# Revisiting Non-separable Binary Classification and its Applications in Anomaly Detection

arXiv: https://arxiv.org/abs/2312.01541 · HTML: https://mlsanity.com/paper/2312.01541/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 44 references that are in the map.
- First posted on arXiv 3 Dec 2023.

## Builds on (most cited first, 10 of 10)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [A Unifying Review of Deep and Shallow Anomaly Detection](https://mlsanity.com/paper/2009.11732/) (2020)
- [Deep Semi-Supervised Anomaly Detection](https://mlsanity.com/paper/1906.02694/) (2019)
- [Deep Anomaly Detection with Deviation Networks](https://mlsanity.com/paper/1911.08623/) (2019)
- [Non-Parametric Outlier Synthesis](https://mlsanity.com/paper/2303.02966/) (2023)
- [DROCC: Deep Robust One-Class Classification](https://mlsanity.com/paper/2002.12718/) (2020)
- [Neural Transformation Learning for Deep Anomaly Detection Beyond Images](https://mlsanity.com/paper/2103.16440/) (2021)
- [Interpretable, Multidimensional, Multimodal Anomaly Detection with Negative Sampling for Detection of Device Failure](https://mlsanity.com/paper/2007.10088/) (2020)
- [Autoencoding Binary Classifiers for Supervised Anomaly Detection](https://mlsanity.com/paper/1903.10709/) (2019)
- [Coin Flipping Neural Networks](https://mlsanity.com/paper/2206.09182/) (2022)

## Built on by (most cited first, 3 of 3)

- [Sparse, self-organizing ensembles of local kernels detect rare statistical anomalies](https://mlsanity.com/paper/2511.03095/) (2025)
- [Bridging Unsupervised and Semi-Supervised Anomaly Detection: A Theoretically-Grounded and Practical Framework with Synthetic Anomalies](https://mlsanity.com/paper/2506.13955/) (2025)
- [Retrofit: Continual Learning with Controlled Forgetting for Binary Security Detection and Analysis](https://mlsanity.com/paper/2511.11439/) (2025)

## Abstract (arXiv)

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.