Persistence-diagram distance between successive neural-network training states correlates with validation accuracy, enabling generalization estimates without a validation set (Gutiérrez-Fandiño et al. 2021)

Gutiérrez-Fandiño, Pérez-Fernández, Armengol-Estapé & Villegas, "Persistent Homology Captures the Generalization of Neural Networks Without A Validation Set" (arXiv:2106.00012, May 2021), represent a network's state at each point in training as a simplicial complex and track its persistence diagram as training proceeds. Their central empirical finding, stated verbatim, is that "the PH diagram distance between consecutive neural network states correlates with the validation accuracy" — across the architectures and datasets they test. Because the signal is read from the topology of the network's own state trajectory, it requires no held-out labeled data, which is the paper's explicit framing: generalization can be estimated without a validation set.

This is a correlational finding, not a formal bound — it should be read as the companion to, not a restatement of, claim-birdal-2021-persistent-homology-dimension-bounds-generalization, which proves generalization error is bounded by a persistent-homology-derived quantity under weaker assumptions than prior fractal-dimension approaches. Both papers rest on the same underlying object: distance between persistence diagrams, standardly computed via the Wasserstein or bottleneck metric.

See observation-persistent-homology-gradient-free-bridge-widrow-byzantine for how this gradient-free diagnostic bridges into cliodynamics via claim-roman-byzantine-trade-network-decoupled-after-1082-chrysobull, and question-tda-neural-net-sampling-artifact-risk for an open question about whether this method inherits sampling-artifact risks documented elsewhere in persistent-homology applications.