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Computing Diffusion State Distance using Green's Function and Heat Kernel on Graphs [article]

Edward Boehnlein, Peter Chin, Amit Sinha, Linyuan Lu
2014 arXiv   pre-print
The diffusion state distance (DSD) was introduced by Cao-Zhang-Park-Daniels-Crovella-Cowen-Hescott [ PLoS ONE, 2013] to capture functional similarity in protein-protein interaction networks.  ...  Based on that, we computed the DSD L_q-distance for Paths, Cycles, Hypercubes, as well as random graphs G(n,p) and G(w_1,..., w_n). We also examined the DSD distances of two biological networks.  ...  It is closely related to the Heat kernel of the graphs (see also [15] ) and the normalized Laplacian. In this paper, we will use Green's function to compute the DSD L q -distance for various graphs.  ... 
arXiv:1410.3168v1 fatcat:vdh5u637wvgvtdtezkwtwlf64a

Robust Multi-body Motion Tracking Using Commute Time Clustering [chapter]

Huaijun Qiu, Edwin R. Hancock
2006 Lecture Notes in Computer Science  
The commute time (i.e. the expected time taken for a random walk to travel between two nodes and return) can be computed from the Laplacian spectrum using the discrete Green's function, and is an important  ...  Our embedding procedure preserves commute time, and is closely akin to kernel PCA, the Laplacian eigenmap and the diffusion map.  ...  Acknowledgements The authors would like to thank João Costeira and Jared Jacobs for generously providing their data and code for this work.  ... 
doi:10.1007/11744023_13 fatcat:gkrxxym4ybdphjcrjctwg535om

Clustering and Embedding Using Commute Times

Huaijun Qiu, Edwin R. Hancock
2007 IEEE Transactions on Pattern Analysis and Machine Intelligence  
spectrum using the discrete Green's function.  ...  Our starting point is the lazy random walk on the graph, which is determined by the heatkernel of the graph and can be computed from the spectrum of the graph Laplacian.  ...  From Equ. 2, the Green's function is an integral of the heat kernel and which can be used to compute the diffusion distance.  ... 
doi:10.1109/tpami.2007.1103 pmid:17848771 fatcat:ipjnswb4znhbxkwct23iycgztu

A kernel view of the dimensionality reduction of manifolds

Jihun Ham, Daniel D. Lee, Sebastian Mika, Bernhard Schölkopf
2004 Twenty-first international conference on Machine learning - ICML '04  
. § © nearest neighborhoods were used for computing the embeddings.  ...  We show how all three algorithms can be described as kernel PCA on specially constructed Gram matrices, and illustrate the similarities and differences between the algorithms with representative examples  ...  We can construct an alternative kernel for LLE that is analogous to the heat kernel for the graph Laplacian by considering the Green's function of é , § ö Ô Á V Õ × Ö 5 q m é Ó D ! .  ... 
doi:10.1145/1015330.1015417 dblp:conf/icml/HamLMS04 fatcat:7wqekpyc3zbh7gabzspuokke3y

A Robust Clustering Algorithm Based on Aggregated Heat Kernel Mapping

Hao Huang, Shinjae Yoo, Hong Qin, Dantong Yu
2011 2011 IEEE 11th International Conference on Data Mining  
This paper aims to improve robustness in clustering algorithms and combat these two limitations based on heat kernel theory.  ...  Heat kernel can statistically depict traces of random walk, so it has an intrinsic connection with diffusion distance, with which we can ensure robustness during any clustering process.  ...  AHK is also known as Green's function [26] , which is closely related to commute distance (CD) or resistance distance.  ... 
doi:10.1109/icdm.2011.15 dblp:conf/icdm/HuangYQY11 fatcat:db2jhldn7nfspbnjkrsgewy5za

Path Planning with Divergence-Based Distance Functions [article]

Renjie Chen, Craig Gotsman, Kai Hormann
2017 arXiv   pre-print
We show that using divergence distances instead of the potential function and other distances has a significant computational advantage, as, following a pre-processing stage, they may be computed up to  ...  We define the divergence distances and compare them to the harmonic potential function and other related distance functions.  ...  The Green's function It is well known that  ... 
arXiv:1708.02845v1 fatcat:uusylb3ph5c3zkwumvm4pt3mby

Diffusion maps

Ronald R. Coifman, Stéphane Lafon
2006 Applied and Computational Harmonic Analysis  
The associated family of diffusion distances, obtained by iterating the Markov matrix, defines multiscale geometries that prove to be useful in the context of data parametrization and dimensionality reduction  ...  The proposed framework relates the spectral properties of Markov processes to their geometric counterparts and it unifies ideas arising in a variety of contexts such as machine learning, spectral graph  ...  Acknowledgments We thank Ann Lee, Mauro Maggioni, Boaz Nadler, Naoki Saito and Amit Singer for useful discussions regarding this work. We also thank Matthias Hein for his comments.  ... 
doi:10.1016/j.acha.2006.04.006 fatcat:a5uslf36lrhyrocar2nezuwcji

