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：无法对异或问题进行线性分类一直是深度学习发展的重要推动力。本文重新审视这一历史问题，并证明异或问题的线性分类确实是可行的。不同于在半空间之间划分数据，我们提出了一种略有不同的范式——等距分离，该方法通过调整SVM目标函数来区分间隔内外的数据。我们的分类器可通过平滑近似集成到神经网络流程中。基于其特性，我们推断等距分离适用于异常检测。为形式化这一概念，我们引入闭合数这一量化指标，用于衡量分类器为异常检测形成封闭决策区域的能力。基于二元分类与异常检测之间的理论联系，我们在监督式异常检测实验中验证了假设，结果表明等距分离能够同时检测已见和未见异常。