
TL;DR
This paper extends the theory of geometric data sets by introducing new distances and compactifications, proving completeness and separability properties, and constructing a natural compactification via al- pyramids, with applications to observable diameters.
Contribution
It introduces al- pyramids for geometric data sets, proving their completeness and separability, and develops a compactification that preserves computability of observable diameters.
Findings
The observable distance $d_{\mathrm{conc}}$ is non-separable on the set of geometric data sets.
The box distance $\Box$ is complete and non-separable.
A natural compactification via al- pyramids is constructed, preserving polynomial-time computability.
Abstract
The observable distance based on measure concentration and the box distance based on collapsing theory are extended to geometric data sets introduced by Hanika--Schneider--Stumme. On the set of isomorphism classes of geometric data sets, is non-separable and is complete and non-separable. We introduce the class of -compact geometric data sets in , for a monoidal subfamily of 1-Lipschitz functions , and prove its -completeness and separability. We then construct a natural compactification of by means of \emph{-pyramids} when contains the clipping family. We further prove a complete limit formula for the observable diameter of…
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