## 2 Related Work  
The current PCA methodologies typically adopt the following two ways to obtain projection vectors and they are the \(l_{p}\) norm based estimator and the rank based estimator. (i) \(l_{p}\) norm based estimator: it models the projection vectors by a function of a \(l_{p}\) norm. In contrast with the classical squared \(l_{2}\) norm, the \(l_{p}\) norm gives much smaller function values to the data objects that are far away from the majority. This can diminish the negative effect caused by outliers. One of the best known and most influential approaches is the \(l_{1}\)-PCA, which produces projection vectors by maximizing the \(l_{1}\)-norm of the projected data. However, owing to singularity of the \(l_{1}\) norm, many different solvers have been proposed. In Reference [13], the \(l_{1}\)-PCA was reformulated as a linear programming problem, which was then solved by the simplex method. The authors in Reference [18] adopted a relaxation approach that turns the binary constraint into a semi-definite constraint and applied the interior point method to obtain the optimal solution. [PERSON] [14] reformulated the \(l_{1}\)-PCA and expressed the optimization problem by the composition constraints of \(l_{1}\) and \(l_{2}\)-norms. This method finds one projection vector at a time. An orthogonalization procedure is incorporated to find the rest of the projection vectors. Later, the PCA with non-greedy \(l_{1}\)-norm maximization [19] was proposed. In contrast with the \(l_{1}\)-PCA, the non-greedy approach can find all required projection vectors at once. [PERSON] et al. [20] directly incorporated a \(l_{1}\)-norm penalty to the original PCA formulation, which performed feature selection for image classification and clustering problems. To find an optimal solution of the \(l_{1}\)-PCA in non-greedy form, [PERSON] et al. [21] proposed a polar expression to the data matrix and expressed the problem in a dual form. Their method can effectively search through all possibilities and output a very high-quality solution. Later, [PERSON] et al. [22] proposed a different solver based on bit-flipping iterations. [PERSON] [23] generalized \(l_{1}\)-PCA by replacing the \(l_{1}\) norm by a general \(l_{p}\) norm. More recently, [PERSON] et al. [24; 25] proposed a novel \(l_{2,1}\)-norm based PCA that does not require data centralization for dimensionality reduction. Variants of \(l_{1}\)-PCA have been proposed and they are formulated as a minimization of a residual function. In References [26] and [27], the authors introduced the \(l_{1}\)-norm residual function and adopted an alternative convex optimization together with divide-and-conquer approach, respectively, to find the projection vectors. Other than \(l_{1}\) norm, [PERSON] et al. [28] proposed the \(l_{2}\)-norm residual function with an optimal mean strategy to automatically obtain the center of the dataset. [PERSON] et al. [29] proposed a residual function of squared \(l_{2}\)-norm to PCA algorithm based on maximum correntropy and solved by a half-quadratic optimization method. More recently, [PERSON] et al. [30] proposed a more general \(l_{p}\)-norm based residual function named as \(l_{2,p}\)-norm PCA for image recognition problems. Other than the above PCA problems, \(l_{p}\) norm based estimators are also widely used in other types of dimensionality approaches including 2 DPCA, LDA, 2 DLDA, tensor and so forth. [PERSON] et al. [31] applied the \(l_{1}\)-norm formulation for 2 DPCA [1]. [PERSON] et al. [32] introduced a probabilistic formulation for the \(l_{1}\)-norm based 2 DPCA with the aid of variational inference. Other than 2 DPCA, it is also widely applied in LDA. [PERSON] et al. [33] and [PERSON] et al. [34] proposed the use of the \(l_{1}\)-norm in the LDA formulation. They replaced the squared \(l_{2}\)-norm for distance between classes and within classes by the \(l_{1}\)-norm. [PERSON] et al. [35] introduced a non-greedy \(l_{1}\)-norm technique for the 2 DLDA.