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'''Diffusion maps''' is a [[machine learning]] algorithm introduced by R. R. Coifman and S. Lafon.<ref name="DifussionMap" /><ref name="Diffusion" /> It computes a family of embeddings 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 other dimensionality reduction methods such as [[principal component analysis]] (PCA) and [[multi-dimensional scaling]] (MDS), diffusion maps is a non-linear method that focuses on discovering the underlying [[manifold]] that the data has been sampled from. By integrating local similarities at different scales, diffusion maps gives a global description of the data-set. Compared with other methods, the diffusion maps algorithm is robust to noise perturbation and is computationally inexpensive.
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==Definition of diffusion maps==
Following,<ref name="Introduction" /> diffusion maps can be defined in four steps.
 
===Connectivity===
Diffusion maps leverages the relationship between [[heat diffusion]] and random walk [[Markov chain]]. 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. Based on this, the connectivity between two data points, <math>x</math> and <math>y</math>, can be defined as the probability of walking from <math>x</math> to <math>y</math> in one step of the random walk. Usually, this probability will be defined as the kernel function on the two points, for example, the popular Gaussian kernel:
 
: <math>
p(x,y)=k(x,y)=e^{-\frac{||x-y||^2}{\alpha}}
</math>
 
More generally, the [[Integral kernel|kernel]] function has the following properties
 
: <math>k(x,y) = k(y,x)</math>
(<math>k</math> is symmetric)
 
: <math> k(x,y) \geq 0 \,\,\forall x,y</math>
(<math>k</math> is positivity preserving).
 
The kernel constitutes the prior definition of the local geometry of 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.
 
===Diffusion process===
With the definition of connectivity between two points, we can define the diffusion matrix <math>L</math> (it is also a version of graph Laplacian matrix)
 
: <math>
L_{i,j}=k(x_i,x_j) \,
</math>
 
Many works use the normalized matrix as
 
: <math>
M=D^{-1}L. \,
</math>
 
We  can define the probabilities for taken from <math>i</math> to <math>j</math> in <math>t</math> steps as:
 
: <math>
p(x_j,t|x_i)=M^t_{ij} \,
</math>
 
As we calculate the probabilities<math>M^{t}</math> for increasing values of <math>t</math>, we observe the data-set at different scales. This is the diffusion process. Diffusion process reveals
the global geometric structure of a data set as paths along the true geometric structure of the data set have high probability.
 
===Diffusion distance===
The diffusion distance at time <math>t</math> 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
 
: <math>
D_{t}(x_i,x_j)^2 =\sum_y (p(y,t|x_i)-p(y,t|x_j))^2 w(y)
</math>
 
here <math>w(y)</math> is a weighted function which can often treated as <math>w(y)=1</math>
 
===Diffusion process===
Now, we can give the definition of diffusion map as:
 
: <math>
\Psi_t(x)=(\lambda_1^t\psi_1(x),\lambda_2^t\psi_2(x),\ldots,\lambda_n^t\psi_n(x))
</math>
 
in,<ref name="Nadler05diffusionmaps" /> it is proved that
 
: <math>
D_t(x_i,x_j)^2=||\Psi_t(x_i)-\Psi_t(x_j)||^2 \,
</math>
 
if we take the first ''k'' eigenvectors and eigenvalues, we get the diffusion map from the original data to a ''k''-dimensional space which is embedded in the original space.
 
==Algorithm==
The basic algorithm framework of diffusion map is as:
 
Step 1. Given the similarity matrix ''L''
 
Step 2. Form the normalized matrix <math>M = D^{-1}L</math>
 
Step 3. Compute the ''k'' largest eigenvalues of <math>M^t</math> and the corresponding eigenvectors
 
Step 4. Use diffusion map to get the embedding matrix ''Y''
 
==Application==
In the paper,<ref name="Nadler05diffusionmaps" /> they showed how to design a kernel that reproduces the diffusion induced by a Fokker-Planck equation. Also, they explained that when the data approximate a manifold, then 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 map gives a global description of the data-set, it can measure the distances between pair of sample points in the manifold the data is embedded. Based on diffusion map, there are many applications, such as spectral clustering, low dimensional representation of images, image segmentation,<ref name="Farbman" /> 3D model segmentation,<ref name="sidi11" /> speaker identification,<ref name="speakerid2011" /> sampling on manifolds and so on.
 
