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| '''Diffusion wavelets''' are a fast multiscale framework for the analysis of functions on discrete (or discretized continuous) structures like graphs, manifolds, and point clouds in Euclidean space. Diffusion wavelets are an extension of classical [[Wavelet|wavelet theory]] from [[harmonic analysis]]. Unlike classical wavelets whose basis functions are predetermined, diffusion wavelets are adapted to the geometry of a given diffusion operator <math>T</math> (e.g., a [[heat kernel]] or a [[random walk]]). Moreover, the diffusion wavelet basis functions are constructed by dilation using the dyadic powers (powers of two) of <math>T</math>. These dyadic powers of <math>T</math> diffusion over the space and propagate local relationships in the function throughout the space until they become global. And if the rank of higher powers of <math>T</math> decrease (i.e., its spectrum decays), then these higher powers become compressible. From these decaying dyadic powers of <math>T</math> comes a chain of decreasing subspaces. These subspaces are the [[Wavelet#Scaling_function|scaling function]] approximation subspaces, and the differences in the subspace chain are the wavelet subspaces. | | I'm Blanche and I live in Scott Creek. <br>I'm interested in Anthropology and Sociology, Board sports and Portuguese art. I like travelling and watching How I Met Your Mother.<br><br>My homepage ... pièces auto réunion; [http://nadejda080878.blog.com/2014/09/28/restoration-the-issues-together-with-your-car-properly/ visit the next document], |
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| Diffusion wavelets were first introduced in 2004 by [[Ronald Coifman]] and Mauro Maggioni at Yale University.<ref>{{cite journal|last=Coifman|first=Ronald|coauthors=Mauro Maggioni|title=Diffusion Wavelets|journal=Applied and Computational Harmonic Analysis|date=May 2008|volume=24|issue=3|pages=329–353|url=http://www.math.duke.edu/~mauro/Papers/DiffusionWavelets.pdf}}</ref>
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| == Algorithm ==
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| This algorithm constructs the scaling basis functions and the wavelet basis functions along with the representations of the diffusion operator <math>T</math> at these scales.
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| In the algorithm below, the subscript notation <math>\Phi_a</math> and <math>\Psi_b</math> represents the scaling basis functions at scale <math>a</math> and the wavelet basis functions at scale <math>b</math> respectively. The notation <math>[\Phi_b]_{\Phi_a}</math> denotes the matrix representation of the scaling basis <math>\Phi_b</math> represented with respect to the basis <math>\Phi_a</math>. Lastly, the notation <math>[T]_{\Phi_a}^{\Phi_b}</math> denotes the matrix represents of the operator <math>T</math>, where the [[row space]] of <math>T</math> is represented with respect to the basis <math>\Phi_a</math>, and the [[column space]] of <math>T</math> is represented with respect to the basis <math>\Phi_b</math>. Otherwise put, the domain of operator <math>T</math> is represented with respect to the basis <math>\Phi_a</math> and the range is represented with respect to the basis <math>\Phi_b</math>. The function <math>QR</math> is a sparse [[QR decomposition]] with <math>\epsilon</math> precision.<ref>{{cite conference|last=Maggioni|first=Mauro|coauthors=Mahadevan, Sridhar|title=Fast Direct Policy Evaluation using Multiscale Analysis of Markov Diffusion Processes|conference=The 23rd International Conference on Machine Learning|year=2006|url=http://www.cs.umass.edu/~mahadeva/papers/icml2006.pdf}}</ref>
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| <code>
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| // Input:
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| // <math>T</math> is the matrix representation of the diffusion operator.
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| // <math>\epsilon</math> is the precision of the QR decomposition, e.g., 1e-6.
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| // <math>J</math> is the maximum number of scale levels (note: this is an <em>optional</em> upper bound, it may converge sooner.)
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| // Output:
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| // <math>\lbrace\Phi_j\rbrace</math> is the set of scaling basis functions indexed by scale <math>j</math>.
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| // <math>\lbrace\Psi_j\rbrace</math> is the set of wavelet basis functions indexed by scale <math>j</math>.
