Border basis computation with gradient-weighted normalization
This work addresses the stability and consistency issues in approximate basis computation for researchers and practitioners working with vanishing ideals, offering an incremental improvement to existing algorithms.
This paper introduces gradient-weighted normalization for approximate border basis computation of vanishing ideals, which improves stability against perturbation and consistency in input point scaling compared to traditional coefficient normalization. It also demonstrates that coefficient normalization can fail with point scaling, a problem addressed by the proposed method.
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.