# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 3 Ground Theory
### Multidimensional Projection  
A multidimensional dataset \(X\), with \(n\) instances \(x_{i}\in\mathbb{R}^{m}\), can be represented as a matrix \(X_{n\times m}\), where each row is a vector \(x_{i}\), and the columns represent the vector attributes. According to [PERSON] [21], high-dimensional datasets are those with \(m\geq 12\) attributes. Since the visual space is limited to three dimensions, when the \(m\)-dimensionality increases, the complexity of representing and interpreting data also increases. In this context, techniques devoted to dimensionality reduction aim to present multidimensional data from high-dimensional spaces as points in dimensionality-reduced spaces. Thus, it is possible to use fewer dimensions to describe the original data, compacting large datasets and reducing the computational effort required in their processing. According to [PERSON] and [PERSON] [22], there are two main categories of dimensionality reduction methods: feature selection and feature extraction. Feature selection identifies the non-significant features related to the analysis task, omitting them to generate a subset containing only useful attributes that can efficiently describe the input data [23]. Based on the assumption that the original dataset may have redundant (or even irrelevant) attributes, the removal process of non-significant features will generate a subset of useful features that automatically result in a reduced data space [24]. In several cases, feature selection is sufficient to carry out data analysis. However, there are situations in which it is challenging to identify a significant attributes subset, limiting the use of feature selection to the pre-processing data phase. As feature selection often does not result in a dataset of two or three dimensions, these methods are not part of this research. (for a comprehensive overview of feature selection methods, see [PERSON] and [PERSON] [23]). On the other hand, feature extraction methods try to reduce data dimensionality by mapping the original dataset \(X\) from an \(m\)-dimensional space to a new dataset \(Y\) of \(p\) dimensions, with \(p<m\). The mapping process uses a single function or a group of transformation functions to map the data, retaining the essential characteristics of the original dataset [25]. When \(p\in\{1,2,3\}\), there is a particular class of multidimensional data mapping techniques called Multidimensional Projections (MDP) [7]. MDP maps multidimensional data to the Cartesian visual space, seeking to preserve some information from the original space in the mapped (projected) space, such as the distance relationships between the data instances [6]. Consequently, it is possible to create a graphical representation from the projected points, to take advantage of the human perceptual abilities to recognize structures or visual patterns based on similarities, particularly those related to groups of elements [9]. Projection techniques have been attracting the attention of the Information Visualization community for some time due to their broad applicability as an analytical tool. Making the mapped data convenient for visualization, tasks that involve distance exploration or neighborhood relations can be streamlined [6]. [PERSON] et al. [26] mathematically defined the multidimensional projection concept as  
**Definition 1**. : _Let \(X\) be a set of \(n\) data objects in \(\mathbb{R}^{m}\), with \(m>3\) and \(\delta:\mathbb{R}^{m}\times\mathbb{R}^{m}\to\mathbb{R}\) a distance measure between the instances in \(\mathbb{R}^{m}\); \(Y\) be a set of \(n\) points in \(\mathbb{R}^{p}\), with \(p\in\{1,2,3\}\) and \(d:\mathbb{R}^{p}\times\mathbb{R}^{p}\to\mathbb{R}\) a distance measure between points in \(\mathbb{R}^{p}\).