Explaining Near-Zero Hessian Eigenvalues Through Approximate Symmetries in Neural Networks

The Hessian of the training loss governs the local geometry of the loss landscape, yet despite existing explanations for its largest eigenvalues, the origin of the vast multitude of vanishingly small eigenvalues remains elusive. We argue that the bulk consists of the weakly lifted pseudo-Goldstone modes of the continuous symmetries of the network parametrization. In deep linear networks these symmetries are exact: they generate flat directions and hence exact zero modes, whose eigenvectors we construct explicitly. Introducing a ReLU nonlinearity as a perturbation, we show that it breaks these symmetries weakly and explicitly. Resolving the spectrum at the level of eigenvectors, we find that the high-curvature directions are orthogonal to the symmetry subspace, while the bulk lies almost entirely within it. We demonstrate the mechanism in a two-layer ReLU student--teacher model and in a network trained on CIFAR-10. A convolutional example demonstrates that the same diagnostic extends beyond fully connected layers. Together, these results link the Hessian bulk to weakly broken symmetries and clarify the origin of near-zero modes.