核主成份分析的MatLab实现

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最近准备往核方法方向发展，于是仔细研读了经典的核主成份分析方法。通过参考别人的源代码，自己实现了KPCA，基本了解了核方法的实质。个人体会亲手实现一下还是很有益的。matlab源代码如下：

function [eigvector, eigvalue,Y] = KPCA(X,r,opts)

% Kernel Principal Component Analysis

% [eigvector, eigvalue,Y] = KPCA(X,r,opts)

% Input:

% X: d*N data matrix;Each column vector of X is a sample vector.

% r: Dimensionality of reduced space (default: d)

% opts: Struct value in Matlab. The fields in options that can be set:

% KernelType - Choices are:

% 'Gaussian' - exp{-gamma(|x-y|^2)}

% 'Polynomial' - (x'*y)^d

% 'PolyPlus' - (x'*y+1)^d

% gamma - parameter for Gaussian kernel

% d - parameter for polynomial kernel

% Output:

% eigvector: N*r matrix;Each column is an embedding function, for a new

% data point (column vector) x, y = eigvector'*K(x,:)'

% will be the embedding result of x.

% K(x,:) = [K(x1,x),K(x2,x),...K(xN,x)]

% eigvalue: The sorted eigvalue of KPCA eigen-problem.

% Y : Data matrix after the nonlinear transform

if nargin<1

error('Not enough input arguments.')

end

[d,N]=size(X);

if nargin<2

r=d;

end

%% Ensure r is not bigger than d

if r>d

r=d;

end;

% Construct the Kernel matrix K

K =ConstructKernelMatrix(X,[],opts);

% Centering kernel matrix

One_N=ones(N)./N;

Kc = K - One_N*K - K*One_N + One_N*K*One_N;

clear One_N;

% Solve the eigenvalue problem N*lamda*alpha = K*alpha

if N>1000 && r

% using eigs to speed up!

opts.disp=0;

[eigvector, eigvalue] = eigs(Kc,r,'la',opts);

eigvalue = diag(eigvalue);

else

[eigvector, eigvalue] = eig(Kc);

eigvalue = diag(eigvalue);

[junk, index] = sort(-eigvalue);

eigvalue = eigvalue(index);

eigvector = eigvector(:,index);

end

if r < length(eigvalue)

eigvalue = eigvalue(1:r);

eigvector = eigvector(:, 1:r);

end

% Only reserve the eigenvector with nonzero eigenvalues

maxEigValue = max(abs(eigvalue));

eigIdx = find(abs(eigvalue)/maxEigValue < 1e-6);

eigvalue (eigIdx) = [];

eigvector (:,eigIdx) = [];

% Normalizing eigenvector

for i=1:length(eigvalue)

eigvector(:,i)=eigvector(:,i)/sqrt(eigvalue(i));

end;

if nargout >= 3

% Projecting the data in lower dimensions

Y = eigvector'*K;

end

function K=ConstructKernelMatrix(X,Y,opts)

% function K=ConstructKernelMatrix(X,Y,opts)

% Usage:

% opts.KernelType='Gaussian';

% K = ConstructKernelMatrix(X,[],opts)

% K = ConstructKernelMatrix(X,Y,opts)

% Input:

% X: d*N data matrix;Each column vector of X is a sample vector.

% Y: d*M data matrix;Each column vector of Y is a sample vector.

% opts: Struct value in Matlab. The fields in options that can be set:

% KernelType - Choices are:

% 'Gaussian' - exp{-gamma(|x-y|^2)}

% 'Polynomial' - (x'*y)^d

% 'PolyPlus' - (x'*y+1)^d

% gamma - parameter for Gaussian kernel

% d - parameter for polynomial kernel

% Output:

% K N*N or N*M matrix

if nargin<1

error('Not enough input arguments.')

end

if (~exist('opts','var'))

opts = [];

else

if ~isstruct(opts)

error('parameter error!');

end

N=size(X,2);

if isempty(Y)

K=zeros(N,N);

else

M=size(Y,2);

if size(X,1)~=size(Y,1)

error('Matrixes X and Y should have the same row dimensionality!');

end;

K=zeros(N,M);

end;

%=================================================

if ~isfield(opts,'KernelType')

opts.KernelType = 'Gaussian';

end

switch lower(opts.KernelType)

case {lower('Gaussian')} % exp{-gamma(|x-y|^2)}

if ~isfield(opts,'gamma')

opts.gamma = 0.5;

end

case {lower('Polynomial')} % (x'*y)^d

if ~isfield(opts,'d')

opts.d = 1;

end

case {lower('PolyPlus')} % (x'*y+1)^d

if ~isfield(opts,'d')

opts.d = 1;

end

otherwise

error('KernelType does not exist!');

end

switch lower(opts.KernelType)

case {lower('Gaussian')}

if isempty(Y)

for i=1:N

for j=i:N

dist = sum(((X(:,i) - X(:,j)).^2));

temp=exp(-opts.gamma*dist);

K(i,j)=temp;

if i~=j

K(j,i)=temp;

end;

end

else

for i=1:N

for j=1:M

dist = sum(((X(:,i) - Y(:,j)).^2));

K(i,j)=exp(-opts.gamma*dist);

end

case {lower('Polynomial')}

if isempty(Y)

for i=1:N

for j=i:N

temp=(X(:,i)'*X(:,j))^opts.d;

K(i,j)=temp;

if i~=j

K(j,i)=temp;

end;

end

else

for i=1:N

for j=1:M

K(i,j)=(X(:,i)'*Y(:,j))^opts.d;

end

case {lower('PolyPlus')}

if isempty(Y)

for i=1:N

for j=i:N

temp=(X(:,i)'*X(:,j)+1)^opts.d;

K(i,j)=temp;

if i~=j

K(j,i)=temp;

end;

end

else

for i=1:N

for j=1:M

K(i,j)=(X(:,i)'*Y(:,j)+1)^opts.d;

end

otherwise

error('KernelType does not exist!');

end

参考文献:

[1]B. Sch¨olkopf, A.J. Smola, and K.-R M¨uller, “Nonlinear component analysis as a kernel eigenvalue problem,”Neural Computation, vol. 10, pp. 1299–1319, 1998.

[2]KERNEL PRINCIPAL COMPONENT ANALYSIS FOR FEATURE REDUCTION IN HYPERSPECTRALE IMAGES ANALYSIS

[3]http://www.cs.uiuc.edu/homes/dengcai2/Data/data.html

KPCA 算法：

引用地址：http://blog.sciencenet.cn/blog-457187-342046.html

注意：

1.如果出现错误：

Undefined function or method 'ConstructKernelMatrix' for input arguments of type 'double'

可以把ConstructKernelMatrix函数另存为一个*.m文件试试。

2.若有其他错误，请回复，共同解决。