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

A standard way of approximating or discretizing a metric space is by takingits Rips complexes. These approximations for all parameters are often boundtogether into a filtration, to which we apply the fundamental group or thefirst homology. We call the resulting object persistence.  Recent results demonstrate that persistence of a compact geodesic locallycontractible space $X$ carries a lot of geometric information. However, bydefinition the corresponding Rips complexes have uncountably many vertices. Inthis paper we show that nonetheless, the whole persistence of $X$ may beobtained by an appropriate finite sample (subset of $X$), and that persistenceof any subset of $X$ is well interleaved with the persistence of $X$. Itfollows that the persistence of $X$ is the minimum of persistences obtained byall finite samples. Furthermore, we prove a much improved Stability theorem forsuch approximations. As a special case we provide for each $r>0$ a density$s>0$, so that for each $s$-dense sample $S \subset X$ the correspondingfundamental group (and the first homology) of the Rips complex of $S$ isisomorphic to the one of $X$, leading to an improved reconstruction result.

Approximations of 1-Dimensional Intrinsic Persistence of Geodesic Spaces and Their Stability