Parameterized topological data analysis
Abstract/Contents
- Abstract
- This dissertation investigates several ways in which topological data analysis can be made more digestible by structuring computations and models. First, we introduce a new method for computing algebraic invariants of diagrams of topological spaces using matrices associated with quiver representations. This computational framework allows for parallel algorithms to compute persistent and zigzag homology in the most general case, with arbitrary induced maps on homology. Next, we extend the classical techniques of acyclic carriers to the filtered setting and demonstrate how these tools can be used to construct interleavings to compare persistent homology of filtered spaces. We introduce a class of geometric complexes parameterized by a cover of a data set and use carriers to analyze the relationship between these complexes to the unparameterized geometric complexes. Finally, we investigate spaces of data generated from sampling small cubes of voxels (patches) from three-dimensional images through the use of a map that captures the direction of the largest variation
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2020; ©2020 |
| Publication date | 2020; 2020 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Nelson, Bradley Jared | |
|---|---|---|
| Degree supervisor | Carlsson, G. (Gunnar), 1952- | |
| Degree supervisor | Taylor, Jonathan E | |
| Thesis advisor | Carlsson, G. (Gunnar), 1952- | |
| Thesis advisor | Taylor, Jonathan E | |
| Thesis advisor | Kerckhoff, Steve | |
| Degree committee member | Kerckhoff, Steve | |
| Associated with | Stanford University, Institute for Computational and Mathematical Engineering. |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Bradley J. Nelson |
|---|---|
| Note | Submitted to the Institute for Computational and Mathematical Engineering |
| Thesis | Thesis Ph.D. Stanford University 2020 |
| Location | electronic resource |
Access conditions
- Copyright
- © 2020 by Bradley Jared Nelson
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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