How to project kernel PCA?

Question

I have an $m\times n$ matrix $X$. To apply a Kernel PCA to my $X$ matrix I need to warp it into a function $K = \Phi(X)$.

The problem here is that $K$ get the size $m \times m$. If I'm doing projection with PCA.

% Average
mu = mean(K);

% Center data
Y = K - mu;

% PCA analysis
abort = input(sprintf('The size of the matrix is %ix%i. Do you want to apply PCA onto it? 1 = Yes, 0 = No: ', size(Y)));
if(abort == 1)
  [U, ~, ~] = svd(Y);
else
  error('Aborted');
end

% Projection
W = U(:, 1:c);
P = W'*Y;

Then the projected matrix $P$ is going to have the size $c \times m$.

Question:

I want the size of the projected matrix to have size $c \times n$. My goal is to turn $X$ into a lower dimension $c$ but keep the remaining columns $n$.

Am I using kernel PCA wrong?

Answer 1

Here is the answer:

  % Create kernel
  disp('Creating kernel');
  K = create_kernel(X, kernel_type, kernel_parameters);
  disp('Done');

  % Do PCA
  disp('Createing PCA of the kernel')
  [~, W] = mi.pca(K, c);

  % We projecting X onto W' instead of K onto W'
  P = W'*X;
  disp('Done');

Fully working example:

https://github.com/DanielMartensson/MataveID/blob/master/examples/kpcaExample.md