# Tropical Support Vector Machines: Evaluations and Extension to Function Spaces

arXiv: https://arxiv.org/abs/2101.11531 · HTML: https://mlsanity.com/paper/2101.11531/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 40 references that are in the map.
- First posted on arXiv 27 Jan 2021.

## Builds on (most cited first, 4 of 4)

- [Particle Dual Averaging: Optimization of Mean Field Neural Networks with Global Convergence Rate Analysis](https://mlsanity.com/paper/2012.15477/) (2020)
- [Generalization bound of globally optimal non-convex neural network training: Transportation map estimation by infinite dimensional Langevin dynamics](https://mlsanity.com/paper/2007.05824/) (2020)
- [Upper Bound of Bayesian Generalization Error in Non-negative Matrix Factorization](https://mlsanity.com/paper/1612.04112/) (2016)
- [Tropical Support Vector Machine and its Applications to Phylogenomics](https://mlsanity.com/paper/2003.00677/) (2020)

## Built on by (most cited first, 3 of 3)

- [Tropical Geometric Tools for Machine Learning: the TML package](https://mlsanity.com/paper/2309.01082/) (2023)
- [Tropical Decision Boundaries for Neural Networks Are Robust Against Adversarial Attacks](https://mlsanity.com/paper/2402.00576/) (2024)
- [Alternating Minimization for Regression with Tropical Rational Functions](https://mlsanity.com/paper/2305.20072/) (2023)

## Abstract (arXiv)

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.