简单易学的机器学习算法——谱聚类(Spectal Clustering)

一、复杂网络中的一些基本概念

1、复杂网络的表示

2、网络簇结构

    网络簇结构(network cluster structure)也称为网络社团结构(network community structure),是复杂网络中最普遍和最重要的拓扑属性之一。网络簇是整个网络中的稠密连接分支,具有同簇内部节点之间相互连接密集,不同簇的节点之间相互连接稀疏的特征。

3、复杂网络的分类

    复杂网络主要分为:随机网络,小世界网络和无标度网络。

二、谱方法介绍

1、谱方法的思想

    在复杂网络的网络簇结构存在着同簇节点之间连接密集,不同簇节点之间连接稀疏的特征,是否可以根据这样的特征对网络中的节点进行聚类,使得同类节点之间的连接密集,不同类别节点之间的连接稀疏?

    在谱聚类中定义了“截”函数的概念,当一个网络被划分成为两个子网络时,“截”即指子网间的连接密度。谱聚类的目的就是要找到一种合理的分割,使得分割后形成若干子图,连接不同的子图的边的权重尽可能低,即“截”最小,同子图内的边的权重尽可能高。

2、“截”函数的具体表现形式   

3、基本“截”函数的弊端

    对于上述的“截”函数,最终会导致不好的分割,如二分类问题:

4、其他的“截”函数的表现形式   

三、Laplacian矩阵

1、Laplacian矩阵的定义

2、度矩阵的定义   

3、Laplacian矩阵的性质

4、不同的Laplacian矩阵

    除了上述的拉普拉斯矩阵,还有规范化的Laplacian矩阵形式:

四、Laplacian矩阵与谱聚类中的优化函数的关系

1、由Laplacian矩阵到“截”函数

    对于二个类别的聚类问题,优化的目标函数为:

定义向量

而已知:

2、新的目标函数   

3、转化到Laplacian矩阵的求解

五、从二类别聚类到多类别聚类

1、二类别聚类

    对于求解出来的特征向量

中的每一个分量

根据每个分量的值来判断对应的点所属的类别:

2、多类别聚类   

六、谱聚类的过程

1、基本的结构   

2、利用相似度矩阵的构造方法   

七、实验代码

1、自己实现的一个

#coding:UTF-8
'''
Created on 2015年5月12日

@author: zhaozhiyong
'''
from __future__ import division
import scipy.io as scio
from scipy import sparse
from scipy.sparse.linalg.eigen import arpack#这里只能这么做,不然始终找不到函数eigs
from numpy import *


def spectalCluster(data, sigma, num_clusters):
    print "将邻接矩阵转换成相似矩阵"
    #先完成sigma != 0
    print "Fixed-sigma谱聚类"
    data = sparse.csc_matrix.multiply(data, data)

    data = -data / (2 * sigma * sigma)
    
    S = sparse.csc_matrix.expm1(data) + sparse.csc_matrix.multiply(sparse.csc_matrix.sign(data), sparse.csc_matrix.sign(data))   
    
    #转换成Laplacian矩阵
    print "将相似矩阵转换成Laplacian矩阵"
    D = S.sum(1)#相似矩阵是对称矩阵
    D = sqrt(1 / D)
    n = len(D)
    D = D.T
    D = sparse.spdiags(D, 0, n, n)
    L = D * S * D
    
    #求特征值和特征向量
    print "求特征值和特征向量"
    vals, vecs = arpack.eigs(L, k=num_clusters,tol=0,which="LM")  
    
    # 利用k-Means
    print "利用K-Means对特征向量聚类"
    #对vecs做正规化
    sq_sum = sqrt(multiply(vecs,vecs).sum(1))
    m_1, m_2 = shape(vecs)
    for i in xrange(m_1):
        for j in xrange(m_2):
            vecs[i,j] = vecs[i,j]/sq_sum[i]
    
    myCentroids, clustAssing = kMeans(vecs, num_clusters)
    
    for i in xrange(shape(clustAssing)[0]):
        print clustAssing[i,0]
    

def randCent(dataSet, k):
    n = shape(dataSet)[1]
    centroids = mat(zeros((k,n)))#create centroid mat
    for j in range(n):#create random cluster centers, within bounds of each dimension
        minJ = min(dataSet[:,j]) 
        rangeJ = float(max(dataSet[:,j]) - minJ)
        centroids[:,j] = mat(minJ + rangeJ * random.rand(k,1))
    return centroids

def distEclud(vecA, vecB):
    return sqrt(sum(power(vecA - vecB, 2))) #la.norm(vecA-vecB)

