We introduce a class of trainable nonlinear operators based on semirings that
are suitable for use in neural networks. These operators generalize the
traditional alternation of linear operators with activation functions in neural
networks. Semirings are algebraic structures that describe a generalised
notation of linearity, greatly expanding the range of trainable operators that
can be included in neural networks. In fact, max- or min-pooling operations are
convolutions in the tropical semiring with a fixed kernel.
We perform experiments where we replace the activation functions for
trainable semiring-based operators to show that these are viable operations to
include in fully connected as well as convolutional neural networks (ConvNeXt).
We discuss some of the challenges of replacing traditional activation functions
with trainable semiring activations and the trade-offs of doing so.

基于半环的可训练非线性运算符类用于神经网络，将传统的线性运算符与激活函数进行了推广，替代了传统激活函数的可训练半环激活函数，并在全连接和卷积神经网络中进行了实验，结果表明这些可训练半环激活函数是可行的。