# Double-Constrained Consensus Clustering with Application to Online Anti-Counterfeiting
## 4 Experimenting with UCI Datasets
### Results
To make the results across the datasets with different constraint sizes comparable, the x-axis is the number of constraints expressed as a percentage. Our first observation was that \(CC\) was between the mean and the maximum of the base algorithms. It outperformed the mean of the base algorithms (except for one data point, with 20% of constraints), often by a clear margin, but its performance was noticeably worse than the maximum performance of the base algorithms regardless of the set of constraints, especially for a large number of them. This is consistent with the findings reported in [7], because in this case, we had, at least for a large number of constraints, two similar base clusterings (according to the PRI) with lower effectiveness (i.e., _ccls_ and _lcvape_) and one dissimilar base clustering with higher effectiveness (i.e., _mpckm_). Apart from the behavior of \(CC\), the main finding observed in Figure 3 was the superior performance of \(CCC\). Not only did it consistently outperform \(CC\) (and thus the mean  
Figure 3: Mean performance of \(CC\) and \(CCC\) (averaged over the four datasets) as a function of the percentage of constraints (on the x-axis) compared to the minimum, average, and maximum performance of the base algorithms. Performance is measured using the F measure (on the y-axis). performance of the base algorithms), with spectacular improvements on almost all data points and an average gain of 5.54%, but it was also almost on a par with the maximum performance of the base algorithms. In particular, it achieved very similar results to those of the best base algorithm for most data points, whereas its performance was worse when constraints accounted for 20% and 50% of the data and (slightly) better when constraints accounted for 40% of the data. We also conducted a paired \(t\)-test to observe the statistical significance of the difference of the means for each of the 40 entries (i.e., 10 constraint sizes by four datasets) that had values from the five algorithms. This analysis confirmed that the difference between \(CCC\) and \(CC\) was statistically significant, whereas the difference between the maximum performance of the base algorithms and \(CCC\) was not statistically significant. These findings suggest that \(CCC\) achieved similar results, on average, to those obtained by choosing the best algorithm for each data point individually (characterized by a specific dataset and number of constraints), without advance knowledge of the relative performance of the available clustering algorithms in each experimental setting. \(CCC\) was thus an effective way of increasing the robustness of constrained clustering with respect to the choice of clustering algorithms and datasets, across the full spectrum of input constraint sizes. It is also important to highlight that the gains achieved by \(CCC\) were obtained from a strong baseline with semi-supervised base algorithms, thus yielding a notable absolute performance of the overall constrained clustering process. If we look at the improvement of \(CCC\) over \(CC\), we can see that the use of constraints made it possible to overcome one main limitation of \(CC\), namely its inability to properly handle a set of base algorithms in which there is one very effective algorithm that is dissimilar to the other base algorithms. Consider the objective function of Expression (7) in this situation: as more and more constraints are applied, the small-valued inter-similarity addend of a candidate partition, similar to the spurious base clustering, can be counterbalanced by its large-valued performance addend, thus promoting the partition. This is yet another advantage of using the constraints in the consensus clustering function, beyond their utilization in the base algorithms.