# Laplacian Eigenmaps Dimensionality Reduction Based on Clustering-Adjusted Similarity
## 4 Results
### Parameter Sensitivity Analysis  
Because we adopted kernel \(k\)-means just as data preprocessing in our approach, so for the determination of \(k_{1}\) value, we make the \(k_{1}\) from 2 to 10, repeat several times kernel \(k\)-means on each \(k_{1}\) value (to avoid local optimal solution), and calculate the current average contour coefficient. Finally, the \(k_{1}\) corresponding to the maximum contour coefficient is selected as the final number of clusters. LE-CAS, LE, and GGLE make use of a \(k\)NN graph to set up the adjacency matrix and the weights between samples, and hence for graph-based dimensionality reduction. To study the sensitivity of these methods to the input value of \(k_{2}\), we increase \(k_{2}\) from 5 to 12, and report the FMI values of these algorithms under each input value of \(k_{2}\) (number of neighbors) in Figure 5.  
From this figure, we can observe that no matter how the \(k_{2}\) value changes, the results of LE-CAS on these datasets are better than those of another two comparing algorithms. Fluctuations mainly happen in LE and GGLE, while LE-CAS gets a relatively smooth curve. GGLE adopts the \(k\)NN graph constructed in the original high-dimensional space to explore the local geometric structure of samples. It uses the \(k_{2}\) nearest neighbors of a sample to seek the linear relationships between \(k_{2}\) nearest neighbors. Therefore, it still focuses on the local manifold, and is sensitive to \(k_{2}\). As \(k_{2}\) increase, the linear relationship between them becomes increasingly complicated, which causes the increase of error rate. LE and LE-CAS also adopt the \(k\)NN graph, LE-CAS additionally uses the global cluster structure of samples to adjust the similarity between neighborhood samples. The performance margin between  
Figure 4: Results of PU on ten datasets (Msplice, W1a, Soccer-sub1, Medelon, FG-NET, ORL, Musk, CNAE-9, SECOM, DrivFace) under different target dimensionalities (\(c\)). Figure 3: Results of F-measure on ten datasets (Msplice, W1a, Soccer-sub1, Medelon, FG-NET, ORL, Musk, CNAE-9, SECOM, DrivFace) under different target dimensionalities (\(c\)). LE-CAS and LE, and the better stability of LE-CAS proves the effectiveness of clustering-adjusted similarity. From these results, we can conclude LE-CAS is robust to noise and can work well under a wide range of input values of \(k_{2}\). Both LE-CAS and GGLE specify edge weight by Gaussian heat kernel function. From Equation (11), we can see that they both rely on a suitable Gaussian heat kernel width \(\sigma_{m}\). \(\sigma_{m}\) should not be too small or too large. If \(\sigma_{m}\) is too small, the similarity in Equation (11) will be close to 0, On the other hand, \(\sigma_{m}\) should not be too big. If \(\sigma_{m}\) is too big, the similarity in Equation (11) will be close to 1. In our previous experiments, we set \(\sigma_{m}\) as the mean of square Euclidean distance between all training instances for both LE-CAS and GGLE. To investigate the sensitivity of \(\sigma_{m}\) on LE-CAS and GGLE, we conduct experiments to investigate the influence of \(\sigma_{m}\). In the following experiments, we increase \(\sigma_{m}\) from \(1\times 10^{-2}\) to \(1\times 10^{5}\) for LE-CAS and GGLE. Other parameter settings are kept the same with previous experiments. Similarly, we run 30 independent experiments for each fixed \(\sigma_{m}\) and report the FMI under each \(\sigma_{m}\) in Figure 6.  
Figure 5: Results (FMI) vs. \(k_{2}\) on ten datasets (Msplice, W1a, Soccer-sub1, Medelon, FG-NET, ORL, Musk, CNAE-9, SECOM, DrivFace). Figure 6: Results (FMI) vs. \(\sigma_{m}\) on ten datasets (Msplice, W1a, Soccer-sub1, Medelon, FG-NET, ORL, Musk, CNAE-9, SECOM, DrivFace). From Figure 6, we can find that LE-CAS outperforms LGC in a wide range of \(\sigma_{m}\). Both GGLE and LE-CAS depend on a suitable \(\sigma_{m}\). The performance of two methods reaches relatively stable with the increase in \(\sigma_{m}\). The FMI of LE-CAS is similar to GGLE when \(\sigma_{m}\) is too small.