Hyperpositive functions, sector bounded functions and a trace formula

What the math gives to ML

The paper gives an explicit matrix-valued parameterization of sector-bounded analytic functions by contractive functions: an affine map converts any contraction sigma into a function F satisfying a prescribed indefinite quadratic inequality. The transferable asset is not the complex analysis itself, but the exact matrix-ball geometry and the ability to enforce positive-real or sector constraints without projecting a learned matrix after every update. This suggests constrained neural layers and recurrent/state-space transitions whose weights are generated from unconstrained parameters through a contractive core and a tunable anisotropic affine transform. The construction is especially promising when stability, bounded gain, or directional conditioning matters more than maximizing unconstrained layer expressivity.