Hilbertian Kahane--Salem--Zygmund Inequalities: Extremizers and Quantitative Gaps

What the math gives to ML

The paper identifies signed multilinear tensors whose operator norm is essentially minimal, with exact equality governed by Hadamard matrices and Hurwitz–Radon families of pairwise anticommuting orthogonal matrices. This gives a directly implementable way to build parameter-free or low-parameter bilinear layers whose induced norm is exactly controlled, rather than relying on spectral normalization or repeated power iterations. The most promising transfer is a norm-preserving signed bilinear feature mixer: precompute a family of structured sign matrices, use them to map two feature vectors to a third, and learn only scalar gains or surrounding projections. Such layers could improve stability in deep multiplicative networks while requiring only fast Hadamard-like matrix-vector products.