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Tropical support vector machines: Evaluations and extension to function spaces.

Ruriko Yoshida¹, Misaki Takamori², Hideyuki Matsumoto³

¹Department of Operations Research, Naval Postgraduate School, Monterey, 93943, CA, USA.

Neural Networks : the Official Journal of the International Neural Network Society

|November 5, 2022

Summary

This summary is machine-generated.

Tropical Support Vector Machines (SVMs) offer robust classification, even with added noise dimensions. This study analyzes their generalization error bounds and demonstrates resilience against the curse of dimensionality.

Keywords:

Extreme value statistics Function spaces Max-plus algebra Supervised learning Tropical geometry

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Area of Science:

Machine Learning
Algebraic Geometry
Computational Neuroscience

Background:

Support Vector Machines (SVMs) are popular supervised learning models using hyperplanes in Euclidean space.
Tropical SVMs adapt this by using tropical hyperplanes and the max-plus algebra tropical metric.
Understanding tropical SVMs' performance and limitations is crucial for advanced classification tasks.

Purpose of the Study:

To determine generalization error bounds for tropical SVMs over the tropical projective torus.
To investigate the robustness of tropical SVMs against the curse of dimensionality.
To explore the application of extreme value statistics to tropical SVMs and tropical distances.

Main Methods:

Derivation of generalization error bounds using Vapnik-Chervonenkis (VC) dimensions.
Numerical and theoretical analysis employing extreme value statistics.
Classification of Gaussian distributions and empirical neuron type datasets.
Definition of tropical SVMs in a function space with the tropical metric.

Main Results:

Generalization error bounds for tropical SVMs were established over the tropical projective torus.
Tropical SVMs demonstrate robustness against the curse of dimensionality, supported by extreme value statistics.
Anomalous scaling behaviors of tropical distances with noise dimensions were identified.
Tropical SVMs were successfully extended to a function space setting.

Conclusions:

Tropical SVMs provide a robust classification method, particularly resilient to high-dimensional data.
Extreme value statistics offer valuable insights into the behavior and scaling of tropical distances.
The generalization of tropical SVMs to function spaces opens new avenues for research.