# Kernel Geometric Mean Metric Learning
## 3 Main Results
### Experimental Results  
The accuracy of each compared algorithm on 15 datasets is shown in Table 3, where the best result on each dataset is shown in boldface. Compared with the other six algorithms, our algorithm achieves the highest accuracy on the eight datasets of pima, vehicle, german, segment, usps, mnist, lymphgraphy and spambase. From the experimental results, we conclude that our proposed KGMML algorithm is more accurate than the GMML algorithm on all 15 datasets. The accuracy of the KGMML algorithm can be improved by roughly 2% to 4% over the GMML algorithm in small datasets, and the improvement is generally smaller for large datasets. In the small dataset, the Hayes-Roth dataset improves accuracy by 18.17%; in the large dataset, the Spambase dataset improves accuracy by 8.13%. The analysis reveals that the accuracy improvement is greater when the number of features is relatively small in both large and small datasets. Table 4 lists the average running time of the GMML algorithm and the KGMML algorithm on the dataset, which shows that the proposed algorithm does not drag down the running efficiency of the original algorithm. All experimental methods are implemented on MATLAB2018b (64-bit), and the simulations are run on a laptop with an Intel Core i5 (2.5 GHz) processor. Figure 3: Varying \(p\) (\(t\) is fixed). Figure 2: Varying \(t\) (\(p\) is fixed). To show the performance advantages of the KGMML algorithm, a score statistic is performed on the KGMML algorithm and other classic algorithms. The scoring process is as follows. Assuming that \(A\) is the result of using the KGMML algorithm on a certain dataset, and \(B\) is the result of uniting the KGMML algorithm on this dataset (that is, using other algorithms), firstly, comparative analysis of \(A\) and \(B\) is performed by using a 5% significance level \(t\) test. In a statistical sense, if \(A>B\), it is considered that the result \(A\) obtained by using the KGMML algorithm on this dataset wins the result \(B\) using other algorithms. Thus, the statistical result of the score on this dataset is recorded as \"1/0/0\". If \(A<B\), the score statistics result is recorded as \"0/0/1\". If \(A=B\), it means that \(A\) and \(B\) are the same in the statistical sense. Thus, it is considered that they are tied and recorded as \"0/1/0\". It is evident that the notation \"5/0/0\" signifies that the outcomes achieved by the KGMML algorithm outperform those of other algorithms across five datasets. On each dataset and the overall score, the statistical results of the KGMML algorithm are listed in Table 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Datasets**} & \multicolumn{6}{c}{**KGMML**} \\ \cline{2-7}  & **GMML** & **DMLMJ** & **LMNN** & **GB-LMNN** & **Euclidean** & **Low-Rank** \\ \hline Pima & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ Vehicle & 0/1/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ German & 1/0/0 & 0/1/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ Segment & 1/0/0 & 0/1/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ Usps & 1/0/0 & 0/1/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ Mnist & 0/1/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ Glasses & 1/0/0 & 0/1/0 & 0/0/1 & 0/0/1 & 0/1/0 & 1/0/0 \\ DNA & 0/1/0 & 0/0/1 & 0/0/1 & 0/1/0 & 1/0/0 & 1/0/0 \\ Heart-Disease & 1/0/0 & 0/1/0 & 1/0/0 & 0/1/0 & 1/0/0 & 1/0/0 \\ Lymphography & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ Liver-Disorders & 1/0/0 & 0/1/0 & 0/1/0 & 1/0/0 & 0/1/0 & 1/0/0 \\ Hayes-Roth & 1/0/0 & 0/0/1 & 1/0/0 & 1/0/0 & 0/0/1 & 1/0/0 \\ Ionosphere & 1/0/0 & 0/1/0 & 0/0/1 & 0/0/1 & 0/0/1 & 1/0/0 \\ Spambase & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 & 0/1/0 \\ Balance & 1/0/0 & 0/1/0 & 1/0/0 & 1/0/0 & 1/0/0 & 1/0/0 \\ \hline Total & 12/3/0 & 5/8/2 & 11/1/3 & 9/2/2 & 11/2/2 & 14/1/0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Win/tie/loss counts of each algorithm between the results obtained the KGMML algorithm and other classical algorithms.