Birdal, Lou, Guibas & Şimşekli (2021) bound neural-network generalization error by the persistent-homology dimension of the training trajectory

Birdal, Lou, Guibas & Şimşekli, "Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks" (NeurIPS 2021; arXiv:2111.13171), make a formal connection between statistical learning theory and topological data analysis (TDA). Building on the observation that the trajectories of iterative optimizers can have fractal structure, they show that a network's generalization error "can be equivalently bounded in terms of a notion called the 'persistent homology dimension' (PHD)," and that, unlike prior fractal-based bounds, their approach "does not require any additional geometrical or statistical assumptions on the training dynamics." PHD is computed directly from the optimization trajectory — the sequence of weight states visited during training — using persistent-homology machinery from algebraic topology, and the paper provides an efficient estimator that scales to modern network sizes.

The bound is derived from training dynamics alone; it does not itself require a held-out validation set, though the paper's headline pitch is the bound's rigor rather than a validation-set-free workflow (contrast the correlational, explicitly validation-set-free framing of claim-gutierrez-fandino-2021-persistence-diagram-distance-tracks-generalization).

This is one anchor of a gradient-free reading of network structure: persistent homology characterizes the shape of a training trajectory with no derivative of a loss function required, in contrast to the gradient-dependent training that claim-widrow-abandoned-multilayer-training-until-1985-backprop shows failing for want of a differentiable nonlinearity. See observation-persistent-homology-gradient-free-bridge-widrow-byzantine for the fuller bridge, and question-tda-neural-net-sampling-artifact-risk for an open robustness question about PHD/PH-diagram-distance estimators.