# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 3 Ground Theory
### Taxonomies Used in the Multidimensional Projection Area  
The application of multidimensional projection techniques results in groups of points embedded in a \(p\)-dimensional space. We define a group, or a class, as a set of instances; however, there are some conceptual differences. A group can be formed by points perceptually close in the visual space, generated by a clustering algorithm that applies some dissimilarity measure in the data space. On the other hand, a class comprises objects that share some intrinsic meaning previously known or revealed by a classification algorithm that describes or distinguishes the objects belonging to the class [7]. Figure 1: Vector normalization is applied to improve the (dis)similarity measure process. The shaded area between \(A\) and \(B\) sequences represents the real difference in magnitude between them. Source: elaborated by the authors. Different approaches have been proposed to map multidimensional data into visual space, and the literature classifies them with some terminologies that share common characteristics. Among the taxonomies found in the literature, we can cite the best known and usually adopted as follows:  
_(1) According to the transformation type_  
Linear techniques are based on transformations that create linear combinations of the data attributes, mapping them into a new space with a reduced dimensionality [9]. Formally, linear methods map the data using a function \(f:X\to Y\), which satisfies the condition \(f(\alpha x_{i}+\beta x_{j})=\alpha f(x_{i})+\beta f(x_{j})\), \(\forall\ x_{i},x_{j}\in X\) and \(\alpha,\beta\in\mathbb{R}\)[25]. Although many MDPs apply linear resources in their processes, few approaches are truly linear, according to the strict mathematical definition [7]. Linear methods are often computationally efficient and relatively simple to develop. Once the transformation is calculated, the mapping of each instance in the Cartesian space is performed with matrix multiplication. Nevertheless, they cannot properly handle complex structures (such as data with nonlinear relationships between its attributes), achieving unsatisfactory results in many real scenarios [34]. This characteristic tends to generate representations with considerable distortions, which is a severe issue for visualization tasks. Nonlinearity arises in many problems, such as physical phenomena modeling, and sometimes functions cannot satisfy linearity conditions that naturally motivate the development of nonlinear projections. Nonlinear techniques aim to minimize an information loss function usually supported by a mechanism able to relate the dissimilarities between the instances in the high dimensional space with the distances between the \(p\)-dimensional points [9]. Linear methods can present good results depending on the data distribution, but they rarely outperform nonlinear techniques in terms of the ability to deal with complex structures [16]. Another problem is the addition of new data subsets in the visual space. Linear techniques require complete reprocessing to introduce a perturbation in the dataset because they are based on matrix operations [9]. On the other hand, nonlinear approaches require only a small number of additional iterations to incorporate new data into the projection. _(II) According to the projection nature_  
The concept of locality is used to distinguish two properties of projections nature: global modeling and local mapping. Global modeling defines methods built in a single transformation [12] and methods that seek to preserve original geometric relationships between all pairs of data instances in the transformed space [6]. That is, global modeling forces close points in the original space to remain close to the projected space, while distant points remain distant in the visual space. Local mapping techniques are also based on neighborhood information, but they seek to preserve relationships considering only the surroundings of a small neighborhood in the \(m\)-dimensional space [6].