In the last decade, the approximate vanishing ideal and its basis construction algorithms have been extensively studied in computer algebra and machine learning as a general model to reconstruct the algebraic variety on which noisy data approximately lie. In particular, the basis construction algorithms developed in machine learning are widely used in applications across many fields because of their monomial-order-free property; however, they lose many of the theoretical properties of computer-algebraic algorithms. In this paper, we propose general methods that equip monomial-order-free algorithms with several advantageous theoretical properties. Specifically, we exploit the gradient to (i) sidestep the spurious vanishing problem in polynomial time to remove symbolically trivial redundant bases, (ii) achieve consistent output with respect to the translation and scaling of input, and (iii) remove nontrivially redundant bases. The proposed methods work in a fully numerical manner, whereas existing algorithms require the awkward monomial order or exponentially costly (and mostly symbolic) computation to realize properties (i) and (iii). To our knowledge, property (ii) has not been achieved by any existing basis construction algorithm of the approximate vanishing ideal.

本文提出了使用梯度的全数值方法来构建近似消失理想的基础算法，以解决现有算法需要使用不方便的单项式顺序或指数级昂贵的计算的问题，并实现了输入平移和缩放的一致输出，实现了非平凡冗余基础的删除。此外，该文针对此前的算法未能解决的问题，实现了数值完整的基础构建方法。

梯度提升近似消失理想