# Dependence Representation Learning with Convolutional Neural Networks and 2D Histograms
## 3 Case Studies
### Correlation Regression
#### 3.2.4 Results  
Quantitative results: Table 2 lists the evaluation scores of each correlation. The scores are reported in terms of the mean squared error (MSE), mean absolute error (MAE), and coefficient of determination (\(R^{2}\)). In the cases of the MSE and MAE, lower values indicate better scores, but higher values are better in the case of \(R^{2}\). For example, if we predict dCor based on a set of feature pairs that have a true dCor of 0.5, then based on the average MAE, the predicted value is approximately in the range of \([0.47,0.53]\). Graphical analysis: we also analyzed the results graphically so that we could present them more intuitively and in more detail in Figure 6. Each point in the figure represents a single true-predicted correlation pair. The \(x\)-coordinate of a point represents the true correlation, and the \(y\)-coordinate is the correlation predicted by the CNN. The closer the points are clustered around the line \(y=x\), the better the performance. As seen in the figure, both sets of results are clustered around the line. One exception is in the case that the true value of dCor is equal to 0.0, when the predicted correlations are highly erroneous. This may be due to clipping the values at zero. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Split** & **Datasets** & **Features** & **Feature Pairs** \\ \hline Training & 159 & 3955 & 372,610 \\ \hline Testing & 13 & 541 & 60,734 \\ \hline Total & 172 & 4496 & 433,344 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of datasets used for correlation regression and feature generation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Correlation** & **MSE** & **MAE** & \(R^{2}\) \\ \hline Distance (dCor) & 0.0033 & 0.0348 & 0.9504 \\ Pearson (pCor) & 0.0036 & 0.0255 & 0.9783 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation regression scores obtained for the test set. Qualitative results: Figure 7 depicts visualizations of the dependence representations learned via the correlation regression technique for the test set. As seen in the figure, the 1743-dimensional representations are reduced into 2-dimensional vectors using (a) t-SNE and (b) PCA in the same way as in Figure 5. Each of the learned representations is plotted as a dot on a 2-dimensional plane, and its corresponding true correlation values is indicated in color. Consequently, if the learned results are correct, then the dependence representations that are nearby in the representation space could produce color-coded clusters since they are displayed in the same, or similar color. In each of the subfigures, the points to the left represent the true dCor value, and points to the right represent the true pCor value. Points that are close to gather have similar correlations. The independent feature pairs (dCor of zero) are clustered with themselves. The formation of coherent clusters of color as a whole indicates that the fundamental characteristics of given correlations have been well learned. Furthermore, the zoomed-in plots in Figure 7b show clusters of point that have a pCor of zero but a nonzero dCor. Figure 6: Scatter plot of the true correlations against the predicted correlation on the test set: (a) dCor regression results; (b) pCor regression results. Note that the subfigures (a,b) have different ranges. Figure 7: _Cont._  
R-unning time analysis: here, we need to consider another issue about running time; as earlier mentioned, the one major drawback of dCor is its high computational complexity. [29] proposed the \(O(n\cdot logn)\) algorithm, but it still takes a long time to compute when the sample size \(n\) is large. Table 3 provides a comprehensive comparison of the running times (in second) of methods for each sample size \(n\). The running times of the fastest method are shown in bold face for each \(n\). All the reported times ware measured using the same machine (Intel Xeon CPU E5-2698 v4 @ 2.20G Hz). We also compare the running time of our CNN-based method (denoted as CorrCNN) on CPU and GPU (GeForce GTX 1080 Ti).