Recurrent Network Redundancy Explored with Schur Coordinates

Simon Dr\"ager· June 18, 2026 View original

Summary

This paper investigates functional redundancy in recurrent neural networks (RNNs) by analyzing their weight space using ordered real Schur coordinates. It identifies task-restricted approximate functional invariances, showing that certain nonnormal Schur couplings can be removed without significant performance loss on specific tasks, while others are crucial.

Recurrent neural networks often possess significant functional redundancy within their weight space. This means that certain modifications to a recurrent matrix might have little effect on the network's output for a given task, while other changes of similar magnitude could severely disrupt its behavior. Researchers explored this phenomenon in one-layer tanh RNNs by employing ordered real Schur coordinates. This mathematical approach helps to separate spectral blocks from directed nonnormal couplings, providing a diagnostic basis for targeted ablations that maintain the network's input and readout maps. The study found that the specific nonnormal Schur couplings that can be removed without significant performance degradation vary depending on the task and the trained solution. This suggests the existence of approximate functional invariances that are specific to the task rather than universal symmetries of the recurrent weight space, offering insights into how RNNs compute.

Why it matters

Understanding these redundancies can lead to more efficient and robust RNN designs, potentially enabling model compression, improved interpretability, and better generalization by identifying and leveraging the critical components of recurrent computations.

How to implement this in your domain

  1. 1Investigate Schur decomposition as a diagnostic tool for analyzing trained RNNs in your applications.
  2. 2Experiment with structured ablations based on Schur coordinates to identify redundant components in your models.
  3. 3Apply insights from functional invariances to optimize RNN architectures for specific tasks.
  4. 4Develop techniques for pruning or compressing RNNs by removing identified non-critical couplings.
  5. 5Use this understanding to improve the robustness of RNNs against perturbations and adversarial attacks.

Who benefits

AI EngineeringMachine Learning ResearchSoftware DevelopmentRobotics

Key takeaways

  • Recurrent neural networks exhibit functional redundancy in their weight space.
  • Schur coordinates provide a method to analyze and identify these redundancies.
  • Task-restricted symmetries allow for targeted removal of non-critical network components.
  • Understanding these invariances can lead to more efficient and robust RNN designs.

Original post by Simon Dr\"ager

"arXiv:2606.18457v1 Announce Type: new Abstract: Recurrent networks can contain substantial functional redundancy in weight space: changing a recurrent matrix may leave the input-output rollout nearly unchanged on a task distribution, while similar-scale changes can destroy the sa…"

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