Discrete Faà di Bruno via Möbius Inversion

What the math gives to ML

The paper supplies an exact finite-difference chain rule for arbitrary compositions, expressed as a sum over set coverings rather than only disjoint partitions. This exposes how an input interaction of order k is assembled from inner-network interactions on subsets of directions and outer-network interactions among those inner differences, including overlap terms that disappear in the infinitesimal limit. A practical transfer is to use low-order Boolean finite differences as a compositional interaction regularizer or diagnostic, separating nonlinearities caused inside a network from nonlinearities caused by its downstream readout. The method is most suitable for small k, where it gives perturbation-scale-aware control without requiring Hessian computation.