SherbyNet - /page/wikibook/entry.mu

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Diffusion map

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Diffusion maps is a <a href="Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a> or <a href="Feature_extraction" class="mw-redirect" title="Feature extraction">feature extraction</a> algorithm introduced by <a href="Ronald_Coifman" title="Ronald Coifman">Coifman</a> and Lafon<a href="#cite_note-PNAS1-1">[1]</a><a href="#cite_note-PNAS2-2">[2]</a><a href="#cite_note-DifussionMap-3">[3]</a><a href="#cite_note-Diffusion-4">[4]</a> which computes a family of <a href="Embedding" title="Embedding">embeddings</a> 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 <a href="Principal_component_analysis" title="Principal component analysis">principal component analysis</a> (PCA), diffusion maps are part of the family of <a href="Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">nonlinear dimensionality reduction</a> methods which focus on discovering the underlying <a href="Manifold" title="Manifold">manifold</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Definition_of_diffusion_maps">Definition of diffusion maps</h2></div>

Following <a href="#cite_note-DifussionMap-3">[3]</a> and,<a href="#cite_note-Introduction-5">[5]</a> diffusion maps can be defined in four steps.

<div class="mw-heading mw-heading3"><h3 id="Connectivity">Connectivity</h3></div>

Diffusion maps exploit the relationship between <a href="Heat_diffusion" class="mw-redirect" title="Heat diffusion">heat diffusion</a> and <a href="Random_walk" title="Random walk">random walk</a> <a href="Markov_chain" title="Markov chain">Markov chain</a>. The basic observation is that if we take a random walk on the data, walking to a nearby data-point is more likely than walking to another that is far away. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,{\mathcal {A}},\mu )}">

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<annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {A}},\mu )}</annotation>

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</math><img src="./d634d210e57700027029694595ffea10410bf0d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.162ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {A}},\mu )}" loading="lazy"> be a <a href="Measure_space" title="Measure space">measure space</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the data set and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> represents the distribution of the points on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

Based on this, the connectivity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> between two data points, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, can be defined as the probability of walking from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> in one step of the random walk. Usually, this probability is specified in terms of a kernel function of the two points: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k:X\times X\rightarrow \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle k:X\times X\rightarrow \mathbb {R} }</annotation>

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</math><img src="./55543ce2ce31286eec495a0db63a421878573892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.241ex; height:2.176ex;" alt="{\displaystyle k:X\times X\rightarrow \mathbb {R} }" loading="lazy">. For example, the popular Gaussian kernel:

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<annotation encoding="application/x-tex">{\displaystyle k(x,y)=\exp \left(-{\frac {||x-y||^{2}}{\epsilon }}\right)}</annotation>

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</math><img src="./18dd439e1017a27044d07396aad77efe3a409ec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.483ex; height:7.509ex;" alt="{\displaystyle k(x,y)=\exp \left(-{\frac {||x-y||^{2}}{\epsilon }}\right)}" loading="lazy"></dd></dl>

More generally, the <a href="Integral_kernel" class="mw-redirect" title="Integral kernel">kernel</a> function has the following properties

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<annotation encoding="application/x-tex">{\displaystyle k(x,y)=k(y,x)}</annotation>

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</math><img src="./acc493aa2beef833f489fcd07d87cbcdfb39815e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.178ex; height:2.843ex;" alt="{\displaystyle k(x,y)=k(y,x)}" loading="lazy"></dd></dl>

(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is symmetric)

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<annotation encoding="application/x-tex">{\displaystyle k(x,y)\geq 0\,\,\forall x,y}</annotation>

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</math><img src="./d101c3ec8440ac105fa36bc4883230da1f2e5bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.386ex; height:2.843ex;" alt="{\displaystyle k(x,y)\geq 0\,\,\forall x,y}" loading="lazy"></dd></dl>

(<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is positivity preserving).

The kernel constitutes the prior definition of the local geometry of the data-set. Since a given kernel will capture a specific feature of the data set, its choice should be guided by the application that one has in mind. This is a major difference with methods such as <a href="Principal_component_analysis" title="Principal component analysis">principal component analysis</a>, where correlations between all data points are taken into account at once.

Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,k)}">

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<annotation encoding="application/x-tex">{\displaystyle (X,k)}</annotation>

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</math><img src="./6b7b2d2d9ac226ee07d76c669118de9a7c23e861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.034ex; height:2.843ex;" alt="{\displaystyle (X,k)}" loading="lazy">, we can then construct a reversible discrete-time Markov chain on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (a process known as the normalized graph Laplacian construction):

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<annotation encoding="application/x-tex">{\displaystyle d(x)=\int _{X}k(x,y)d\mu (y)}</annotation>

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</math><img src="./e60fa5bc1e2b92a3dcb268e98b6d2c42d2c6536b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.887ex; height:5.676ex;" alt="{\displaystyle d(x)=\int _{X}k(x,y)d\mu (y)}" loading="lazy"></dd></dl>

and define:

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<annotation encoding="application/x-tex">{\displaystyle p(x,y)={\frac {k(x,y)}{d(x)}}}</annotation>

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</math><img src="./ee36277439ae231e246aeb07f12f1e24187e2938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:17.061ex; height:6.509ex;" alt="{\displaystyle p(x,y)={\frac {k(x,y)}{d(x)}}}" loading="lazy"></dd></dl>

Although the new normalized kernel does not inherit the symmetric property, it does inherit the positivity-preserving property and gains a conservation property:

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<annotation encoding="application/x-tex">{\displaystyle \int _{X}p(x,y)d\mu (y)=1}</annotation>

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</math><img src="./e815da22e5ab8aecf85ef6fb3240c5668b17c79a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.653ex; height:5.676ex;" alt="{\displaystyle \int _{X}p(x,y)d\mu (y)=1}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Diffusion_process">Diffusion process</h3></div>

From <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle p(x,y)}</annotation>

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</math><img src="./089e91a1824e14cebc8e8d04dc652c61b3008e0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle p(x,y)}" loading="lazy"> we can construct a transition matrix of a Markov chain (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">) on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle p(x,y)}</annotation>

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</math><img src="./089e91a1824e14cebc8e8d04dc652c61b3008e0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle p(x,y)}" loading="lazy"> represents the one-step transition probability from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{t}}">

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<annotation encoding="application/x-tex">{\displaystyle M^{t}}</annotation>

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</math><img src="./afc143312b1a3f60d76a503ed60cf2cedd026e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.325ex; height:2.509ex;" alt="{\displaystyle M^{t}}" loading="lazy"> gives the t-step transition matrix.

We define the diffusion matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> (it is also a version of graph <a href="Laplacian_matrix" title="Laplacian matrix">Laplacian matrix</a>)

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<annotation encoding="application/x-tex">{\displaystyle L_{i,j}=k(x_{i},x_{j})\,}</annotation>

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</math><img src="./d11063e105d2d75b02aa4f5a3fb04f3607c47bdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.426ex; height:3.009ex;" alt="{\displaystyle L_{i,j}=k(x_{i},x_{j})\,}" loading="lazy"></dd></dl>

We then define the new kernel

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<annotation encoding="application/x-tex">{\displaystyle L_{i,j}^{(\alpha )}=k^{(\alpha )}(x_{i},x_{j})={\frac {L_{i,j}}{(d(x_{i})d(x_{j}))^{\alpha }}}\,}</annotation>

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</math><img src="./dad61f2b8b82dd6e943f07e2acb0175786a03487.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.066ex; height:6.343ex;" alt="{\displaystyle L_{i,j}^{(\alpha )}=k^{(\alpha )}(x_{i},x_{j})={\frac {L_{i,j}}{(d(x_{i})d(x_{j}))^{\alpha }}}\,}" loading="lazy"></dd></dl>

or equivalently,

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<annotation encoding="application/x-tex">{\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }\,}</annotation>

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</math><img src="./8f771e6b9403ec1139cc640c747b948b7d7e5d25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.189ex; height:2.843ex;" alt="{\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }\,}" loading="lazy"></dd></dl>

where D is a diagonal matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i,i}=\sum _{j}L_{i,j}.}">

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<annotation encoding="application/x-tex">{\displaystyle D_{i,i}=\sum _{j}L_{i,j}.}</annotation>

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</math><img src="./4a3ad8c79782989841ec1f1ffeb0530b70351e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.753ex; height:5.843ex;" alt="{\displaystyle D_{i,i}=\sum _{j}L_{i,j}.}" loading="lazy">

We apply the graph Laplacian normalization to this new kernel:

