## 2 Related Work
### Flexible Manifold Embedding (FME) for Semi-Supervised Classification  
Let \(\mathbf{X}\in\mathbb{R}^{d\times n}\) denote the data matrix containing \(n\) samples, each of dimensionality \(d\), where \(l\) samples are labelled and \(u=n-l\) samples have no labels. Assume that the data matrix is ordered such that the first \(l\) samples have labels and the unlabelled samples range from \(l+1\) to \(l+u\). In addition, in classification problems with \(C\) classes, the corresponding labels are ordered in a binary label matrix \(\mathbf{Y}=[\mathbf{Y}_{l},\mathbf{Y}_{u}]=[\mathbf{y}_{1};\mathbf{y}_{2}; \ldots;\mathbf{y}_{l};\mathbf{y}_{l+1};\ldots;\mathbf{y}_{l+u}]\in\mathbb{R}^{ n\times C}\) such that \(\mathbf{y}_{i}\) is a \(C\) dimensional vector that determines the class for the samples \(\mathbf{x}_{i}\). For any sample \(\mathbf{x}_{i}\) belonging to class \(c\), the \(c^{th}\) element of \(\mathbf{y}_{i}\) is one, and the remainder is zero. The Flexible Manifold Embedding (FME) method was introduced in [31] and consists of a unified framework for semi-supervised and unsupervised manifold learning that can handle data from nonlinear manifolds. For semi-supervised learning, FME enables mappings for unseen data points through a linear regression function that is able to handle the data from the nonlinear manifold by modeling the regression residual. It is useful for both semi-supervised settings: (i) transductive and (ii) inductive. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \hline
**X** & Training data samples \(\in\mathbb{R}^{d\times n}\) \\  & \(\mathbf{X}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{\ell},\mathcal{X}_{\ell+1}, \ldots,\mathbf{x}_{n}]\) \\ \(\ell\) & Number of labelled samples \\ \(u\) & Number of unlabelled samples (\(u=n-\ell\)) \\ \(d\) & Dimensionality of data \\ \(n\) & Number of training samples/images \\ \(C\) & Total number of classes \\ \(\mathbf{F}\) & Prediction label matrix \(\in\mathbb{R}^{n\times C}\) or Prediction score vector \(\in\mathbb{R}^{n}\) \\ \(\mathbf{Y}\) & Binary label matrix \(\in\mathbb{R}^{n\times C}\) or ground-truth score vector \(\in\mathbb{R}^{n}\) \\ \(\mathbf{U}\) & Indicator diagonal Matrix \(\in\mathbb{R}^{n\times n}\) \\ \(\mathbf{Q}\) & Projection Matrix \(\in\mathbb{R}^{d\times d}\) or Projection vector \(\in\mathbb{R}^{d}\) \\ \(\mathbf{b}\) & Bias vector \(\in\mathbb{R}^{c}\) or bias scalar \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The primary notations used in the paper. In the inductive setting, the framework is able to predict the label of unseen test samples. Many semi-supervised methods are transductive, meaning that they cannot handle unseen test samples. The FME method attempts to simultaneously find the optimal prediction label matrix \(\mathbf{F}\) by minimizing the following function:  
\[\min_{\mathbf{F},\mathbf{W},\mathbf{b}}\,Tr\left(\mathbf{F}^{T}\,\mathbf{L}\, \mathbf{F}\right)+Tr\left(\mathbf{F}-\mathbf{Y}\right)^{T}\mathbf{U}\left( \mathbf{F}-\mathbf{Y}\right)+\mu\left(||\mathbf{Q}||^{2}+\gamma\,||\mathbf{X} ^{T}\mathbf{Q}+\mathbf{1}\mathbf{b}^{T}-\mathbf{F}||^{2}\right) \tag{1}\]  
where the first and second terms are the label smoothness and label fitness terms, respectively. \(\mathbf{U}\) is the diagonal matrix, where \(U_{ii}\) is zero for the unlabelled samples and a non-zero value for the labelled samples. The third term consists of the regularization of the projection matrix (\(\mathbf{Q}\)) and the linear regression model (provided by the projection matrix \(\mathbf{Q}\) and the shift vector \(\mathbf{b}\)). By seeking a small regression error (last term), FME seeks a nonlinear representation that is close to a linear model. This makes the model flexible. The Laplacian graph \(\mathbf{L}=\mathbf{D}-\mathbf{W}\), where \(\mathbf{D}\) is the diagonal degree matrix, and \(\mathbf{W}\) the similarity matrix. The parameters \(\mu\) and \(\gamma\) refer to balancing terms and \(||.||^{2}\) denotes the \(\ell_{2}\) norm of a matrix.