,<ref name="speakerid2011" /> sampling on manifolds, anomaly detection,<ref name="Mishne" /><ref>{{Cite journal|last1=Shabat|first1=Gil|last2=Segev|first2=David|last3=Averbuch|first3=Amir|date=2018-01-07|title=Uncovering Unknown Unknowns in Financial Services Big Data by Unsupervised Methodologies: Present and Future trends|url=http://proceedings.mlr.press/v71/shabat18a.html|journal=KDD 2017 Workshop on Anomaly Detection in Finance|language=en|volume=71|pages=8–19}}</ref> image inpainting,<ref name="Gepshtein" /> revealing brain resting state networks organization <ref name="Margulies_et_al_2016">{{cite journal▼
| isbn = 978-1-4799-0356-6
▲ }}</ref> and identification,<ref name="speakerid2011" /> sampling on manifolds, anomaly detection,<ref name="Mishne" /><ref>{{Cite journal|last1=Shabat|first1=Gil|last2=Segev|first2=David|last3=Averbuch|first3=Amir|date=2018-01-07|title=Uncovering Unknown Unknowns in Financial Services Big Data by Unsupervised Methodologies: Present and Future trends|url=http://proceedings.mlr.press/v71/shabat18a.html|journal=KDD 2017 Workshop on Anomaly Detection in Finance|language=en|volume=71|pages=8–19}}</ref> image inpainting,<ref name="Gepshtein" /> revealing brain resting state networks organization <ref name="Margulies_et_al_2016">{{cite journal
| last1 = Margulies
| first1 = Daniel S.
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| doi = 10.1073/pnas.1608282113
| pmid = 27791099
| pmc = 5098630
| bibcode = 2016PNAS..11312574M
| doi-access = free
}}</ref> and so on.
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| volume = 21
| pages = 5–30
| doi-access=
| s2cid = 17160669
<ref name="Diffusion">{{cite thesis
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| year = 2013
| title = Multiscale Anomaly Detection Using Diffusion Maps
| journal = IEEE Journal of Selected Topics in Signal Processing
| volume = 7
| issue = 1
| |||