Spurious Vanishing Problem in Approximate Vanishing Ideal

Approximate vanishing ideal, which is a new concept from computer algebra, is a set of polynomials that almost take a zero value for a set of given data points. The introduction of approximation to exact vanishing ideal has played a critical role in capturing the nonlinear structures of noisy data by computing the approximate vanishing polynomials. However, approximate vanishing has a theoretical problem, which has given rise to the spurious vanishing problem that any polynomial turns into an approximate vanishing polynomial by coefficient scaling. In this paper, we propose a first general method that enables many recent 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 as early basis construction algorithms. We further propose a method that takes advantages of 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 acceleration. As a result of the experiments, it can be shown that the proposed method overcomes the spurious vanishing problem and significantly increases the accuracy of classification.