{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace an unconstrained recurrent or state-space update with a delayed continuous-time hidden-state block and constrain its local closed-loop Jacobian using an output-to-output dissipativity LMI. The certificate bounds amplification from external perturbations, such as corrupted observations, injected hidden-state noise, or delayed-input errors, to the task output. Training rejects or penalizes parameter updates for which the certified gain becomes too large.",
 "formulas_latex": [
  "$$\\dot h(t)=A h(t)+A_1h(t-h_u)+A_2h(t-h_y)+B r(t),\\qquad p(t)=C_p h(t)+D_p r(t),\\quad q(t)=C_r h(t)+D_r r(t).$$",
  "$$\\begin{bmatrix}A^{\\mathsf T}P+PA\u0026PB\\\\B^{\\mathsf T}P\u00260\\end{bmatrix}-\\begin{bmatrix}C_r^{\\mathsf T}\\\\D_r^{\\mathsf T}\\end{bmatrix}\\begin{bmatrix}C_r\u0026D_r\\end{bmatrix}+\\gamma\\begin{bmatrix}C_p^{\\mathsf T}\\\\D_p^{\\mathsf T}\\end{bmatrix}\\begin{bmatrix}C_p\u0026D_p\\end{bmatrix}\\preceq0,\\qquad P\\succeq0,\\ \\gamma\u003e0.$$$$",
  "$$\\dot V(h_t)+\\|q(t)\\|_2^2-\\gamma\\|p(t)\\|_2^2\\le0\\ \\Longrightarrow\\ \\int_0^T\\|q(t)\\|_2^2dt\\le\\gamma\\int_0^T\\|p(t)\\|_2^2dt+V(h_0).$$",
  "$$\\min_{P\\succeq0,\\,\\gamma}\\ \\gamma\\quad\\text{subject to}\\quad \\operatorname{LMI}_{\\mathrm{LKF}}(A,A_1,A_2,B,C_p,D_p,C_r,D_r,h_u,h_y,P,\\gamma)\\preceq0.$$$$"
 ],
 "id": 2799,
 "implementation": "1. Integration point: implement a continuous-time recurrent or state-space layer, \\(\\dot h=f_\\theta(h,h_{\\mathrm{delayed}},x)\\), with FIFO buffers for independently delayed input and observation channels. At training checkpoints, linearize the hidden dynamics around minibatch trajectories and obtain \\(A,A_1,A_2,B\\) using automatic differentiation. Let \\(p\\) be logits or task features and \\(q\\) be the injected-noise output. Initially use a linear recurrent core with a nonlinear readout so the certificate is exact for the core. 2. Pseudocode: simulate the delayed buffers; compute the Jacobian matrices; assemble a first-order Padé augmented state for each delay pair, or call a delay-dependent LKF semidefinite program; solve for the smallest feasible \\(\\gamma\\); add \\(\\lambda\\max(0,\\gamma-\\gamma_{\\max})^2\\) to the loss, or reject an optimizer step when the SDP is infeasible. Use bisection over \\(\\gamma\\) and warm-start \\(P\\) between checkpoints. 3. Computed from the paper: the storage-function dissipation inequality, LMI certificate, and certified gain bound. Estimated empirically: local Jacobians, effective delays, and whether the certificate holds across sampled states; global certification requires known Jacobian bounds over the activation region. 4. First cheap experiment: compare a vanilla delayed GRU or linear SSM, a spectral-normalized SSM, and this certified block on sequential MNIST or a delayed-copy task with additive hidden-state perturbations. Sweep the delay \\(h\\) and perturbation amplitude. The quantitative prediction is a sharp feasibility boundary \\(h_c\\): below \\(h_c\\), the SDP returns finite \\(\\gamma\\) and measured amplification remains below \\(\\sqrt{\\gamma}\\); above \\(h_c\\), feasibility is lost or \\(\\gamma\\) rises rapidly. For the Padé approximation, the measured boundary should agree with direct delayed simulation within approximately 20%, and tightening the LMI objective should monotonically reduce measured perturbation amplification.",
 "math_summary": "Consider the linearized delayed hidden-state model \\(\\dot h(t)=A h(t)+A_1h(t-h_u)+A_2h(t-h_y)+B r(t)\\), with performance output \\(p(t)=C_p h(t)+D_p r(t)\\) and perturbation-monitor output \\(q(t)=C_r h(t)+D_r r(t)\\). Here \\(h\\) is the hidden state, \\(r\\) is an external perturbation, \\(h_u,h_y\\ge0\\) are independent delays, and the matrices are Jacobians or output maps. The paper searches for a positive semidefinite storage matrix \\(P\\) and gain parameter \\(\\gamma\u003e0\\) satisfying a delay-dependent Lyapunov–Krasovskii LMI. In the delay-free or Padé-augmented case, the core dissipativity condition is \\(\\begin{bmatrix}A^{\\mathsf T}P+PA\u0026PB\\\\B^{\\mathsf T}P\u00260\\end{bmatrix}-\\begin{bmatrix}C_r^{\\mathsf T}\\\\D_r^{\\mathsf T}\\end{bmatrix}\\begin{bmatrix}C_r\u0026D_r\\end{bmatrix}+\\gamma\\begin{bmatrix}C_p^{\\mathsf T}\\\\D_p^{\\mathsf T}\\end{bmatrix}\\begin{bmatrix}C_p\u0026D_p\\end{bmatrix}\\preceq0\\). This implies a dissipation inequality \\(\\dot V+\\|q\\|^2-\\gamma\\|p\\|^2\\le0\\), so \\(\\gamma\\) bounds the relevant output-to-output energy amplification under the paper's convention. For a nonlinear network this is a local or region-wise certificate unless Jacobian bounds are established over the full activation domain.",
 "math_tags": [
  "control-theory",
  "dynamical-systems",
  "optimization",
  "linear-algebra",
  "numerical-analysis"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "optimizer",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.28969",
  "arxiv_url": "https://arxiv.org/abs/2608.28969",
  "summary_what_math_gives_to_ml": "The paper provides a transferable dissipativity mechanism for quantifying how measurement and actuation delays amplify external perturbations: a delay-dependent Lyapunov–Krasovskii functional yields LMIs certifying an upper bound on output-to-output gain. The direct neural-network transfer is a delayed recurrent or state-space block whose local Jacobian is constrained by the same LMI, rather than relying only on zero-delay spectral-radius constraints. The certificate gives a measurable stability margin: increase the allowed delay until the LMI becomes infeasible, while monitoring the certified perturbation gain. A cheaper fixed-delay implementation uses Padé augmentation to convert delays into a finite-dimensional augmented state.",
  "title": "Performance Analysis of Time-Delay Systems under External Perturbations Using Output-to-Output Gain",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 7,
  "novelty": 7,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Delay-Gain Certified Recurrent Block",
 "url": "https://synthcore.org/idea/2799/delay-gain-certified-recurrent-block",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}