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<annotation encoding="application/x-tex">{\displaystyle M=({D}^{(\alpha )})^{-1}L^{(\alpha )},\,}</annotation>

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</math><img src="./b6a0fea0b7a6812c8be803e2fc4e5e4a4a316880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.351ex; height:3.343ex;" alt="{\displaystyle M=({D}^{(\alpha )})^{-1}L^{(\alpha )},\,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{(\alpha )}}">

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<annotation encoding="application/x-tex">{\displaystyle D^{(\alpha )}}</annotation>

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</math><img src="./5a668379806f10821865cc874d3419df5533c4fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.488ex; height:2.843ex;" alt="{\displaystyle D^{(\alpha )}}" loading="lazy"> is a diagonal matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {D}_{i,i}^{(\alpha )}=\sum _{j}L_{i,j}^{(\alpha )}.}">

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<annotation encoding="application/x-tex">{\displaystyle {D}_{i,i}^{(\alpha )}=\sum _{j}L_{i,j}^{(\alpha )}.}</annotation>

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</math><img src="./56589dd7ec6249c8e47c1781c907d134c49a6d7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.122ex; height:6.009ex;" alt="{\displaystyle {D}_{i,i}^{(\alpha )}=\sum _{j}L_{i,j}^{(\alpha )}.}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle p(x_{j},t|x_{i})=M_{i,j}^{t}\,}</annotation>

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</math><img src="./2e0a7d56f201cb1bc2eaab3870428ef062075128.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:17.631ex; height:3.509ex;" alt="{\displaystyle p(x_{j},t|x_{i})=M_{i,j}^{t}\,}" loading="lazy"></dd></dl>

One of the main ideas of the diffusion framework is that running the chain forward in time (taking larger and larger powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">) reveals the geometric structure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> at larger and larger scales (the diffusion process). Specifically, the notion of a cluster in the data set is quantified as a region in which the probability of escaping this region is low (within a certain time t). Therefore, t not only serves as a time parameter, but it also has the dual role of scale parameter.

The eigendecomposition of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{t}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle M^{t}}</annotation>

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</math><img src="./afc143312b1a3f60d76a503ed60cf2cedd026e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.325ex; height:2.509ex;" alt="{\displaystyle M^{t}}" loading="lazy"> yields

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle M_{i,j}^{t}=\sum _{l}\lambda _{l}^{t}\psi _{l}(x_{i})\phi _{l}(x_{j})\,}</annotation>

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</math><img src="./e85da55908c09c56ff6daba812cf9852580f9423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.928ex; height:5.509ex;" alt="{\displaystyle M_{i,j}^{t}=\sum _{l}\lambda _{l}^{t}\psi _{l}(x_{i})\phi _{l}(x_{j})\,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{l}\}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \{\lambda _{l}\}}</annotation>

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</math><img src="./27f7b81cabd2c04912a11d5de3911be2d5806d6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.403ex; height:2.843ex;" alt="{\displaystyle \{\lambda _{l}\}}" loading="lazy"> is the sequence of eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\psi _{l}\}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \{\psi _{l}\}}</annotation>

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</math><img src="./80de2384f8bfac74911bd4f03b91726390f1e62b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.561ex; height:2.843ex;" alt="{\displaystyle \{\psi _{l}\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\phi _{l}\}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \{\phi _{l}\}}</annotation>

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</math><img src="./7f567ca4e5048d126a2bbd3dfac91f2b2ba58ed3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.433ex; height:2.843ex;" alt="{\displaystyle \{\phi _{l}\}}" loading="lazy"> are the biorthogonal left and right 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.

<div class="mw-heading mw-heading4"><h4 id="Parameter_α_and_the_diffusion_operator">Parameter α and the diffusion operator</h4></div>

The reason to introduce the normalization step involving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =1}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =1}</annotation>

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</math><img src="./03d67a45a44be8b8f15e99b7def2b0cf0aba1717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =1}" loading="lazy"> and the diffusion operator approximates the <a href="Laplace%E2%80%93Beltrami_operator" title="Laplace–Beltrami operator">Laplace–Beltrami operator</a>. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0.5}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =0.5}</annotation>

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</math><img src="./d1152b74ec1d0be1ceabb8d63499e80ed2d2479f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.558ex; height:2.176ex;" alt="{\displaystyle \alpha =0.5}" loading="lazy"> and the resulting Markov chain approximates the <a href="Fokker%E2%80%93Planck_equation" title="Fokker–Planck equation">Fokker–Planck diffusion</a>. With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation>

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</math><img src="./30cc00f65bbc630448311dd2dc82e7ce5e90985a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}" loading="lazy">, it reduces to the classical graph Laplacian normalization.

