Quadratic Perturbations of Markov Systems

What the math gives to ML

The paper introduces a nonlinear perturbation of finite Markov dynamics in which transition coefficients depend quadratically on inner products of probability vectors. The transferable asset is a simplex-preserving nonlinear operator: the condition on perturbation magnitudes gives a sufficient margin ensuring that perturbed transition coefficients remain nonnegative, while zero-sum perturbations preserve normalization. A practical neural-network use is a recurrent probability-state layer whose transition matrix is modulated by similarities between parallel latent distributions, with coefficients constrained so every update remains on the simplex. This is most promising for mixture routing, multi-agent latent states, or probabilistic memory rather than generic hidden-state recurrence.