@mastersthesis{petrov2024dissertation,
  title      = {Persistent Homology for the Analysis of Stratified Spaces},
  author     = {Petrov, Alice},
  school     = {University of Oxford},
  year       = {2024},
  url_paper  = {https://alicepetrov.github.io/assets/pdf/papers/2024/petrov2024dissertation.pdf},
  abstract   = {This dissertation investigates the application of persistent homology to the analysis of stratified spaces, focusing on word embeddings as a case study. Stratified spaces present unique challenges for traditional topological data analysis techniques, particularly in identifying and analyzing singularities. To address these challenges, we discuss persistent intersection homology and extend the concepts of kernel, image, and cokernel persistence. We also present a novel approach to analyzing singularities using bifiltrations and sliding window (co)kernel diagrams. In addition, we discuss multiparameter persistence. These tools allow for a more nuanced analysis, enabling the differentiation of various types of singularities, such as those associated with polysemous words in word embeddings.
We demonstrate the practical utility of these theoretical advancements through computational pipelines and experiments on word embeddings, specifically analyzing how dimensionality reduction techniques influence the topology of local neighborhoods. The results indicate that our methods can reveal significant structural differences in word embeddings, offering new insights into their topological properties.
This work advances the theoretical understanding of persistent homology in stratified spaces and opens new avenues for applying these techniques to other complex, high-dimensional datasets. We conclude with a discussion of potential future research directions, including extending these methods to multifiltrations, exploring different invariants, and broader applications beyond word embeddings.},
  keywords   = {Computational Topology}
}

{"_id":"hmJdxovu8DfLwcLMT","bibbaseid":"petrov-persistenthomologyfortheanalysisofstratifiedspaces-2024","author_short":["Petrov, A."],"bibdata":{"bibtype":"mastersthesis","type":"mastersthesis","title":"Persistent Homology for the Analysis of Stratified Spaces","author":[{"propositions":[],"lastnames":["Petrov"],"firstnames":["Alice"],"suffixes":[]}],"school":"University of Oxford","year":"2024","url_paper":"https://alicepetrov.github.io/assets/pdf/papers/2024/petrov2024dissertation.pdf","abstract":"This dissertation investigates the application of persistent homology to the analysis of stratified spaces, focusing on word embeddings as a case study. Stratified spaces present unique challenges for traditional topological data analysis techniques, particularly in identifying and analyzing singularities. To address these challenges, we discuss persistent intersection homology and extend the concepts of kernel, image, and cokernel persistence. We also present a novel approach to analyzing singularities using bifiltrations and sliding window (co)kernel diagrams. In addition, we discuss multiparameter persistence. These tools allow for a more nuanced analysis, enabling the differentiation of various types of singularities, such as those associated with polysemous words in word embeddings. We demonstrate the practical utility of these theoretical advancements through computational pipelines and experiments on word embeddings, specifically analyzing how dimensionality reduction techniques influence the topology of local neighborhoods. The results indicate that our methods can reveal significant structural differences in word embeddings, offering new insights into their topological properties. This work advances the theoretical understanding of persistent homology in stratified spaces and opens new avenues for applying these techniques to other complex, high-dimensional datasets. We conclude with a discussion of potential future research directions, including extending these methods to multifiltrations, exploring different invariants, and broader applications beyond word embeddings.","keywords":"Computational Topology","bibtex":"@mastersthesis{petrov2024dissertation,\n title = {Persistent Homology for the Analysis of Stratified Spaces},\n author = {Petrov, Alice},\n school = {University of Oxford},\n year = {2024},\n url_paper = {https://alicepetrov.github.io/assets/pdf/papers/2024/petrov2024dissertation.pdf},\n abstract = {This dissertation investigates the application of persistent homology to the analysis of stratified spaces, focusing on word embeddings as a case study. Stratified spaces present unique challenges for traditional topological data analysis techniques, particularly in identifying and analyzing singularities. To address these challenges, we discuss persistent intersection homology and extend the concepts of kernel, image, and cokernel persistence. We also present a novel approach to analyzing singularities using bifiltrations and sliding window (co)kernel diagrams. In addition, we discuss multiparameter persistence. These tools allow for a more nuanced analysis, enabling the differentiation of various types of singularities, such as those associated with polysemous words in word embeddings.\nWe demonstrate the practical utility of these theoretical advancements through computational pipelines and experiments on word embeddings, specifically analyzing how dimensionality reduction techniques influence the topology of local neighborhoods. The results indicate that our methods can reveal significant structural differences in word embeddings, offering new insights into their topological properties.\nThis work advances the theoretical understanding of persistent homology in stratified spaces and opens new avenues for applying these techniques to other complex, high-dimensional datasets. We conclude with a discussion of potential future research directions, including extending these methods to multifiltrations, exploring different invariants, and broader applications beyond word embeddings.},\n keywords = {Computational Topology}\n}","author_short":["Petrov, A."],"key":"petrov2024dissertation","id":"petrov2024dissertation","bibbaseid":"petrov-persistenthomologyfortheanalysisofstratifiedspaces-2024","role":"author","urls":{" paper":"https://alicepetrov.github.io/assets/pdf/papers/2024/petrov2024dissertation.pdf"},"keyword":["Computational Topology"],"metadata":{"authorlinks":{}},"downloads":3},"bibtype":"mastersthesis","biburl":"https://raw.githubusercontent.com/alicepetrov/alicepetrov.github.io/main/_bibliography/papers.bib","dataSources":["2Eox22rXM4CLdebT7"],"keywords":["computational topology"],"search_terms":["persistent","homology","analysis","stratified","spaces","petrov"],"title":"Persistent Homology for the Analysis of Stratified Spaces","year":2024,"downloads":3}