The classes of bivariate Schur and Herglotz matrix-valued rational functions: realizations, symmetrizations, and related determinantal representations

What the math gives to ML

The paper gives finite-dimensional state-space realizations for multivariate rational maps whose positivity or contractivity is certified by explicit matrix inequalities. The transferable asset is the positive-real/Herglotz realization: a rational feature mixer can be parameterized so that its symmetric part is positive semidefinite over an entire positive-input domain, rather than merely being stable at sampled training points. A particularly promising neural adaptation is to use such a map inside an implicit resolvent layer, whose accretivity gives a built-in nonexpansive stability mechanism. This could provide a compact alternative to unconstrained token mixers or state-space blocks, especially when feature-dependent rational filtering is desired.