How to implement the Gaussian radial basis function in MATLAB ?

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charu shree
charu shree 2023-3-20
编辑： Torsten 2023-3-21
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.

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采纳的回答

Torsten
Torsten 2023-3-21
编辑：Torsten 2023-3-21
Ran in:
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
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Torsten
Torsten 2023-3-21
编辑：Torsten 2023-3-21
@Matt J
Thank you for the correction.
Is it somehow obvious that the matrix will be positive-definite ?

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Matt J
Matt J 2023-3-21
编辑：Matt J 2023-3-21
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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