Graph-Spectral Flow Matching for Multivariate Time Series Anomaly Detection

2026-09-29

AI Summary

Standard flow matching neglects variable dependencies, introducing bias in multivariate time series anomaly detection. To address this limitation, this work proposes GRASP, a framework that innovatively integrates graph structures with flow matching. Specifically, it constructs closed-form spectral paths via hyperbolic interpolation to optimize probability distributions, and incorporates the principle of least action alongside multi-source weighted velocity prediction for efficient anomaly detection, supported by theoretical guarantees of Laplacian basis invariance. Extensive experiments on four benchmark datasets demonstrate that GRASP significantly outperforms existing models, thoroughly validating the effectiveness of the proposed spectral path construction and weighting mechanisms.

Abstract

Multivariate time series anomaly detection typically relies on evaluating discrepancies between observations and outputs produced by models trained on normal data. An alternative perspective is to characterize the distribution of normal data through the generative dynamics, i.e., the velocity field, of flow matching models. However, standard flow matching typically adopts linear probability paths that overlook dependencies among variables, leading to a misalignment with the structured data distribution. To address this issue, we propose GRASP, a flow matching framework with a graph-spectral path for multivariate time series anomaly detection. GRASP incorporates graph structure into the probability path by minimizing a fixed-endpoint action that combines kinetic energy with graph Dirichlet energy. This formulation yields a closed-form path based on graph-frequency-dependent hyperbolic interpolation. A velocity predictor trained on normal data then detects anomalies using weighted velocity discrepancies aggregated across source samples, flow times, and graph frequencies. Theoretically, we establish that GRASP is invariant to the choice of Laplacian eigenbasis and decompose its expected oracle anomaly score into bounded endpoint uncertainty and graph-frequency-weighted Fisher discrepancy. Experiments on four benchmarks demonstrate the superior anomaly detection performance of GRASP and validate the effectiveness of its graph-spectral path and weighting mechanism.

Problem

• Multivariate Time Series
• Anomaly Detection
• Flow Matching
• Variable Dependencies

Innovation

• Flow Matching
• Graph-Spectral Path
• Multivariate Time Series
• Anomaly Detection
• Dirichlet Energy