Tropical support vector machines: Evaluations and extension to function spaces. (January 2023)

• Record Type: Journal Article
• Authors: Yoshida, Ruriko; Takamori, Misaki; Matsumoto, Hideyuki; Miura, Keiji
• Is Part Of: Neural networks. Volume 157(2023)
• Page Start: 77
• Page End: 89
• Publication Date: 2023-01
• Subjects: Extreme value statistics -- Function spaces -- Max-plus algebra -- Supervised learning -- Tropical geometry; Neural computers -- Periodicals; Neural networks (Computer science) -- Periodicals; Neural networks (Neurobiology) -- Periodicals; Nervous System -- Periodicals; Ordinateurs neuronaux -- Périodiques; Réseaux neuronaux (Informatique) -- Périodiques; Réseaux neuronaux (Neurobiologie) -- Périodiques; Neural computers; Neural networks (Computer science); Neural networks (Neurobiology); Periodicals; 006.32
• DOI: 10.1016/j.neunet.2022.10.002
• Languages: English
• ISSNs: 0893-6080

Abstract

Abstract: Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperplane under the tropical metric with the max-plus algebra. In this paper, first we show generalization error bounds of tropical SVMs over the tropical projective torus. While the generalization error bounds attained via Vapnik–Chervonenkis (VC) dimensions in a distribution-free manner still depend on the dimension, we also show numerically and theoretically by extreme value statistics that the tropical SVMs for classifying data points from two Gaussian distributions as well as empirical data sets of different neuron types are fairly robust against the curse of dimensionality. Extreme value statistics also underlie the anomalous scaling behaviors of the tropical distance between random vectors with additional noise dimensions. Finally, we define tropical SVMs over a function space with the tropical metric. Highlights: We obtain generalization error bounds of tropical SVMs via the VC dimensions. We demonstrate that the tropical SVMs are robust against the curse of dimensionality. We define tropical SVMs over a function space to enable the classification of curves.