How is the error term of Radial Basis Function (Network) found?

Asked Jul 04 '18 at 18:57

Active Jul 04 '18 at 18:57

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Such as in:

$$\min_W I(W)=\frac{1}{2N} \sum_{i=1}^N ||W g(x_i)-y_i||^2+ \color{red}{\frac{\beta}{2} \sum_{j,k} |W_{jk}|^2}$$

Then how does one discover the error term?

$W$ is the coefficient matrix (for which to solve), $g$ is the kernel, $\beta$ is regularization coefficient.

asked Jul 04 '18 at 18:57

mavavilj

I would recommend asking the Data Science website if you don't get answers here. I don't know anything about RBFs, but my guess would be this: it's just $L_2$ regularization applied to the error term, which you get by adding the coefficient $\beta > 0$ to each of the diagonal entries of the $W^TW$ matrix, but this is just a guess from what I know of ridge regression. – Clarinetist Jul 04 '18 at 19:42
@Clarinetist And why does one consider $W^T W$? – mavavilj Jul 04 '18 at 19:51
I reread your question, and actually... I think I'm wrong. My guess is that it's actually the matrix $\mathbf{g}^{T}\mathbf{g}$, where $\mathbf{g}$ is the matrix of $\mathbf{g}(x_i)$s. But for all I know, I could be completely wrong. To understand the perspective I'm coming from with this, you should read this answer. – Clarinetist Jul 04 '18 at 19:54
The motivation for adding that error term is to make the matrix $\mathbf{g}^{T}\mathbf{g}$ invertible by adding $\beta >0$ to its diagonals. – Clarinetist Jul 04 '18 at 19:55

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