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 space. While the generalization error bounds attained via VC dimensions in a distribution-free manner still depend on the dimension, we also show 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 and discuss the Gaussian function space as an example.
翻译:支持矢量机(SVMs)是使用超高飞机在赤道空间进行分类的最受欢迎的监管学习模型之一。与SVMs相似,热带SVMs在热带指标下用热带超高空数据点进行分类,使用最大增代数。在本文中,我们首先展示热带SVMs在热带射电空间上的典型误差。虽然通过VC尺寸以无分布方式实现的笼统误差仍然取决于该尺寸,但我们从极端值统计数据中也显示,热带SVMs对两个高斯分布的数据点进行分类的热带SVMs以及不同类型神经的经验性数据集对维度的诅咒相当活跃。极端值统计数据也是热带随机矢量之间热带距离的异常缩放行为和额外噪音尺寸的基础。最后,我们用热带指标界定一个功能空间上的热带SVMss,并讨论高斯功能空间作为例子。