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Graded persistence diagrams and persistence landscapes

We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.Comment: accepted for publication in Discrete and Computational Geometr

The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

Block persistence

We define a block persistence probability $p_l(t)$ as the probability that the order parameter integrated on a block of linear size $l$ has never changed sign since the initial time in a phase ordering process at finite temperature T<T_c. We argue that p_l(t)\sim l^{-z\theta_0}f(t/l^z) in the scaling limit of large blocks, where \theta_0 is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent \theta for large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large n model, and generically it can be used to determine easily \theta_0 from simulations of coarsening models. We also argue that \theta_0 and the scaling function do not depend on temperature, leading to a definition of \theta at finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. We also discuss conserved models for which different scaling are shown to arise depending on the value of the autocorrelation exponent \lambda. We illustrate our discussion by extensive numerical results. We also comment on the relation between this method and an alternative definition of \theta at finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)].Comment: Revtex, 18 pages (multicol.sty), 15 eps figures (uses epsfig), submitted to Eur. Phys. J.

A worldwide correlation of lactase persistence phenotype and genotypes

Background: The ability of adult humans to digest the milk sugar lactose - lactase persistence - is a dominant Mendelian trait that has been a subject of extensive genetic, medical and evolutionary research. Lactase persistence is common in people of European ancestry as well as some African, Middle Eastern and Southern Asian groups, but is rare or absent elsewhere in the world. The recent identification of independent nucleotide changes that are strongly associated with lactase persistence in different populations worldwide has led to the possibility of genetic tests for the trait. However, it is highly unlikely that all lactase persistence-associated variants are known. Using an extensive database of lactase persistence phenotype frequencies, together with information on how those data were collected and data on the frequencies of lactase persistence variants, we present a global summary of the extent to which current genetic knowledge can explain lactase persistence phenotype frequency. Results: We used surface interpolation of Old World lactase persistence genotype and phenotype frequency estimates obtained from all available literature and perform a comparison between predicted and observed trait frequencies in continuous space. By accommodating additional data on sample numbers and known false negative and false positive rates for the various lactase persistence phenotype tests (blood glucose and breath hydrogen), we also apply a Monte Carlo method to estimate the probability that known lactase persistence-associated allele frequencies can explain observed trait frequencies in different regions. Conclusion: Lactase persistence genotype data is currently insufficient to explain lactase persistence phenotype frequency in much of western and southern Africa, southeastern Europe, the Middle East and parts of central and southern Asia. We suggest that further studies of genetic variation in these regions should reveal additional nucleotide variants that are associated with lactase persistence