## 3 Proposed Method
### The Proposed VAEGA Model  
This section presents our proposed VAEGA model in detail. The model is applied to reduce the dimensionality challenges encountered by outlier detection in high-dimensional data. While focusing on improving the effectiveness of the outlier detection model from the perspective of data subspace, the heuristic optimization algorithm is used to obtain the abnormal subspace of outliers to solve the problem that high-dimensional outlier detection models generally lack interpretability. The VAEGA model is defined as a model that combines a variational autoencoder and a genetic algorithm for the purpose of detecting outliers in high-dimensional data and searching for abnormal subspaces. Figure 5 shows the framework of the model. Figure 4: Genetic algorithm process. The VAEGA model is divided into two modules. The first part is the VAE model, which executes the process of quickly filtering out the outliers. In the training phase, the VAE module uses a neural network to build a variational encoder \(p_{\theta}(z|x)\) and decoder \(q_{\phi}(x|z)\). The unlabeled data are used as input for iterative optimization to learn the best encoding and decoding scheme and obtain the mean \(\mu\), and standard deviation \(\sigma\), of the input data \(x\). This step effectively represents the distribution of high-dimensional input data and ameliorates the representation of the high-dimensional data. For real-valued sample data, assuming that the distribution function is a multi-dimensional Gaussian distribution, its indicators are shown in Equations (5)-(7) as follows:  
\[p(z)=N(0,I) \tag{5}\]  
\[q_{\phi}(z|x)=N(\mu_{e},\sigma_{e}^{2}) \tag{6}\]  
\[p_{\theta}(x|z)=N(\mu_{d},\sigma_{d}^{2}) \tag{7}\]  
To optimize the weight parameters, we fit the data's true probability distribution with a standard normal distribution. Through the use of stochastic gradient descent and backpropagation, the loss function of the model converged to a stable minimum value. The loss function of the VAE neural network is shown in Equation (8), and Equation (9) shows the _KL_ divergence used to measure the two probability distributions. \[L(\theta,\phi)=-\|x-\mu_{d}\|^{2}+KL(N(\mu_{e},\sigma_{e})||N(0,I)) \tag{8}\]  
\[KL(N(X)||\epsilon)=\frac{1}{2}\Big{(}tr\Big{[}\sigma_{e}^{2}+\mu_{e}^{T}\mu_{ e}-R_{d}-log\Big{|}\sigma_{e}^{2}\Big{|}\Big{]}\Big{)} \tag{9}\]  
where a random variable \(X\) is normally distributed with a mean \(\mu_{e}\) and standard deviation \(\sigma_{e}\), \(\epsilon=N(0,I)\), \(t_{r}\) is the trace of the matrix, and \(R_{d}\) is the input data dimension. In Equation (8), the first item depicts the space between the reconstructed data and the input data. As a \"reconstruction item\" it tends to produce high performance in the encoding and decoding schemes. Regarding the second item, the KL divergence can be viewed as a penalty that regulates the hidden space. Figure 5: VAEGA model framework. It is important that we sample the latent vector \(z\) from \(q_{\phi}(z|x)\), which obeys the Gaussian distribution \(z\sim q_{\phi}(z|x)=N(\mu,\sigma^{2})\). The mean \(\mu\) and variance \(\sigma\) of \(z\) can be computed by the model, and the neural network relies on this sampling process to reverse optimize the variational autoencoder model. In this case, backpropagation for the model is impossible because sampling is not derivable. We then take into account that the results of sampling are derivable, which can be overcome by employing the reparameterization trick. The random variable \(z\) is sampled from the standard normal distribution \(N(0,1)\) instead of the original distribution, and then the random variable \(q_{\phi}(z|x)\) undergoes the following transformation, as shown in Equation (10). \[z=h_{\phi}(\varepsilon,x) \tag{10}\]  
From \(\varepsilon\) to \(z\) involves linear operations, which are derivable. The distribution of \(\varepsilon\) is deterministic and does not need to be learned.