# Anomaly Detection in High-Dimensional Time Series Data with Scaled Bregman Divergence
## 1 Introduction
Though significant progress has been made in anomaly detection algorithms, several challenges remain. Most existing anomaly detection methods require learning the underlying distribution of the data to differentiate between normal and anomalous data, which is challenging, particularly in cases where the data distribution is high-dimensional, non-stationary, or multimodal. Accurately capturing these distributions often demands complex models and large amounts of data, which may not always be available. Furthermore, the assumption of a specific distribution may not hold true for all datasets, potentially limiting the effectiveness of these methods. Although directly estimating individual distributions is challenging in high-dimensional data, estimating the ratio of the densities between training and test data [29; 30] is comparatively more feasible. Density ratio estimation-based approaches circumvent the need for explicit distribution estimation by focusing on the relative likelihood of data points under different conditions. Techniques such as density ratio estimation [30; 31] and ratio matching [32] methods offer an alternative for anomaly detection in complex, high-dimensional data since they simplify the problem by estimating the ratio between normal and anomalous data distributions. Density ratio estimation involves directly estimating the ratio between the probability densities of two distributions, typically corresponding to normal (inlier) and anomalous (outlier) data. This approach avoids the explicit estimation of individual densities and focuses on identifying deviations between two distributions. Several techniques have been developed for density ratio estimation, including the Kullback-Leibler Importance Estimation Procedure (KLIEP) [29], unconstrained least-squares importance fitting (uLSIF) [31], Relative uLSIF (RuLSIF) [30], Bregman divergence-based [33], RobustRealNVP [34], and Pearson-like scaled Bregman divergence-based (PLsBD) [35] methods. In the previous studies [30, 35, 36], these approaches have been successfully applied to identify change points in time series data, a problem closely related to anomaly detection. Change-point detection, like anomaly detection, involves identifying significant shifts in data behavior over time. The existing density ratio estimation methods utilize various divergence measures such as the Kullback-Leibler (KL), Pearson, and \(\alpha\)-relative Pearson divergence to quantify the dissimilarity between data distributions and detect the change points or anomalies. For example, [PERSON] et al. 's work [37] applied uLSIF to identify anomalies; as we discussed above, these divergence functions may introduce severe unboundedness issues under some conditions. Moreover, most density ratio estimation methods adopt a squared error (\(L_{2}\)) loss as the optimization objective to estimate the density ratio [30, 37, 38, 39]. This approach is widely favored for its simplicity and computational efficiency in convex optimization problems. However, its sensitivity to outliers and limitations in handling non-Gaussian noise can affect performance in certain applications. To address the limitations of existing anomaly detection methods, we propose a scaled Bregman divergence-based anomaly detection approach with density ratios, inspired by our recent change-point detection work [35]. Compared to existing density ratio estimation-based methods, the proposed scaled Bregman divergence-based approach, using both \(L_{1}\) and \(L_{2}\) loss functions, addresses the unboundedness issue, ensuring stable divergence calculations. Additionally, compared to deep learning-based methods, our approach is more computationally efficient, particularly in handling high-dimensional datasets with limited data availability. The flexibility of scale-Bregman divergence can improve the accuracy and reliability of anomaly detection, particularly in high-dimensional or complex data.