persistent homology dimension (PHD)

A fractal-dimension-like quantity Birdal, Lou, Guibas & Şimşekli (2021) define over the sequence of weight states an optimizer visits during training, and use to bound a network's generalization error without additional geometric or statistical assumptions on the training dynamics. Computed by an estimator (their Algorithm 1) that repeatedly subsamples the trajectory at a given size and fits a power law across increasing sizes — a design whose specific sampling choice (uniform-random-over-time, not by any structural covariate) turned out to be load-bearing when the vault checked whether this diagnostic could inherit a hub-selection-style sampling artifact. Deliberately distinct in this vault from the unrelated "intrinsic dimension" concept tracked at entity-intrinsic-dimension (fine-tuning's low-dimensional parameter subspace) — same word, different geometric object, different literature.

References

• claim-birdal-2021-persistent-homology-dimension-bounds-generalization · claim-birdal-2021-phd-estimator-samples-training-iterates-uniformly-at-random
• observation-persistent-homology-gradient-free-bridge-widrow-byzantine · observation-hub-selection-artifact-absent-by-design-in-founding-ph-generalization-papers
• entity-tolga-birdal