# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 4 Multidimensional Projection Approaches and Domains
### Local Affine Multidimensional Projection  
Following Section 3.4, projection techniques that apply linear transformations are based on one function that maps high-dimensional data into the visual space. This function can be very generic in dealing with all data characteristics, even when it is optimized to treat each data subset, such as PLMP (see Section 4.9). On the other hand, most of the nonlinear transformations depend on dissimilarity matrices whose calculation demands high computational complexity. Some techniques project representative instances to guide the process to outperform complexity-related issues, but it demands the choice of several samples to execute the initial projection. In this context, [PERSON] et al. [12] developed the Local Affine Multidimensional Projection (LAMP). This method also uses representative instances (control points) to guide the projection at the first step. It then maps the remaining instances locally through interpolation using a family of orthogonal affine mappings--one linear function for each instance to be projected. More specifically, LAMP executes a process similar to LSP (see Section 4.8), where \(s\) representative samples are randomly selected from a dataset \(X\) and projected on the visual space with Force Scheme (see Section 4.6). With this initial information from control points' positions, each remaining instance \(x_{i}\in X\) will be mapped with a linear function \(f_{X}(p)=pM+t\), that matrix \(M\) and vector \(t\) are unknowns (by taking partial derivatives with respect to \(t\) equal to zero, one can write \(t\) in terms of \(M\)) that minimizes:  
\[\sum_{i=1}^{s}\alpha_{i}\|\bar{x}_{i}M-\bar{y}_{i}\|^{2},\quad\text{such that}\quad M^{T}M=I, \tag{28}\]  
being \(\bar{x}_{i}=x_{i}-\frac{\sum_{i}a_{i}\bar{x}_{i}}{\alpha}\), and \(\bar{y}_{i}=y_{i}-\frac{\sum_{i}a_{i}y_{i}}{\alpha}\), respectively, a control point and its projection, and \(\alpha_{i}=\frac{1}{\|\bar{x}_{i}-\bar{x}\|^{2}}\) a weight that leads the control points to influence other data instances, where \(\alpha=\sum_{i}\alpha_{i}\). Finally, the matrix M is going to be computed from the product of the left and right singular vectors derived from the SVD of the vector column (\(\bar{x}\)) \(\times\) vector line (\(\bar{y}\)) (for further information, please see Section 3.1 in [12]). Consequently, the more a point has similarities to \(x_{i}\), the more it affects Equation (28), which reinforces local characteristics of the LAMP process. Following [PERSON] et al. [12], the orthogonality restriction imposed on Equation (28) guarantees an isometric transformation, avoiding scaling and shear effects. Moreover, the restriction reduced the high error levels generated during the projection of control points and propagated during the other transformations, keeping the (inevitable) distortions as small as possible. The authors rewrote Equation (28) into matrix notation and generated a _Procrustes Orthogonal problem_[73], a well-known mathematical problem of matrix approximation. This equation can be expressed as \(\|AM-B\|_{F}\), \(\quad\text{tal que}\quad M^{T}M=I\), being \(\|\cdot\|_{F}\) the Frobenius norm and:  
\[A= \begin{bmatrix}\sqrt{\alpha_{1}}\bar{x}_{1}\\ \sqrt{\alpha_{2}}\bar{x}_{2}\\ \vdots\\ \sqrt{\alpha_{s}}\bar{y}_{s}\end{bmatrix} ,\quad B= \begin{bmatrix}\sqrt{\alpha _{1}}\bar{y}_{1}\\ \sqrt{\alpha_{2}}\bar{y}_{2}\\ \vdots\\ \sqrt{\alpha_{s}}\bar{y}_{s}\end{bmatrix} . \tag{29}\]  
Finally, the matrix equation is solved using SVD, being \(M=UV\) and \(A^{T}B=UDV\), which demands \(O(s)\) operations. As a result, the position of data points is obtained with \(y=f_{X}(x)=(x-\bar{x})M+\bar{y}\). LAMP was classified as a hybrid projection because its behavior can be local or global depending on the number of control points [6], which can be seen as a positive correlation, i.e., the more control points, the more global will be the mapping. [PERSON] et al. [12] performed tests that revealed LAMP generates high-quality layouts even with a small number of control points.