On Energy Laws and Stability of First-Subdiagonal Pade Approximants for Linear Seminegative Problems

What the math gives to ML

The paper supplies a constructive stability mechanism for rational discretizations of dissipative but potentially non-normal linear dynamics: an exact discrete energy identity based on the operator-induced form [w,v]_L and an explicit Cholesky factorization of the Padé energy matrix. This can be transferred to neural dynamical blocks by replacing explicit residual updates with first-subdiagonal Padé maps whose step size is contractive for seminegative generators. The most practical experiment is a small learned linear or channel-mixing residual block, parameterized so its symmetric part is negative semidefinite, and compared against Euler, Cayley, and unconstrained residual updates at equal depth and compute.