Tropical Support Vector Machines: Evaluations and Extension to Function Spaces

TL;DR

This paper evaluates tropical SVMs, providing generalization error bounds, analyzing their robustness against high dimensionality, and extending the concept to function spaces with tropical metrics.

Contribution

It offers the first theoretical and empirical analysis of tropical SVMs' generalization bounds and robustness, and introduces their extension to function spaces.

Findings

Tropical SVMs have generalization error bounds dependent on dimension.

They are robust against the curse of dimensionality in high-dimensional data.

Tropical SVMs can be extended to function spaces using tropical metrics.

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.