Unverified 2026

Pseudospectral Stability Regularizer for Stable SSMs

Implementation & benchmark of arXiv:2608.06272 — A solution to the inverse generator problem and related questions

Usefulness6/10

Difficulty6/10

Novelty6/10

Dynamics Stability Accuracy Generalization Ssm Rnn Training dynamics Initialization Dynamical systems Linear algebra Functional analysis Numerical analysis

Source paper: A solution to the inverse generator problem and related questions arXiv:2608.06272 ⓘ · analyzed Aug 31, 2026

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 Not peer-reviewed.

Idea description

Replace eigenvalue-only stability checks for a continuous-time recurrent or state-space layer with an explicit finite-horizon transient-growth test. Penalize state matrices that have small spectral decay but large induced norms of exp(tA), exp(tA^{-1}), or their discretized transition operators. This targets the paper's phenomenon in which a system is exponentially stable in continuous time yet numerically and inversely unstable because its eigenbasis is highly conditional.

Formulas

$$\lVert e^{tA_n}\rVert \leq \left(1+C\frac{\alpha}{1-\alpha}\right)e^{-\omega_n t},\qquad t\geq 0,$$

$$\frac{1}{C}n^{\alpha}\leq \lVert e^{\pi A_n^{-1}}\rVert \leq n^{\alpha},$$

$$\mathcal{R}(A)=\frac{1}{|\mathcal{T}|}\sum_{t\in\mathcal{T}}\left[\log\lVert e^{tA}\rVert\right]_+^2+\lambda_{\mathrm{inv}}\left[\log\lVert e^{\tau A^{-1}}\rVert-\rho\right]_+^2,$$

$$A=V\operatorname{diag}(\lambda_1,\ldots,\lambda_d)V^{-1},\qquad \Phi_h(A)=e^{hA}\ \text{or}\ \Phi_h(A)=\left(I-\tfrac{h}{2}A\right)^{-1}\left(I+\tfrac{h}{2}A\right).$$

Mathematical statement

Theorem 1.1 states that for every alpha in (0,1), there are invertible diagonalizable matrices A_n in M_{2n}(C), a decreasing sequence omega_n in (0,1), and an absolute constant C such that ||A_n|| <= 1 + C alpha/(1-alpha), ||A_n^{-1}|| <= (1/sqrt(omega_n))(1 + C alpha/(1-alpha)), and ||e^{tA_n}|| <= (1 + C alpha/(1-alpha))e^{-omega_n t} for all t >= 0. Nevertheless, (1/C)n^alpha <= ||e^{pi A_n^{-1}}|| <= n^alpha. Here ||.|| is the Euclidean operator norm, e^{tA_n} is the matrix exponential, omega_n is the forward decay-rate parameter, alpha controls the growth associated with the nonorthogonal eigenbasis, and n is half the matrix dimension. The result shows that forward exponential decay does not imply bounded inverse dynamics or small transient amplification. The neural-network adaptation uses these operator norms as directly measurable stability quantities for a learned state matrix A.

Implementation notes

1. Integration point: apply this to the latent transition matrix A of a continuous-time SSM, linear RNN, or residual dynamical block. Parameterize A so that Re(lambda_i) <= -epsilon, but do not treat this spectral constraint as sufficient. During training, add R(A) to the task loss and evaluate the same diagnostic on the discretization Phi_h(A) actually used in the forward pass. 2. Pseudocode: given A, choose T={h,2h,...,T_max} and tau in {h,2h}; for each t in T compute E_t=expm(tA), estimate s_t=||E_t||_2 by 5-10 power iterations, and accumulate max(0,log(s_t))^2. If A is invertible, solve AX=I instead of explicitly forming A^{-1}, compute E_inv=expm(tau X), estimate s_inv=||E_inv||_2, and add lambda_inv*max(0,log(s_inv)-rho)^2. Backpropagate through these operations for small state dimension, or compute the penalty every K optimizer steps with a detached norm estimate. 3. The paper supplies the separation between forward decay and inverse amplification; C, alpha, and omega_n are not assumed known for a learned A. Estimate operator norms empirically by power iteration and estimate the spectral margin using eigendecomposition or Arnoldi iteration. Also monitor ||Phi_h(A)^k|| over a short rollout because Crank-Nicolson-like discretizations can expose transient growth. 4. First experiment: train a one- or two-layer S4-style SSM and a vanilla linear RNN on sequential MNIST and the adding problem, comparing eigenvalue-only stabilization against the proposed penalty at equal parameter count and FLOPs. Record task loss, gradient norm, maximum finite-horizon state amplification, and performance under longer sequences or perturbed inputs. Success means comparable short-horizon training but lower max ||Phi_h(A)^k||, fewer exploding gradients, and better extrapolation to sequence lengths 2-4 times those used in training.

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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