# Markov-Embedded Affinity Learning with Connectivity Constraints for Subspace Clustering
## 6 Experiments
### Research Methodology  
In this subsection, we introduce the details of the research methodology. We seek to verify the effectiveness of the proposed method by raising the following questions. * Can the proposed method achieve excellent clustering performance in various real application scenarios? * Can high-order neighbor information improve the clustering performance of the proposed method? * Does the high-order neighbor manifold structure enable the method to effectively handle nonlinear data? * Can the connectivity constraint guarantee the group structure of the affinity matrix? * Can the learned affinity matrix preserve the key information of the original data? To answer these questions, we design a comprehensive series of experiments. First, we validate the clustering performance of the algorithm, including its accuracy, speed, sensitivity to parameters, and the ability to handle nonlinear data. In particular, the proposed method is compared with several state-of-the-art baseline methods regarding their clustering performance on three types of real application data sets using five clustering evaluation metrics. The speed of the proposed method is validated using a convergence analysis and the clustering time cost in Sections 6.3.3 and 6.3.5. An algorithm that is insensitive to the parameters can be applied to more potential scenarios. Therefore, we conduct parameter sensitivity experiments in Section 6.3.7. In Section 6.3.2, the proposed method is compared with the baseline methods on a nonlinear data set. Second, we verify the impact of the high-order neighborhood information, the connectivity constraints, and the novel manifold term on the clustering performance of the proposed algorithm in Section 6.3.6. Third, we validate the ability of the learned affinity matrix of the proposed algorithm to capture key information. Generally, the number of block diagonals in the affinity matrix is equal to the number of clusters in the data. Therefore, we visualize the learned representation matrix in Section 6.3.4. Meanwhile, to validate the ability of the affinity matrix to preserve neighbor information, we demonstrate the changes in the distribution of the original data and clustered data in Section 6.3.4. In fact, by identifying key features, the data can be effectively partitioned into different clusters. In Section 6.3.8, by utilizing the self-expressive property, we visually compare the reconstructed images of \(X\) and \(XZ\) in order to demonstrate the ability of the affinity matrix to preserve key features. The details of the five evaluation metrics are as follows [24, 25, 26]. Accuracy (ACC) is defined as  
\[\frac{\sum_{i=1}^{n}\delta(map(r_{i}),l_{i})}{n}, \tag{22}\]  
where \(r_{i}\) and \(l_{i}\) represent the predicted and true labels of the data point \(x_{i}\), respectively. \(map(r_{i})\) maps each cluster label \(r_{i}\) to the equivalent label from the data set by permutation such that (22) can be maximized, and \(\delta(x,y)\) is the delta function, which returns 1 when \(x=y\) or 0 otherwise. Normalized mutual information (NMI) is defined as  
\[\frac{\sum_{i=1}^{M}\sum_{j=1}^{M}n_{i,j}log\frac{n_{i,j}}{n_{i,j}}}{\sqrt{( \sum_{i=1}^{M}n_{i}log\frac{n_{i}}{n})(\sum_{j=1}^{M}\hat{n_{j}}log\frac{n_{i}} {n})}}, \tag{23}\]  
where \(n_{i}\) and \(\hat{n_{j}}\) denote the sizes of the \(i\)-th cluster and \(j\)-th class, respectively. \(n_{i,j}\) denotes the number of data points that are present in the intersection between them, and \(M\) is the number of clusters. Purity (PUR) is defined as  
\[\frac{1}{n}\sum_{i=1}^{M}\max(n_{i}^{j}), \tag{24}\]  
where \(\max(n_{i}^{j})\) is the number of data points in the \(j\)-th cluster that belong to the \(i\)-th class. PUR measures the extent to which each cluster contains data points from primarily one class. More details about these measures can be found in [27]. The F-score (FS) is defined as  
\[\frac{(1+\rho^{2})\times Precision\times Recall}{\rho^{2}\times Precision+Recall}, \tag{25}\]  
where \(Precision=TP/(TP+FP)\) and \(Recall=TP/(TP+FN)\).