Scale-Slack Homology Consistency

Integrate this at the output of a point-cloud or graph encoder, before the task head. For each input batch, create two stochastic views using augmentations such as point dropout, jitter, edge masking, or feature masking. Let z_i^(1), z_i^(2)∈R^d be embeddings of corresponding vertices, and compute δ=max_i||z_i^(1)-z_i^(2)||_2 after matching vertices; use a detached copy of δ when constructing filtration scales so the model cannot reduce the penalty merely by manipulating the scale. Use this MVP loop: (1) encode both views; (2) choose a small set E={ε_1,...,ε_m} from pairwise-distance quantiles in the first view; (3) compute H_0 and H_1 persistence diagrams of Vietoris–Rips complexes at these scales, using a sparse k-nearest-neighbor graph and a capped maximum simplex dimension; (4) compare the diagram or persistence-rank representation from view 1 at ε with view 2 at ε+δ, padding the latter scale to the next available grid point; (5) add λ_top L_top to the supervised loss; (6) optionally clip the regularizer to pairs with δ<ε/2, because ω(ε)=max(ε−2δ,0) otherwise becomes zero. The exact mathematical quantities are the filtration inclusion and δ scale slack; persistence diagrams, distance quantiles, and weights w are engineering approximations. First test a small PointNet or DGCNN on ModelNet40 and a two-layer GCN on Cora, comparing supervised training with and without the regularizer under clean data and stronger geometric perturbations. Track task accuracy, short-lived H_1 bars, connected-component consistency, and corruption accuracy. A successful signal is unchanged or improved clean accuracy with substantially smaller topology disagreement and better corruption robustness at equal training FLOPs. Use coefficients in Z_2 so boundary matrices are simple, and begin with batches of 16–32 objects and only H_0/H_1 to keep persistence computation cheap.