Fast generation of spectrally-shaped disorder, on the sphere

What the math gives to ML

The paper provides a constructive spectral optimization mechanism for point sets on the sphere: represent a weighted point density with spherical harmonics, then optimize the harmonic power spectrum to impose target pair correlations. Its transferable asset is a fast differentiable repulsion or spectral-shaping regularizer for neural representations constrained to a hypersphere, such as class prototypes, token embeddings, spherical codebooks, or manifold samples. The most promising implementation is to penalize selected spherical-harmonic modes of normalized embeddings while retaining a low-cost real-space repulsion term, with a measurable prediction that the empirical angular power spectrum follows the prescribed target and that low-mode suppression produces hyperuniform-like variance scaling.