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Emptiness of fibers of point-cloud persistence maps

Determine necessary and sufficient conditions on a full barcode D under which the fiber (preimage) of D under the point-cloud persistence maps PH^{VR} (Vietoris–Rips) or PH^{Cech} (Čech) from the configuration space of ordered point clouds PC to full barcodes is empty; equivalently, characterize exactly which full barcodes D have (PH^{VR})^{-1}(D)=∅ or (PH^{Cech})^{-1}(D)=∅.

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Background

The paper studies the fibers (level sets) of the persistence maps that send an ordered point cloud P in Rd to its full barcode via either the Vietoris–Rips or Čech filtration. Understanding when these fibers are empty is a fundamental inverse problem: it asks which full barcodes can arise from some point cloud and which cannot.

Oudot and Solomon previously formulated this inverse question for point-cloud persistence fibers. The present paper notes that the question remains unresolved and highlights the broader lack of structural understanding of these fibers when nonempty.

References

Oudot and Soloman posed the first problem about level sets of the persistence maps $PH$ and $PH$, they asked when they are empty . While this question is still open, in fact very little is known about the structure of these level sets when they are nonempty.

— Fibers of point cloud persistence (2411.08201 - Beers et al., 12 Nov 2024) in Related Work (Section 1.2)