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| <math>\lbrace\Phi_j\rbrace, \lbrace\Psi_j\rbrace \leftarrow \text{function DiffusionWaveletTree} ( T , \epsilon , J ):</math>
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| <math>\textbf{for } j\leftarrow 0 \text{ to } J-1</math>:
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| <math>[\Phi_{j+1}]_{\Phi_j}, [T^{2^j}]_{\Phi_j}^{\Phi_{j+1}} \leftarrow QR\left([T^{2^j}]_{\Phi_j}^{\Phi_{j}}, \epsilon\right)</math>
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| <math>[T^{2^{j+1}}]_{\Phi_{j+1}}^{\Phi_{j+1}} \leftarrow \left([T^{2^j}]_{\Phi_j}^{\Phi_{j+1}} [\Phi_{j+1}]_{\Phi_j}\right)^2</math>
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| <math>[\Psi_j]_{\Phi_j} \leftarrow QR\left(I_{\langle\Phi_j\rangle}-[\Phi_{j+1}]_{\Phi_j}\left([\Phi_{j+1}]_{\Phi_j}\right)^*, \epsilon\right)</math>
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| <math>\textbf{end for}</math>
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| </code>
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| == Applications ==
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| ===Mathematics===
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| Diffusion wavelets are of general interest in mathematics. Specifically, they allow for the direct calculation of the [[Green′s function]] and the inverse [[Laplacian matrix|graph Laplacian]].
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| ===Computer science===
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| Diffusion wavelets have been used extensively in computer science, especially in machine learning. They have been applied to the follow fields:
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| * solving [[Markov decision process]]es and [[Markov chains]] for machine learning,<ref>{{cite journal|last=Mahadevan|first=Sridhar|title=Learning Representation and Control in Markov Decision Processes|journal=Foundations and Trends in Machine Learning|year=2008|volume=1|issue=4}}</ref>
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| * [[transfer learning]],<ref>{{cite journal|last=Wang|first=Chang|coauthors=Mahadevan, Sridhar|title=Multiscale Manifold Alignment|journal=Univ. of Massachusetts Technical Report|year=2010|issue=UM-CS-2010-049|url=http://www.cs.umass.edu/~mahadeva/papers/UM-CS-2010-049.pdf}}</ref>
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| * value function approximation in [[reinforcement learning]],<ref>{{cite journal|last=Mahadevan|first=Sridhar|coauthors=Maggioni, Mauro|title=Value Function Approximation using Diffusion Wavelets and Laplacian Eigenfunctions|journal=Advances in Neural Information Processing Systems|year=2006|url=http://www.cs.umass.edu/~mahadeva/papers/nips-paper1-v5.pdf}}</ref>
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| * [[dimensionality reduction]],<ref>{{cite journal|last=Wang|first=Chang|coauthors=Mahadevan, Sridhar|title=Multiscale Dimensionality Reduction with Diffusion Wavelets|journal=Univ. of Massachusetts Technical Report|year=2009|issue=UM-CS-2009-030|url=http://www.cs.umass.edu/~mahadeva/papers/TR-2009-DP.pdf}}</ref>
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| * mesh compression for 3D graphics,<ref>{{cite conference|last=Mahadevan|first=Sridhar|title=Adaptive Mesh Compression in 3D Computer Graphics using Multiresolution Manifold Learning|conference=The 24th International Conference on Machine Learning|year=2007|url=http://www.cs.umass.edu/~mahadeva/papers/sridhar-icml07.pdf}}</ref>
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| * [[topic model]] analysis of document corpora.<ref>{{cite conference|last=Wang|first=Chang|coauthors=Mahadevan, Sridhar|title=Multiscale Analysis of Document Corpora Based on Diffusion Models|conference=The 21st International Joint Conference on Artificial Intelligence|year=2009|url=http://www.cs.umass.edu/~chwang/papers/IJCAI-2009-TD.pdf}}</ref>
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| * relation extraction. <ref>{{cite conference|last=Wang|first=Chang|coauthors=James Fan, Aditya A. Kalyanpur, David Gondek|title=Relation Extraction with Relation Topics|conference=The 2011 Conference on Empirical Methods in Natural Language Processing|year=2011|url=http://acl.eldoc.ub.rug.nl/mirror/D/D11/D11-1132.pdf}}</ref>
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| == See also ==
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| * [[Wavelets]]
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| ==References==
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| <references/>
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| ==External links==
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| *[http://www.math.duke.edu/~mauro/code.html Mauro Maggioni's MATLAB code implementation]
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| *[http://www.cs.umass.edu/~chwang/dwt.html Chang Wang's diffusion wavelet page]
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| [[Category:Wavelets]]
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I'm Blanche and I live in Scott Creek.
I'm interested in Anthropology and Sociology, Board sports and Portuguese art. I like travelling and watching How I Met Your Mother.
My homepage ... pièces auto réunion; visit the next document,