On the Strengthening of Topological Signals in Persistent Homology through Vector Bundle Based Maps

Persistent homology is a relatively new tool from topological data analysis that has transformed, for many, the way data sets (and the information contained in those sets) are viewed. It is derived directly from techniques in computational homology but has the added feature that it is able to capture structure at multiple scales. One way that this multi-scale information can be presented is through a barcode. A barcode consists of a collection of line segments each representing the range of parameter values over which a generator of a homology group persists. A segment’s length relative to the lenght of other segments is an indication of the strength of a corresponding topological signal. In this paper, we consider how vector bundles may be used to re-embed data as a means to improve the topological signal. As an illustrative example, we construct maps of tori to a sequence of Grassmannians of increasing dimension. We equip the Grassmannian with the geodesic metric and observe an improvement in barcode signal strength as the dimension of the Grassmannians increase.