How to implement the Gaussian radial basis function in MATLAB ?

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Hello all, I am dealing with the following optimization problem
where is a column vector of dimension , is also a column vector of dimension , is matrix of dimension , is row vector of and is the Gaussian radial basis function, where is the variance.
My query is how to code this Gaussian radial basis function (RBF) ?
Any help in this regard will be highly appreciated.

Accepted Answer

Torsten
Torsten on 21 Mar 2023
Edited: Torsten on 21 Mar 2023
Before you start the optimization, check whether K in your case is positive-definite.
Lt = 10;
p = rand(Lt,4);
sigma = 1.0;
for l = 1:Lt
for j = 1:Lt
K(l,j) = exp((norm(p(l,:)-p(j,:)))^2/(2*sigma^2));
end
end
K
K = 10×10
1.0000 1.0558 1.2332 1.1752 1.1144 1.7121 1.4345 1.4970 1.3174 1.4356 1.0558 1.0000 1.1059 1.1852 1.0453 1.4602 1.3978 1.2603 1.1884 1.3441 1.2332 1.1059 1.0000 1.4993 1.1769 1.7198 1.2359 1.2245 1.2960 1.8679 1.1752 1.1852 1.4993 1.0000 1.3279 1.1864 1.5427 1.3336 1.2247 1.0896 1.1144 1.0453 1.1769 1.3279 1.0000 1.6242 1.4504 1.5273 1.1608 1.4210 1.7121 1.4602 1.7198 1.1864 1.6242 1.0000 1.7290 1.2760 1.2118 1.1176 1.4345 1.3978 1.2359 1.5427 1.4504 1.7290 1.0000 1.5769 1.2315 1.9993 1.4970 1.2603 1.2245 1.3336 1.5273 1.2760 1.5769 1.0000 1.4254 1.5342 1.3174 1.1884 1.2960 1.2247 1.1608 1.2118 1.2315 1.4254 1.0000 1.2412 1.4356 1.3441 1.8679 1.0896 1.4210 1.1176 1.9993 1.5342 1.2412 1.0000
eig(K)
ans = 10×1
-1.7051 -0.8915 -0.6203 -0.3001 0.0081 0.0171 0.0346 0.0637 0.1039 13.2897
  6 Comments
Torsten
Torsten on 21 Mar 2023
Edited: Torsten on 21 Mar 2023
Thank you for the correction.
Is it somehow obvious that the matrix will be positive-definite ?

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More Answers (1)

Matt J
Matt J on 21 Mar 2023
Edited: Matt J on 21 Mar 2023
If you have the Statistics Toolbox, you can avoid a loop by using pdist2
Lt = 10;
p = rand(Lt,4);
sigma = 1.0;
K=exp(-pdist2(p,p).^2 / 2/sigma^2);

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