Quadratic Convexification of a Square Truncated by a Hyperbola
arXiv:2608.26639
2026
Architecture
1 ideas extracted · analyzed Aug 29, 2026
What the math gives to ML
The paper gives an exact description of quadratic functions that are nonnegative on a bounded, nonconvex product-constrained region. Its transferable asset is the parameterized family of tangent quadratics associated with the hyperbola x1 x2 = 1: these functions encode reciprocal interaction geometry using only low-degree features and have explicit nonnegativity and contact properties. A practical neural-network transfer is a small quadratic feature or regularization module for positive feature pairs, gates, scales, or attention factors. The idea is most relevant when multiplicative structure is expected; it is not a general-purpose optimizer or architecture replacement.
Ideas from this paper
Unverified
2026
Add a bank of quadratic features encoding tangent contact with the reciprocal manifold x1 x2 = 1, rather than forcing a generic MLP to discover this interaction from arbitrary monomials. For positive bounded feature pairs, each feature is nonnegative and becomes exactly zero at a selected reciprocal operating point. The module can be used either as an input feature expansion or as a regularizer encouraging learned gates and scales to follow a reciprocal geometry.
Useful5/10
Difficulty3/10
Novelty7/10