A simplex-based measure of symmetry

What the math gives to ML

The paper supplies a computable convex-geometric diagnostic for the depth required to construct highly simplex-like polytopes from repeated Minkowski sums and convex-hull-of-union operations, the two operations induced by piecewise-linear ReLU constructions. Its key transferable asset is the explicit obstruction \(\rho_{\Delta}(P)\le 2^d-1\): a target polytope with large simplex asymmetry cannot be represented by a shallow compositional architecture of this type. The most practical neural-network use is an architecture-selection and training diagnostic for convex piecewise-linear MLPs, based on estimating \(\rho_\Delta\) on the convex hull of sampled gradients or latent states rather than blindly increasing width.