Euler Characteristics of Random Manifolds

What the math gives to ML

The paper gives an exact combinatorial identity for the expected Euler characteristic of a random codimension-one level complex in a simplicial host: \(\mathbb{E}[\chi(H)]=2-2K(G)-\chi(G)\). The transferable asset is a cheap global topological target determined entirely by the host complex's f-vector, which can serve as a calibration loss for neural scalar fields on meshes or graphs. A practical adaptation is to perturb network-produced vertex scores, extract random threshold interfaces, and regularize their empirical Euler characteristic toward the host-predicted value; this tests whether learned fields produce topologically plausible decision boundaries without requiring persistent-homology computations.