Date of Completion

4-26-2019

Embargo Period

4-26-2019

Keywords

topological data analysis, homology inference, persistent homology, computational geometry, algebraic topology

Major Advisor

Donald R. Sheehy

Associate Advisor

Parasara Sridhar Duggirala

Associate Advisor

Thomas J. Peters

Associate Advisor

Alexander Russell

Field of Study

Computer Science and Engineering

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

In this dissertation we introduce novel techniques to infer the shape of a geometric space from local data. The major contribution of this research is that the techniques aim to minimize the hypotheses of the sample and the space compared to existing results in the field.

We first prove an equivalence between the two standard geometric sampling methods: adaptive and uniform. From this we create a collection of topological interleavings derived from this equivalence, which we use to compute the homology of an arbitrary Euclidean compact set only assuming a bound on the weak feature size of the domain and a sample of an arbitrary reference set for sampling.

Next we provide an algorithm to check for k-coverage of sensor network from a collection of finite sensors using computable structures built upon from sensors' local data. This algorithm generalizes the original Topological Coverage Criterion by De Silva and Ghrist to domains with non-smooth boundaries among other relaxed assumptions. We then prove that if one has coverage of the sensor network, there exists subsampling parameters such that one can infer the homology of the entire domain.

We culminate the dissertation by proving an approximation of the Persistent Nerve Theorem assuming a relaxation of the standard topological cover assumption. Namely, given such a cover exists, there is a tight bottleneck distance between the persistence diagrams of the nerve filtration and the space filtration, generalizing the Persistent Nerve Theorem. This results provides information about the shape of a subdivisions of triangulations with no global geometric data provided, as well as broadens the applications of nerves to cases there are not nice sampling guarantees or there exists approximation errors during triangulation.

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