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

Relative cell complexes in closure spaces

We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Cech's closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the constructions of cell complexes by a finite sequences of closed cell attachments, which attach arbitrarily many cells at a time, agree. Likewise, the constructions of finite CW complexes relative to a compactly generated weak Hausdorff space also agree. On the other hand, we give examples showing that the constructions of finite-dimensional CW complexes, CW complexes of finite type, and finite relative CW complexes need not agree.Comment: 8 pages, expanded the main theorem to an if and only if statement, more descriptive title, more detailed abstract, clarified connections to related wor