Anomaly as Non-Conformity via Training-Free Graph Laplacian Energy Minimization
Authors: Jungwook Seo, Minjeong Kim, Younkwan Lee, Seungho Shin, Sungyong Baik
Organizations: Dept. of Artificial Intelligence, Hanyang University · Global Technology Research, Samsung Electronics · Dept. of Data Science, Hanyang University
Abstract
Detecting subtle visual anomalies in images remains challenging, particularly when only normal samples are available a priori. Such unsupervised anomaly detection is typically solved by measuring feature similarity of a query patch to a memory of normal patches. However, similarity alone does not reveal how strongly a query patch violates the structure of the normal feature manifold. We propose a training-free Laplacian graph energy optimization formulation, named ANoCo that scores Anomaly by the cost of Non-Conformity of a query patch to align with a fixed normal manifold. For each query patch, we construct a bipartite query to normal graph weighted by cosine affinity, explicitly removing query-query and normal-normal edges to prevent evidence dilution. We formulate anomaly scoring as a convex Laplacian energy with anchored normal nodes, and solve in closed form. In particular, we do not use the optimized features themselves-the anomaly score is the magnitude of the update required to satisfy normality constraints, reframing the graph Laplacian as a non-conformity operator rather than a smoothing prior. The proposed method introduces no learnable parameters, message passing, or sampling, and has complexity comparable to a single linear solve. Across standard benchmarks, it delivers strong image-level AUROC, stable localization maps, and improved robustness over prior methods, demonstrating the effectiveness of using optimization-induced feature drift as anomaly measure.
Reconstruction-based methods are a cornerstone of unsupervised image anomaly detection, but they remain vulnerable to \emph{outlier leakage}, where standard mean squared error (MSE) loss drives the model to faithfully reconstruct anomalous patterns. We propose a Non-linear Reconstruction Loss that applies a sigmoid-based squashing function to suppress high-magnitude features, preventing outliers from dominating optimization while preserving sensitivity to normal patterns. In addition, we introduce a statistical calibration scheme that selects the scaling factor k from the confidence interval (CI) of the normal feature distribution, enabling data-driven control of the suppression strength. Our approach achieves competitive or superior anomaly detection performance compared to state-of-the-art methods, reaching 99.0% Image-AUROC and 97.3% Pixel-AUROC on MVTec-AD, and 95.3% Image-AUROC and 99.0% Pixel-AUROC on VisA. These results indicate that non-linear gradient suppression is an effective mechanism for mitigating outlier leakage and improving anomaly localization in unified industrial inspection settings. The implementation is available at https://github.com/mintii13/Statistical-Non-linear-Reconstruction-Loss.git.
Graph anomaly detection methods aim to distinguish anomalous nodes. While prior methods characterize anomalies through increased variation in the spectral energy distributions, they overlook those that result in decreased variation, i.e., camouflaged anomalies that appear normal. We show that this type of anomaly persists across multiple datasets and remains undetectable by existing spectral approaches. To address this limitation, we propose a node-level spectral energy formulation that is fully compatible with message passing and enables the detection of camouflaged anomalies. Building on this formulation, we introduce an energy-aware graph learning framework that models spectral shifts through energy-driven message passing in both static and time-series graphs. Besides, our unified architecture extends to temporal settings without introducing specialized sequence modules, enabling efficient learning under long sliding windows. Extensive experiments on large-scale benchmarks demonstrate the effectiveness and scalability of our approach.
Memory-based anomaly detection is attractive because it localizes defects from normal images without training a decoder or synthesizing pseudo anomalies. However, most memory methods still use the memory bank as a nearest-neighbor lookup table: a test patch is treated as normal if it has one nearby normal anchor. This hard retrieval view is vulnerable to false-normal matches and does not test whether the patch is consistently supported by a local normal neighborhood. We propose ProCon, a training-free framework that turns memory retrieval into decoder-free reconstruction. ProCon softly projects each test patch onto nearby normal memory vectors and uses the projection residual as anomaly evidence. To stabilize this residual, it constructs seed-perturbed layer-wise memories, aggregates bank residuals by a median, and fuses depth-specific residual maps by layer consensus. ProCon requires no decoder training, backbone fine-tuning, learned fusion weights, or pseudo-anomaly supervision. Across MVTec-AD, VisA, and Real-IAD under the single-category evaluation protocol, ProCon achieves strong image- and pixel-level performance under seven standard metrics, including image AUROC scores of 99.8%, 99.2%, and 93.2%, respectively. Ablations show that the gains come from replacing hard retrieval with soft normal projection and stabilizing the residuals through memory and depth consensus. The code is available at https://github.com/jw-chae/Procon