Tropical Support Vector Machine and its Applications to Phylogenomics
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Most data in genome-wide phylogenetic analysis (phylogenomics) is essentially multidimensional, posing a major challenge to human comprehension and computational analysis. Also, we cannot 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) over the space of phylogenetic trees. 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 open sectors. We show that we can formulate hard margin tropical SVMs and soft margin tropical SVMs as linear programming problems. In addition, we show the necessary and sufficient conditions for each data point to be separated and an explicit formula for the optimal solution for the feasible linear programming problem. Based on our theorems, we develop novel methods to compute tropical SVMs and computational experiments show our methods work well. We end this paper with open problems.
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