can you please explain the logic of this code

function alphaKrow = mkpca(data,sigma,numev) 
% -----------------------------------------------------------------------
% global K
[n, d] = size(data);
% n : number of data points
% d : dimension of data points
sigma = 2.5;
% kernel matrix:
K = zeros(n,n);
% kernel parameter:
param = 0.5/(sigma*sigma);
%
fprintf('computing kernel matrix K\n');
for i=1:n
    for j=i:n
        K(i,j) = kernel(data(i,:),data(j,:),param); 
        K(j,i) = K(i,j); 
    end
end
% correct K for non-zero center of data in feature space:
Krow = sum(K,1)/n;
Ksum = sum(Krow)/n;
for i=1:n
    for j=1:n 
        K(i,j) = K(i,j) - Krow(i) - Krow(j) + Ksum; 
    end
end
opts.disp = 0; 
[alpha,lambda] = eigs(K);
% normalize alpha: 
alpha = alpha * inv(sqrt(lambda));
% compute some helper vectors: 
alphaKrow = Krow * alpha;
K = imresize(K,[256,256]); 
% -----------------------------------------------------------------------
% 
function k = kernel(x,y,param) 
% ----------------------------------------------------------------------- 
%
diff = x-y;
k = exp(-(diff * diff')*param);

Answer 1

[Hari]

Hi Priyanka,

I understand that you are seeking an explanation of the logic behind the given code, which implements a kernel PCA (Principal Component Analysis) using a Gaussian kernel.

Initialization and Parameters:

The function "mkpca" takes data, sigma, and numev as inputs. Here, data represents the input data points, sigma is a parameter for the Gaussian kernel, and numev is presumably the number of eigenvectors to compute.

The number of data points n and the dimensionality d of the data are determined using [n, d] = size(data).

The kernel parameter param is calculated as 0.5/(sigma*sigma) for use in the Gaussian kernel.

Kernel Matrix Calculation:

A kernel matrix K is initialized as an n x n matrix. It is used to store the pairwise kernel values between all data points.

The nested loops compute the Gaussian kernel between each pair of data points using the kernel function and store the symmetric values in K(i,j) and K(j,i).

Centering the Kernel Matrix:

The kernel matrix K is centered to ensure the data is mean-centered in the feature space.

Krow is the mean of each row of K, and Ksum is the mean of Krow.

The nested loops adjust each element of K to ensure the data is centered in the feature space.

Eigenvalue Decomposition:

The "eigs" function is used to compute the eigenvectors (alpha) and eigenvalues (lambda) of the centered kernel matrix K.

The eigenvectors (alpha) are then normalized by multiplying with the inverse square root of their corresponding eigenvalues.

Helper Vector Calculation:

"alphaKrow" is computed as the product of Krow and alpha, which can be useful for projecting new data points into the PCA space.

The kernel matrix K is resized to 256x256 using imresize, although this step seems unrelated to the PCA logic and might be for visualization purposes.

Kernel Function:

The kernel function calculates the Gaussian kernel value between two data points x and y using the formula exp(-(diff * diff')*param).

Refer to the documentation of "eigs" function for more details on eigenvalue decomposition: https://www.mathworks.com/help/matlab/ref/eigs.html

Refer to the documentation of "imresize" function for more details on image resizing: https://www.mathworks.com/help/matlab/ref/imresize.html

Hope this helps!