Mutiscale Mapper: A Framework for Topological Summarization of Data and Maps

Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. textsfMapper, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the codomain of the map. This cover, via a pullback operation to the domain, produces a simplicial complex connecting the data points. The resulting view of the data through a cover of the codomain offers flexibility in analyzing the data. However, it offers only a view at a fixed scale at which the cover is constructed. Inspired by the concept, we explore a notion of a tower of covers which induces a tower of simplicial complexes connected by simplicial maps, which we call em multiscale mapper. We study the resulting structure, its stability, and design practical algorithms to compute its associated persistence diagrams efficiently. Specifically, when the domain is a simplicial complex and the map is a real-valued piecewise-linear function, the algorithm can compute the exact persistence diagram only from the 1-skeleton of the input complex. For general maps, we present a combinatorial version of the algorithm that acts only on vertex sets connected by the 1-skeleton graph, and this algorithm approximates the exact persistence diagram thanks to a stability result that we show to hold. We also relate the multiscale mapper with the vCech complexes arising from a natural pullback pseudometric defined on the input domain.