## 1 Introduction
Among the commonly utilized unsupervised methods, there are distance-based techniques such as K-nearest neighbors [12; 13; 14; 15; 16]. This approach measures the similarity between data points based on their distances, and anomalies are identified as points that significantly deviate from their neighboring data points. Clustering techniques, like K-means, group similar data points together and consider points that do not belong to any cluster as potential anomalies [17; 18; 19]. Classification techniques, such as one-class support vector machines (SVMs), create a model of normal behavior and classify instances that fall outside the learned boundaries as anomalies [20; 21; 22]. Probabilistic methods analyze the statistical properties of the data and identify instances that have low probability under the learned distribution as anomalies [23]. However, traditional unsupervised methods have faced limitations over time. One significant challenge is the exponential increase in the dimensionality and length of the acquired measurements in modern network systems. As the complexity of the data grows, traditional techniques struggle to effectively capture the intricate patterns and correlations present in the time series network data. Additionally, these methods may not perform optimally in capturing the temporal correlation across different time steps, which is crucial for accurate anomaly detection. The dynamic nature of network systems requires models to consider the sequential dependencies and changes over time, which traditional techniques may overlook [24]. As a result, there has been a shift towards employing deep learning-based unsupervised anomaly detection methods. These approaches leverage the power of deep neural networks to automatically learn representations and infer correlations between time series [25; 26; 27; 28]. Recurrent neural networks (RNNs) [29; 30; 31], such as the long short-term memory (LSTM) network [32; 33], have shown promise in capturing long-term dependencies and temporal correlations in time series data. These can effectively model sequential information and have been successfully applied to anomaly detection tasks. Additionally, generating adversarial networks (GANs) have been explored in this domain [34]. GANs can learn to generate synthetic data that closely resemble the normal behavior of the network system, allowing for the detection of deviations from this learned distribution as anomalies. Despite the advantages of deep learning-based methods, they also face their own set of challenges. The training of RNNs can be computationally expensive and time-consuming due to the sequential nature of the network. GANs, on the other hand, can suffer from issues such as pattern collapse and non-convergence during training, which can impact their performance. Furthermore, deep learning-based methods often struggle with noisy data in multivariate time series, as noise can introduce false positives and reduce the accuracy of anomaly detection. Addressing these challenges is an active area of research. To the best of our knowledge, ref. [25] proposed a novel model called MSCRED that addresses the challenges of anomaly detection, root cause identification, and anomaly severity interpretation in multivariate time series data. MSCRED specifically focuses on capturing the correlation between different variables in the time series, enabling more accurate anomaly detection. By jointly considering these tasks, MSCRED offers a comprehensive solution for analyzing complex network systems. Similarly, ref. [28] introduced OmniAnomaly, a multivariate time series recurrent neural network designed for stochastic anomaly detection. OmniAnomaly utilizes reconstruction probabilities to determine anomalies, providing a reliable and interpretable measure of anomaly severity. With its stochastic nature, OmniAnomaly is well suited for capturing uncertainties in time series data. However, one common limitation shared by these methods is the lack of consideration for the time cost associated with training.