Geometric Realizations of Capability Spaces of Large Models and Zigzag Persistent Homology
Abstract
We propose a topological model for capability changes during large-model training. The model represents knowledge states and preregistered capability probes as subcomplexes of a finite regular CW complex. We prove that attaching a single n-cell can only create a class in Hn or kill a class in Hn−1. With the reliability threshold and the training checkpoint as two parameter axes, capability gains and losses make the spaces along the training axis nonnested. Union or intersection bridges produce zigzag persistence modules, and we distinguish checkpoint, transition, and bridge-sensitive classes. A finite four-stage example is computed by boundary matrices. The framework records algebraic changes under a chosen encoding, and a homological change is called a candidate capability emergence only after robustness tests, null-model comparisons, and independent behavioral validation.
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BibTeX
@misc{liu2026geometric,
title = {{Geometric Realizations of Capability Spaces of Large Models and Zigzag Persistent Homology}},
author = {Yu-Zhe Liu and Shengda Liu and Jianyong Pi and Chao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04374},
eprint = {2610.04374},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04374},
}
RIS
TY - UNPB
TI - Geometric Realizations of Capability Spaces of Large Models and Zigzag Persistent Homology
AU - Yu-Zhe Liu
AU - Shengda Liu
AU - Jianyong Pi
AU - Chao Zhang
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04374
UR - https://arxiv.org/abs/2610.04374
AB - We propose a topological model for capability changes during large-model training. The model represents knowledge states and preregistered capability probes as subcomplexes of a finite regular CW complex. We prove that attaching a single n-cell can only create a class in H_n or kill a class in H_n−1. With the reliability threshold and the training checkpoint as two parameter axes, capability gains and losses make the spaces along the training axis nonnested. Union or intersection bridges produce zigzag persistence modules, and we distinguish checkpoint, transition, and bridge-sensitive classes. A finite four-stage example is computed by boundary matrices. The framework records algebraic changes under a chosen encoding, and a homological change is called a candidate capability emergence only after robustness tests, null-model comparisons, and independent behavioral validation.
ER -