Publication Date

Summer 2017

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

Advisor

Timothy Hsu

Keywords

Algebraic Topology, Data Science, Local Homology, Point Cloud Data, Simplicial Homology, Topological Data Analysis

Subject Areas

Mathematics; Applied mathematics; Computer science

Abstract

We have invented a method that uses the mathematical idea of local homology to calculate the local dimension (one-dimensional, two-dimensional, mixed dimension of varying types) of an underlying object in two-dimensional space from a point cloud approximation.

Motivated by the need to find an efficient method for computing local homology, we define a new mathematical object -- the Local Complex, and some variations of this idea -- that we show to be exactly related to the Vietoris-Rips complex under some settings. This concept captures the essence of the local homology of point cloud data at any scale. We provide a computationally tractable heuristic -- the simplex arc projection on a particular variation of the Local Complex -- the Acute Local Complex -- to produce yet another object, simply called the simplex arc projections. Homology computation for this latter class of objects is then described. This homology is then related to the true local homology, by example. This relationship is not tight (as we show by counter-example), and conditions under which it holds are the subject of future extensions to this research. Our method also includes mechanisms for detecting whether a particular dimensionality analysis is mathematically more meaningful (in technical terms, ``persistent'') and not just a function of a choice of parameters. We also provide implementation and experimental results on synthetic data as well as a subset of the Sloan Digital Sky Survey data.

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