Fourth-Moment Geometry of Rademacher Sums

arXiv:2608.17802v1 Announce Type: new Abstract: Let arepsilon_1,ldots,arepsilon_n be independent Rademacher signs and let a=(a_1,ldots,a_n)inR^n satisfy the normalization below. For the normalized Rademacher sum, we determine how its higher moments depend on the fourth-order mass. Combining a sharp fixed-q moment envelope with a separate argument below the convexity threshold gives the Gaussian stability inequality for the full range pgeq4 of this linear-in-q bound. The same fourth-order framework determines the sharp finite dimensional L_p/L_4 Khintchine constant for pgeq5, with the flat coefficient vector as the extremizer. These results settle the conjectures of Jakimiuk and of Baranski, Murawski, Nayar, and Oleszkiewicz stated below. We also prove Jakimiuk's conjectured quadratic stability estimate at p=3. The resulting bounds retain information about sparsity and effective dimension, with applications to Rademacher random projections and randomly signed errors; those applications are not developed further here. Their Laplace-transform form also gives coefficient-sensitive tail bounds. The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol.

On the Rademacher Complexity of Graph Neural Networks: Unifying Expressivity and Geometry

Source: arXiv cs.LG | 2026-08-19

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