Riccati-Gated Observation Skipping

1. Exact integration point: use a recurrent state-space model with latent update z_(t+1) = f_theta(z_t,u_t) and an expensive observation pathway e_t = Encoder_phi(y_t). Run the observation pathway only when the scheduler requests an update; otherwise roll out f_theta for d steps. Maintain a diagonal covariance P for the first implementation, then test a low-rank-plus-diagonal version. 2. Pseudocode: at every scheduling time, evaluate the latent Jacobian A using automatic differentiation, estimate H from the observation head, and estimate Q from the moving covariance of one-step latent residuals z_(t+1) - f_theta(z_t,u_t). Estimate R from sensor residuals or learn a positive diagonal parameter. For each candidate d from 1 through d_max, compute A_d by repeated multiplication, accumulate Q_d using Q_d <- Q_d + A^(d-1) Q (A^(d-1))^T, solve the Riccati equation by fixed-point iteration, and select the largest feasible d. During skipped steps, propagate P <- A P A^T + Q. After an observation, apply P_plus <- P - P H^T (H P H^T + R)^(-1) H P. 3. Taken directly from the paper are the equivalent matrices, covariance propagation rule, Riccati equation, and covariance-threshold decision. Estimated empirically are local Jacobians, Q, R, and the threshold tau; tau can be calibrated from validation prediction-error quantiles. 4. First cheap experiment: train a GRU or diagonal neural state-space model on Moving-MNIST or Lorenz-96 sequences, using an observation encoder at every frame as baseline. Compare fixed strides d = 1, 2, 4, and 8 against Riccati-gated skipping at equal latent dimensions. Measure encoder calls, rollout error, and empirical latent error covariance. The quantitative prediction is that the selected stride decreases sharply as the largest eigenvalue of A approaches one, and predicted versus empirical g(P_d) should agree within approximately 20 percent when local linearization is accurate. If measured g(P_d) exceeds tau, the next observation interval must use a smaller d.