Moment comparisons, Sudakov inequalities and entropy of centroid bodies

What the math gives to ML

The paper provides explicit comparisons between moments of any gauge under an isotropic log-concave distribution and under a Gaussian reference, with a dimension-dependent but nonasymptotic moment window. The transferable asset is a principled way to constrain feature-norm tails after whitening: low and moderate moments should remain within a controlled factor of Gaussian moments, while excessively large orders are avoided. This suggests a cheap activation regularizer that monitors a small set of feature moments and penalizes departures from the theorem's admissible window, with the theorem serving as a calibration target rather than an unconditional guarantee because neural activations need not be log-concave.