Second-order information, in the form of Hessian- or Inverse-Hessian-vector products, is a fundamental tool for solving optimization problems. Recently, there has been a tremendous amount of work on utilizing this information for the current compute and memory-intensive deep neural networks, usually via coarse-grained approximations (such as diagonal, blockwise, or Kronecker-factorization). However, not much is known about the quality of these approximations. Our work addresses this question, and in particular, we propose a method called ‘WoodFisher’ that leverages the structure of the empirical Fisher information matrix, along with the Woodbury matrix identity, to compute a faithful and efficient estimate of the inverse Hessian. Our main application is to the task of compressing neural networks, where we build on the classical Optimal Brain Damage/Surgeon framework (LeCun et al., 1990; Hassibi and Stork, 1993). We demonstrate that WoodFisher significantly outperforms magnitude pruning (isotropic Hessian), as well as methods that maintain other diagonal estimates. Further, even when gradual pruning is considered, our method results in a gain in test accuracy over the state-of-the-art approaches, for standard image classification datasets such as CIFAR-10, ImageNet. We also propose a variant called ‘WoodTaylor’, which takes into account the first-order gradient term, and can lead to additional improvements. An important advantage of our methods is that they allow us to automatically set the layer-wise pruning thresholds, avoiding the need for any manual tuning or sensitivity analysis.