# Adaptive Clustering via Symmetric Nonnegative Matrix Factorization of the Similarity Matrix
## 6 Experimentation  
The experimentation has been performed in JAVA on a Macintosh with a 4 GHz Intel 8 core Xeon W processor running MAC OS MOJAVE 10.14.4 on synthetic datasets of points in \(\mathbb{R}^{2}\). It intends to prove that the proposed heuristics is reliable. In fact, with regard to a multistart strategy, the proposed heuristics allows for time-saving, but one could expect that during the elimination of the less promising roads, roads leading to final good solutions can also be lost. To verify that this does not happen, the experimentation aims to test the effectiveness of the algorithm. Initially, an extensive computation has been carried out on selected datasets of points, to determine suitable values for the parameters to be used in the subsequent computation. Namely:* the tolerances for GCD and Sym_NMF are set \(\eta_{1}=10^{-3}\) and \(\eta_{2}=10^{-4}\), respectively,
* the constants for CL index are set \(p=4\) and \(c=100\),
* the number of the starting random matrices for Kfix is set \(q=8\),
* the number of iterations for a single segment of Kfix is set \(\lambda=10\), hence \(\
u_{\max}=10\),
* the bounds for Control are set \(t_{\min}=3\,\lambda\) and \(t_{\max}=20\,\lambda\),
* for the definition of \(\sigma^{(k)}\) we have set \(\widehat{\sigma}=0.04\) and \[\sigma_{0}^{(k)}=\widehat{\sigma}\text{ for }k\leq 5,\;\sigma_{0}^{(k)}= \widehat{\sigma}/2\text{ for }5<k\leq 10,\;\sigma_{0}^{(k)}=\widehat{\sigma}/4\text{ for }10<k\leq 20,\] \[\sigma_{0}^{(k)}=\widehat{\sigma}/8\text{ for }20<k\leq 40,\;\sigma_{0}^{ (k)}=\widehat{\sigma}/16\text{ for }k>40.\] The triplets considered in the experiments, varying \(k\), are \[\sigma^{(k)}=\{\sigma_{i}^{(k)}=\sigma_{0}^{(k)}/2^{i},\;i=0,\,1,\,2\}.\] (10)  
Regarding the value \(\widehat{\sigma}\), we have taken into account that the quantities \(\|\mathbf{p}_{i}-\mathbf{p}_{r}\|_{2}^{2}/\mu\) in (1) are not greater than \(1\). The choice \(\widehat{\sigma}=0.04\) makes the lowest value \(\exp(-1/\sigma)\) of \(e_{i,r}\) comparable with the machine precision. The datasets used in the first two experiments are suggested in [25]. The points are normally distributed with different variances around a given number of centers. The first experiment concerns the clustering of the three datasets of Figure 1: dataset _Subcluster_ (SC) has five clusters, and four of them are subclusters forming two pairs of clusters, respectively; dataset _Skewdistribution_ (SD) has three clusters with skewed dispersions; dataset _Differentdensity_ (DD) has clusters with different densities. Each dataset is generated with 1000 points. We applied Kvar to each dataset with \(k\) varying in the interval [2; 15]. Figure 2 shows the piecewise linear plots versus \(k\) of the DB index (dashed line), DB** index (solid line), and CL index (dotted line) for the three datasets. The CL plot is normalized in order to fit the size of the figure. For the dataset SC, the DB index suggests choosing \(k=3\), while the DB** index suggests both \(k=3\) and \(k=5\), with a weak preference for the former. The plot of the CL index, which is zero for \(k\leq 5\), shows that the clusters in this range of \(k\) are well-separated, while for \(k\geq 6\) the increase of the plot suggests that some clusters are improperly subdivided. Figure 1: Synthetic datasets SC, SD, and DD of points in \(\mathbb{R}^{2}\). Figure 2: Plots versus \(k\) of DB (dashed line), DB** (solid line), and CL (dotted line) for SC, SD, and DD. For both datasets SD and DD, the DB and DB** indexes indicate the same value, 3, for \(k\). The CL plot, which is not zero even for small values of \(k\), reveals the presence of close points in different clusters. The difficulty encountered in the assignment of these points to the clusters suggests that such points could be considered as noise. In any case, using the information gathered from the CL plot, the computation should stop before \(k\) becomes too large. The purpose of the second test is precisely to see how the procedure behaves in the presence of noise.