Quiver theory, zigzag homology and deep learning
Abstract/Contents
- Abstract
- Topological data analysis deals with the study of data using techniques from algebraic topology. It has emerged as a popular and powerful technique in studying data in recent times. In this dissertation, we will introduce a new perspective on algorithms for topological data analysis (TDA). The new framework will allow us to generalize existing algorithms to new problems and also provide novel methods to parallelize the algorithms to compute zigzag persistent homology. We will first introduce how quiver representations can be used to reduce the problem of computing persistent and zigzag homology to that of computing a canonical form similar to the eigenvalue decomposition or the Jordan normal form. Then we will describe an algorithm to compute the canonical form of any type A quiver representation. We will show how the algorithm can be expressed as a sequence of matrix factorizations and matrix passing steps performed on the graph underlying the quiver representation. We will conclude by looking at an example of how TDA can be used along with deep learning, specifically we will show how zigzag homology performs better than persistent homology in the case of the MNIST dataset
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 | Dwaraknath, Anjan |
|---|---|
| Degree supervisor | Carlsson, G. (Gunnar), 1952- |
| Degree supervisor | Gerritsen, Margot (Margot G.) |
| Thesis advisor | Carlsson, G. (Gunnar), 1952- |
| Thesis advisor | Gerritsen, Margot (Margot G.) |
| Thesis advisor | Guibas, Leonidas J |
| Degree committee member | Guibas, Leonidas J |
| Associated with | Stanford University, Institute for Computational and Mathematical Engineering |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Anjan Dwaraknath |
|---|---|
| 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 Anjan Dwaraknath
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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