UP - logo
FMF in IMFM, Matematična knjižnica, Ljubljana (MAKLJ)
  • Tropical coordinates on the space of persistence barcodes : a dissertation submitted to the Department of Mathematics and the Committee on Graduate Studies of Stanford University in partial fulfillment of the requirements for the degree of doctor of philosophy
    Kališnik Verovšek, Sara, 1986-
    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 ▫$S_n$▫. 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.
    Vrsta gradiva - disertacija ; neleposlovje za odrasle
    Založništvo in izdelava - Stanford : [S. Kališnik Verovšek], 2016
    Jezik - angleški
    COBISS.SI-ID - 17666905

Signatura – lokacija, inventarna št. ... Status izvoda Rezervacija
Skladišče-Jadranska 19

0000011161/0000000165 Skladišče-Jadranska 19

11161/165 		prosto - za čitalnico
loading ...
loading ...
loading ...

Vnos na polico