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Most data in genome-wide phylogenetic analysis (phylogenomics) is essentiallymultidimensional, posing a major challenge to human comprehension andcomputational analysis. Also, we can not directly apply statistical learningmodels in data science to a set of phylogenetic trees since the space ofphylogenetic trees is not Euclidean. In fact, the space of phylogenetic treesis a tropical Grassmannian in terms of max-plus algebra. Therefore, to classifymulti-locus data sets for phylogenetic analysis, we propose tropical supportvector machines (SVMs). Like classical SVMs, a tropical SVM is a discriminativeclassifier defined by the tropical hyperplane which maximizes the minimumtropical distance from data points to itself in order to separate these datapoints into sectors (half-spaces) in the tropical projective torus. Both hardmargin tropical SVMs and soft margin tropical SVMs can be formulated as linearprogramming problems. We focus on classifying two categories of data, and westudy a simpler case by assuming the data points from the same category ideallystay in the same sector of a tropical separating hyperplane. For hard margintropical SVMs, we prove the necessary and sufficient conditions for twocategories of data points to be separated, and we show an explicit formula forthe optimal value of the feasible linear programming problem. For soft margintropical SVMs, we develop novel methods to compute an optimal tropicalseparating hyperplane. Computational experiments show our methods work well. Weend this paper with open problems.

Tropical Support Vector Machine and its Applications to Phylogenomics