# Dependence Representation Learning with Convolutional Neural Networks and 2D Histograms
## 1 Introduction  
Understanding the statistical relationships between random variables (or features) is a fundamental problem in the fields of statistics and machine learning. Researchers usually visualize the features of a dataset (e.g., using scatter plots) to understand the dependencies between variables. For example, in supervised learning, researchers use visualizations of the features to understand which features are relevant or irrelevant to the values of the target variables, and how the relevant features are related to the target variables. They then select learning algorithms based on the dependence types. However, if the number of features is large, interpretation of visualizations is cumbersome and time-consuming. Therefore, many dependence measures have been developed to automatically detect dependency relationships. A number of information-theoretic measures have been invented based on the Shannon entropy [1]. Mutual information (MI) [2] and its variants have been widely used to analyze categorical or discrete features [3]. Various types of correlations have been developed to help categorize the relationships between continuous features: [PERSON]'s correlation coefficient (pCor), [PERSON]'s rank, [PERSON]'s tau, distance correlation (dCor) [4], maximal information coefficient [5], the randomized dependence coefficient [6], and etc. Various applications of dependency measures have been developed. [PERSON] et al. (2020) used dCor to screen the features of gene expression data. [PERSON] et al. (2020) proposed a variant of MI to measure interactions between features to aid feature selection. [PERSON] et al. (2020) applied MI to the representation learning of generative adversarial networks (GANs). [PERSON] et al. (2020) presented an automated feature generation algorithm based on dCor called AutoLearn. Existing dependency measures are subject to three main limitations. Firstly, they do not consider dependency types. For example, in Figure 1, the dCor of all feature pairs is 0.5, but the types of dependence are obviously different. Secondly, these measures are human-readable scalars rather than machine-readable vectors. Thirdly, they are not optimized to specific applications. These three problems can be solved by using visualizations as inputs to convolutional neural networks (CNNs). Firstly, dependency types can easily be classified based on visualizations, as shown in Figure 1. The visualizations, such as scatter plots, can be regarded as images. Advances in deep learning-based computer vision have proven CNNs to be effective for processing high-dimensional images. CNNs are frequently used for image classification ([PERSON] et al., 2015), object detection ([PERSON] et al., 2015), and image generation ([PERSON] et al., 2015). Secondly, neural networks are also capable of learning machine-readable representations; for example, word representations ([PERSON] et al., 2015) from textual data have achieved great success in natural language processing ([PERSON] et al., 2015). Likewise, dependence representations can be learned by CNNs if an appropriate learning method is applied. Thirdly, when applying an end-to-end (feature learning) approach, CNNs can learn task-specific representations from visualizations, and thus they can be optimized for a specific task (such as text recognition ([PERSON] et al., 2015), speech recognition ([PERSON] et al., 2015), and self-driving cars ([PERSON] et al., 2015)) based on dependency measures. Basically, the usage of CNNs is motivated by the fact that they can extract significant features from visualizations at different hierarchical levels. Thanks to the above advantages of CNNs, the hand-crafted representations may be replaced with convolutions. A few previous studies used deep neural networks (DNNs) for dependence estimation; for example, [PERSON] et al. (2020) used DNNs for estimation of mutual information between high dimensional continuous random variables. [PERSON] et al. (2020) proposed an unsupervised learning method, called Deep InfoMax, for representation learning.