The approximate basis computation of vanishing ideals has recently been extensively studied and applied both in computer algebra and data-driven applications such as machine learning. However, the symbolic computation and the dependency on the monomial ordering remain an essential gap between the two fields. In the present paper, we propose the first efficient monomial-agnostic approximate basis computation of vanishing ideals, where polynomials are manipulated without any information of monomials, which can be implemented in a fully-numerical manner and thus desirable for data-driven applications. In particular, we propose the gradient normalization, which achieves not only the first efficient and monomial-agnostic normalization of polynomials but also brings fascinating properties such as translation- and scaling-consistency, which are not realized by any existing basis computation algorithms. During the basis computation, the gradient of polynomials at given points are proven to be efficiently and exactly obtained without performing differentiation. By exploiting the gradient information, we further propose a basis set reduction method to remove redundant polynomials in a monomial-agnostic manner. A regularization method using gradient is also proposed for avoiding overfitting of the basis set to given points.

中文翻译：

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消失理想的单项不可知性计算

消失的理想的近似基础计算最近已得到广泛研究，并应用于计算机代数和数据驱动的应用程序（例如机器学习）中。但是，符号计算和对单项式顺序的依赖性仍然是这两个领域之间的重要差距。在本文中，我们提出了消失的理想的第一个有效的单项不可知近似基础计算，其中多项式在没有单项式的任何信息的情况下进行操作，可以以全数值方式实现，因此对于数据驱动的应用程序是理想的。特别是，我们提出了梯度归一化，它不仅可以实现多项式的第一个有效且与多项式无关的归一化，而且还可以带来令人着迷的特性，例如平移和缩放一致性，这是任何现有的基础计算算法都无法实现的。在基础计算过程中，证明了在不进行微分的情况下有效且精确地获得了给定点处的多项式的梯度。通过利用梯度信息，我们进一步提出了一种基集约简方法，以单项不可知的方式去除冗余多项式。还提出了使用梯度的正则化方法，以避免将基集过度拟合到给定点。