# Getting over High-Dimensionality: How Multidimensional Projection Methods Can Assist Data Science
## 4 Multidimensional Projection Approaches and Domains
### TopoMap  
Following the topological approach, [PERSON] et al. [56] developed TopoMap, a projection method devoted to mapping data from a high-dimensional space to a visual space preserving the global 0-dimensional topological persistence diagram defined by a Rips filtration over the \(m\)-dimensional data. Furthermore, TopoMap ensures that Rips filtrations generate the same connected components when applied to the original and the projected data by preserving the minimal spanning tree (MST) of the original data embedding in the projection [84].  
Figure 3: Comparing UMAP and t-SNE applied to some well-known datasets. Point colors represent data labels in each dataset. Source: [PERSON] et al. [29].  
Geometric relations such as distances or the proximity relationship between data instances are not the only attractive property to be preserved in a projection. According to TopoMap's authors, particular structures such as clusters and outliers could be more reliable and meaningful if the mapping reveals some topological invariants such as connected components and loops.  
In contrast to dissimilarity-based projections, TopoMap is based on the non-differentiable 0-homology topological persistence evolution of cycles in the simplicial complexes resulting from a Euclidean distance-based Rips filtration [85]. Given a set of \(m\)-dimensional data instances \(P\), TopoMap builds a complete weighted graph over \(P\), weighting each edge (\(p_{i}\), \(p_{j}\)) with the Euclidean distance between the corresponding endpoints, \(d(p_{i},p_{j})\). The Rips filtration [86] grows a high-dimensional ball around the data instances, adding an edge (simplexes) to the filtration, one at a time, when two (or more) balls intersect each other. The addition of each new simplex can change the topology. The ordered set of changing edges is the MST of edges computed over the graph. TopoMap projects the data while preserving the evolution of cycles, such that the topological filtrations over the high-dimensional and projected data generate the same connected components (0-cycle) at the same instances of the respective filtrations.  
TopoMap is a different approach from Isomap and UMAP. Isomap does not consider 0-homology groups, which are therefore not preserved. UMAP is based on category theory [29] and TopoMap focuses on persistent homology. Topological methods have been used to evaluate the projection process [87]. Since TopoMap bears theoretical guarantees, a side contribution is that it can be applied as an analytical tool helping to probe and illustrate how other projection methods behave, especially regarding distortions and cluster preservation. In this context, [PERSON] et al. [56] demonstrated that clusters visualized in a t-SNE layout, in fact, tend to correspond only to pieces of clusters present in the \(m\)-dimensional data.  
TopoMap can be used with an alternative distance metric than Euclidean distance. However, changing the metric requires computing the MST using this new metric, which means the running time for computing the MST will degenerate TopoMap complexity to \(O(n^{2})\) due to the computation of the distance matrix.