Towards statistical topology: homology, persistent homology and persistence landscapes
Peter Bubenik (May 22, 2012)
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One of the principal uses of topology is to patch together local quantitative data to obtain global qualitative information not readily accessible to other methods. While the early development of topology was largely driven by applications, many later advances were motivated by strictly mathematical concerns. Now the field of applied topology is returning topology to its roots, adapting some of the later advances in topological methods to current questions in applications. I will survey some of the central constructions in topological data analysis, introducing homology and persistent homology.
There is a clear need to combine these tools with statistical analysis. However there are difficulties in doing so, as the space of the usual topological descriptor is not a manifold. I define a new topological descriptor, the persistence landscape, whose definition allows for the calculation of means and standard deviations, laws of large numbers, central limit theorems and hypothesis testing.