Two-Layer Convolutional Autoencoders Trained on Normal Data Provably Detect Unseen Anomalies

Anomaly detection refers to the techniques that identify (probably unseen) rare or suspicious data that deviate significantly from the pre-defined normal data (Chalapathy & Chawla, 2019; Ruff et al., 2021). Empirical studies have observed that generative models trained on normal data tend to produce larger reconstruction errors when reconstructing anomalies. Based on this observation, researchers have developed various anomaly detection methods, referred to as reconstruction-based anomaly detection (RBAD) (Lv et al., 2024; Li et al., 2024) in the literature. Despite the empirical success of RBAD, the theoretical understanding of RBAD is still limited. This paper provides a theoretical analysis of RBAD. We analyze the training dynamics of a 2-layer convolutional autoencoder and introduce the cone set of the features. We prove that the cone sets of the normal features would absorb the (convolutional) kernels of the autoencoder during training and use these absorbed kernels to reconstruct the inputs. The absorbed kernels are more aligned with the normal features, which explains the cause of the reconstruction error gap between the normal data and the anomalies. Synthesized experiments are provided to validate our theoretical findings. We also visualize the training dynamics of the autoencoder on real-world data, demonstrating our proposed cone set intuition.