<div class="mw-heading mw-heading3"><h3 id="Diffusion_distance">Diffusion distance</h3></div>

The diffusion distance at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> between two points can be measured as the similarity of two points in the observation space with the connectivity between them. It is given by

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<annotation encoding="application/x-tex">{\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{y}{\frac {(p(y,t|x_{i})-p(y,t|x_{j}))^{2}}{\phi _{0}(y)}}}</annotation>

</semantics>

</math><img src="./30edf6c1d5b8a9bd1d51fc76ded107573c7530f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.075ex; height:7.343ex;" alt="{\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{y}{\frac {(p(y,t|x_{i})-p(y,t|x_{j}))^{2}}{\phi _{0}(y)}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{0}(y)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{0}(y)}</annotation>

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</math><img src="./b69c32be50f6198aa238ece8eff6b853efabf476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.404ex; height:2.843ex;" alt="{\displaystyle \phi _{0}(y)}" loading="lazy"> is the stationary distribution of the Markov chain, given by the first left eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Explicitly:

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<annotation encoding="application/x-tex">{\displaystyle \phi _{0}(y)={\frac {d(y)}{\sum _{z\in X}d(z)}}}</annotation>

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</math><img src="./86fa5191959fb997c02722e324a9097e33e497bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.791ex; height:6.676ex;" alt="{\displaystyle \phi _{0}(y)={\frac {d(y)}{\sum _{z\in X}d(z)}}}" loading="lazy"></dd></dl>

Intuitively, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{t}(x_{i},x_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle D_{t}(x_{i},x_{j})}</annotation>

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</math><img src="./3c88ef5122be9e37f69ad5af925c346ebe59b0c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.962ex; height:3.009ex;" alt="{\displaystyle D_{t}(x_{i},x_{j})}" loading="lazy"> is small if there is a large number of short paths connecting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>

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</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy">. There are several interesting features associated with the diffusion distance, based on our previous discussion that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> also serves as a scale parameter:

<ol><li>Points are closer at a given scale (as specified by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{t}(x_{i},x_{j})}">

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<annotation encoding="application/x-tex">{\displaystyle D_{t}(x_{i},x_{j})}</annotation>

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</math><img src="./3c88ef5122be9e37f69ad5af925c346ebe59b0c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.962ex; height:3.009ex;" alt="{\displaystyle D_{t}(x_{i},x_{j})}" loading="lazy">) if they are highly connected in the graph, therefore emphasizing the concept of a cluster.</li>

<li>This distance is robust to noise, since the distance between two points depends on all possible paths of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> between the points.</li>

<li>From a machine learning point of view, the distance takes into account all evidences linking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>

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</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy">, allowing us to conclude that this distance is appropriate for the design of inference algorithms based on the majority of preponderance.<a href="#cite_note-DifussionMap-3">[3]</a></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Diffusion_process_and_low-dimensional_embedding">Diffusion process and low-dimensional embedding</h3></div>

The diffusion distance can be calculated using the eigenvectors by

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<annotation encoding="application/x-tex">{\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{l}\lambda _{l}^{2t}(\psi _{l}(x_{i})-\psi _{l}(x_{j}))^{2}\,}</annotation>

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</math><img src="./0af9d57bc7f554583d2c727fecf5b7fab8167f02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.41ex; height:5.509ex;" alt="{\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{l}\lambda _{l}^{2t}(\psi _{l}(x_{i})-\psi _{l}(x_{j}))^{2}\,}" loading="lazy"></dd></dl>

So the eigenvectors can be used as a new set of coordinates for the data. The diffusion map is defined as:

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{t}(x)=(\lambda _{1}^{t}\psi _{1}(x),\lambda _{2}^{t}\psi _{2}(x),\ldots ,\lambda _{k}^{t}\psi _{k}(x))}</annotation>

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</math><img src="./80a7d21cc151e690051a42bda5fcaa060c2f2ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.31ex; height:3.176ex;" alt="{\displaystyle \Psi _{t}(x)=(\lambda _{1}^{t}\psi _{1}(x),\lambda _{2}^{t}\psi _{2}(x),\ldots ,\lambda _{k}^{t}\psi _{k}(x))}" loading="lazy"></dd></dl>

Because of the spectrum decay, it is sufficient to use only the first k eigenvectors and eigenvalues.

