  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.
