When the noncommutative AM-GM inequality holds

arXiv:2610.04874v1 Announce Type: cross Abstract: In this note, we prove that the noncommutative AM-GM inequality holds if nge 2lceil m/2 rceil^2. The motivation comes from counterexamples constructed in [De Sa, Random reshuffling is not always better, NeurIPS2020]. The proof constructs a vertex measure based on the Chebyshev nodes on the Boolean cube to extract the distinct indices. The main difficulty is that the measure is not positive on non-integer nodes. The key technique comes from [Grigoriev, Complexity of Positivstellensatz proofs for the knapsack, Computational Complexity (2001)] and eventually transforms the problem into a quadrature estimate.

Source: arXiv cs.LG | 2026-10-06