## 2 Related Work  
### AE  
AE is a neural network architecture composed of stacked layers, often employing nonlinear activation functions to model complex, nonlinear relationships within the data. With this capability, AE is extensively employed for dimensionality reduction and feature extraction in high-dimensional datasets, providing a foundation for subsequent machine learning tasks. A standard AE primarily consists of three components: an encoder, a latent representation, and a decoder. The encoder maps high-dimensional input data into a more compact, lower-dimensional representation known as the latent representation. Subsequently, the decoder utilizes this latent representation to reconstruct the original input while minimizing information loss. The information loss between the original input and the reconstructed output is referred to as the reconstruction error. Figure 1 illustrates a standard AE model.  
The encoder is responsible for mapping the original input data \(\mathbf{x}\), typically represented as an \(m\)-dimensional vector (\(\mathbf{x}\in\mathbb{R}^{m}\)), into a lower-dimensional latent representation, often denoted as \(\mathbf{z}\). The encoding process can be mathematically expressed as follows:  
\[\mathbf{z}=g(\mathbf{W}_{e}\mathbf{x}+\mathbf{b}_{e}) \tag{1}\]  
where \(\mathbf{W}_{e}\) is the weight matrix with dimensions \(p\times m\), \(\mathbf{b}_{e}\) is the bias vector with dimensions \(p\), and \(g\) is an activation function. The dimensionality of the latent space \(\mathbf{z}\) is specified as \(p\), where \(p\) is a positive integer, representing the reduced dimensionality of the input data.  
However, recent research has challenged the necessity of a bottleneck in anomaly detection models. [PERSON] and [PERSON] [26] have demonstrated that the latent space dimensionality does not necessarily need to be smaller than the dimensionality of the input data. Specifically, removing the bottleneck by either expanding the latent space or introducing skip connections can lead to improved performance in anomaly detection tasks. This finding suggests that non-bottlenecked autoencoders, which allow the model to learn more complex representations, may outperform their bottlenecked counterparts in certain contexts, particularly in tasks where a higher expressivity of the model is beneficial for detecting anomalies. Our study will also employ this overcomplete architecture Autoencoder in the subsequent sections.  
The decoder utilizes the latent representation \(\mathbf{z}\) to reconstruct the original input as accurately as possible. The reconstructed output is denoted by \(\mathbf{\hat{x}}\), while the difference between \(\mathbf{\hat{x}}\) and the original input \(\mathbf{x}\) is referred to as the reconstruction error. Equation (2) formalizes the decoding process, where \(f\) is the activation function, \(\mathbf{W}_{d}\) is the weight matrix with dimensions \(m\times p\), \(\mathbf{b_{d}}\) represents the bias vector with dimensions \(m\), and \(\mathbf{\hat{x}}\) denotes the reconstructed input data. Equation (3) describes the computation of the reconstruction error, where L represents the reconstruction error, and m denotes the dimensionality of the original data, corresponding to the number of variables.  
\[\mathbf{\hat{x}}=f(\mathbf{W}_{d}\mathbf{z}+\mathbf{b_{d}}) \tag{2}\]  
\[L(\mathbf{x},\mathbf{\hat{x}})=\frac{1}{m}\sum_{i=1}^{m}(\mathbf{\hat{x}}_{i}-x_{i})^{2} \tag{3}\]