Spectral Characterisation of Covariance Existence in Markov-Switching Affine Recurrences

What the math gives to ML

The paper isolates a finite-dimensional mean-square stability condition for affine systems whose linear dynamics switch according to a finite-state Markov chain. The key transferable asset is that almost-sure stability of random matrix products is insufficient to prevent exploding second moments; the relevant object is an explicit block Kronecker operator whose spectral radius controls variance. This gives neural architectures with discrete routing or mode switching a computable stability certificate, rather than relying only on gradient-based empirical stabilization. The most direct transfer is to recurrent, state-space, or mixture-of-experts modules: constrain the mode-transition Kronecker operator during training and optionally solve its linear moment equations to initialize or normalize hidden-state covariances.