Log-Corrector Perron Regularization

(1) Integration point: apply this to a positive sparse recurrent transition or continuous-time state-space model. Parameterize each off-diagonal transition rate as $r_\nu^\pm(x)=\operatorname{softplus}(a_\nu^\pm(x))+\epsilon$ and the diagonal term as $q_x$; the recurrent state update can be $h_{t+1}=\exp(\Delta\mathcal M)h_t$ or a positive Euler approximation. Maintain one auxiliary corrector vector $\phi$ over lattice/channel positions, and subtract its mean after every update to remove the additive gauge freedom. (2) Pseudocode: compute dplus=roll(phi,-1)-phi and dminus=roll(phi,+1)-phi; compute F=q+sum(rplus*exp(dplus)+rminus*exp(dminus)); compute Lambda_hat=logsumexp(tau*F)/tau; add lambda_reg*softplus(Lambda_hat-rho)**2 to the task loss; backpropagate through the network parameters; then perform 1-5 auxiliary Adam steps on phi minimizing Lambda_hat, followed by phi=phi-mean(phi). The displayed $F_x$ is the paper's local Perron quotient and $\widehat\Lambda_\tau$ is the smooth approximation to its maximum. (3) The mathematical quantity is computed directly from local sparse shifts; no eigendecomposition is needed during training. Periodically validate it with 20-50 power iterations on the explicit transition matrix, checking that the optimized certificate approaches the measured principal eigenvalue. Start with $\tau=10$ and increase it if the smooth maximum is too loose. (4) First experiment: train a small positive one-dimensional lattice SSM on sequential MNIST or a copy-memory task, comparing unconstrained training, power-iteration spectral regularization, and this corrector penalty at equal regularization overhead. Use lattice sizes 64 and 256 and sequence horizons of 1,000-10,000 steps. Success means fewer exploding or vanishing gradients, a smaller gap between the certificate and measured growth rate, equal or better long-horizon accuracy, and lower wall-clock cost than repeatedly estimating the spectral radius.