  Multidimensional Projection is a fundamental tool for high-dimensional data
analytics and visualization. With very few exceptions, projection techniques
are designed to map data from a high-dimensional space to a visual space so as
to preserve some dissimilarity (similarity) measure, such as the Euclidean
distance for example. In fact, although adopting distinct mathematical
formulations designed to favor different aspects of the data, most
multidimensional projection methods strive to preserve dissimilarity measures
that encapsulate geometric properties such as distances or the proximity
relation between data objects. However, geometric relations are not the only
interesting property to be preserved in a projection. For instance, the
analysis of particular structures such as clusters and outliers could be more
reliably performed if the mapping process gives some guarantee as to
topological invariants such as connected components and loops. This paper
introduces TopoMap, a novel projection technique which provides topological
guarantees during the mapping process. In particular, the proposed method
performs the mapping from a high-dimensional space to a visual space, while
preserving the 0-dimensional persistence diagram of the Rips filtration of the
high-dimensional data, ensuring that the filtrations generate the same
connected components when applied to the original as well as projected data.
The presented case studies show that the topological guarantee provided by
TopoMap not only brings confidence to the visual analytic process but also can
be used to assist in the assessment of other projection methods.
