# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 4 Multidimensional Projection Approaches and Domains
### Uniform Manifold Approximation for Dimension Reduction  
Uniform Manifold Approximation and Projection (UMAP) [29] is a multidimensional projection technique with a solid base on geometry and topology theories. UMAP is a manifold learning method that relies on Riemannian geometry and algebraic topology to perform dimensional reduction [77]. The authors highlighted that UMAP is scalable, which turns it suitable in real applications with large datasets. Furthermore, UMAP presents results as good as t-SNE (currently one of the widely used projection techniques) in visualization quality; nonetheless, it better preserves global data structures and is faster than t-SNE. In addition, UMAP was developed with machine learning applications in mind [8].  
To perform data mapping, UMAP considers the following assumptions: (i) data are uniformly distributed in a manifold, and (ii) the manifold is locally connected, i.e., there are no isolated points. [PERSON] et al. [29] used manifold approximations and patching associated with local fuzzy simplicial set [78] to construct a topological representation for original and projected spaces. Following this step, UMAP minimizes the cross-entropy function between the original topology and the projected one.  
The authors presented mathematical bases and a practical and computational vision of the process. UMAP is described as steps that construct and manipulate a weighted graph in the latter. The first step creates a graph, considering the weight function as  
\[w(x_{i},x_{ij})=\exp\left(\frac{-\max(0,d(x_{i},x_{ij})-\rho_{i})}{\sigma_{i}} \right), \tag{32}\]  
\(\rho_{i}=\min\{d(x_{i},x_{ij})|1\leq j\leq k,d(x_{i},x_{ij})>0\}\) and \(\sigma_{i}\) being calculated to satisfy:  
\[\sum_{j=1}^{k}\exp\left(\frac{-\max(0,d(x_{i},x_{ij})-\rho_{i})}{\sigma_{i}} \right)=log_{2}(k).\] (a matrix generated with the previous weight function and the operator \(\circ\) of the Hadamard product [79].  
UMAP applies a Force Directed approach to place the graph nodes in the projected space. The algorithm uses attraction and repulsion functions, respectively, defined as  
\[\frac{-2 ab\|y_{\mathrm{i}}-y_{\mathrm{j}}\|_{2}^{2(b-1)}}{1+\|y_{\mathrm{i}}-y _{\mathrm{j}}\|_{2}^{2}}w(x_{i},x_{j})(y_{\mathrm{i}}-y_{\mathrm{j}})\quad \text{ and } \tag{34}\]  
\[\frac{2b}{(\epsilon+\|y_{\mathrm{i}}-y_{\mathrm{j}}\|_{2}^{2})(1+a\|y_{ \mathrm{i}}-y_{\mathrm{j}}\|_{2}^{2b})}(1-w(x_{i},x_{j})(y_{\mathrm{i}}-y_{ \mathrm{j}})), \tag{35}\]  
where \(a\) and \(b\) are hyperparameters, with \(\epsilon\) being a small number to avoid division by zero (the authors used \(\epsilon=0.001\)). [PERSON] et al. [29] highlighted that the initial projection could be randomly generated; however, they used a spectral layout to initialize the projection. As a result, it provides faster convergence and superior algorithm stability.  
In the last projection step, UMAP minimizes with gradient descent a smooth approximation of the strength between two elements (the original data instance and the projected one), which is defined as \(\Phi(x,y)=(1+a(\|x-y\|_{2}^{2})^{b})^{-1}\). The parameters \(a\) and \(b\) are defined with a nonlinear least-square that fits one function dependent on the original instance and projected point position and a parameter that defines the desired separation of the points on transformed space.  
As stated by its authors, UMAP provides highly similar visual results to t-SNE, and for this reason, it is often seen as a modern, faster, and more scalable alternative to t-SNE, providing visually similar outputs [56]. Figure 3 depicts a comparison between UMAP and t-SNE applied to well-known datasets, namely COIL20 [80], MNIST [81], Fashion MNIST [82], and Google News [83]. One can figure out that UMAP results are comparable with t-SNE ones, segregating and maintaining the same data structures.