# Anomaly Detection in High-Dimensional Time Series Data with Scaled Bregman Divergence
## 1 Introduction  
Anomaly detection (AD), also known as outlier detection, is the process of identifying data points, patterns, or events that deviate significantly from the expected or normal behavior within a dataset. Anomaly detection has a broad range of applications, including network security (e.g., detecting unauthorized access or cybersecurity threats) [1], fraud detection (e.g., unusual transactions in banking or insurance) [2], medical diagnostics (e.g., identifying tumors in MRI scans) [3], quality control in manufacturing (e.g., equipment faults), and healthcare and environmental monitoring (e.g., irregularities in vital signs of patients or unusual weather) [4]. Numerous anomaly detection methods based on statistical and machine learning algorithms have been developed and successfully applied across various fields [5, 6]. These techniques aim to identify abnormal patterns within diverse types of datasets, including time series data. Most existing anomaly detection techniques are generally categorized into supervised, semisupervised, and unsupervised approaches. The review by [PERSON] et al. [7] provides a detailed discussion of various methods. Here, we briefly introduce some popular techniques for completeness. Supervised methods, also referred to as classification methods [8, 9, 10], rely on labeled training data to construct predictive models. Commonsupervised anomaly detection algorithms include Support Vector Machines (SVMs), k-Nearest Neighbors (k-NN), Bayesian Networks, Decision Trees, and Neural Networks. For example, [PERSON] et al. [11] proposed an intrusion detection system using Kohonen Neural Networks and SVM to identify anomalous patterns in network traffic. Similarly, SVMs have been combined with Decision Trees for intrusion detection [12] and Denial-of-Service (DoS) attack detection [13]. Neural Network-based approaches, such as Multi-Layer Perceptrons (MLPs) or Artificial Neural Networks, have also been employed to differentiate between normal and abnormal behaviors. For instance, Radial Basis Function Neural Networks (RBF) [14] were utilized to learn multiple local clusters for well-known attacks. These methods can also integrate results from multiple classifiers to enhance performance, making them particularly well suited for problems involving large datasets. Unsupervised anomaly detection methods identify anomalies based on the intrinsic characteristics of the data, without requiring any labeled training data or prior knowledge of which points are anomalous. These methods are based on the assumption that the majority of data points are normal, while only a small fraction are anomalous. Common unsupervised anomaly detection algorithms include clustering-based methods such as K-Means [15], Fuzzy C-Means [16], and the unsupervised niche clustering-based anomaly detection algorithm [17], as well as probabilistic approaches like the Expectation-Maximization (EM) Meta algorithm [18]. Deep learning-based techniques, such as Generative Adversarial Network (GAN)-based [19; 20], variational autoencoder (VAE)-based [21; 22; 23], transformer-based [24; 25; 26], and diffusion model-based [27] anomaly detection approaches, have garnered significant attention due to their ability to model complex data distributions and identify subtle anomalies. Popular generative model-based tools for anomaly detection include MAD-GAN [20], VAE-GAN [23], UTRAD [26], etc. However, these methods often require a large amount of data for effective training, which can be a significant limitation in scenarios where data availability is restricted. This reliance on extensive training data poses challenges in applications with limited labeled or high-quality datasets. The semisupervised anomaly detection methods, for example, GANomaly [28], typically use a small set of labeled data, which provide information about normal or anomalous patterns, alongside a larger set of unlabeled data to learn the underlying data distribution and detect deviations.