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DOI: 10.4230/LIPIcs.SoCG.2022.61

From Geometry to Topology: Inverse Theorems for Distributed Persistence

Authors: Elchanan Solomon, Alexander Wagner, and Paul Bendich

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

Abstract

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.

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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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DOI: 10.4230/LIPIcs.SoCG.2016.65

Geometric Models for Musical Audio Data

Authors: Paul Bendich, Ellen Gasparovic, John Harer, and Christopher Tralie

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract

We study the geometry of sliding window embeddings of audio features that summarize perceptual information about audio, including its pitch and timbre. These embeddings can be viewed as point clouds in high dimensions, and we add structure to the point clouds using a cover tree with adaptive thresholds based on multi-scale local principal component analysis to automatically assign points to clusters. We connect neighboring clusters in a scaffolding graph, and we use knowledge of stratified space structure to refine our estimates of dimension in each cluster, demonstrating in our music applications that choruses and verses have higher dimensional structure, while transitions between them are lower dimensional. We showcase our technique with an interactive web-based application powered by Javascript and WebGL which plays music synchronized with a principal component analysis embedding of the point cloud down to 3D. We also render the clusters and the scaffolding on top of this projection to visualize the transitions between different sections of the music.

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Paul Bendich, Ellen Gasparovic, John Harer, and Christopher Tralie. Geometric Models for Musical Audio Data. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 65:1-65:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bendich_et_al:LIPIcs.SoCG.2016.65,
  author =	{Bendich, Paul and Gasparovic, Ellen and Harer, John and Tralie, Christopher},
  title =	{{Geometric Models for Musical Audio Data}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{65:1--65:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.65},
  URN =		{urn:nbn:de:0030-drops-59577},
  doi =		{10.4230/LIPIcs.SoCG.2016.65},
  annote =	{Keywords: Geometric Models, Audio Analysis, High Dimensional Data Analysis, Stratified Space Models}
}

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From Geometry to Topology: Inverse Theorems for Distributed Persistence

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Geometric Models for Musical Audio Data

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