# Fusing Local and Global Information for One-Step Multi-View Subspace Clustering
## 5 Experiments
### Experiment Results and Analysis
However, there is a large gap between these methods and our approach, probably because they separate the representation learning and clustering processes, leading to suboptimal clustering results. Compared with similar multi-view information fusion methods like GMC and LMVSC, our approach achieves a more impressive performance. This is mainly because we fuse graph learning into the self-expression model to jointly explore local and global structural information in the data. More information is used to serve the clustering task and therefore better performance is obtained. Compared with the partition-based models to construct multi-view subspace clustering for joint optimization methods like COMVSC, mPAC, and PMSC, our approach achieves more impressive performance. This is mainly because we make the clustering structure of the multi-view data revealed while generating the common subspace representation under the rank constraint of Laplacian matrix. Thus, the end-to-end clustering approach facilitates a better clustering performance. Compared with the MVSCLG method, the overall results from the nine data sets show that our method achieves the best results on all evaluation criteria, except for the Caltech101, Cornell and Wikipedia data sets. It states that using rank constraint can obtain an ideal graph matrix fitting for direct clustering. Our method also has superiorities in terms of running time and memory usage, which will be discussed in the next subsection. Thus, our method is more effective than MVSCLG. Compared with these state-of-the-art clustering methods, the proposed method can achieve a more impressive clustering performance. The main reason is that it considers the local and global information from the original multi-view data to learn the subspace representation. Moreover, the proposed LGOMSC is an end-to-end model, which fuses the construction of subspace representation, multi-view information fusion, and clustering into a seamless whole. The purpose for this is to dig into their potential correlations. **Statistical Analysis. ** To demonstrate the statistical properties of our proposed method, we conducted the Friedman test and Nemenyi post-hoc test. The Friedman test assumes that all the \(k\) compared methods hold the same performance on \(H\) data sets. Specifically, this model performance evaluation consists of the following two main steps. In the first step, first, sort all methods on each data set from high to low according to the clustering performance index and assign corresponding ordinal values (e.g., 1, 2,...), and then calculate each method on all data sets average rank. In particular, the ordinal values are averaged if the performance of the two methods is the same. Finally, \(\Gamma_{\chi^{2}}\) and \(\Gamma_{F}\) are calculated, and their mathematical expressions are as follows:  
\[\Gamma_{F}=\frac{(H-1)\Gamma_{\chi^{2}}}{H(k-1)-\Gamma_{\chi^{2}}} \tag{14}\]  
where \(\Gamma_{\chi^{2}}=\frac{12H}{k(k+1)}(\sum_{i=1}^{k}r_{i}^{2}-\frac{k(k+1)^{2}} {4}).r_{i}\) represents the average rank of the \(i\)-th method over all data sets. Besides, \(\Gamma_{F}\) obeys the F-distribution with the degree of freedom \(k\)\(-\)1 and \((k\)\(-\)1)(\(H\)\(-\)1). The correctness of the hypothesis is eliminated by comparing the \(\Gamma_{F}\) with its corresponding threshold (the thresholds of the Friedman test can be calculated by \(qf(1-\alpha,k-1,(k-1)(H-1))\) in R programming language). If the hypothesis is rejected, this indicates a significant difference in the performance of the compared methods. A further Nemenyi post-hoc test is then required to further distinguish between the methods. In the second step, the Nemenyi post-hoc test calculates the critical distance by Equation (15) to reflect the difference between the average ordinal results of various methods. \[CD=q_{a}\sqrt{\frac{k(k+1)}{6H}} \tag{15}\]  
where \(q_{a}\) can be calculate by \(qtukey(1-\alpha,k,Inf)/\mathcal{S}qrt(2)\) in R programming language. In our case, the number of compared methods, \(k\), equals 13 and \(H\) equals 9.