arXiv:2607.18128 2026 Dynamics 2 ideas extracted · analyzed Aug 30, 2026
The paper isolates a sharp difference between acyclic and cyclic feedback in entropy-regularized equilibria: a nonnegative causal graph with longest path L gives only polynomial sensitivity in temperature, while a positive feedback cycle creates exponential susceptibility through the Perron mode. Its linear response is represented by a matrix-exponential resolvent, making the spectral radius of the feedback operator the key operational quantity. This suggests temperature control and Jacobian regularization for recurrent attention, iterative routing, and deep residual stacks, where positive feedback between logits and representations can cause unstable sharpening or mode collapse. The most practical transfer is to estimate a nonnegative local feedback operator and keep its temperature-scaled spectral gain below a stability threshold, while separately testing whether deliberately enhanced cycles improve specialization.
Use an online estimate of the positive feedback gain among logits, routing probabilities, and representations to adjust the softmax temperature. Increase temperature when the estimated cyclic gain approaches the instability regime, preventing exponential amplification and router collapse without globally weakening all layers.
Penalize short positive feedback cycles in an iterative neural module by suppressing products of absolute Jacobian blocks around the cycle. This targets the mechanism responsible for exponential temperature sensitivity rather than merely penalizing the total Jacobian norm, allowing strong feed-forward paths while controlling recurrent amplification.