TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data

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