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1. Nearly-Doubling Spaces of Persistence Diagrams

   Sheehy, Donald R. (January 2022, 38th International Symposium on Computational Geometry (SoCG 2022))

   Goaoc, Xavier and (Ed.)

   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. 

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2. Computing Bottleneck Distance for 2-D Interval Decomposable Modules

   Dey, Tamal K; Xin, Cheng (June 2018, j34th International Symposium on Computational Geometry (SoCG 2018))

   Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $$1$$-D persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most $$n$$-D persistence modules, $n>1$$, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called {\em $$2$$-D interval decomposable} modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the bottleneck distance for these modules from indecomposables, which bounds the interleaving distance from above, and give another algorithm to compute a new distance called {\em dimension distance} that bounds it from below. 

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3. Meta-diagrams for 2-parameter Persistence

   Clause, N.; Dey, T. K.; Memoli, F.; Wang, B. (June 2023, LIPIcs, Volume 258, SoCG 2023, Complete Volume)

   We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module M indexed by a bifiltration of n simplices in O(n^3) time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has O(n^4) time complexity. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of M from O(n^4) to O(n^3). In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistent diagram in the 1-parameter setting. 

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4. Robust Persistence Diagrams using Reproducing Kernels

   Vishwanath, Siddharth; Fukumizu, Kenji; Kuriki, Satoshi and (January 2020, Advances in neural information processing systems)

   Larochelle, H.; Ranzato, M.; Hadsell, R.; Balcan, M.F.; and Lin, H. (Ed.)

   Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams are very sensitive to perturbations in the input space. In this work, we develop a framework for constructing robust persistence diagrams from superlevel filtrations of robust density estimators constructed using reproducing kernels. Using an analogue of the influence function on the space of persistence diagrams, we establish the proposed framework to be less sensitive to outliers. The robust persistence diagrams are shown to be consistent estimators in the bottleneck distance, with the convergence rate controlled by the smoothness of the kernel — this, in turn, allows us to construct uniform confidence bands in the space of persistence diagrams. Finally, we demonstrate the superiority of the proposed approach on benchmark datasets. 

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5. Fast and Exact Convex Hull Simplification

   https://doi.org/10.4230/LIPIcs.FSTTCS.2021.26  

   Klimenko, Georgiy; Raichel, Benjamin (October 2021, Foundations of Software Technology and Theoretical Computer Science (FSTTCS))

   Bojanczyk, Mikolaj; Chekuri, Chandra (Ed.)

   Given a point set P in the plane, we seek a subset Q ⊆ P, whose convex hull gives a smaller and thus simpler representation of the convex hull of P. Specifically, let cost(Q,P) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q,P) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q,P), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log²n) time algorithm and an O(n log³n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n^2.5302) time algorithm when minimizing k and an O(min{n^2.5302, kn^2.376}) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case. 

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