  Most data in genome-wide phylogenetic analysis (phylogenomics) is essentially
multidimensional, posing a major challenge to human comprehension and
computational analysis. Also, we can not directly apply statistical learning
models in data science to a set of phylogenetic trees since the space of
phylogenetic trees is not Euclidean. In fact, the space of phylogenetic trees
is a tropical Grassmannian in terms of max-plus algebra. Therefore, to classify
multi-locus data sets for phylogenetic analysis, we propose tropical support
vector machines (SVMs). Like classical SVMs, a tropical SVM is a discriminative
classifier defined by the tropical hyperplane which maximizes the minimum
tropical distance from data points to itself in order to separate these data
points into sectors (half-spaces) in the tropical projective torus. Both hard
margin tropical SVMs and soft margin tropical SVMs can be formulated as linear
programming problems. We focus on classifying two categories of data, and we
study a simpler case by assuming the data points from the same category ideally
stay in the same sector of a tropical separating hyperplane. For hard margin
tropical SVMs, we prove the necessary and sufficient conditions for two
categories of data points to be separated, and we show an explicit formula for
the optimal value of the feasible linear programming problem. For soft margin
tropical SVMs, we develop novel methods to compute an optimal tropical
separating hyperplane. Computational experiments show our methods work well. We
end this paper with open problems.