def kMeans(dataSet, k):
    m = shape(dataSet)[0]
    clusterAssment = mat(zeros((m,2)))#create mat to assign data points to a centroid, also holds SE of each point
    centroids = randCent(dataSet, k)
    clusterChanged = True
    while clusterChanged:
        clusterChanged = False
        for i in range(m):#for each data point assign it to the closest centroid
            minDist = inf; minIndex = -1
            for j in range(k):
                distJI = distEclud(centroids[j,:],dataSet[i,:])
                if distJI < minDist:
                    minDist = distJI; minIndex = j
            if clusterAssment[i,0] != minIndex: clusterChanged = True
            clusterAssment[i,:] = minIndex,minDist**2
        #print centroids
        for cent in range(k):#recalculate centroids
            ptsInClust = dataSet[nonzero(clusterAssment[:,0].A==cent)[0]]#get all the point in this cluster
            centroids[cent,:] = mean(ptsInClust, axis=0) #assign centroid to mean 
    return centroids, clusterAssment


if __name__ == '__main__':
    # 导入数据集
    matf = 'E://data_sc//corel_50_NN_sym_distance.mat'
    dataDic = scio.loadmat(matf)
    data = dataDic['A']
    # 谱聚类的过程
    spectalCluster(data, 20, 18)

2、网上提供的一个Matlab代码

function [cluster_labels evd_time kmeans_time total_time] = sc(A, sigma, num_clusters)
%SC Spectral clustering using a sparse similarity matrix (t-nearest-neighbor).
%
%   Input  : A              : N-by-N sparse distance matrix, where
%                             N is the number of data
%            sigma          : sigma value used in computing similarity,
%                             if 0, apply self-tunning technique
%            num_clusters   : number of clusters
%
%   Output : cluster_labels : N-by-1 vector containing cluster labels
%            evd_time       : running time for eigendecomposition
%            kmeans_time    : running time for k-means
%            total_time     : total running time

%
% Convert the sparse distance matrix to a sparse similarity matrix,
% where S = exp^(-(A^2 / 2*sigma^2)).
% Note: This step can be ignored if A is sparse similarity matrix.
%
disp('Converting distance matrix to similarity matrix...');
tic;
n = size(A, 1);

if (sigma == 0) % Selftuning spectral clustering
  % Find the count of nonzero for each column
  disp('Selftuning spectral clustering...');
  col_count = sum(A~=0, 1)';
  col_sum = sum(A, 1)';
  col_mean = col_sum ./ col_count;
  [x y val] = find(A);
  A = sparse(x, y, -val.*val./col_mean(x)./col_mean(y)./2);
  clear col_count col_sum col_mean x y val;
else % Fixed-sigma spectral clustering
  disp('Fixed-sigma spectral clustering...');
  A = A.*A;
  A = -A/(2*sigma*sigma);
end

% Do exp function sequentially because of memory limitation
num = 2000;
num_iter = ceil(n/num);
S = sparse([]);
for i = 1:num_iter
  start_index = 1 + (i-1)*num;
  end_index = min(i*num, n);
  S1 = spfun(@exp, A(:,start_index:end_index)); % sparse exponential func
  S = [S S1];
  clear S1;
end
clear A;
toc;

%
% Do laplacian, L = D^(-1/2) * S * D^(-1/2)
%
disp('Doing Laplacian...');
D = sum(S, 2) + (1e-10);
D = sqrt(1./D); % D^(-1/2)
D = spdiags(D, 0, n, n);
L = D * S * D;
clear D S;
time1 = toc;

%
% Do eigendecomposition, if L =
%   D^(-1/2) * S * D(-1/2)    : set 'LM' (Largest Magnitude), or
%   I - D^(-1/2) * S * D(-1/2): set 'SM' (Smallest Magnitude).
%
disp('Performing eigendecomposition...');
OPTS.disp = 0;
[V, val] = eigs(L, num_clusters, 'LM', OPTS);
time2 = toc;

%
% Do k-means
%
disp('Performing kmeans...');
% Normalize each row to be of unit length
sq_sum = sqrt(sum(V.*V, 2)) + 1e-20;
U = V ./ repmat(sq_sum, 1, num_clusters);
clear sq_sum V;
cluster_labels = k_means(U, [], num_clusters);
total_time = toc;

%
% Calculate and show time statistics
%
evd_time = time2 - time1
kmeans_time = total_time - time2
total_time
disp('Finished!');
function cluster_labels = k_means(data, centers, num_clusters)
%K_MEANS Euclidean k-means clustering algorithm.
%
%   Input    : data           : N-by-D data matrix, where N is the number of data,
%                               D is the number of dimensions
%              centers        : K-by-D matrix, where K is num_clusters, or
%                               'random', random initialization, or
%                               [], empty matrix, orthogonal initialization
%              num_clusters   : Number of clusters
%
%   Output   : cluster_labels : N-by-1 vector of cluster assignment
%
%   Reference: Dimitrios Zeimpekis, Efstratios Gallopoulos, 2006.
%              http://scgroup.hpclab.ceid.upatras.gr/scgroup/Projects/TMG/

