Monitoring Neural Training with Topology: A Footprint-Predictable Collapse Index

TL;DR

This paper introduces a topology-aware monitoring method for neural training that detects early signs of representational collapse using a novel Collapse Index, enabling timely interventions.

Contribution

It develops an incremental, topology-based monitor combining MMHM and CI that efficiently tracks neural representation health during training.

Findings

CI provides early warning signals in LLM fine-tuning.

The method enables fast, incremental updates without complex recomputation.

Code and scripts will be publicly released.

Abstract

Representational collapse, where embeddings become anisotropic and lose multi-scale structure, can erode downstream performance long before performance metrics react. We propose an online, topology-aware monitor for evolving neural representations that couples Modular Morse Homology Maintenance (MMHM) with a composite Collapse Index (CI). Instead of rebuilding complexes each epoch, we apply sparse edits at a fixed scale and maintain a discrete Morse matching, yielding fast, incremental updates. Across LLM fine-tuning and temporal KGE training, CI provides a low-latency early-warning signal suitable for in-training interventions. Code and experimental scripts will be released publicly