Trajectory-Certified Contractive RNN

1. Integration point: build a gated or plain RNN with hidden state x, but expose its recurrent linear map and activation feedback explicitly. For example, use x_{k+1}=Ax_k+B Phi(Cx_k+Du_k), setting w=Phi(v) and d=u. Apply the certificate during initialization, architecture search, or checkpoint selection; alternatively add a penalty when the certificate margin becomes negative. Use activations with known slope sectors, such as ReLU with [alpha,beta]=[0,1], hard-swish with a verified sector, or tanh with [-1,1]. 2. Pseudocode: collect a trajectory under Gaussian or multisine inputs with input Hankel matrix persistently exciting of order at least 2(n_x+1); record (x,w,d,v,e) if hidden states are available, otherwise reconstruct x with deterministic subspace identification. Form trajectory matrices and solve an SDP over P>0, lambda>=0, and gamma^2 enforcing the lifted dissipation LMI above. Normalize P by trace(P)=n_x. During training, periodically resolve the SDP on a short calibration trajectory and reject updates for which the smallest LMI eigenvalue is nonnegative. 3. Computed from the paper: the quadratic constraint, Lyapunov or dissipativity inequality, PE requirement, and similarity-invariant state reconstruction. Estimated empirically: trajectory matrices, hidden-state realization, activation sector if not analytically known, and the smallest eigenvalue margin. For noisy data, replace strict inequalities by a slack and bootstrap trajectories to estimate certificate confidence. 4. First cheap experiment: compare a vanilla Elman RNN, spectral-normalized RNN, and this certified RNN on sequential MNIST or adding-problem sequences, using identical parameter counts. Sweep a scalar recurrent gain s multiplying A, and independently inject Gaussian input disturbances. The quantitative prediction is a sharp certificate boundary at the largest s for which the SDP is feasible; beyond it, hidden-state perturbations should stop decaying and eventually grow. For certified models, estimate ||delta x_k|| and fit its decay; the LMI predicts geometric decay governed by the largest certified contraction factor rho_P<1, while the measured disturbance-to-output gain should remain below the SDP value gamma up to finite-sample error. Require the data-driven certificate to agree with a model-based SDP within 20 percent on a synthetic known RNN before testing accuracy.