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Weight Magnitude and Direction Play Dissociable Roles in Grokking

Truong Xuan Khanh· July 9, 2026 View original

Summary

This research introduces "cross-trajectory chimera interventions" to show that weight direction carries transferable circuit identity in grokking, while weight magnitude primarily influences the susceptibility to overwriting. This dissociates their roles in how neural networks learn and generalize.

This paper delves into the mechanisms behind "grokking," the phenomenon where neural networks generalize long after achieving high training accuracy. The authors introduce a novel experimental technique called "cross-trajectory chimera interventions." This method involves taking two independently trained networks, splitting their weight vectors into norm (magnitude) and unit direction, and then recombining one run's norm with another's direction before continuing training. Through these interventions, the study reveals that weight direction carries a distinct, transferable circuit identity. Implanting a donor network's weight direction into a recipient network, while using the recipient's weight norm, consistently drives the recipient to adopt the donor's specific circuit solution. In contrast, weight magnitude primarily exerts a modest, distributed delay effect and does not convey any identity signal. The transfer of circuit identity is found to be threshold-like, predicted by the recipient's norm, indicating that norm governs how easily a learned identity can be overwritten.

Why it matters

Understanding the distinct roles of weight magnitude and direction provides deeper insights into how neural networks learn, generalize, and form specific internal representations, which can inform the design of more robust and interpretable AI models.

How to implement this in your domain

  1. 1Consider the implications of weight initialization and regularization strategies on the learned circuit identity and generalization.
  2. 2Explore methods to explicitly control or manipulate weight direction during training to guide model learning towards desired solutions.
  3. 3Investigate how these findings apply to transfer learning scenarios, particularly in fine-tuning pre-trained models.
  4. 4Develop diagnostic tools to analyze the evolution of weight magnitude and direction in your own deep learning models.

Who benefits

AI/ML ResearchDeep Learning EngineeringAcademia

Key takeaways

  • Weight direction in neural networks carries a transferable circuit identity during grokking.
  • Weight magnitude primarily influences the susceptibility of a learned identity to being overwritten.
  • Cross-trajectory chimera interventions are a novel method for studying causal portability across network runs.
  • These findings offer deeper insights into the mechanisms of generalization in deep learning.

Original post by Truong Xuan Khanh

"arXiv:2607.06628v1 Announce Type: cross Abstract: Which properties of a partially trained network are causally portable to a different, independently trained network? Single-trajectory interventions show necessity within one run, not portability across runs. We introduce cross-tr…"

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