## 3 Evaluation
### Experimental Setup  
We used an autoencoder architecture for feature extraction. The autoencoder architecture vary from dataset to another. A summary of the autoencoder architecture, batch size and number of epochs used for each dataset is presented in Table 2. For some datasets, we increased the number of epochs to test for potential overfitting when the model keeps training. Because of this, the model overfitting is subject to the model's complexity, especially, in the case of small datasets [69]. However, as can be seen from Figure 3, the reconstruction loss of the AE remains smooth as the model keeps training. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **MNIST** &
\begin{tabular}{c} **Fashion-** \\ **MNIST** \\ \end{tabular} & **USPS** & **MSTAR** & **WDBC** \\ \hline No. of Training & 596 & 600 & 725 & 698 & 398 \\ No. of Testing Samples & 69,404 & 69,400 & 8573 & 587 & 171 \\ No. of Classes & 10 & 10 & 10 & 3 & 2 \\ Input Size & 28 \(\times\) 28 & 28 \(\times\) 28 & 16 \(\times\) 16 & 128 \(\times\) 128 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The number of training and testing samples, number of classes, and input sizes for each dataset. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **MNIST** &
\begin{tabular}{c} **Fashion-** \\ **MNIST** \\ \end{tabular} & **USPS** & **MSTAR** & **WDBC** \\ \hline Flattened Input Size & 784 & 784 & 256 & 16,384 & 30 \\ AE Architecture & 512-256-512 & 512-256-512 & 512-128-512 & 256-64-256 & 32-16-32 \\ Batch Size & 50 & 50 & 50 & 50 & 10 \\ No. of Epochs & 1200 & 1200 & 600 & 100 & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Autoencoder architecture, batch sizes and number of epochs for each dataset. For MLP regressors architecture, we used only one hidden layer of 50 hidden units with ReLU activation function and an output layer of only one output node with linear activation, and we used an adaptive learning rate \(\gamma\) initialized at 0.001. The use of the ReLU activation function stemmed from the advantages of ReLU over sigmoidal functions such as sparsity representation and to avoid the saturation problem of sigmoidal activation functions which cause the vanishing gradient problem [70]. We adopted the same MLP architecture for all MLP regressors in the feature mapping network. We adopted an Adam optimizer for backpropagation of the Mean-Squared Error (MSE) for both the autoencoder and the MLP regressors used in the feature mapping network for the ease of hyperparameter tuning. Figure 3 depicts the MSE loss of the autoencoder for each of the five datasets. For the code and implementation details, please refer to Supplementary Materials. As per our proposed method, we took the Fashion-MNIST dataset as an example of applying our algorithm. We trained the autoencoder with its corresponding architecture (784-512-256-512-784) on the 600 training images (60 samples per class), then we created  
Figure 3: Autoencoder error loss vs. epochs on the five datasets: (**a**) MNIST; (**b**) Fashion-MNIST; (**c**) USPS; (**d**) MSTAR; (**e**) WDBC. the Cartesian Product set as in Algorithm 1 using the 600 images in two sets \(Tr_{1}\) and \(Tr_{2}\) of size \(60\times 59\times 10=35,400\) data samples. We then extracted features of \(Tr_{1}\) and \(Tr_{2}\) using our trained autoencoder resulting in two matrices of features \(Z_{1}\) and \(Z_{2}\), both of size \([35,400\times 256]\). Next, we trained 256 MLP regressors \(\{MLP_{1},\ MLP_{2},\ \ldots,\ MLP_{256}\}\) using \(Z_{1}\) as input and \(Z_{2}\) as target variables, where each MLP regressor takes as input the whole feature vector \(z_{i}\in Z_{1}\) and learns to output one target feature \(z_{i}^{s}\in Z,\ s=1,\ 2,\ \ldots,\ 256,\ i=0,1,\ldots,35,400\). To speed up the training process, we distributed the training of the 256 MLP regressors over eight processes that run in parallel such that each process corresponds to \(256/8=16\) MLP regressors.