arXiv:2608.18561 2026 Dynamics 2 ideas extracted · analyzed Sep 1, 2026
The paper introduces the robust instability radius, defined as the smallest norm of a stable diagonal heterogeneous perturbation that can stabilize an otherwise unstable network. This is a transferable distance-to-stability mechanism for recurrent and state-space neural layers, where diagonal perturbations represent channel-wise gains or leaks. The most useful implementation is a training monitor and regularizer based on adversarial diagonal perturbations of the hidden-state Jacobian. Its falsifiable signature is a stability transition when the estimated perturbed spectral radius crosses one, together with exponential perturbation decay below that boundary.
Apply the paper's distance-to-stabilization concept to the Jacobian of a recurrent or state-space neural layer. Estimate the smallest channel-wise diagonal perturbation that makes the local hidden-state dynamics contractive, then penalize models whose estimated radius is below a target margin.
When a recurrent or graph coupling matrix is approximately rank one, replace expensive full spectral monitoring with a scalar small-gain controller. Adapt a residual mixing coefficient so that the dominant coupled mode remains below a prescribed contraction threshold.