Geometry-Induced Hodge Stars on Rips and Dowker--Rips Complexes

What the math gives to ML

The paper provides a concrete way to restore metric information to Vietoris–Rips and Dowker–Rips complexes through positive diagonal simplex weights, without changing the underlying topology. The key transferable object is the weighted Hodge Laplacian, whose kernel remains topologically determined while its nonzero spectrum responds to simplex geometry; this gives neural networks a geometry-aware operator on higher-order data rather than an unweighted combinatorial propagation rule. A particularly practical adaptation is to compute Euclidean simplex-volume weights on a point-cloud Rips complex and use the resulting sparse Hodge Laplacian as a fixed or learnably rescaled convolution operator. This is most promising for simplicial or point-cloud networks where ordinary Hodge layers treat a long, thin simplex and a well-shaped simplex identically.