# Anomaly Detection in High-Dimensional Time Series Data with Scaled Bregman Divergence
## 2 Materials and Methods
### Algorithm and Computational Complexity Analysis
#### 2.3.1 Anomaly Detection Algorithm
Both theoretical complexity analysis and real-world data evaluations demonstrate that the \(L_{1}\) loss-based method is more computationally efficient than the \(L_{2}\) loss-based methods, while maintaining comparable performance in high-dimensional data analysis. Although the proposed scaled Bregman divergence-based anomaly detection algorithm effectively resolves the unboundedness issue and outperforms the uLSIF method; however, it has several limitations. Currently, the method assumes that the input data are complete and does not account for missing values. However, many datasets may contain \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Method** & **Loss** & \(\alpha\) & **Accuracy** & **Precision** & **F1 Score** & **Time (s)** \\ \hline \multirow{2}{*}{uLSIF} & L1 & 0.00 & 0.6591 & 0.3219 & 0.2683 & 73.56 \\  & L2 & 0.00 & 0.7283 & 0.5000 & 0.0196 & 91.56 \\ \hline \multirow{6}{*}{RuLSIF} & L1 & 0.01 & 0.7228 & 0.4000 & 0.0727 & 73.73 \\  & L1 & 0.10 & 0.7255 & 0.4865 & 0.2628 & 73.59 \\  & L1 & 0.50 & 0.6576 & 0.3194 & 0.2674 & 72.54 \\  & L1 & 0.90 & 0.6576 & 0.3194 & 0.2674 & 72.18 \\ \cline{2-7}  & L2 & 0.01 & 0.7204 & 0.3400 & 0.0568 & 95.74 \\  & L2 & 0.10 & 0.7284 & 0.5007 & 0.2648 & 94.14 \\  & L2 & 0.50 & 0.6576 & 0.3194 & 0.2674 & 94.61 \\  & L2 & 0.90 & 0.6576 & 0.3194 & 0.2674 & 94.28 \\ \hline \multirow{6}{*}{PLsBD} & L1 & 0.01 & 0.7255 & 0.4444 & 0.0734 & 71.79 \\  & L1 & 0.10 & 0.7310 & 0.5161 & 0.2443 & 70.94 \\ \cline{1-1}  & L1 & 0.50 & 0.6576 & 0.3194 & 0.2674 & 69.72 \\ \cline{1-1}  & L1 & 0.90 & 0.6576 & 0.3194 & 0.2674 & 70.40 \\ \cline{1-1} \cline{2-7}  & L2 & 0.01 & 0.7213 & 0.3583 & 0.0497 & 89.53 \\ \cline{1-1}  & L2 & 0.10 & 0.7308 & 0.5136 & 0.2660 & 90.84 \\ \cline{1-1}  & L2 & 0.50 & 0.6576 & 0.3194 & 0.2674 & 91.69 \\ \cline{1-1}  & L2 & 0.90 & 0.6576 & 0.3194 & 0.2674 & 89.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance metrics (accuracy, precision, F1 score) and computation time for the diabetes dataset analysis. missing values, necessitating the use of imputation or pre-processing techniques to handle incomplete observations. If not applied carefully, these methods can introduce biases or inaccuracies. Moreover, the proposed method involves several pre-processing steps, such as kernel matrix computation and parameter optimization, which can be computationally intensive for high-dimensional data or large datasets. These steps may also require careful tuning to ensure robustness and effectiveness. To address these limitations, future work could incorporate advanced imputation methods [44; 45; 46; 47] and develop more efficient computational techniques to streamline the pre-processing pipeline. This integration with the scaled Bregman divergence framework could enhance robustness and scalability, enabling effective anomaly detection in high-dimensional datasets with missing values. Recently, deep learning-based methods, including GAN-based [19; 20; 23; 48], VAE-based [21; 22; 49], transformer-based [24; 25; 50], and diffusion model-based [27; 51; 52] anomaly detection techniques, have gained significant popularity. These approaches excel in capturing complex patterns within high-dimensional datasets, modeling intricate data distributions, and effectively identifying subtle anomalies. However, these methods typically require a large amount of labeled data for effective training, which can be a significant limitation in scenarios where labeled data are scarce. Additionally, deep learning approaches are computationally intensive due to the need for training Deep Neural Networks, which involves iterative gradient-based optimization over a large number of parameters. This computational demand makes them less practical for real-time or resource-constrained applications. In contrast, our proposed method is computationally more efficient, requiring only a small number of parameters to tune.