## 2 Preliminaries
### Foundations of Label Distribution Learning  
We can formalize a LDL problem as a set of \(m\) training samples \(S=\{(x_{1},D_{1}),...,(x_{m},D_{m})\}\), where \(x_{i}=\{x_{i1},x_{i2},...,x_{iq}\}\) is a \(q\)-dimensional vector. For each instance \(x_{i}\), the label distribution is denoted by \(D_{i}=\{d_{x_{i}}^{y_{1}},d_{x_{i}}^{y_{2}},...,d_{x_{i}}^{y_{c}}\}\) where \(y_{j}\in\Upsilon|j\in\{j,\cdots,c\}\), such that \(Y=\{y_{1},\cdots,y_{c}\}\) denotes the complete set of labels. The constant \(c\) is the number of labels and \(d_{x_{i}}^{y_{j}}\) is the description degree of the particular \(j\)th label \(y_{j}\) for a particular \(i\)th instance \(x_{i}\). According to the definition, for each \(x_{i}\) the description degree should meet the constraints \(d_{x_{i}}^{y_{j}}\in[0,1]\) and \(\sum_{j=1}^{c}d_{x_{i}}^{y_{j}}=1\).  
The solution to an LDL problem can be addressed from several perspectives. According to the selected approach, the algorithm to be developed may differ significantly, either a brand-new algorithm designed especially to deal with LDL constraints, or an adaptation of already available classification algorithms, reformulated to work with those constraints. The LDL study presented in [9] suggested six algorithms grouped in three categories. The first one is Problem Transformation (PT), a simple way to convert an LDL problem into a Single-Label Learning or SLL [36] problem. PT transforms the training samples into weighted single-label examples. Thus, any SLL algorithm may be applied. Two representative algorithms are PT-Bayes and PT-SVM. The second one is Algorithm Adaptation (AA), in which the algorithms are tailored to existing learning algorithms to handle directly with the label distribution. Two suitable algorithms were presented: AA-\(k\)NN, an adaptation of the well-known \(k\)-nearest neighbors method [37], and AA-BP, a three-layer backpropagation neural network. Finally, Specialized Algorithms (SAs), in contrast to the indirect strategy of PT and AA, directly match the LDL problem. SA-IIS and SA-BFGS are two specialized algorithms that learn by optimizing an energy function that is based on the maximum entropy model.  
Subsequent works have successfully improved the results achieved by these original algorithms through different approaches. The methods LDLogitBoost and AOSO-LDLogitBoost proposed in [19], are a combination of the boosting method and the logistic regression applied to LDL framework. Deep Label Distribution Learning (DLDL) [18] and LDL based on Ensemble Neural Networks (ENN-LDL) [38] are two examples of success in the application of neural networks in LDL. Modelled on differentiable decision trees [39], an end-to-end strategy LDL forests proposed in [17] was used as the basis for Structured Random Forest (StructRF) [20]. BC-LDL [40] and DBC-LDL [41] use the binary coding strategies to address with the large-scale LDL problem. Classification with LDL (LDL4C) [42] is also an interesting approach when the learned label distribution model is considered as a classification model. Feature selection on LDL [43] shows encouraging results by applying feature selection on LDL problems.