Thus we get the diffusion map from the original data to a k-dimensional space which is embedded in the original space.

In <a href="#cite_note-Nadler05diffusionmaps-6">[6]</a> it is proved that

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<annotation encoding="application/x-tex">{\displaystyle D_{t}(x_{i},x_{j})^{2}\approx ||\Psi _{t}(x_{i})-\Psi _{t}(x_{j})||^{2}\,}</annotation>

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</math><img src="./857072cfff9cfab24af9adbb10b4255d8f432927.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.24ex; height:3.509ex;" alt="{\displaystyle D_{t}(x_{i},x_{j})^{2}\approx ||\Psi _{t}(x_{i})-\Psi _{t}(x_{j})||^{2}\,}" loading="lazy"></dd></dl>

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

<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

The basic algorithm framework of diffusion map is as:

Step 1. Given the similarity matrix L.

Step 2. Normalize the matrix according to parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }}</annotation>

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</math><img src="./057c385558ddc2d87c55da1c191b3211782dd49f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.802ex; height:2.843ex;" alt="{\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }}" loading="lazy">.

Step 3. Form the normalized matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=({D}^{(\alpha )})^{-1}L^{(\alpha )}}">

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<annotation encoding="application/x-tex">{\displaystyle M=({D}^{(\alpha )})^{-1}L^{(\alpha )}}</annotation>

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</math><img src="./386deb545917b28a3167b629b954720ffb4f4d00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.317ex; height:3.343ex;" alt="{\displaystyle M=({D}^{(\alpha )})^{-1}L^{(\alpha )}}" loading="lazy">.

Step 4. Compute the k largest eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{t}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M^{t}}</annotation>

</semantics>

</math><img src="./afc143312b1a3f60d76a503ed60cf2cedd026e0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.325ex; height:2.509ex;" alt="{\displaystyle M^{t}}" loading="lazy"> and the corresponding eigenvectors.

Step 5. Use diffusion map to get the embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi _{t}}</annotation>

</semantics>

</math><img src="./5a87f0d64f2e13593293205ba5daa798d55ec592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.634ex; height:2.509ex;" alt="{\displaystyle \Psi _{t}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Application">Application</h2></div>

In the paper <a href="#cite_note-Nadler05diffusionmaps-6">[6]</a> Nadler et al. showed how to design a kernel that reproduces the diffusion induced by a <a href="Fokker%E2%80%93Planck_equation" title="Fokker–Planck equation">Fokker–Planck equation</a>. They also explained that, when the data approximate a manifold, one can recover the geometry of this manifold by computing an approximation of the <a href="Laplace%E2%80%93Beltrami_operator" title="Laplace–Beltrami operator">Laplace–Beltrami operator</a>. 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 <a href="Facial_recognition_system" title="Facial recognition system">face recognition</a>,<a href="#cite_note-vmrs-7">[7]</a> <a href="Spectral_clustering" title="Spectral clustering">spectral clustering</a>, low dimensional representation of images, image segmentation,<a href="#cite_note-Farbman-8">[8]</a> 3D model segmentation,<a href="#cite_note-sidi11-9">[9]</a> speaker verification<a href="#cite_note-spk-10">[10]</a> and identification,<a href="#cite_note-speakerid2011-11">[11]</a> sampling on manifolds, anomaly detection,<a href="#cite_note-Mishne-12">[12]</a><a href="#cite_note-13">[13]</a> image inpainting,<a href="#cite_note-Gepshtein-14">[14]</a> revealing brain resting state networks organization <a href="#cite_note-Margulies_et_al_2016-15">[15]</a> and so on.

Furthermore, the diffusion maps framework has been productively extended to <a href="Complex_networks" class="mw-redirect" title="Complex networks">complex networks</a>,<a href="#cite_note-16">[16]</a> revealing a functional organisation of networks which differs from the purely topological or structural one.

<ul><li><a href="Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">Nonlinear dimensionality reduction</a></li>

<li><a href="Spectral_clustering" title="Spectral clustering">Spectral clustering</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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