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The VC Dimension of Metric Balls Under Fréchet and Hausdorff Distances

Authors Anne Driemel, Jeff M. Phillips, Ioannis Psarros

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Author Details

Anne Driemel
  • University of Bonn, Germany
Jeff M. Phillips
  • University of Utah, Salt Lake City, USA
Ioannis Psarros
  • National & Kapodistrian University of Athens, Greece

Funding

  • Driemel, Anne: Anne Driemel thanks the Hausdorff Center for Mathematics for their generous support and the Netherlands Organization for Scientific Research (NWO) for support under Veni Grant 10019853.
  • Phillips, Jeff M.: Jeff Phillips thanks his support from NSF CCF-1350888, ACI-1443046, CNS-1514520, CNS-1564287, and IIS-1816149. Part of the work was completed while visiting the Simons Institute for Theory of Computing.
  • Psarros, Ioannis: This research is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme << Human Resources Development, Education and Lifelong Learning >> in the context of the project "Strengthening Human Resources Research Potential via Doctorate Research" (MIS-5000432), implemented by the State Scholarships Foundation (IKY).

Acknowledgements

We thank Peyman Afshani for useful discussions on the topic of this paper. We also thank the organizers of the 2016 NII Shonan Meeting "Theory and Applications of Geometric Optimization" where this research was initiated.

Cite AsGet BibTex

Anne Driemel, Jeff M. Phillips, and Ioannis Psarros. The VC Dimension of Metric Balls Under Fréchet and Hausdorff Distances. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.28

Abstract

The Vapnik-Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in R^d and the sets {R} are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Computational geometry
Keywords
  • VC dimension
  • Fréchet distance
  • Hausdorff distance

