1,548 research outputs found
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
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
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
Separated Lie models and the homotopy Lie algebra
A simply connected topological space X has homotopy Lie algebra \pi_*(\Omega
X) \tensor \Q. Following Quillen, there is a connected differential graded
free Lie algebra (dgL) called a Lie model, which determines the rational
homotopy type of X, and whose homology is isomorphic to the homotopy Lie
algebra. We show that such a Lie model can be replaced with one that has a
special property we call separated. The homology of a separated dgL has a
particular form which lends itself to calculations.Comment: Final version. To appear in the Journal of Pure and Applied Algebra.
Added connections to the radical of the homotopy Lie algebra and the
Avramov-Felix conjecture. Added examples of wedges of spheres of any
"thickness" and connected sums of products of spheres. 15 page
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