# A Two-Phase Approach for Semi-Supervised Feature Selection
## 2 Existing Feature Selection Techniques  
The problem of feature selection can be formulated as follows: Given a dataset \(X\subset R^{p}\) (i.e., each \(x_{i}\in X\) has \(p\) features), we have to select a subset of features of size \(q\) that leads to the smallest (or highest as the case may be) value with respect to some criterion. Let \(F\) be the given set of features and \(F\) the selected set of features of cardinality \(m\), \(F\)\(\subseteq\)\(F\). Let the feature selection criterion for the dataset \(X\) be represented by \(J(F,X)\) (lower value of \(J(.)\) indicates a better selection). When the training instances are labeled, we can use the label information, but in the case of unlabeled data, this cannot be done [29].  
In [PERSON] et al. [29], a new approach to unsupervised feature selection preserving the topology of the data is proposed. Here, the genetic algorithm (GA) has been used to select a subset of features by taking the Sammon stress/error as the fitness function. The dataset with the reduced set of features is then evaluated using classification (1-nearest neighbor (1-NN)) and clustering (K-means) techniques. The correlation coefficient between the proximity matrices of the original dataset and the reduced one is also computed to check how well-preserved the topology of the dataset is in the reduced dimension. In feature selection, the filter model [30; 31; 32], wrapper model [33], and embedded model [34] are three main categories. The filter model relies on the features' properties with certain evaluation metrics. In the wrapper model, it evaluates feature sets via the model's prediction accuracy with a combination of features. For the embedded model, the feature selection part and the learning part interact with each other. Though the wrapper model and embedded model can achieve effectively selected features in certain cases, the computational cost is high in application [35].  
Supervised feature selection relies on the classification labels, and the typical models are the Fisher metric [36] and [PERSON]'s correlation coefficient [37]. Unsupervised feature selection is based on feature similarity or local information. Laplacian score [38] is a typical unsupervised feature selection model, which measures the geometrical properties in the feature sets. Utilizing both the labeled data and unlabeled data is a method to achieve optimal feature subsets, which is the focus of semi-supervised feature selection [39]. In [40], both labeled and unlabeled data are trained via the spectral analysis to establish a regularization framework, and the authors demonstrate that the unlabeled data can be helpful for feature selection. In [41], label propagation is conducted and a wrapper-type forward semi-supervised feature selection framework is proposed.  
[PERSON] et al. in [35] used [PERSON]'s correlation coefficients to measure the feature-to-feature, as well as the feature-to-label, information. The coefficients are trained with the labeled and the unlabeled data, and the combination of this two-fold information is performed with max-relevance and min-redundancy criteria. The experiments are applied on several real-life applications [35]. Despite having labeled or unlabeled datasets individually, it is natural to have a mix of both, viz. labeled and unlabeled, in a single dataset. Such types of semi-supervised datasets can be made available intentionally or unintentionally. For the former case, labeling datasets may cost a large amount due to the processes involved to obtain labels after several experiments. In the latter case, labels are missing or doubtful or as good as not available. Some literature can be seen in [PERSON] [42]. In this paper, we propose a two-phase approach to find the missing labels of patterns of a dataset that contains some labeled data (patterns).