Rank-One Small-Gain Recurrent Controller

Integration point: place the controller in an RNN, state-space layer, graph neural layer, or learned optimizer whose update contains a dense coupling matrix A. Every few optimizer steps, estimate the leading singular vectors using one or two power iterations. Normalize u and v, compute the dominant scalar coefficient c = v-transpose u, and estimate the residual norm with a few power iterations on A minus u v-transpose. Choose gamma by a one-dimensional grid search or analytic clipping so that q_gamma = L_phi times the absolute value of 1 minus gamma plus gamma c, plus the residual norm, remains below a target q_target less than one. Then apply x_next = phi(((1 minus gamma) I plus gamma u v-transpose) x plus B u_input). Use L_phi equal to one for tanh or ReLU as a conservative bound, and use the active-mask estimate for a less conservative ReLU bound. The paper contributes the rank-one reduction and small-gain condition; the rank-one factorization and residual gain are estimated from current weights. First experiment: compare this controller with unconstrained recurrence and standard spectral normalization on sequential MNIST and the adding problem. Sweep q_target over 0.8, 0.9, 0.99, and 1.05. Measure the ratio of perturbation norms at successive steps and compare it with q_target. Prediction: perturbations decay exponentially when the estimated q is below one, with log decay slope close to log(q); long-horizon instability should appear sharply as q crosses one.