Zigzag persistent homology tracks how topological features (p-dimensional holes) of a real pretrained LLM's representation point-cloud persist and evolve across layers, and the resulting topological-stability signature identifies layers safe to prune

Gardinazzi, Viswanathan, Panerai, Ansuini, Cazzaniga & Biagetti (2024) introduce zigzag persistent homology as a tool for tracking topological features of the layer-by-layer representation point-cloud in real pretrained language models (Pythia family, up to Pythia-6.9B), a genuine topological-data-analysis measurement applied directly to natural model activations rather than a toy or synthetic setting. By connecting successive layers' point clouds into a zigzag filtration, the persistence of topological features (independent loops/holes) across the network's depth becomes measurable, and layers whose removal would least disturb this persistent topological structure are flagged as safe candidates for pruning — turning a purely descriptive topological measurement into a practical model-compression criterion validated against downstream task performance. Clears scope on quantified geometric-shape measurement (persistent homology of the representation point-cloud) on a real pretrained LLM's natural activations; purely observational-diagnostic (the pruning application is a downstream use of the measurement, not a geometry-tied steering intervention). See [[manifolds]] for the topological-measurement precedent this shares with `a-141-hypothesis-automated-audit-of-scgpt-and-geneformer-finds-significant-persistent-homology-in-nearly-every-layer-and-a-cca-canonical-correlation-of-0-80-between-the-two-models` and `magai-ayzenberg-2022-persistent-homology-and-intrinsic-dimension-of-the-last-layer-data-manifold-track-generalization-in-cnns`.