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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from injective. We therefore propose that, in many cases, the collection of persistence diagrams of many small subsets of X is a better invariant. This invariant, which we call "distributed persistence," is perfectly parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of metric spaces (with the quasi-isometry distance) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is globally bi-Lipschitz. This is a much stronger property than simply being injective, as it implies that the inverse image of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the inverse Lipschitz constant depends on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying simple covering properties. These theoretical results are complemented by synthetic experiments demonstrating the use of distributed persistence in practice.

Elchanan Solomon, Alexander Wagner, and Paul Bendich. From Geometry to Topology: Inverse Theorems for Distributed Persistence. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{solomon_et_al:LIPIcs.SoCG.2022.61, author = {Solomon, Elchanan and Wagner, Alexander and Bendich, Paul}, title = {{From Geometry to Topology: Inverse Theorems for Distributed Persistence}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {61:1--61:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.61}, URN = {urn:nbn:de:0030-drops-160690}, doi = {10.4230/LIPIcs.SoCG.2022.61}, annote = {Keywords: Applied Topology, Persistent Homology, Inverse Problems, Subsampling} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT), both of which apply to shapes embedded in Euclidean space. The contribution of this paper is to define topological transforms for abstract metric measure spaces. Our proposed pipeline is to pre-compose the PHT or ECT with a Euclidean embedding derived from the eigenfunctions and eigenvalues of an integral operator. To that end, we define and study an integral operator called the distance kernel operator, and demonstrate that it gives rise to stable and quasi-injective topological transforms. We conclude with some numerical experiments, wherein we compute and compare the eigenfunctions and eigenvalues of our operator across a range of standard 2- and 3-manifolds.

Clément Maria, Steve Oudot, and Elchanan Solomon. Intrinsic Topological Transforms via the Distance Kernel Embedding. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{maria_et_al:LIPIcs.SoCG.2020.56, author = {Maria, Cl\'{e}ment and Oudot, Steve and Solomon, Elchanan}, title = {{Intrinsic Topological Transforms via the Distance Kernel Embedding}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {56:1--56:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.56}, URN = {urn:nbn:de:0030-drops-122145}, doi = {10.4230/LIPIcs.SoCG.2020.56}, annote = {Keywords: Topological Transforms, Persistent Homology, Inverse Problems, Spectral Geometry, Algebraic Topology, Topological Data Analysis} }