We introduce a fast algorithm for entry-wise evaluation of the Gauss-Newton Hessian (GNH) matrix for the fully-connected feed-forward neural network. The algorithm has a precomputation step and a sampling step. While it generally requires $O(Nn)$ work to compute an entry (and the entire column) in the GNH matrix for a neural network with $N$ parameters and $n$ data points, our fast sampling algorithm reduces the cost to $O(n+d/epsilon^2)$ work, where $d$ is the output dimension of the network and $epsilon$ is a prescribed accuracy (independent of $N$). One application of our algorithm is constructing the hierarchical-matrix (H-matrix) approximation of the GNH matrix for solving linear systems and eigenvalue problems. It generally requires $O(N^2)$ memory and $O(N^3)$ work to store and factorize the GNH matrix, respectively. The H-matrix approximation requires only $O(N r_o)$ memory footprint and $O(N r_o^2)$ work to be factorized, where $r_o ll N$ is the maximum rank of off-diagonal blocks in the GNH matrix. We demonstrate the performance of our fast algorithm and the H-matrix approximation on classification and autoencoder neural networks.
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