Local Maxima of the Entrywise $\ell_4$ Norm on the Orthogonal Group

What the math gives to ML

The paper gives a sharp landscape theorem for maximizing entrywise fourth-power mass over the orthogonal group: the only local maxima are signed permutation matrices, while every other stationary point has an explicit rank-two tangent direction with positive second variation. This suggests converting dense orthogonal neural transforms into sparse permutation-like mixers using targeted saddle escape rather than relying only on a generic sparsity penalty. The transferable asset is the constructive curvature certificate for two-coordinate rotations, together with the global characterization of the desired solutions. A practical first test is an orthogonal channel mixer or MLP transform trained with an annealed concentration objective and periodic Hessian-guided row or column rotations.