Curvature Graph Mergeability: Predicting Model Compatibility from Per-Layer Hessian Geometry

Abstract

Merging independently trained neural networks --- via weight averaging, model soups, or task-vector arithmetic --- is increasingly central to large-scale model reuse, yet the success of any merge hinges on whether two parameter vectors occupy the same loss basin, a property currently certified only by expensive multi-point interpolation barriers. We introduce Curvature Graph Mergeability (CGM), a scoring framework that predicts the interpolation barrier from only cheap, per-layer Hessian curvature features at each endpoint. CGM extracts two scalar curvature statistics per layer using Hutchinson's trace estimator and the power method, organizes them into a chain-structured factor graph whose latent compatibility states encode qualitative regimes of layer alignment, and computes a mergeability score from the negative variational free energy of the chain. For deep linear networks under balanced factorization, we prove that the interpolation barrier is upper-bounded by a multilinear function of per-layer curvature mismatch, and we identify a diagonal-dominance condition under which the score is monotone in mismatch; the bound is stated as an implicit joint bound on the per-layer mismatch vector, with the regime of validity made explicit. Across four meta-training architectures and four held-out OOD architectures (ResNet-110, ViT-B/ImageNet, WRN-28-10, 3-layer LSTM), CGM attains Spearman ?=0.86 on leave-models-out splits and ?=0.81 pooled OOD, outperforming zero-order distance, CKA, Fisher overlap, REPAIR, midpoint-barrier, and architecture-agnostic curvature-MLP baselines at Holm--Bonferroni-corrected p<0.01. It is approached only by Git Re-Basin + barrier (?=0.90), which costs ~14,500× more per pair; CGM closes the gap to permutation-based merging at 35× amortized speedup over direct 11-point barrier when screening 50 candidates, with negligible memory overhead. Per-layer posterior marginals additionally pinpoint specific layers that act as merge bottlenecks.