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.

无法线性分类XOR激发了深度学习的很多研究。我们重新审视这个古老的问题并证明线性分类XOR是可能的。我们提出了一种稍微不同的范式，即等式分离，将SVM目标适应于区分数据在或超过边界之外。我们的分类器可以与神经网络流水线集成，并进行平滑逼近。根据其属性，我们直觉等式分离适用于异常检测。为了形式化这个概念，我们引入了封闭数字，这是对分类器形成封闭决策区域的容量的定量度量。基于这个二元分类与异常检测之间的理论联系，我们在监督异常检测实验中测试了我们的假设，结果显示等式分离可以检测到已知和未知的异常。

非可分二元分类及其在异常检测中的应用再思考