## 3 Experiments and Discussions
### Semi-Supervised Classification for Facial Images  
In this section, we utilize the FMDR for facial image semi-supervised classification. A conventional way to evaluate the effectiveness of the dimensionality reduction is to classify the data in the low-dimensional space. As the effectiveness of dimensionality reduction improves, the discriminability of the resulting low-dimensional data becomes stronger, thereby leading to higher classification accuracy. In this experiment, to smooth the bias, we conducted cross-validation that contains 100 repeated trials, randomly partitioning 20% of the data as labeled samples. The labeled and unlabeled samples are assembled into a unified data matrix, serving as input for all semi-supervised algorithms. The performances of semi-supervised classification for facial images is presented in Table 1. The highest values are indicated in bold font. The experimental results indicate that FMDR achieves the best performance in the majority of cases. Specifically, FMDR gains the highest values in AR, with 97.14% Acc, 97.87% Pre, 97.29% Rec and 53.91% F1 score. Also, for YaleB, FMDR achieves 76.18% Acc, 77.32% Pre, 78.29% Rec and 39.68% F1 score. The samples of those two datasets exhibit significant large area occlusions, such as masks, sunglasses, scarves and shadows, which are difficult to handle in the original domain and invalidate the existing algorithms. The superiority of our algorithm on these datasets demonstrates its ability to accurately extract key information from samples in the frequency domain, thereby mitigating the interference of occlusions and improving recognition. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multicolumn{8}{c}{**Acc**} \\ \cline{2-10}  & **FMDR** & \(\text{KNN}_{semi}\) & \(\text{KMeans}_{semi}\) & **RPCA** & **NMF** & **LE** & **ABNMTF** & **DANMF** \\ \hline AT\&T & **79.81** & 73.89 & 57.00 & 33.02 & 50.31 & 34.35 & 78.35 & 51.95 \\ ORL & **64.44** & 45.59 & 42.25 & 32.33 & 31.85 & 31.85 & 41.30 & 33.60 \\ AR & **97.14** & 77.00 & 38.46 & 50.23 & 52.23 & 50.23 & 67.62 & 61.27 \\ Yale & **64.44** & 45.59 & 28.48 & 32.85 & 31.85 & 31.85 & 41.30 & 33.60 \\ YaleB & **76.18** & 43.30 & 32.14 & 31.38 & 32.63 & 28.07 & 45.56 & 29.05 \\ \hline \hline \multicolumn{10}{c}{Pre} \\ \cline{2-10}  & FMDR & \(\text{KNN}_{semi}\) & \(\text{KMeans}_{semi}\) & RPCA & NMF & LE & ABNMTF & DANMF \\ \hline AT\&T & 82.81 & 75.80 & 71.23 & 28.31 & 45.00 & 30.92 & 81.56 & **88.49** \\ ORL & **62.76** & 58.24 & 60.05 & 46.71 & 31.87 & 40.99 & 55.79 & 46.54 \\ AR & **97.87** & 75.54 & 42.62 & 48.58 & 50.34 & 53.58 & 70.15 & 63.11 \\ Yale & **62.76** & 58.24 & 35.09 & 46.71 & 31.87 & 40.99 & 55.79 & 46.54 \\ YaleB & **77.32** & 41.46 & 48.30 & 32.96 & 31.24 & 31.99 & 57.47 & 29.77 \\ \hline \hline \multicolumn{10}{c}{Rec} \\ \cline{2-10}  & FMDR & \(\text{KNN}_{semi}\) & \(\text{KMeans}_{semi}\) & RPCA & NMF & LE & ABNMTF & DANMF \\ \hline AT\&T & **79.12** & 79.12 & 62.64 & 31.58 & 48.17 & 35.97 & 78.47 & 55.78 \\ ORL & **62.11** & 52.20 & 46.63 & 34.36 & 29.61 & 38.33 & 47.61 & 31.84 \\ AR & **97.29** & 78.43 & 42.76 & 49.63 & 48.41 & 51.04 & 67.77 & 63.40 \\ Yale & **62.11** & 52.20 & 30.82 & 34.36 & 29.61 & 38.33 & 47.61 & 31.84 \\ YaleB & **78.29** & 43.41 & 33.12 & 29.84 & 31.45 & 28.84 & 46.04 & 29.77 \\ \hline \hline \multicolumn{10}{c}{F1 score} \\ \cline{2-10}  & FMDR & \(\text{KNN}_{semi}\) & \(\text{KMeans}_{semi}\) & RPCA & NMF & LE & ABNMTF & DANMF \\ \hline AT\&T & **40.19** & 39.35 & 34.01 & 14.54 & 23.13 & 16.73 & 41.23 & 35.77 \\ ORL & **32.37** & 29.28 & 24.10 & 22.05 & 15.99 & 22.34 & 25.23 & 18.45 \\ AR & **53.91** & 43.07 & 20.59 & 26.71 & 26.15 & 28.84 & 47.39 & 42.81 \\ Yale & **32.37** & 29.28 & 17.86 & 22.05 & 15.99 & 22.34 & 25.23 & 18.45 \\ YaleB & **39.68** & 21.50 & 19.23 & 15.68 & 16.42 & 16.06 & 34.01 & 23.38 \\ \hline \hline \end{tabular}  
* The highest values are emphasized in bold font.  
\end{table} 
Table 1: The performances of semi-supervised classification for facial images.