Tropical coordinates on the space of persistence barcodes
Abstract/Contents
- Abstract
- In the last two decades applied topologists have developed numerous methods for `measuring' and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology. This adaptation of classical homology assigns a barcode, i.e. a collection of intervals with endpoints on the real line, to a finite metric space. Unfortunately, barcodes are not well-adapted for use by practitioners in machine learning tasks. In this dissertation, I identify classes of max-plus polynomials and tropical rational functions that can be used as coordinates on the space of barcodes. All of these are stable with respect to standard distance functions (bottleneck distance, Wasserstein distances) used on the barcode space. I demonstrate how these coordinates can be used by combining persistent homology with SVM to classify numbers from the MNIST dataset. In order to identify functions on the barcode space, I find generators for the semirings of tropical polynomials, max-plus polynomials and tropical rational functions invariant under the action of the symmetric group. The fundamental theorem of ordinary symmetric polynomials has an equivalent in the tropical and max-plus semirings. There are interesting differences if we consider the tropical polynomial semiring with nr variables that come in n blocks of r variables each and are permuted by the symmetric group. As opposed to the ordinary polynomial case, the semiring of r-symmetric tropical polynomials is not finitely generated, but the semiring of r-symmetric tropical rational functions is.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2016 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Kališnik Verovšek, Sara |
|---|---|
| Associated with | Stanford University, Department of Mathematics. |
| Primary advisor | Carlsson, G. (Gunnar), 1952- |
| Thesis advisor | Carlsson, G. (Gunnar), 1952- |
| Thesis advisor | Brumfiel, Gregory W |
| Thesis advisor | Kerckhoff, Steve |
| Advisor | Brumfiel, Gregory W |
| Advisor | Kerckhoff, Steve |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Sara Kališnik Verovšek. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2016. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2016 by Sara Kalisnik Verovsek
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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