Low-Rank and Structured Sparse Tensor Decomposition for Anomaly Detection in Multivariate Functional Data

2026-10-02

AI Summary

This study addresses the challenge of anomaly detection in multivariate functional data characterized by strong correlations, with a focus on precisely identifying local deviations and structured faults. To this end, two unsupervised sparse tensor decomposition methods are proposed. Normal behavioral patterns are represented via low-rank CANDECOMP/PARAFAC (CP) decomposition, and two novel algorithms, ES-CP and FG-Lasso, are developed by incorporating element-wise L1-norm and fiber-group Lasso regularization, respectively. Efficient optimization is achieved through alternating minimization combined with closed-form update rules. Experimental results demonstrate that the proposed methods significantly outperform traditional baselines such as TRPCA in high-dimensional process fault localization. Notably, FG-Lasso achieves the highest macro F1-score of 0.85 in a forging case study.

Abstract

Multivariate functional data arise in many modern manufacturing systems, where multiple sensors record densely sampled process trajectories. Monitoring such data is challenging because nominal variation is strongly correlated across samples, sensors, and time, while faults may appear either as isolated deviations or as structured departures concentrated within a limited number of sensor-specific temporal trajectories. We propose two unsupervised sparse tensor decomposition methods that preserve this multimode structure. Entrywise Sparse CP Decomposition (ES-CP) uses an entrywise \(\ell_1\) penalty to identify localized anomalies, whereas Fiberwise Sparse-Group Lasso CP Decomposition (FG-Lasso) combines entrywise and fiberwise penalties to detect both localized deviations and anomalies concentrated within temporal fibers. Both methods represent nominal process behavior through a low-rank CP decomposition and are estimated using alternating optimization with closed-form sparse-component updates. Two simulation studies evaluate performance under different fault structures, signal severities, noise levels, and missing observations. FG-Lasso attains or ties the highest macro F$_1$ score in almost all settings in the first study and achieves the highest macro F$_1$ score. In a multichannel forging-process case study, FG-Lasso and ES-CP obtain macro F$_1$ scores of 0.85 and 0.82, respectively, compared with 0.69 or lower for TRPCA and PCA-based anomaly detectors. The results demonstrate that explicitly matching the sparse penalty to the anticipated fault structure improves both anomaly detection and fault localization in high-dimensional functional processes.

Problem

• Anomaly Detection
• Multivariate Functional Data
• Tensor Decomposition
• Fault Localization
• Manufacturing Systems

Innovation

• Tensor Decomposition
• Anomaly Detection
• Sparse-Group Lasso
• Multivariate Functional Data
• Low-Rank CP Decomposition