arXiv:2607.09152 2026 Architecture 1 ideas extracted · analyzed Aug 30, 2026
The paper supplies a constructive functional calculus for bounded accretive, possibly nonnormal operators: operator-monotone functions are implemented as positive mixtures of resolvents rather than through eigendecomposition. This is transferable to neural layers that apply fractional or other monotone matrix functions to learned nonnormal operators, while preserving a positive-real-part constraint that makes the shifted linear solves well behaved. The most direct experiment is an accretive fractional-power feature transform, approximated by a small number of batched shifted solves and compared with eigendecomposition-based matrix powers and unconstrained matrix-function layers.
Parameterize a learned feature-space operator as accretive but not necessarily symmetric, then apply its fractional power through a finite positive mixture of shifted resolvents. This provides a matrix-function layer that can represent directional and rotational interactions while avoiding unstable eigendecomposition of nonnormal matrices.