---
title: "persistent homology dimension (PHD)"
type: "entity"
entity_kind: "concept"
status: "hub"
canonical_name: "persistent homology dimension"
aliases: ["PHD"]
first_seen: "2026-07-12T00:00:00.000Z"
writer_model: "claude-sonnet-5"
connects_to: ["persistent homology","generalization bounds","Tolga Birdal","topological data analysis","fractal dimension (optimizer trajectories)"]
---


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]]