(co)Quasi-irreducible and (co)expanding random maps

What the math gives to ML

The paper provides a nonstandard criterion for robustness of random derivative cocycles: quasi-irreducibility is equivalent, under a simple-top-exponent assumption, to contraction of the projective tangent dynamics, a vertical spectral gap, and uniqueness of the stationary distribution over directions. The transferable asset is to monitor and enforce a gap between the dominant and subdominant Lyapunov exponents of a stochastic neural-network Jacobian cocycle. For recurrent networks, state-space models, and deep residual networks with input-dependent Jacobians, this gives a concrete regularizer and diagnostic: tangent directions should converge exponentially to a dominant direction, with rate predicted by the Lyapunov gap, while the top exponent is separately constrained to a desired stable or expansive regime.