skip to main content
10.1145/1998196.1998205acmconferencesArticle/Chapter ViewAbstractPublication PagessocgConference Proceedingsconference-collections
research-article

Witnessed k-distance

Published: 13 June 2011 Publication History

Abstract

Distance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary power distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation.

References

[1]
N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete and Computational Geometry, 22(4):481--504, 1999.
[2]
S. Arya and D. Mount. Computational geometry: proximity and location. Handbook of Data Structures and Applications, pages 63.1--63.22, 2005.
[3]
F. Bolley, A. Guillin, and C. Villani. Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces. Probability Theory and Related Fields, 137(3):541--593, 2007.
[4]
F. Chazal, D. Cohen-Steiner, and A. Lieutier. A sampling theory for compact sets in Euclidean space. Discrete and Computational Geometry, 41(3):461--479, 2009.
[5]
F. Chazal, D. Cohen-Steiner, and Q. Mérigot. Geometric inference for probability measures. Preprint (INRIA RR-6930 v2), 2010.
[6]
F. Chazal and S. Oudot. Towards persistence-based reconstruction in Euclidean spaces. Proceedings of the ACM Symposium on Computational Geometry, pages 232--241, 2008.
[7]
K. Clarkson. Nearest-neighbor searching and metric space dimensions. Nearest-Neighbor Methods for Learning and Vision: Theory and Practice, pages 15--59, 2006.
[8]
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete and Computational Geometry, 37(1):103--120, 2007.
[9]
D. Cohen-Steiner, H. Edelsbrunner, and D. Morozov. Vines and vineyards by updating persistence in linear time. In Proceedings of the ACM Symposium on Computational Geometry, pages 119--126, 2006.
[10]
S. Dasgupta. Learning mixtures of Gaussians. In Proceedings of the IEEE Symposium on Foundations of Computer Science, page 634, 1999.
[11]
T. Dey and S. Goswami. Provable surface reconstruction from noisy samples. Computational Geometry, 35(1--2):124--141, 2006.
[12]
H. Edelsbrunner. The union of balls and its dual shape. Discrete and Computational Geometry, 13:415--440, 1995.
[13]
H. Edelsbrunner and J. Harer. Persistent homology -- a survey. Surveys on Discrete and Computational Geometry. Twenty Years Later, pages 257--282, 2008.
[14]
P. Indyk. Nearest neighbors in high-dimensional spaces. Handbook of Discrete and Computational Geometry, pages 877--892, 2004.
[15]
B. Kloeckner. Approximation by finitely supported measures. Preprint (arXiv:1003.1035), 2010.
[16]
P. Niyogi, S. Smale, and S. Weinberger. A topological view of unsupervised learning from noisy data. Preprint, 2008.
[17]
P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete and Computational Geometry, 39(1):419--441, 2008.
[18]
C. Villani. Topics in Optimal Transportation. American Mathematical Society, 2003.

Cited By

View all
  • (2012)Down the Rabbit HoleProceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science10.1109/FOCS.2012.31(430-439)Online publication date: 20-Oct-2012

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences
SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry
June 2011
532 pages
ISBN:9781450306829
DOI:10.1145/1998196
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 13 June 2011

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. computational topology
  2. geometric inference
  3. power distance

Qualifiers

  • Research-article

Conference

SoCG '11
SoCG '11: Symposium on Computational Geometry
June 13 - 15, 2011
Paris, France

Acceptance Rates

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)10
  • Downloads (Last 6 weeks)0
Reflects downloads up to 22 Feb 2025

Other Metrics

Citations

Cited By

View all
  • (2012)Down the Rabbit HoleProceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science10.1109/FOCS.2012.31(430-439)Online publication date: 20-Oct-2012

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Figures

Tables

Media

Share

Share

Share this Publication link

Share on social media