A finite-order characterization of entrywise positivity preservers
arXiv:2608.15904
2026
Architecture
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper gives a finite-dimensional certificate for when a scalar function preserves positive semidefiniteness under entrywise application: nonnegative derivatives through order n-1 together with positive semidefiniteness of an Euler-Hankel matrix. This enables neural architectures to learn nonlinear transformations of positive Gram, covariance, or similarity matrices without destroying their PSD structure, while being less restrictive than requiring a globally nonnegative power-series expansion. The most direct transfer is a constrained learnable scalar activation applied entrywise to fixed-width Gram matrices, with the theorem's conditions enforced on the observed positive interval. A second use is a PSD-safe learnable similarity kernel for attention or graph propagation.
Ideas from this paper
Unverified
2026
Replace an unconstrained entrywise nonlinearity on a positive Gram or covariance matrix by a learned scalar function satisfying the paper's finite-order positivity-preserver conditions. The transformed matrix remains PSD for matrices of the target width n, allowing nonlinear Gram propagation without eigenvalue clipping or projection.
Useful6/10
Difficulty5/10
Novelty7/10
Unverified
2026
Use the finite-order characterization to learn a nonlinear similarity function for token, patch, or graph-node Gram matrices while preserving PSD by construction or by a differentiable certificate loss. This creates a kernelized attention or graph-readout mechanism in which nonlinear affinity transformations cannot introduce indefinite similarity geometry.
Useful5/10
Difficulty6/10
Novelty6/10