Spurious Vanishing Problem in Approximate Vanishing Ideal
Spurious Vanishing Problem in Approximate Vanishing Ideal
复制标题
DOI：
10.1109/access.2019.2958648
复制
发表时间：
2019-01
期刊：
影响因子：
3.9
通讯作者：
Hiroshi Kera;Yoshihiko Hasegawa
中科院分区：
文献类型：
--
作者：
Hiroshi Kera;Yoshihiko Hasegawa
近似零理想是计算机代数中的一个概念，它研究扰动数据点背后的代数变量。为了捕捉扰动点的非线性结构，引入对精确消失理想的近似起着关键作用。然而，这种近似在近似零理想的基构造中也产生了一个理论问题--伪零问题，即所得到的基多项式可能仅仅因为小的系数而近似零。在本文中，我们首先提出了一种通用的方法，使各种基构造算法能够克服虚假消失问题。特别是，我们将系数归一化与基于多项式的基构造相结合，这不需要对单项式进行适当的排序来处理基构造。我们进一步提出了一种方法，该方法利用了基构造的迭代性质，从而可以避免系数归一化的计算代价。此外，还提出了进一步加速的系数截断方法。实验表明，该方法克服了伪消去问题，在保持相当甚至更低的分类误差的同时，使特征向量变得更短。
Approximate vanishing ideal is a concept from computer algebra that studies the algebraic varieties behind perturbed data points. To capture the nonlinear structure of perturbed points, the introduction of approximation to exact vanishing ideals plays a critical role. However, such an approximation also gives rise to a theoretical problem—the spurious vanishing problem—in the basis construction of approximate vanishing ideals; namely, obtained basis polynomials can be approximately vanishing simply because of the small coefficients. In this paper, we propose a first general method that enables various basis construction algorithms to overcome the spurious vanishing problem. In particular, we integrate coefficient normalization with polynomial-based basis constructions, which do not need the proper ordering of monomials to process for basis constructions. We further propose a method that takes advantage of the iterative nature of basis construction so that computationally costly operations for coefficient normalization can be circumvented. Moreover, a coefficient truncation method is proposed for further accelerations. From the experiments, it can be shown that the proposed method overcomes the spurious vanishing problem, resulting in shorter feature vectors while sustaining comparable or even lower classification error.