Border basis computation with gradient-weighted normalization

TL;DR

This paper introduces gradient-weighted normalization for approximate border basis computation, improving stability and scaling consistency over traditional coefficient normalization, with minimal algorithm modifications.

Contribution

It proposes a novel data-dependent normalization method inspired by machine learning, enhancing stability and robustness in polynomial basis computations.

Findings

Gradient-weighted normalization improves stability against perturbations.

Scaling of input points affects the success of basis computation.

The method maintains computational complexity while enhancing robustness.

Abstract

Normalization of polynomials plays a vital role in the approximate basis computation of vanishing ideals. Coefficient normalization, which normalizes a polynomial with its coefficient norm, is the most common method in computer algebra. This study proposes the gradient-weighted normalization method for the approximate border basis computation of vanishing ideals, inspired by recent developments in machine learning. The data-dependent nature of gradient-weighted normalization leads to better stability against perturbation and consistency in the scaling of input points, which cannot be attained by coefficient normalization. Only a subtle change is needed to introduce gradient normalization in the existing algorithms with coefficient normalization. The analysis of algorithms still works with a small modification, and the order of magnitude of time complexity of algorithms remains unchanged. We also prove that, with coefficient normalization, which does not provide the scaling consistency property, scaling of points (e.g., as a preprocessing) can cause an approximate basis computation to fail. This study is the first to theoretically highlight the crucial effect of scaling in approximate basis computation and presents the utility of data-dependent normalization.