A kernel proof of the De Cock-De Moor Lyapunov identity
arXiv:2608.24405
2026
Regularization
1 ideas extracted · analyzed Aug 29, 2026
What the math gives to ML
The paper proves a non-obvious spectral identity connecting cross-Gramian canonical correlations to the product of controllability and observability Gramians: the matrices P^{-1} R Q^{-1} R^T and (I+PQ)^{-1} have identical characteristic polynomials. This gives a direct algebraic way to measure how strongly a dynamical system's reachable state directions overlap with its observable directions, without explicitly computing principal angles. The most transferable use is in recurrent or state-space neural layers, where this spectrum can become a differentiable regularizer against hidden modes that are unreachable, unobservable, or badly conditioned.
Ideas from this paper
Unverified
2026
Add a spectral regularizer to a linear state-space or recurrent layer that controls the overlap between its controllable and observable state directions. The regularizer uses the paper's identity to monitor eigenvalues of (I+PQ)^{-1}, equivalently the squared canonical correlations between reachable and observable subspaces, and penalizes degenerate or overly concentrated spectra.
Useful5/10
Difficulty5/10
Novelty6/10