Implement Gradient-Weighted Normalization in Quadratic-Program-Based Approximate Ideal Methods

Develop and analyze implementations of gradient-weighted normalization for algorithms that compute approximately vanishing ideals using simple quadratic programming formulations, such as least-squares problems, which do not rely on eigenvalue problems or singular value decomposition. Determine the algorithmic formulation, normalization constraints, and theoretical properties (e.g., correctness and stability) of such adaptations in the quadratic-program-based setting.

Background

Gradient-weighted normalization weights polynomial coefficients by the magnitudes of gradients of their terms evaluated at the data points and has been integrated into term-aware approximate border basis computations by replacing standard eigenvalue or SVD steps with generalized eigenvalue problems (ABM+GWN).

Beyond eigen/SVD-based approaches, there exist methods for computing approximately vanishing ideals that solve simple quadratic programs (e.g., least-squares formulations). The authors explicitly note that adapting gradient-weighted normalization to these quadratic-program-based methods has not yet been carried out and is deferred to future work, highlighting a concrete unresolved implementation and analysis task.