Embedding data points problem

Hello. I wrote this following which I have theta data column points (1024 in my case)and finds the eigenvectors of the Laplace-Beltrami operator using density invariant normalization.

theta=0:2*pi/1023:2*pi;
t=-1.5:3/511:1.5;
P=my_data(theta,t);
B=(1/size(P,2))*sum(P,2);
U=zeros(size(P));
for l=1:size(P,2)
    U(:,l)=B;
end
X=P-U;
epsilon=0.01
W=zeros(size(P,2),size(P,2));
T=zeros(size(P,2),size(P,2));
for i=1:size(P,2)
    for j=1:size(P,2)
        W(i,j)=exp(-norm(X(:,i)-P(:,j))/(sqrt(2*epsilon)));
    end
end
a=(sum(W,2))';
D=zeros(size(W,1),size(W,2));
for k=1:size(W,2)
    D(k,k)=a(k);
end
T=inv(D)*W*inv(D);
b=(sum(T,2))';
D2=zeros(size(W,1),size(W,2));
for m=1:size(W,2)
    D2(m,m)=b(m);
end
Anorm=inv(D2)*T;
G=(D2^(-0.5))*T*(D2^(-0.5));
[V,gamma]=eig(G);

Now I need to make embedding of each column of X onto the first 3 and 4 non trivial eigenvectors excluding V(:,1) but I don’t know how to do it. Help is greatly appreciated!

Answer 1

[Shlok]

Hi Desiree,

To embed each column of “X” onto the eigenvectors, the approach is to extract out the desired eigenvectors and perform matrix multiplication with them. To achieve this, follow the following steps:

• Exclude the first eigenvector “V(:,1)” using indexing, and then extract next 3 and 4 eigenvectors.

V_non_trivial = V(:, 2:end); 
V_first_3 = V_non_trivial(:, 1:3); 
V_first_4 = V_non_trivial(:, 1:4); 

• Now, project the data by multiplying “X” with the transpose of selected eigenvectors to obtain the embedding.

embedding_3 = V_first_3' * X; 
embedding_4 = V_first_4' * X; 

This is how we can embed the data points (columns of X) into a lower-dimensional space using the first 3 or 4 non-trivial eigenvectors.

To know more about indexing in MATLAB, refer to the following documentation link:

https://www.mathworks.com/company/technical-articles/matrix-indexing-in-matlab.html