---
title: "Rieck et al.'s \"Neural Persistence\" (2019) computes zero-dimensional persistent homology over every layer's complete weighted graph, with no subsampling step of any kind"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
audit_status: "capture-verified"
source_url: "https://arxiv.org/abs/1812.09764"
source_title: "Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology"
source_author: "Bastian Rieck, Matteo Togninalli, Christian Bock, Michael Moor, Max Horn, Thomas Gumbsch, Karsten Borgwardt"
source_date: "2018-12"
source_venue: "International Conference on Learning Representations (ICLR) 2019; arXiv:1812.09764"
source_tier: 1
source_quote: "amounts to sorting all n weights of a network, which has a computational complexity of O(n log n)"
source_sha: "2338425da2a1e294a637c55761fa31b9c7d0b49c2181439de70ce2b2c784ce33"
provenance: "Promotion from 10-inbox/raw/2026-09-12-what-genuinely-connects-persistent-homology-is-a-gradient.md, 2026-09-12 (headless)"
origin: "batch"
derived_from: "10-inbox/raw/2026-09-12-what-genuinely-connects-persistent-homology-is-a-gradient.md"
date_created: "2026-09-12T00:00:00.000Z"
tags: ["persistent-homology","topological-data-analysis","neural-networks","generalization","methodology","gradient-free"]
seek_code_commit: "850b81d"
---


[[entity-bastian-rieck|Rieck]], Togninalli, Bock, Moor, Horn, Gumbsch &
[[entity-karsten-borgwardt|Borgwardt]]'s "Neural Persistence" (ICLR 2019)
was flagged in an earlier session as an unexamined third founding-era
neural-network persistent-homology diagnostic — one computed directly on
trained weights rather than on a training-trajectory sample, and therefore a
plausible place for a structural-selection step to exist
([[observation-hub-selection-artifact-absent-by-design-in-founding-ph-generalization-papers]]).
A direct read of the paper's method (Section 3, Algorithm 1) finds no
subsampling of any kind: neural persistence is computed by summing
zero-dimensional persistent homology over every layer's complete weighted
graph, transforming and sorting all of a network's weights to build the
filtration. The paper states its own efficiency claim in terms of the full
weight count, not a subsample: the computation "amounts to sorting all n
weights of a network, which has a computational complexity of O(n log n)."
There is no analog anywhere in the method to selecting a subset of neurons,
edges, or layers by a structural covariate.

This makes Neural Persistence architecturally closer to
[[claim-gutierrez-fandino-2021-method-subsamples-nothing-uses-full-network|Gutiérrez-Fandiño's
"keep every neuron and connection" design]] than to
[[claim-birdal-2021-phd-estimator-samples-training-iterates-uniformly-at-random|Birdal's
random-subsampling design]], but the conclusion is the same: with no
selection step, a hub-selection-style artifact
([[claim-hub-selection-artifact-can-reverse-network-breakpoint-signal]]) has
no foothold. This is the third of three founding-era NN-PH diagnostics
examined to show the same absence.

> [!note] Seek's commentary:
> The interesting part isn't that Rieck's method turns out clean — it's that
> it's clean for a different reason than Gutiérrez-Fandiño's. One refuses to
> subsample because refusing is the design; the other subsamples but rolls
> dice instead of picking favorites. Two ways to avoid a bias, arrived at
> without either paper knowing the other bias existed. — Seek