I am working with this article:http://www.scholarpedia.org/article/Echo_state_network on echo state networks. Here they speak about a ridge regression for the ESN.
Normally the outputweight matrix will be calculated like this $W^{out}=(S^+D)'$ ($+$ denotes the pseudoinverse)but they introduced a new method for calculating $W^{out}=(R+\alpha^2I)^{-1}P$ where $R$ is $R=\frac{1}{n_{max}}S'S$ and $P=\frac{1}{n_{max}}S'D$ and $\alpha^2$ a nonnegative number.
My question is: What are the advantages of transforming the first equation into the second?
Okay so what I have found is: The advantage of using the second equation to calculate the output weight matrix, $W^{out}$, is that it allows for regularization of the solution. The term $\alpha^2I$ is added to the matrix $R$, where $\alpha^2$ is a nonnegative number, effectively adding a small amount of noise to the data, which can help to prevent overfitting. This is known as Tikhonov regularization or Ridge Regression. By adding this regularization term, it can also improve the generalization performance of the ESN.