Statistical topological data analysis using persistence landscapes

We define a new topological summary for data that we call the persistence landscape. In contrast to the standard topological summaries, the barcode and the persistence diagram, it is easy to combine with statistical analysis, and its associated computations are much faster. This summary obeys a Strong Law of Large Numbers and a Central Limit Theorem. Under certain finiteness conditions, this allows us to calculate approximate confidence intervals for the expected total squared persistence. With these results one can use t-tests for statistical inference in topological data analysis. We apply these methods to numerous examples including random geometric complexes, random clique complexes, and Gaussian random fields. We also show that this summary is stable and gives lower bounds for the bottleneck distance and the Wasserstein distance.