Tropical neural networks and its applications to classifying phylogenetic trees

TL;DR

This paper introduces tropical neural networks that embed phylogenetic trees into Euclidean space, enabling effective classification and interpretation, with proven universality and practical implementation in TensorFlow 2.

Contribution

It proposes a novel tropical embedding layer and neural network architecture tailored for phylogenetic trees, extending neural network applicability beyond Euclidean inputs.

Findings

Tropical neural networks are universal approximators.

The method successfully classifies influenza virus sequences.

TensorFlow 2 implementation is provided with considerations for weight initialization.

Abstract

Deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical projective torus. Here we propose tropical embedding to transform a vector in the tropical projective torus to a vector in the Euclidean space via the tropical metric. We introduce a tropical neural network where the first layer is a tropical embedding layer and the following layers are the same as the classical ones. We prove that this neural network with the tropical metric is a universal approximator and we derive a backpropagation rule for deep neural networks. Then we provide TensorFlow 2 codes for implementing a tropical neural network in the same fashion as the classical one, where the weights initialization problem is considered according to the extreme value statistics. We apply our method to empirical data including sequences of hemagglutinin for influenza virus from New York. Finally we show that a tropical neural network can be interpreted as a generalization of a tropical logistic regression.