==See also==
*[[Nonlinear dimensionality reduction]]
 
==References==
 
{{Reflist|
refs=
 
<ref name="Nadler05diffusionmaps">{{cite journal
|last      = Nadler
|first      = Boaz
|coauthors  = Stéphane Lafon ; Ronald R. Coifman ; Ioannis G. Kevrekidis
|year      = 2005
|title      = Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker–Planck Operators
|url        = http://www.wisdom.weizmann.ac.il/~nadler/Publications/dm_nips05.pdf
|journal    = in Advances in Neural Information Processing Systems 18
}}</ref>
 
<ref name="Farbman">{{cite journal
|last      = Zeev
|first      = Farbman
|coauthors  = Fattal Raanan ;Lischinski Dani
|year      = 2010
|title      = Diffusion maps for edge-aware image editing
|url        = http://doi.acm.org/10.1145/1882261.1866171
|journal    = ACM Trans. Graph
|volume    = 29
|pages      = 145:1–145:10
}}</ref>
 
<ref name="DifussionMap">{{cite journal
|last      = Coifman
|first      =  R.R.
|coauthors  =  S. Lafon.
|year      = 2006
|title      = Diffusion maps
|url        = http://www.pnas.org/content/102/21/7426.long
|journal    = Applied and Computational Harmonic Analysis
}}</ref>
 
<ref name="Diffusion">{{cite journal
|last      = Lafon
|first      = S.S.
|year      = 2004
|title      = Diffusion Maps and Geometric Harmonics
|journal    = PhD thesis, Yale University
}}</ref>
<!-- unused
<ref name="Macrostate">{{cite journal
|last      = Korenblum
|first      = D.
|coauthors  = Shalloway D.
|year      = 2003
|title      = Macrostate Data Clustering
|url        = http://pre.aps.org/abstract/PRE/v67/i5/e056704
|journal    = Physical Review E
|volume    = 67
|page      = 056704
}}</ref>-->
 
<ref name="Introduction">{{cite journal
|last      = J De
|first      = Porte
|coauthors  =  B M; Hereman, W; Walt, S J Van Der
|year      = 2008
|title      = An Introduction to Diffusion Maps
|journal    =Techniques
}}</ref>
 
<ref name="sidi11">{{cite conference
|last      = Oana
|first      = Sidi
|coauthors  = Oliver van Kaick ; Yanir Kleiman ; Hao Zhang ; Daniel Cohen-Or
|year      = 2011
|title      = Unsupervised Co-Segmentation of a Set of Shapes via Descriptor-Space Spectral Clustering
|journal    =ACM Transactions on Graphics
}}</ref>
 
<ref name="speakerid2011">{{cite journal
|last    = Michalevsky
|first  = Yan
|coauthors = Ronen Talmon ; Israel Cohen
|year  = 2011
|title  = Speaker Identification Using Diffusion Maps
|conference = European signal processing conference 2011
|url = http://www.eurasip.org/Proceedings/Eusipco/Eusipco2011/papers/1569414913.pdf
}}</ref>
}}
 
[[Category:Machine learning algorithms]]

Latest revision as of 12:35, 23 September 2014

Bear in mind the film Field of Dreams, 1989? Kevin Costner plays an Iowa farmer whose dream is to be a baseball player, anything his father wanted for h...

Youve constructed a internet site and are extremely proud to be counted amongst the ranks. Now youre expecting individuals to come for a visit. In truth if this is your 1st internet site youre most likely pondering possibly Ill get a few guests or possibly a thousand guests. Would it surprise you if I told you that you are going to get quite handful of guests?

Keep in mind the movie Field of Dreams, 1989? Kevin Costner plays an Iowa farmer whose dream is to be a baseball player, anything his father wanted for his personal life. In the movie he hears voices from players that have passed on. What he hears becomes an aphorism nevertheless used today, if you construct it he (they) will come. To compare additional info, consider peeping at: [http://www.purevolume.com/romanticsleepfsa/posts/7839363/How+To+Improve+The+Benefits+Of+Green+Tea+ PureVolume� | We"re Listening To You]. Which means if he built the ball field his father, shoeless Joe and other baseball players from Chicago would come and he would get his likelihood. Nevertheless this is not the case with creating a site. However know 1 knows youre there and you wont hear any voices telling exactly where to go from here. Sooner or later the search engines will pick it up and you will get a couple of visitors that will trickle in but how to get the thousands to come is the question.

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Yes its time to get out the paper and pen like you did in college. Folks want information. They want to understand something new. They want to really feel they have some inside data that no a single else has. This is the information age, are you going to give them what they want?

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