Efficient computation of simplicial weights for provably faithful TOPF

Develop efficient algorithms to compute simplicial weights for the symmetric weighted Hodge Laplacian L_k^w = W^{1/2} B_{k-1} B_{k-1}^⊤ W^{1/2} + W^{-1/2} B_k B_k^⊤ W^{-1/2} within the TOPF pipeline, so that the resulting harmonic representatives and their aggregated point-level features are provably the most faithful topological point features for a given point cloud and its α-/Vietoris–Rips filtration.

Background

TOPF projects persistent homology generators to the harmonic subspace of a simplicial complex using the (possibly weighted) Hodge Laplacian, and then aggregates simplex-level values to point-level features. The paper introduces two heuristic weighting schemes—one based on the count of incident higher-dimensional simplices (w_Δ) and another based on effective Hodge resistance (w_R)—to improve robustness and homogeneity of harmonic representatives.

While these heuristics are practical, the authors emphasize the need for principled, efficient computation of simplicial weights that come with provable guarantees of producing the most faithful topological point features. Achieving such guarantees would directly strengthen the theoretical foundation and practical reliability of TOPF across diverse datasets.