English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We develop a general framework for the probabilistic analysis of randomfinite point clouds in the context of topological data analysis. We extend thenotion of a barcode of a finite point cloud to compact metric spaces. Such abarcode lives in the completion of the space of barcodes with respect to thebottleneck distance, which is quite natural from an analytic point of view. Asan application we prove that the barcodes of i.i.d. random variables sampledfrom a compact metric space converge to the barcode of the support of theirdistribution when the number of points goes to infinity. We also examine morequantitative convergence questions for uniform sampling from compact manifolds,including expectations of transforms of barcode valued random variables inBanach spaces. We believe that the methods developed here will serve as usefultools in studying more sophisticated questions in topological data analysis andrelated fields.

Geometric and Probabilistic Limit Theorems in Topological Data Analysis