2018/01/29 by Hiroshi Kera, Kera, Hiroshi, Yoshihiko Hasegawa +1 · 2 citations
Computer Science · Mathematics · #Commutative Algebra and Its Applications #FOS: Computer and information sciences #Machine Learning (stat.ML) #Polynomial and algebraic computation #stat.ML
paper · pdf · doi:10.48550/arxiv.1801.09367
11 pages; AAAI'18
arxiv created 2018/01/29 · openalex publication_date 2018/01/29 · arxiv updated 2018/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.