## 1 Introduction  
Dimension reduction (DR) techniques are used to obtain new feature sets in a subspace from high-dimensional original data. This helps to omit the redundant data and noise from the input and provides advantages, especially in terms of computational time and data storage [1, 2]. Manifold learning is a special kind of technique for DR, which assumes that there is a low-dimensional manifold on high-dimensional data and its purpose is to find this manifold [3]. Manifold learning is applied to different application domains such as face recognition, medical image segmentation, and object recognition [4]. Furthermore, an increase in the availability of hyperspectral imaging (HSI) data and advanced computing has led to the popularity of manifold learning techniques in this research area [5]. Manifold learning aims to preserve local or global geometric relations and not lose the graph topology of high-dimensional data [6, 5, 7]. Isometric feature mapping (Isomap), locally linear embedding (LLE), and Laplacian eigenmaps (LE) are the most widely utilized nonlinear manifold learning methods. While Isomap is a global approach [8], LE [9] and LLE [10] retain the local geometry of data. Local approaches are initiated by obtaining the nearest neighborhood of each data point instead of retaining overall topology. There are more robust domain-specific manifold learning methods in the literature. In this study, an effective local manifold representation that makes use of spatial and spectral information is proposed [11]. Traditional nonlinear manifold learning algorithms take all previous samples and new samples as input for the learning process. This is known as an out-of-sample problem and is not practical for classification purposes including face recognition and HSI tasks [3]. Nevertheless, there are some studies on generalizations and extensions of out-of-sample data for nonlinear manifold learning that use polynomial mapping [3], the Nystrom approach [12], kernel Hilbert space [13], and sparse grid functions [14]. Additionally, linearized manifold learning algorithms can make embedding generalizations from the learned linear projections for new samples. There are various linear manifold learning algorithms in the literature. Locality preserving projection (LPP) and neighborhood preserving embedding (NPE) linearize the LE and LLE, respectively [15; 16]. Linearization provides applicability for out-of-sample problems, while LLE applies convex optimization to obtain a weight matrix and LE employs a heat kernel function for this purpose. Orthogonal locality preserving projection (OLPP) and orthogonal neighborhood preserving embedding (ONPE) are the same with LPP and NPE, respectively, except for orthogonality restrictions [17; 18]. Transformation matrices are restricted to be orthogonal [3]. These algorithms have been analyzed on hyperspectral images in a study by [19]. The aforementioned methods use an unsupervised approach. If the labeled data is scarce, semi-supervised techniques can also be used. One study presented a multitask regression framework to enhance model generalization and the quality of class indicators of unlabeled samples [20]. In addition, some supervised manifold learning approaches have also been proposed. They use class information of the input and aim to optimize similarities between and within class neighbors, while preserving the geometric structure of the data. They can be applied as linear or nonlinear. Approaches using class information which redefine the distances between the samples in LE and LLE have been shown in some studies [21; 22]. Furthermore, linear operator mappings for the LPP have been proposed for better class separability [23; 24; 25]. Similarly, a specific method has been developed which was applied during the construction of the adjacency graph step in the NPE for the same purpose [26].