Approximate Spectral Decomposition of Fisher Information Matrix for Simple ReLU Networks

We argue the Fisher information matrix (FIM) of one hidden layer networks with the ReLU activation function. Let $W$ denote the $d \times p$ weight matrix from the $d$-dimensional input to the hidden layer consisting of $p$ neurons, and $v$ the $p$-dimensional weight vector from the hidden layer to the scalar output. We focus on the FIM of $v$, which we denote as $I$. When $p$ is large, under certain conditions, the following approximately holds. 1) There are three major clusters in the eigenvalue distribution. 2) Since $I$ is non-negative owing to the ReLU, the first eigenvalue is the Perron-Frobenius eigenvalue. 3) For the cluster of the next maximum values, the eigenspace is spanned by the row vectors of $W$. 4) The direct sum of the eigenspace of the first eigenvalue and that of the third cluster is spanned by the set of all the vectors obtained as the Hadamard product of any pair of the row vectors of $W$. We confirmed by numerical simulation that the above is approximately correct when the number of hidden nodes is about 10000.