调和矩阵

图论中,调和矩阵harmonic matrix),也称拉普拉斯矩阵拉氏矩阵Laplacian matrix)、离散拉普拉斯discrete Laplacian),是矩阵表示。[1]

调和矩阵也是拉普拉斯算子离散化。换句话说,调和矩阵的缩放极限拉普拉斯算子。它在机器学习物理学中有很多应用。

定义编辑

若G是简单,G有n个顶点,A是邻接矩阵,D是度数矩阵,则调和矩阵[1]

 

 

动机编辑

这跟拉普拉斯算子有什么关系?若f 是加权图G的顶点函数,则[2]

 

w是边的权重函数。u、v是顶点。f = (f(1), ..., f(n)) 是n维的矢量。上面泛函也称为Dirichlet泛函。[3]

接续矩阵编辑

而且若K是接续矩阵(incidence matrix),则[2]

 

 

Kf 是f 的图梯度。另外,特征值满足

 

举例编辑

度数矩阵 邻接矩阵 调和矩阵
       

其他形式编辑

对称正规化调和矩阵编辑

 

 

注意[4]

 

随机漫步编辑

 

 

动力学微分方程编辑

例如,离散的冷却定律使用调和矩阵[5]

 

使用矩阵矢量

 

 

解是

 

 

 

平衡举动编辑

 的时候,

 

 

 

 

MATLAB代码编辑

N = 20;%The number of pixels along a dimension of the image
A = zeros(N, N);%The image
Adj = zeros(N*N, N*N);%The adjacency matrix

%Use 8 neighbors, and fill in the adjacency matrix
dx = [-1, 0, 1, -1, 1, -1, 0, 1];
dy = [-1, -1, -1, 0, 0, 1, 1, 1];
for x = 1:N
   for y = 1:N
       index = (x-1)*N + y;
       for ne = 1:length(dx)
           newx = x + dx(ne);
           newy = y + dy(ne);
           if newx > 0 && newx <= N && newy > 0 && newy <= N
               index2 = (newx-1)*N + newy;
               Adj(index, index2) = 1;
           end
       end
   end
end

%%%BELOW IS THE KEY CODE THAT COMPUTES THE SOLUTION TO THE DIFFERENTIAL
%%%EQUATION
Deg = diag(sum(Adj, 2));%Compute the degree matrix
L = Deg - Adj;%Compute the laplacian matrix in terms of the degree and adjacency matrices
[V, D] = eig(L);%Compute the eigenvalues/vectors of the laplacian matrix
D = diag(D);

%Initial condition (place a few large positive values around and
%make everything else zero)
C0 = zeros(N, N);
C0(2:5, 2:5) = 5;
C0(10:15, 10:15) = 10;
C0(2:5, 8:13) = 7;
C0 = C0(:);

C0V = V'*C0;%Transform the initial condition into the coordinate system 
%of the eigenvectors
for t = 0:0.05:5
   %Loop through times and decay each initial component
   Phi = C0V.*exp(-D*t);%Exponential decay for each component
   Phi = V*Phi;%Transform from eigenvector coordinate system to original coordinate system
   Phi = reshape(Phi, N, N);
   %Display the results and write to GIF file
   imagesc(Phi);
   caxis([0, 10]);
   title(sprintf('Diffusion t = %3f', t));
   frame = getframe(1);
   im = frame2im(frame);
   [imind, cm] = rgb2ind(im, 256);
   if t == 0
      imwrite(imind, cm, 'out.gif', 'gif', 'Loopcount', inf, 'DelayTime', 0.1); 
   else
      imwrite(imind, cm, 'out.gif', 'gif', 'WriteMode', 'append', 'DelayTime', 0.1);
   end
end
 
GIF:离散拉普拉斯过程,使用拉普拉斯矩阵

应用编辑

参考文献编辑

  1. ^ 1.0 1.1 Weisstein, Eric W. Laplacian Matrix. mathworld.wolfram.com. [2020-02-14]. (原始内容存档于2019-12-23) (英语). 
  2. ^ 2.0 2.1 Muni Sreenivas Pydi (ముని శ్రీనివాస్ పైడి)'s answer to What's the intuition behind a Laplacian matrix? I'm not so much interested in mathematical details or technical applications. I'm trying to grasp what a laplacian matrix actually represents, and what aspects of a graph it makes accessible. - Quora. www.quora.com. [2020-02-14]. 
  3. ^ 3.0 3.1 Shuman, David I.; Narang, Sunil K.; Frossard, Pascal; Ortega, Antonio; Vandergheynst, Pierre. The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains. IEEE Signal Processing Magazine. 2013-05, 30 (3): 83–98 [2020-02-14]. ISSN 1053-5888. doi:10.1109/MSP.2012.2235192. (原始内容存档于2020-01-11). 
  4. ^ Chung, Fan. Spectral Graph Theory. American Mathematical Society. 1997 [1992] [2020-02-14]. ISBN 978-0821803158. (原始内容存档于2020-02-14). 
  5. ^ Newman, Mark. Networks: An Introduction. Oxford University Press. 2010. ISBN 978-0199206650. 

阅读编辑

  • T. Sunada. Chapter 1. Analysis on combinatorial graphs. Discrete geometric analysis. P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev (编). 'Proceedings of Symposia in Pure Mathematics 77. 2008: 51–86. ISBN 978-0-8218-4471-7. 
  • B. Bollobás, Modern Graph Theory, Springer-Verlag (1998, corrected ed. 2013), ISBN 0-387-98488-7, Chapters II.3 (Vector Spaces and Matrices Associated with Graphs), VIII.2 (The Adjacency Matrix and the Laplacian), IX.2 (Electrical Networks and Random Walks).