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.
Highlights
Discovering nonlinear structure behind data is a common task across various fields, such as machine learning, computer vision, and systems biology
Nonlinear feature vectors of data are constructed for classifications [3], [6], [7]; independent signals are estimated for blind source separation tasks [8], [9]; nonlinear dynamical systems are reconstructed from noisy observations [10]; and so forth [11], [12]
We propose two efficient methods for coefficient normalization, which is computationally costly to introduce into polynomial-based basis construction algorithms
Summary
Discovering nonlinear structure behind data is a common task across various fields, such as machine learning, computer vision, and systems biology. An emerging concept from computer algebra for this task is the approximate vanishing ideal [1], [2], which is defined as a set of polynomials that almost take a zero value, i.e., approximately vanish, for any point in data. An approximate vanishing polynomial g ∈ Iapp(X ) holds X as its approximate roots, which implies g reflects the nonlinear structure underlying X. Computing the basis set of approximate vanishing ideal has been attracting a lot of attention [1], [3]–[5]; such basis vanishing polynomials describe a system that has X as approximate common roots, The associate editor coordinating the review of this manuscript and approving it for publication was Chao Tong. Nonlinear feature vectors of data are constructed for classifications [3], [6], [7]; independent signals are estimated for blind source separation tasks [8], [9]; nonlinear dynamical systems are reconstructed from noisy observations [10]; and so forth [11], [12]
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