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We consider deep classifying neural networks. We expose a structure in thederivative of the logits with respect to the parameters of the model, which isused to explain the existence of outliers in the spectrum of the Hessian.Previous works decomposed the Hessian into two components, attributing theoutliers to one of them, the so-called Covariance of gradients. We show thisterm is not a Covariance but a second moment matrix, i.e., it is influenced bymeans of gradients. These means possess an additive two-way structure that isthe source of the outliers in the spectrum. This structure can be used toapproximate the principal subspace of the Hessian using certain "averaging"operations, avoiding the need for high-dimensional eigenanalysis. Wecorroborate this claim across different datasets, architectures and samplesizes.

Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians