Extended generalized permutahedra, and cointeracting bialgebras

What the math gives to ML

The paper supplies a combinatorial-geometric way to turn submodular set functions into polyhedral allocation objects whose regions are indexed by preorders, together with an explicit decomposition into faces and tangent cones. The transferable asset is not the Hopf/cointeraction formalism itself, but the resulting greedy, piecewise-linear structure: a score vector selects an ordered region, while submodularity enforces diminishing returns and coupled resource constraints. A promising neural-network use is a submodular mixture-of-experts router that replaces independent top-k decisions with globally budgeted expert allocations, retaining a fast greedy implementation and interpretable routing regions. The first experiment should focus on this router because it is directly implementable and has measurable load-balancing and capacity benefits.