Spurious Vanishing Problem in Approximate Vanishing Ideal

TL;DR

This paper addresses the spurious vanishing problem in approximate vanishing ideals by proposing a normalization-based method that improves basis construction, resulting in more meaningful features for data analysis.

Contribution

It introduces a novel coefficient normalization technique for basis construction algorithms that overcomes the spurious vanishing problem without requiring monomial ordering.

Findings

The proposed method effectively overcomes spurious vanishing.

It produces shorter feature vectors with comparable or better classification accuracy.

Normalization accelerates basis construction and improves feature quality.

Abstract

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.