Diffusion map: Difference between revisions

Revision as of 06:46, 1 October 2021

Latest revision as of 16:25, 13 June 2025

(15 intermediate revisions by 12 users not shown)

Line 1:

{{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.]]

'''Diffusion maps''' is a [[dimensionality reduction]] or [[feature extraction]] algorithm introduced by [[Ronald Coifman| Coifman]] and Lafon<ref name="PNAS1" /><ref name="PNAS2" /><ref name="DifussionMap" /><ref name="Diffusion" /> which computes a family of [[Embedding|embeddingsembedding]]s of a data set into Euclidean space (often low-dimensional) whose coordinates can be computed from the eigenvectors and eigenvalues of a diffusion operator on the data. The Euclidean distance between points in the embedded space is equal to the "diffusion distance" between probability distributions centered at those points. Different from linear dimensionality reduction methods such as [[principal component analysis]] (PCA), diffusion maps isare part of the family of [[nonlinear dimensionality reduction]] methods which focus on discovering the underlying [[manifold]] that the data has been sampled from. By integrating local similarities at different scales, diffusion maps give a global description of the data-set. Compared with other methods, the diffusion map algorithm is robust to noise perturbation and computationally inexpensive.

==Definition of diffusion maps==

Line 82 ⟶ 83:

</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 rightleft and leftright eigenvectors respectively.

Due to the spectrum decay of the eigenvalues, only a few terms are necessary to achieve a given relative accuracy in this sum.

====Parameter <math>\alpha</math>α and the Diffusiondiffusion Operatoroperator====

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===

Line 123 ⟶ 124:

: <math>

D_t(x_i,x_j)^2= \approx||\Psi_t(x_i)-\Psi_t(x_j)||^2 \,

</math>

so the Euclidean distance in the diffusion coordinates approximates the diffusion distance.

Line 141 ⟶ 142:

==Application==

In the paper <ref name="Nadler05diffusionmaps" /> Nadler et. al. showed how to design a kernel that reproduces the diffusion induced by a [[Fokker–Planck equation]]. They also explained that, when the data approximate a manifold, one can recover the geometry of this manifold by computing an approximation of the [[Laplace–Beltrami operator]]. This computation is completely insensitive

to the distribution of the points and therefore provides a separation of the statistics and the geometry of the

data. Since diffusion maps give a global description of the data-set, they can measure the distances between pairs of sample points in the manifold in which the data is embedded. Applications based on diffusion maps include [[facial recognition system|face recognition]],<ref name="vmrs">{{cite journal

Line 156 ⟶ 157:

| 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 journalbook

| last1 = Barkan

| first1 = Oren

| last2 = Aronowitz

| first2 = Hagai

| journal title = Proceedings of the2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

| chapter-url = http://smartfp7.eu/sites/default/files/field/files/page/DIFFUSION%20MAPS%20FOR%20PLDA-BASED%20SPEAKER%20VERIFICATION.pdf

| titlechapter = Diffusion maps for PLDA-based speaker verification

| 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">https://www.pnas.org/content/113/44/12574.short</ref> and so on.

| 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">https://www.pnas.org/content/113/44/12574.short</ref>{{cite and so on.journal

| last1 = Margulies

| first1 = Daniel S.

| last2 = Ghosh

| first2 = Satrajit S.

| last3 = Goulas

| first3 = Alexandros

| last4 = Falkiewicz

| first4 = Marcel

| last5 = Huntenburg

| first5 = Julia M.

| last6 = Langs

| first6 = Georg

| last7 = Bezgin

| first7 = Gleb

| last8 = Eickhoff

| first8 = Simon B.

| last9 = Castellanos

| first9 = F. Xavier

| last10 = Petrides

| first10 = Michael

| last11 = Jefferies

| first11 = Elizabeth

| last12 = Smallwood

| first12 = Jonathan

| title = Situating the default-mode network along a principal gradient of macroscale cortical organization

| journal = Proceedings of the National Academy of Sciences

| pages = 12574–12579

| volume = 113

| issue = 44

| year = 2016

| doi = 10.1073/pnas.1608282113

| pmid = 27791099

| pmc = 5098630

| bibcode = 2016PNAS..11312574M

| doi-access = free

}}</ref> and so on.

Furthermore, the diffusion maps framework has been productively extended to [[complex networks]],<ref>{{cite journal

| last1 = De Domenico

| first1 = Manlio

| url = https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.168301

| title = Diffusion geometry unravels the emergence of functional clusters in collective phenomena

| journal = Physical Review Letters

| volume = 118

| issue = 16

| pages = 168301

| year = 2017

| doi = 10.1103/PhysRevLett.118.168301

| pmid = 28474920

| arxiv = 1704.07068

| bibcode = 2017PhRvL.118p8301D

| s2cid = 2638868

}}</ref> revealing a functional organisation of networks which differs from the purely topological or structural one.

==See also==

Line 260 ⟶ 316:

| volume = 21

| pages = 5–30

| doi-access= free

| s2cid = 17160669

}}</ref>

<ref name="Diffusion">{{cite thesis

Line 338 ⟶ 395:

| year = 2013

| title = Multiscale Anomaly Detection Using Diffusion Maps

| journal = IEEE Journal of Selected Topics in Signal Processing

| volume = 7

| issue = 1