Implementing the (polynomial) "kernel trick" in matlab

Given p = 2 vectors x in R^n, y in R^n, how to compute the kernel of order m > 2, which is of the form

A = [ones(n,1), x, y, x.*y, x.^2, y.^2, x.^2 *y, ..., x.^m, y.^m]  
q = nchoosek(p+m, m)

A in R^(nxq)

For example, given

x=[1 2]'
y=[2 1]'
m = 3;

Output:

A = [1 1 2 1 4 1 8;1 2 1 4 1 8 1]

Answer 1

[Jaynik]

Hi Ashwin,

You are on the right path to implement the polynomial kernel trick. Here is a sample code that can be further enhanced to code the solution:

function A = polynomial_kernel(x, y, m)
    n = length(x);
    p = 2; % Since we have two vectors x and y
    q = nchoosek(p + m, m);
    A = ones(n, q);
    
    col = 2;
    for i = 1:m
        for j = 0:i
            A(:, col) = (x.^(i-j)) .* (y.^j);
            col = col + 1;
        end
    end
end

For each combination of degrees,we need coumpute the term and fill in the matrix A. i represents the total degree of the polynomial term and j represents the degree of y in the term while (i - j) is the degree of x.

Hope this helps!