Exploring Linear Neural Network Models for Anomaly Detection: From Mathematical Foundations to Real-World Deployment

Abstract

The problem of anomaly detection plays an important role in many tasks related to cybersecurity, financial fraud prevention, and industry, and modern anomaly detection approaches rely predominantly on deep and nonlinear network architectures, such as convolutional and recurrent autoencoders, GANs (generative adversarial networks), and transformers. The current review provides an insight into the problem of using linear neural networks — networks comprised only of affine transformations without any nonlinear activation function — for anomaly detection. We discuss the fundamental theory behind the linear neural networks which is different from the one used for nonlinear networks, including the equivalence of linear autoencoders to PCA and the absence of spurious local minima in linear networks. We then catalogue the principal architectures used for anomaly detection with linear models, including reconstruction-based linear autoencoders, the Principal Component Classifier, multivariate statistical process control charts, and recent linear forecasting backbones such as DLinear. The core of the review surveys industrial case studies spanning network intrusion detection, credit card fraud detection, chemical process monitoring on the Tennessee Eastman benchmark, and predictive maintenance in industrial IoT, drawing out recurring patterns in why practitioners choose linear models. A comparative analysis contrasts linear and nonlinear approaches on expressiveness, computational cost, interpretability, and robustness, and the review closes with open challenges and directions for hybrid linear–nonlinear systems.