The exclusive or (xor) function is one of the simplest examples that illustrate why nonlinear feedforward networks are superior to linear regression for machine learning applications. We review the xor representation and approximation problems and discuss their solutions in terms of probabilistic logic and associative copula functions. After briefly reviewing the specification of feedforward networks, we compare the dynamics of learned error surfaces with different activation functions such as RELU and tanh through a set of colorful three-dimensional charts. The copula representations extend xor from Boolean to real values, thereby providing a convenient way to demonstrate the concept of cross-validation on in-sample and out-sample data sets. Our approach is pedagogical and is meant to be a machine learning prolegomenon.

翻译：异或（xor）函数是说明为何非线性前馈网络在机器学习应用中优于线性回归的最简单范例之一。我们回顾了异或表示与近似问题，并从概率逻辑与关联连接函数的角度探讨其解决方案。在简要回顾前馈网络规范后，通过一系列彩色三维图表对比了采用RELU和tanh等不同激活函数时学习误差曲面的动态特性。连接函数表示将异或从布尔值扩展至实数值，从而为在样本内和样本外数据集上演示交叉验证概念提供了便捷途径。本方法兼具教学性，旨在作为机器学习的导论性论述。