Beyond Unit-Circle Eigenvalues: Invariant Bases for Stable State Space Dynamics
Jiaqian Zhu, Yang Zhang, Junhua Ding, Xiaowei Yu
Abstract
Diagonal state space models process long sequences at linear cost, yet the same transition is applied at every step of a rollout, so an unconstrained recurrence holds no quantity fixed and the state decays or grows without bound. Current designs constrain one quantity, and preserving one does not guarantee preservation of the others, so a model can hold its norm to machine precision while momentum drifts by hundreds of percent. We claim that the conserved quantities of a diagonal recurrence are layered, that each layer requires a strictly stronger condition on the operator, and that a unit-modulus spectrum reaches only the outermost. Acting on the claim, we enforce the conditions required for every layer by construction. The generator of the Kronecker-Cayley transition (KC) is a graph Laplacian over object subspaces, $\mathbf{H}=\mathbf{L}\otimes I_m$, advanced by a Cayley transform, and the step stays diagonal at $O(N^3+N^2m)$ cost. The conserved set is exactly a $2m^2$-dimensional family of quadratic forms, held at every parameter value, so the quantities and the dynamics the construction reaches are fixed before training. On four conservation benchmarks, KC holds total mass at or near the reference floor while every diagonal baseline loses it by orders of magnitude. We further compare against mass correction and find that two correctors fail once the underlying prediction diverges, which shows that assembly rather than correction is necessary for a guarantee that holds across the rollout. A free modal basis trained under a conservation penalty conserves only by freezing its spectrum, whereas the assembled operator conserves with every mode but one free to rotate. Beyond these benchmarks, KC holds the linear momentum of three $N$-body systems and, at the operator level, the conserved quantity in molecular dynamics, chemical kinetics, pharmacokinetics and power-grid swing dynamics.