Multi-type Galton-Watson processes in dynamical environments

What the math gives to ML

The paper studies multi-type branching processes whose reproduction law changes along a deterministic dynamical environment, so population growth is governed by products of time-dependent nonnegative mean matrices rather than by a single matrix. Its transferable mechanism is a cocycle-level phase boundary: the sign of the asymptotic logarithmic growth rate of these matrix products separates uniformly subcritical, critical, and uniformly supercritical behavior. In neural networks, the same construction can monitor or constrain products of time-dependent recurrent Jacobians, providing a quantitative long-horizon stability certificate that is more faithful than checking each layer's spectral radius independently.