## 5 Results  
### Reconstruction Task  
The Mean Squared Error (MSE) and Mean Absolute Error (MAE) are employed to evaluate the reconstruction performance, which reflects the model's ability to capture the essential features of the data. The Mean Squared Error (MSE) is defined as follows:  
\[\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\hat{x}_{i})^{2} \tag{12}\]  
MSE is calculated as the mean of the square of the differences between the original data and the data that has been reconstructed. Here, \(x_{i}\) denotes the original data point, \(\hat{x}_{i}\) denotes the corresponding reconstructed data point, and \(n\) is the total number of samples. This metric is more sensitive to larger errors, which are made bigger by squaring. It helps to identify significant reconstruction distortions. In activity recognition, MSE capturesthe model's ability to remember the details of an action and is more sensitive to large-magnitude action features. Mean Absolute Error (MAE) is defined as follows:  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}|x_{i}-\hat{x}_{i}| \tag{13}\]  
MAE calculates the mean absolute difference between the original and reconstructed data. It uses \(x_{i}\) for the original data points, \(\hat{x}_{i}\) for the reconstructed data points, and \(n\) for the total number of samples. MAE is less sensitive to outliers and provides a more balanced picture of overall reconstruction quality compared to MSE. These two error metrics work well together to provide a thorough assessment of how well the reconstruction is carried out. MSE focuses on capturing important details, while MAE shows how accurate the reconstruction is overall.  
As shown in Table 1, CoTD-VAE outperforms LSTM-AE, \(\beta\)-VAE, and CVAE across both MSE and MAE metrics. An interesting observation from the result is that Transformer-AE exhibits significantly lower MSE and MAE values compared to other models. Overfitting is potentially suggested by this. Despite the notably low reconstruction error, its generalization ability was further assessed through downstream prediction tasks.  
To qualitatively assess the reconstruction fidelity of CoTD-VAE, we have designed Figure 3 to present a focused, narrative-driven analysis. Channel 4 serves as an example of high-fidelity reconstruction. Here, the reconstructed signal (red curve) closely tracks the original signal's (blue curve) primary peaks and troughs, demonstrating the model's capability to preserve core signal dynamics when the underlying pattern is clear. In contrast, Channel 5 highlights a more sophisticated capability: event capture. The model successfully identifies and reconstructs the significant, sharp dip around the t = 60 mark, proving that it can distinguish meaningful transient events from random noise. Finally, Channel 1 offers a transparent view of the model's limitations and inductive biases. Faced with high-frequency oscillations, the model produces a significantly smoothed reconstruction. This behavior, a direct consequence of our Trend Smoothness Loss, shows the model's tendency to function as a denoising filter, which is beneficial for trend extraction but comes at the cost of losing fine-grained texture. This balanced, qualitative evidence provides a much more nuanced and convincing assessment of CoTD-VAE's capabilities than a simple visual inspection of all channels would allow.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Model** & **MSE (\(\times\)10\({}^{-3}\))** & **MAE (\(\times\)10\({}^{-2}\))** \\ \hline LSTM-AE & 3.311 & 3.837 \\ Transformer-AE & 0.438 & 1.589 \\ \(\beta\)-VAE & 6.616 & 5.145 \\ CVAE & 5.776 & 4.852 \\ CoTD-VAE & 3.267 & 3.422 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance on the UCI dataset reconstruction task.