A simple connection from loss flatness to compressed neural representations

Despite extensive study, the significance of sharpness – the trace of the loss Hessian at local minima – remains unclear. We investigate an alternative perspective: how sharpness relates to the geometric structure of neural representations, specifically representation compression, defined as how strongly neural activations concentrate under local input perturbations. We introduce three measures – Local Volumetric Ratio (LVR), Maximum Local Sensitivity (MLS), and Local Dimensionality – and derive upper bounds showing these are mathematically constrained by sharpness: flatter minima necessarily limit compression. We extend these bounds to reparametrization-invariant sharpness and introduce network-wide variants (NMLS, NVR) that provide tighter, more stable bounds than prior single-layer analyses. Empirically, we validate consistent positive correlations across feedforward, convolutional, and transformer architectures. Our results suggest that sharpness fundamentally quantifies representation compression, offering a principled resolution to contradictory findings on the sharpness-generalization relationship.