Discrete Green's Functions

Fan Chung, S.-T. Yau
2000 Journal of combinatorial theory. Series A  
Green's functions can be used to deal with diffusion-type problems on graphs, such as chipfiring, load balancing, and discrete Markov chains.  ...  We study discrete Green's functions and their relationship with discrete Laplace equations. Several methods for deriving Green's functions are discussed.  ...  The heat kernel H of 1 and the heat kernel h of P are related in a nice way [11] : H t (x, y)=-? &1 (v r ) h t (v 0 , v r ).  ... 
doi:10.1006/jcta.2000.3094 fatcat:svgoe2su3vdmnh7oq5btiuzs6i

A Novel Material-Aware Feature Descriptor for Volumetric Image Registration in Diffusion Tensor Space [chapter]

Shuai Li, Qinping Zhao, Shengfa Wang, Tingbo Hou, Aimin Hao, Hong Qin
2012 Lecture Notes in Computer Science  
and local distance definition based on PDF formulated in DT space.  ...  We rigorously formulate a novel probability density function (PDF) based distance metric to devise a compact local feature descriptor supporting invariance of full 3D orientation and isometric deformation  ...  This research is supported in part by National Natural Science Foundation of China (No.61190120, No.61190121 and No.61190125) and NSF grants IIS-0949467, IIS-1047715, and IIS-1049448.  ... 
doi:10.1007/978-3-642-33765-9_36 fatcat:vydmziiumneuzgnzdf46o22nwy

Heat kernels and zeta functions on fractals

Gerald V Dunne
2012 Journal of Physics A: Mathematical and Theoretical  
On fractals, spectral functions such as heat kernels and zeta functions exhibit novel features, very different from their behaviour on regular smooth manifolds, and these can have important physical consequences  ...  for both classical and quantum physics in systems having fractal properties.  ...  Acknowledgements: I gratefully acknowledge support from the US DOE grant DE-FG02-92ER40716, and collaboration with Eric Akkermans and Sasha Teplyaev.  ... 
doi:10.1088/1751-8113/45/37/374016 fatcat:mbnkgxopcfegxdz5ly5bnmugpm

Regularization on Graphs with Function-adapted Diffusion Processes

Arthur D. Szlam, Mauro Maggioni, Ronald R. Coifman
2008 Journal of machine learning research  
Harmonic analysis and diffusion on discrete data has been shown to lead to state-of-the-art algorithms for machine learning tasks, especially in the context of semi-supervised and transductive learning  ...  In this paper we present a method for modifying the given geometry so the function(s) to be studied are smoother with respect to the modified geometry, and thus more amenable to treatment using harmonic  ...  Acknowledgments The authors would like to thank Francis Woolfe and Triet Le for helpful suggestions on how to improve the manuscript, and to James C.  ... 
dblp:journals/jmlr/SzlamMC08 fatcat:dkyrmqncsrhifoqyeti52vr2ju

Shape Analysis Using the Auto Diffusion Function

K. Gȩbal, J. A. Baerentzen, H. Aanaes, R. Larsen
2009 Computer graphics forum (Print)  
We describe the ADF and its properties in detail and compare it to other choices of scalar functions on manifolds.  ...  Scalar functions defined on manifold triangle meshes is a starting point for many geometry processing algorithms such as mesh parametrization, skeletonization, and segmentation.  ...  Acknowledgments The 3D models used are provided courtesy of the Princeton and Aim@Shape repositories.  ... 
doi:10.1111/j.1467-8659.2009.01517.x fatcat:6xraszi3ajbrzak4vkij7cohfa

Solving Inverse PDE Problems using Grid-Free Monte Carlo Estimators [article]

Ekrem Fatih Yılmazer, Delio Vicini, Wenzel Jakob
2022 arXiv   pre-print
Modeling physical phenomena like heat transport and diffusion is crucially dependent on the numerical solution of partial differential equations (PDEs).  ...  They solve an adjoint form of the problem and exploit arithmetic invertibility to compute gradients using a constant amount of memory and linear time complexity.  ...  ACKNOWLEDGMENTS We thank Rohan Sawhney and Dario Seyb for sharing the implementation of their primal solvers with us.  ... 
arXiv:2208.02114v1 fatcat:u7wbqq4d6nbbdn45yknibr4oiq

Geodesics in heat

Keenan Crane, Clarisse Weischedel, Max Wardetzky
2013 ACM Transactions on Graphics  
The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently  ...  The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems.  ...  Over time heat spreads out over the rest of the domain and can be described by a function k t,x (y) called the heat kernel, which measures the heat transferred from a source x to a destination y after  ... 
doi:10.1145/2516971.2516977 fatcat:wbk5u3p64vd5dhg7ucxryovhga

Solving Local Linear Systems with Boundary Conditions Using Heat Kernel Pagerank

Fan Chung, Olivia Simpson
2015 Internet Mathematics  
With an efficient algorithm for approximating Dirichlet heat kernel pagerank, our local linear solver algorithm computes an approximate local solution with multiplicative and additive error ϵ by performing  ...  We introduce the method of using the Dirichlet heat kernel pagerank vector to approximate local solutions to linear systems in the graph Laplacian satisfying given boundary conditions over a particular  ...  The authors would like to thank the anonymous reviewers for their comments and suggestions.  ... 
doi:10.1080/15427951.2015.1009522 fatcat:evydxrwai5cyzcobu4wgsijfpa
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