Nested-Cone Latent Dynamics

Integrate this into a small latent state-space model or GRU. Represent each hidden state by an ellipsoid \(E_t=\{c_t+P_tv:\|v\|_2\leq1\}\), where \(c_t\) is the center and \(P_t\) is a diagonal positive radius matrix. The network receives \((c_t,P_t,u_t)\) and proposes the next center and radius. For each training sequence, sample boundary points \(x_i=c_{t+1}+P_{t+1}v_i\), propagate them through the transition \(F_\theta(x_i,u_t)\), and penalize predecessor-inclusion violations with \(\max(0,\|F_\theta(x_i,u_t)-c_t\|_{P_t^{-1}}-1+m)\), where \(m>0\) is a chosen margin and \(\|x\|_{P^{-1}}=\|P^{-1}x\|_2\). Add a contraction penalty \(\max(0,\|P_{t+1}\|_2/\|P_t\|_2-\rho)\), targeting \(\rho<1\). The training loop is: encode \(E_0\); propose \(E_{t+1}\); sample ellipsoid boundary points; propagate them; compute inclusion and contraction penalties; decode the center; backpropagate. The paper supplies the nested-set mechanism; empirical estimation supplies \(\rho\), inclusion margins, and local Lipschitz constants. The first cheap experiment should use a 2D nonlinear pendulum or Lorenz forecasting task with a GRU baseline, comparing one-step and 100-step rollouts under Gaussian input noise and random weight perturbations. Estimate \(r_t=\operatorname{diam}(E_{t+1})/\operatorname{diam}(E_t)\). The prediction is \(\operatorname{diam}(E_t)\approx\rho^t\operatorname{diam}(E_0)\) when the inclusion margin is positive, with a sharp increase in rollout error when the measured contraction ratio crosses \(r=1\) or when \(\min_t m_t\leq0\). Test whether perturbation error agrees within 20 percent with \((L_u\varepsilon_u+L_\theta\varepsilon_\theta)/(1-r)\).