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DOI: 10.4230/LIPIcs.SoCG.2018.31

Graph Reconstruction by Discrete Morse Theory

Authors: Tamal K. Dey, Jiayuan Wang, and Yusu Wang

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract

Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.

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Tamal K. Dey, Jiayuan Wang, and Yusu Wang. Graph Reconstruction by Discrete Morse Theory. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2018.31,
  author =	{Dey, Tamal K. and Wang, Jiayuan and Wang, Yusu},
  title =	{{Graph Reconstruction by Discrete Morse Theory}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{31:1--31:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.31},
  URN =		{urn:nbn:de:0030-drops-87443},
  doi =		{10.4230/LIPIcs.SoCG.2018.31},
  annote =	{Keywords: graph reconstruction, discrete Morse theory, persistence}
}

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DOI: 10.4230/LIPIcs.SoCG.2017.23

Declutter and Resample: Towards Parameter Free Denoising

Authors: Mickael Buchet, Tamal K. Dey, Jiayuan Wang, and Yusu Wang

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract

In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases. Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms are effective in practice.

Cite as

Mickael Buchet, Tamal K. Dey, Jiayuan Wang, and Yusu Wang. Declutter and Resample: Towards Parameter Free Denoising. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buchet_et_al:LIPIcs.SoCG.2017.23,
  author =	{Buchet, Mickael and Dey, Tamal K. and Wang, Jiayuan and Wang, Yusu},
  title =	{{Declutter and Resample: Towards Parameter Free Denoising}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.23},
  URN =		{urn:nbn:de:0030-drops-72133},
  doi =		{10.4230/LIPIcs.SoCG.2017.23},
  annote =	{Keywords: denoising, parameter free, k-distance,compact sets}
}

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Documents authored by Wang, Jiayuan

Graph Reconstruction by Discrete Morse Theory

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Declutter and Resample: Towards Parameter Free Denoising

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