{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace an unconstrained recurrent or latent-dynamics transition with a transition whose local Jacobian is contractive in a learned or fixed positive-definite metric P. Add a hinge penalty whenever the induced P-norm of the Jacobian exceeds a target factor eta\u003c1, so perturbations and differences between trajectories decay after each step.",
 "formulas_latex": [
  "$$\\|x^{(1)}(t+1)-x^{(2)}(t+1)\\|_{P}\\leq\\eta\\|x^{(1)}(t)-x^{(2)}(t)\\|_{P},\\qquad \\|v\\|_{P}=\\sqrt{v^{\\top}Pv},\\quad 0\u003c\\eta\u003c1.$$",
  "$$J_{\\theta}(x)^{\\top}P J_{\\theta}(x)\\preceq \\eta^{2}P\\quad\\Longleftrightarrow\\quad \\left\\|P^{1/2}J_{\\theta}(x)P^{-1/2}\\right\\|_{2}\\leq\\eta.$$",
  "$$\\mathcal{L}_{\\mathrm{contr}}(x)=\\left[\\left\\|P^{1/2}J_{\\theta}(x)P^{-1/2}\\right\\|_{2}^{2}-\\eta^{2}\\right]_{+}^{2},\\qquad \\mathcal{L}=\\mathcal{L}_{\\mathrm{task}}+\\lambda\\,\\mathbb{E}_{x\\sim\\mathcal{D}}[\\mathcal{L}_{\\mathrm{contr}}(x)].$$"
 ],
 "id": 307,
 "implementation": "Integrate this at the recurrent or state-space transition, not at the readout: for an RNN use h_{t+1}=f_theta(h_t,u_t), and for a latent world model use z_{t+1}=f_theta(z_t,a_t). Start with P=I for a cheap baseline, then optionally parameterize P=LL^T+epsilon I and normalize trace(P)=d to prevent the penalty from being defeated by metric scaling. For each minibatch, collect hidden states x from teacher-forced or replay trajectories. Estimate the largest singular value of the transformed Jacobian T=P^(1/2)J_theta(x)P^(-1/2) by 2-5 power iterations using Jacobian-vector products and vector-Jacobian products; do not materialize J. Use this update: inputs are x, target eta, metric P, and coefficient lambda; initialize a random unit vector v; repeatedly compute w=T v, set v=w/(||w||_2+epsilon); after the iterations set s=||T v||_2; accumulate hinge=(max(0,s^2-eta^2))^2; optimize task_loss+lambda*mean(hinge). Detach the power-iteration vector between iterations initially for stability, then test full differentiation. The paper supplies the contraction inequality and target factor; the Jacobian norm, sampled states, and any P update are estimated numerically. Begin with a 1-2 layer GRU on sequential MNIST or a small linear-control-inspired latent prediction task, comparing an ordinary GRU, spectral-normalized GRU, and this penalty at equal parameter count and FLOPs. Measure validation accuracy, gradient norms, multi-step rollout error, sensitivity to perturbed initial states, and the fraction of sampled states violating the bound. Success means fewer exploding or unstable rollouts and lower long-horizon prediction error at comparable one-step loss, without materially reducing short-horizon accuracy.",
 "math_summary": "Definition 1 of the paper calls a discrete-time system x(t+1)=f_t(x(t)) strongly contracting in the norm ||v||_P=sqrt(v^T P v), with P=P^T positive definite, when every pair of trajectories satisfies ||x^(1)(t+1)-x^(2)(t+1)||_P \u003c= eta ||x^(1)(t)-x^(2)(t)||_P for eta in (0,1). For a differentiable neural transition f_theta, let J_theta(x)=partial f_theta(x)/partial x. A sufficient local differential condition is J_theta(x)^T P J_theta(x) \u003c= eta^2 P, equivalently ||P^(1/2)J_theta(x)P^(-1/2)||_2 \u003c= eta. Enforcing this inequality on states sampled from training trajectories transfers the paper's contraction guarantee approximately to a neural transition; if the inequality holds everywhere on an invariant state domain, the resulting closed-loop latent dynamics are globally contractive on that domain.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "linear-algebra",
  "optimization"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "world-model",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2607.00383",
  "arxiv_url": "https://arxiv.org/abs/2607.00383",
  "summary_what_math_gives_to_ml": "The paper's transferable asset is a certificate for making an implicitly defined feedback map contractive in a non-Euclidean metric, rather than merely penalizing parameter norms or spectral radii. This can be transplanted to recurrent, state-space, and latent world-model transitions by constraining the one-step Jacobian of the learned state update, yielding bounded sensitivity to initialization and perturbations. The most practical first implementation is a sampled Jacobian contraction regularizer with a fixed or learned positive-definite metric P; the full multiplier/Lur'e machinery could later support nonsmooth or implicit neural blocks. The likely benefit is improved long-horizon stability and less exploding sensitivity, at the cost of potentially reducing dynamical expressivity.",
  "title": "Regularized Model Predictive Control via Contractivity and Implicit Lur'e Analysis",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "generalization",
  "accuracy"
 ],
 "title": "P-Metric Contractive State Transition",
 "url": "https://synthcore.org/idea/307/p-metric-contractive-state-transition",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}