Approximate Vanishing Ideal via Data Knotting

TL;DR

This paper introduces a novel approach to approximate the vanishing ideal that is robust to noisy data and maintains strong algebraic structure, improving efficiency in classification tasks.

Contribution

It proposes a new data knotting method to find polynomials that approximately vanish on data and nearly exactly on data knots, enhancing algebraic structure and computational efficiency.

Findings

Fewer and lower-degree polynomials were discovered compared to existing methods.

The method accelerated classification runtime without loss of accuracy.

It effectively handles noisy data while preserving algebraic properties.

Abstract

The vanishing ideal is a set of polynomials that takes zero value on the given data points. Originally proposed in computer algebra, the vanishing ideal has been recently exploited for extracting the nonlinear structures of data in many applications. To avoid overfitting to noisy data, the polynomials are often designed to approximately rather than exactly equal zero on the designated data. Although such approximations empirically demonstrate high performance, the sound algebraic structure of the vanishing ideal is lost. The present paper proposes a vanishing ideal that is tolerant to noisy data and also pursued to have a better algebraic structure. As a new problem, we simultaneously find a set of polynomials and data points for which the polynomials approximately vanish on the input data points, and almost exactly vanish on the discovered data points. In experimental classification tests, our method discovered much fewer and lower-degree polynomials than an existing state-of-the-art method. Consequently, our method accelerated the runtime of the classification tasks without degrading the classification accuracy.