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.

翻译：深度神经网络在输入向量位于欧几里得空间时展现出卓越性能。然而，当输入为可表示为热带射影环面上向量的系统发育树时，经典神经网络表现不佳。本文提出热带嵌入方法，通过热带度量将热带射影环面上的向量变换至欧几里得空间。我们构建了一种热带神经网络，其首层为热带嵌入层，后续层与经典网络结构一致。理论证明该热带度量神经网络具有通用逼近性质，并推导出深度神经网络的反向传播规则。随后提供与经典实现方式一致的TensorFlow 2代码框架，其权重初始化基于极端值统计理论。我们将该方法应用于纽约州流感病毒血凝素序列等实证数据，最终证明热带神经网络可视为热带逻辑回归的泛化形式。