The space of persistence diagrams under bottleneck distance is known to have infinite doubling dimension. Because many metric search algorithms and data structures have bounds that depend on the dimension of the search space, the high-dimensionality makes it difficult to analyze and compare asymptotic running times of metric search algorithms on this space. We introduce the notion of nearly-doubling metrics, those that are Gromov-Hausdorff close to metric spaces of bounded doubling dimension and prove that bounded $k$-point persistence diagrams are nearly-doubling. This allows us to prove that in some ways, persistence diagrams can be expected to behave like a doubling metric space. We prove our results in great generality, studying a large class of quotient metrics (of which the persistence plane is just one example). We also prove bounds on the dimension of the $k$-point bottleneck space over such metrics. The notion of being nearly-doubling in this Gromov-Hausdorff sense is likely of more general interest. Some algorithms that have a dependence on the dimension can be analyzed in terms of the dimension of the nearby metric rather than that of the metric itself. We give a specific example of this phenomenon by analyzing an algorithm to compute metric nets, a useful operation on persistence diagrams.

@inproceedings{sheehy22nearly,
    title = {Nearly-Doubling Spaces of Persistence Diagrams},
    author = {Donald R. Sheehy and Siddharth Sheth},
    booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
    pages = {60:1--60:15},
    doi =		{10.4230/LIPIcs.SoCG.2022.60},
    year = {2022}}