Statistical topological data analysis using persistence landscapes

We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries. Viewed as a random variable with values in a Banach space, this summary obeys a strong law of large numbers and a central limit theorem. We show how a number of standard statistical tests can be used for statistical inference using this summary. We also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and Wasserstein distances.Comment: 26 pages, final version, to appear in Journal of Machine Learning Research, includes two additional examples not in the journal version: random geometric complexes and Erdos-Renyi random clique complexe

Homological Algebra for Persistence Modules

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself.Comment: 41 pages, accepted by Foundations of Computational Mathematic

Simplicial models for concurrency

We model both concurrent programs and the possible executions from one state to another in a concurrent program using simplices. The latter are calculated using necklaces of simplices in the former.Comment: 12 pages, Section 4 from v1 omitted since quasi-category equivalences are too strong: they induce equivalences of path categorie

Palmated Antlers of Moose May Serve as A Parabolic Reflector of Sounds

It has been postulated that the excellent sense of hearing in moose is mostly due to: (1) the large surface of the external ear, (2) better stereophony due to the large distance between ears, (3) independently movable, extremely adjustable pinna, and (4) the amplification of sounds reflected by the palms of the antlers. The last factor, possible reflection of sounds into pinna by the palm of the antlers, was tested in this study on a large antler trophy of Alaskan moose. The reception of a standard tone, broadcast from the frontally placed speaker, was recorded by a sound level meter located in an artificial moose ear. Three locations of the ear, as positioned relative to the speaker, e.g., frontward, sideward, and backward, were tested. The weakest reception was recorded in the backward position of the ear. If the sound pressure measured in the frontward position was set as 100%, the sound pressure in the backward position was 79%. The strongest reception was recorded when the artificial ear was positioned toward the center of the antler palm. In this position, the sound pressure was 119% relative to the frontward position. These findings strongly indicate that the palm of moose antlers may serve as an effective, parabolic reflector which increases the acoustic pressure of the incoming sound