Sharp stability for the (B)-theorem

What the math gives to ML

The paper identifies a sharp spectral signature of near-equality in the Gaussian (B)-theorem: the covariance eigenvalues of the Gaussian restricted to a symmetric convex body are polarized, with each principal variance close either to 0 or to 1. This suggests a differentiable alternative to ordinary low-rank regularization: force latent representations or stochastic gates to separate into nearly inactive directions and nearly full-variance directions, producing an adaptive subspace rather than uniformly shrinking all dimensions. The most practical transfer is a covariance-spectrum regularizer combined with a rank or memory budget, tested in bottleneck layers, mixture-of-experts routers, or latent diffusion models. The theorem itself applies only to Gaussian restrictions of symmetric convex sets, so the neural version should be treated as a falsifiable inductive bias rather than a guarantee for arbitrary activations.