Unverified 2026

Contour-reduced parametric SSM

Implementation & benchmark of arXiv:2608.05363 — Contour integral methods and model order reduction for parametric linear control systems

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Architecture Speedup Memory Stability Ssm Inference speedup World model Numerical analysis Linear algebra Approximation theory Dynamical systems

Source paper: Contour integral methods and model order reduction for parametric linear control systems arXiv:2608.05363 ⓘ · analyzed Aug 31, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace repeated full-dimensional matrix-exponential or ODE solves in a conditioned continuous-time state-space layer with contour quadrature evaluated in a projection basis. The same reduced basis and contour nodes can serve many conditioning vectors, while shifted reduced resolvents provide a stable and differentiable approximation over a prescribed time window.

Formulas

$$E(\mu)\dot{x}(t,\mu)=A(\mu)x(t,\mu)+B(\mu)u(t),\qquad y(t,\mu)=C(\mu)x(t,\mu)+D(\mu)u(t).$$

$$z_\Gamma(s)=\begin{cases}\ell_1(s),&s\leq-\frac{\pi}{2},\\ z(s),&-\frac{\pi}{2}\leq s\leq\frac{\pi}{2},\\ \ell_2(s),&s\geq\frac{\pi}{2},\end{cases}\qquad z(s)=A_1\cos s+\mathrm{i}A_2\sin s+A_3,$$

$$\widehat{y}(t,\mu)=D(\mu)u(t)+\sum_{j=1}^{N}w_j e^{z_jt}C_r(\mu)\bigl(z_jE_r-A_r(\mu)\bigr)^{-1}\left[E_rx_{0,r}+B_r(\mu)U(z_j)\right],\qquad x_{0,r}=W^Tx_0.$$

Mathematical statement

The paper starts from the descriptor system $$E(\mu)\dot{x}(t,\mu)=A(\mu)x(t,\mu)+B(\mu)u(t),\qquad y(t,\mu)=C(\mu)x(t,\mu)+D(\mu)u(t),$$ where $E(\mu)$ is invertible, $x\in\mathbb{R}^{n}$ is the state, $u\in\mathbb{R}^{m}$ is the input, $y\in\mathbb{R}^{p}$ is the output, and $\mu$ is a parameter vector. In the Laplace domain, the state is represented with the resolvent $(sE(\mu)-A(\mu))^{-1}$. The inverse Laplace integral is approximated by quadrature nodes $z_j=z_\Gamma(s_j)$ and weights $w_j$, where the paper's contour has an elliptic middle segment $z(s)=A_1\cos s+\mathrm{i}A_2\sin s+A_3$, joined to two rays. Here $A_1,A_2,A_3$ define the ellipse, $\mathrm{i}$ is the imaginary unit, $s_j$ are quadrature abscissas, and $w_j$ include the contour derivative and quadrature weights. Projection matrices $V,W\in\mathbb{R}^{n\times r}$, with $W^TV=I_r$, define $E_r=W^TEV$, $A_r(\mu)=W^TA(\mu)V$, $B_r(\mu)=W^TB(\mu)$, and $C_r(\mu)=C(\mu)V$. The expensive full resolvent is replaced by the reduced resolvent $(z_jE_r-A_r(\mu))^{-1}$, where $r\ll n$.

Implementation notes

Integrate this at the forward pass of a continuous-time SSM or latent world-model layer, replacing the operation that computes the hidden state with a matrix exponential or adaptive ODE solver. Let the conditioning vector $\mu$ be a task embedding or per-example control vector, and parameterize $A(\mu)=A_0+\sum_k\mu_kA_k$, with analogous parameterizations for $B$ and $C$. Choose a time window $t\in[T,\Lambda T]$ and construct $N$ contour nodes $z_j=z_\Gamma(s_j)$ and weights $w_j$ from the displayed elliptic contour. Build a projection basis $V$ offline from Krylov or Arnoldi vectors generated by shifted systems $(z_jE-A(\mu_q))^{-1}B(\mu_q)$ and $(z_jE-A(\mu_q))^{-1}Ex_0$ for sampled conditioning vectors $\mu_q$. Use $W=V$ for a Galerkin implementation, or compute a biorthogonal $W$ satisfying $W^TV=I_r$. At runtime form $E_r$, $A_r(\mu)$, $B_r(\mu)$, and $C_r(\mu)$. For each contour node, solve the small dense system $(z_jE_r-A_r(\mu))q_j=E_rx_{0,r}+B_r(\mu)U(z_j)$, then accumulate $\widehat{y}(t,\mu)=D(\mu)u(t)+\sum_jw_je^{z_jt}C_rq_j$. Cache nodes, weights, and any factorization shared across examples; automatic differentiation can backpropagate through the reduced solves. Estimate the basis empirically with greedy residual maximization over sampled $\mu$, using residual $\|(z_jE-A(\mu))Vq_j-[Ex_0+B(\mu)U(z_j)]\|$. First test a 256-dimensional conditioned SSM with a 16-dimensional parameter vector on synthetic damped linear trajectories and sequential MNIST. Compare against exact matrix exponentials and a fourth-order adaptive ODE solver, sweeping $r\in\{8,16,32,64\}$ and $N\in\{16,32,64\}$. Success requires matched output error below $10^{-4}$, lower memory and latency, and at least 2x lower solve cost at $r=32$ without worse validation loss.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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