Statistical topology using persistence landscapes

We define a new descriptor for persistent homology, which we call the persistence landscape, for the purpose of facilitating statistical inference. This descriptor may be thought of as an embedding of the usual descriptors, barcodes and persistence diagrams, into a space of functions, which inherits an $L^p$ norm. We show that the corresponding metric is topologically equivalent to the (p+1)-Wasserstein distance, and that this metric space is complete and separable. We prove a stability theorem for persistence landscapes. For p=2, we show that the Frechet mean of persistence landscapes is the pointwise mean, and that the Frechet variance is the integral of the pointwise variances. Furthermore, the sample mean of persistence landscapes converges pointwise to the mean of the underlying distribution, and there is a corresponding central limit theorem.