Aclass of incrementally scattering-passive nonlinear systems

What the math gives to ML

The paper provides a constructive recipe for making nonlinear state-space systems incrementally contractive while retaining an input-output energy inequality, by subtracting a whole-space maximal monotone damping operator from the generator of a scattering-passive linear system. This can be transferred to recurrent or state-space neural layers: use a passive linear transition plus an implicit maximal-monotone nonlinear damping step, giving a certificate that differences between trajectories cannot grow faster than differences in their inputs. The most practical implementation is an implicit Euler or resolvent recurrent cell, where the nonlinear term is the subgradient of a convex potential and can be solved with a few fixed-point or proximal iterations. A lower-risk alternative is to use the paper's energy inequality as a regularizer for existing recurrent and state-space architectures.