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

Alexander Kalinowski

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arXiv · 2604.26984

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

Alexander Kalinowski

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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

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Alexander Kalinowski. 2026-04-28. Monitoring Neural Training with Topology: A Footprint-Predictable Collapse Index. https://arxiv.org/abs/2604.26984

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