The inability to linearly classify XOR has motivated much of deep learning. We revisit this age-old problem and show that linear classification of XOR is indeed possible. Instead of separating data between halfspaces, we propose a slightly different paradigm, equality separation, that adapts the SVM objective to distinguish data within or outside the margin. Our classifier can then be integrated into neural network pipelines with a smooth approximation. From its properties, we intuit that equality separation is suitable for anomaly detection. To formalize this notion, we introduce closing numbers, a quantitative measure on the capacity for classifiers to form closed decision regions for anomaly detection. Springboarding from this theoretical connection between binary classification and anomaly detection, we test our hypothesis on supervised anomaly detection experiments, showing that equality separation can detect both seen and unseen anomalies.

翻译：无法线性分类异或问题（XOR）推动了深度学习的发展。我们重新审视这一经典问题，并证明异或的线性分类实际上是可行的。不同于在半个空间之间分割数据，我们提出了一种略有不同的范式——相等性分离，该方法通过调整支持向量机（SVM）的目标函数来区分数据位于间隔内或间隔外。我们的分类器可通过平滑近似集成到神经网络流水线中。基于其特性，我们推断相等性分离适用于异常检测。为形式化这一概念，我们引入了闭合数，这是一种量化分类器在异常检测中形成闭合决策区域能力的度量标准。借助二元分类与异常检测之间的这一理论关联，我们在监督式异常检测实验中验证了假设，结果表明相等性分离能够同时检测到已知和未知的异常。