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 takes 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 is giving rise to the spurious vanishing problem that any polynomial turns into an approximate vanishing polynomial by coefficient scaling. In the present paper, we propose a general method that enables many basis construction methods to overcome this problem. Furthermore, a coefficient truncation method is proposed that balances the theoretical soundness and computational cost. The experiments show that the proposed method overcomes the spurious vanishing problem and significantly increases the accuracy of classification.