%
% Parameter setting
%
iter = 0;
qold = inf;
threshold = 0.001;

%
% Check if with initial centers
%
if strcmp(centers, 'random')
  disp('Random initialization...');
  centers = random_init(data, num_clusters);
elseif isempty(centers)
  disp('Orthogonal initialization...');
  centers = orth_init(data, num_clusters);
end

%
% Double type is required for sparse matrix multiply
%
data = double(data);
centers = double(centers);

%
% Calculate the distance (square) between data and centers
%
n = size(data, 1);
x = sum(data.*data, 2)';
X = x(ones(num_clusters, 1), :);
y = sum(centers.*centers, 2);
Y = y(:, ones(n, 1));
P = X + Y - 2*centers*data';

%
% Main program
%
while 1
  iter = iter + 1;

  % Find the closest cluster for each data point
  [val, ind] = min(P, [], 1);
  % Sum up data points within each cluster
  P = sparse(ind, 1:n, 1, num_clusters, n);
  centers = P*data;
  % Size of each cluster, for cluster whose size is 0 we keep it empty
  cluster_size = P*ones(n, 1);
  % For empty clusters, initialize again
  zero_cluster = find(cluster_size==0);
  if length(zero_cluster) > 0
    disp('Zero centroid. Initialize again...');
    centers(zero_cluster, :)= random_init(data, length(zero_cluster));
    cluster_size(zero_cluster) = 1;
  end
  % Update centers
  centers = spdiags(1./cluster_size, 0, num_clusters, num_clusters)*centers;

  % Update distance (square) to new centers
  y = sum(centers.*centers, 2);
  Y = y(:, ones(n, 1));
  P = X + Y - 2*centers*data';

  % Calculate objective function value
  qnew = sum(sum(sparse(ind, 1:n, 1, size(P, 1), size(P, 2)).*P));
  mesg = sprintf('Iteration %d:\n\tQold=%g\t\tQnew=%g', iter, full(qold), full(qnew));
  disp(mesg);

  % Check if objective function value is less than/equal to threshold
  if threshold >= abs((qnew-qold)/qold)
    mesg = sprintf('\nkmeans converged!');
    disp(mesg);
    break;
  end
  qold = qnew;
end

cluster_labels = ind';


%-----------------------------------------------------------------------------
function init_centers = random_init(data, num_clusters)
%RANDOM_INIT Initialize centroids choosing num_clusters rows of data at random
%
%   Input : data         : N-by-D data matrix, where N is the number of data,
%                          D is the number of dimensions
%           num_clusters : Number of clusters
%
%   Output: init_centers : K-by-D matrix, where K is num_clusters
rand('twister', sum(100*clock));
init_centers = data(ceil(size(data, 1)*rand(1, num_clusters)), :);

function init_centers = orth_init(data, num_clusters)
%ORTH_INIT Initialize orthogonal centers for k-means clustering algorithm.
%
%   Input : data         : N-by-D data matrix, where N is the number of data,
%                          D is the number of dimensions
%           num_clusters : Number of clusters
%
%   Output: init_centers : K-by-D matrix, where K is num_clusters

%
% Find the num_clusters centers which are orthogonal to each other
%
Uniq = unique(data, 'rows'); % Avoid duplicate centers
num = size(Uniq, 1);
first = ceil(rand(1)*num); % Randomly select the first center
init_centers = zeros(num_clusters, size(data, 2)); % Storage for centers
init_centers(1, :) = Uniq(first, :);
Uniq(first, :) = [];
c = zeros(num-1, 1); % Accumalated orthogonal values to existing centers for non-centers
% Find the rest num_clusters-1 centers
for j = 2:num_clusters
  c = c + abs(Uniq*init_centers(j-1, :)');
  [minimum, i] = min(c); % Select the most orthogonal one as next center
  init_centers(j, :) = Uniq(i, :);
  Uniq(i, :) = [];
  c(i) = [];
end
clear c Uniq;

个人的一点认识:谱聚类的过程相当于先进行一个非线性的降维,然后在这样的低维空间中再利用聚类的方法进行聚类。

欢迎大家一起讨论,如有问题欢迎留言,欢迎大家转载。

参考

1、从拉普拉斯矩阵说到谱聚类(http://blog.csdn.net/v_july_v/article/details/40738211)

2、谱聚类(spectral clustering)(http://www.cnblogs.com/FengYan/archive/2012/06/21/2553999.html)

3、谱聚类算法(Spectral Clustering)(http://www.cnblogs.com/sparkwen/p/3155850.html)

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