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References

  1. Peyman Afshani and Anne Driemel. On the complexity of range searching among curves. CoRR, arXiv:1707.04789v1, 2017. URL: http://arxiv.org/abs/1707.04789.
  2. Peyman Afshani and Anne Driemel. On the complexity of range searching among curves. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 898-917, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.58.
  3. S. Agarwal, B. Mozafari, A. Panda, H. Milner, S. Madden, and I. Stoica. BlinkDB: queries with bounded errors and bounded response times on very large data. In EuroSys, 1993. Google Scholar
  4. Yohji Akama, Kei Irie, Akitoshi Kawamura, and Yasutaka Uwano. VC Dimension of Principal Component Analysis. Discrete &Computational Geometry, 44:589-598, 2010. Google Scholar
  5. Helmut Alt, Bernd Behrends, and Johannes Blömer. Approximate matching of polygonal shapes. Annals of Mathematics and Artificial Intelligence, 13(3):251-265, September 1995. URL: http://dx.doi.org/10.1007/BF01530830.
  6. Martin Anthony and Peter L. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, 1999. Google Scholar
  7. Maria Astefanoaei, Paul Cesaretti, Panagiota Katsikouli, Mayank Goswami, and Rik Sarkar. Multi-resolution sketches and locality sensitive hashing for fast trajectory processing. In International Conference on Advances in Geographic Information Systems (SIGSPATIAL 2018), volume 10, 2018. Google Scholar
  8. Julian Baldus and Karl Bringmann. A Fast Implementation of Near Neighbors Queries for FréChet Distance (GIS Cup). In Proceedings of the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, SIGSPATIAL'17, pages 99:1-99:4, 2017. Google Scholar
  9. Anselm Blumer, A. Ehrenfeucht, David Haussler, and Manfred K. Warmuth. Learnability and the Vapnik-Chervonenkis Dimension. Journal of the ACM, 36:929-965, 1989. Google Scholar
  10. Hervé Brönnimann and Michael T. Goodrich. Almost Optimal Set Covers in Finite VC-Dimension. Discrete &Computational Geometry, 1995. Google Scholar
  11. Kevin Buchin, Yago Diez, Tom van Diggelen, and Wouter Meulemans. Efficient trajectory queries under the Fréchet distance (GIS Cup). In Proc. 25th Int. Conference on Advances in Geographic Information Systems (SIGSPATIAL), pages 101:1-101:4, 2017. Google Scholar
  12. Bernard Chazelle and Emo Welzl. Quasi-Optimal Range Searching in Spaces of Finite VC-Dimension. Discrete and Computational Geometry, 4:467-489, 1989. Google Scholar
  13. Monika Csikos, Andrey Kupavskii, and Nabil H. Mustafa. Optimal Bounds on the VC-dimension. arXiv:1807.07924, 2018. URL: http://arxiv.org/abs/1807.07924.
  14. Mark De Berg, Atlas F Cook, and Joachim Gudmundsson. Fast Fréchet queries. Computational Geometry, 46(6):747-755, 2013. Google Scholar
  15. Mark de Berg and Ali D. Mehrabi. Straight-Path Queries in Trajectory Data. In WALCOM: Algorithms and Computation - 9th Int. Workshop, WALCOM 2015, Dhaka, Bangladesh, February 26-28, 2015. Proceedings, pages 101-112, 2015. URL: http://dx.doi.org/10.1007/978-3-319-15612-5_10.
  16. Anne Driemel, Amer Krivošija, and Christian Sohler. Clustering time series under the Fréchet distance. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 766-785, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch55.
  17. Anne Driemel, Jeff M. Phillips, and Ioannis Psarros. The VC dimension of metric balls under Fréchet and Hausdorff distances. CoRR, arXiv:1903.03211, 2019. URL: http://arxiv.org/abs/1903.03211.
  18. Anne Driemel and Francesco Silvestri. Locally-sensitive hashing of curves. In 33st International Symposium on Computational Geometry, SoCG 2017, pages 37:1-37:16, 2017. Google Scholar
  19. Fabian Dütsch and Jan Vahrenhold. A Filter-and-Refinement- Algorithm for Range Queries Based on the Fréchet Distance (GIS Cup). In Proc. 25th Int. Conference on Advances in Geographic Information Systems (SIGSPATIAL), pages 100:1-100:4, 2017. Google Scholar
  20. Ioannis Z. Emiris and Ioannis Psarros. Products of Euclidean Metrics and Applications to Proximity Questions among Curves. In Proc. 34th Int. Symposium on Computational Geometry (SoCG), volume 99 of LIPIcs, pages 37:1-37:13, 2018. Google Scholar
  21. Alexander Gilbers and Rolf Klein. A new upper bound for the VC-dimension of visibility regions. Computational Geometry: Theory and Applications, 74:61-74, 2014. Google Scholar
  22. Paul W. Goldberg and Mark R. Jerrum. Bounding the Vapnik-Chervonenkis Dimension of Concept Classes Parameterized by Real Numbers. Machine Learning, 18:131-148, 1995. Google Scholar
  23. Joachim Gudmundsson and Michiel Smid. Fast algorithms for approximate Fréchet matching queries in geometric trees. Computational Geometry, 48(6):479-494, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2015.02.003.
  24. Sariel Har-Peled. Geometric Approximation Algorithms. American Mathematical Society, Boston, MA, USA, 2011. Google Scholar
  25. Lingxiao Huang, Shaofeng Jiang, Jian Li, and Xuan Wu. Epsilon-Coresets for Clustering (with Outliers) in Doubling Metrics. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 814-825. IEEE, 2018. Google Scholar
  26. Sarang Joshi, Raj Varma Kommaraju, Jeff M. Phillips, and Suresh Venkatasubramanian. Comparing Distributions and Shapes Using the Kernel Distance. In ACM SoCG, 2011. Google Scholar
  27. Marek Karpinski and Angus Macintyre. Polynomial bounds for VC dimension of sigmoidal neural networks. In STOC, 1995. Google Scholar
  28. Elmar Langetepe and Simone Lehmann. Exact VC-dimension for L1-visibility of points in simple polygons. arXiv:1705.01723, 2017. URL: http://arxiv.org/abs/1705.01723.
  29. Frank Olken. Random Sampling in Databases. PhD thesis, University of California at Berkeley, 1993. Google Scholar
  30. Norbert Sauer. On the Density of Families of Sets. Journal of Combinatorial Theory Series A, 13:145-147, 1972. Google Scholar
  31. Saharon Shelah. A Combinatorial Problem; Stability and Order for Models and Theories in Infinitary Languages. Pacific Journal of Mathematics, 41(1), 1972. Google Scholar
  32. Pavel Valtr. Guarding Galleries Where No Point Sees a Small Area. Israel Journal of Mathematics, 104:1-16, 1998. Google Scholar
  33. Vladimir Vapnik and Alexey Chervonenkis. On the Uniform Convergence of Relative Frequencies of Events to their Probabilities. Theory of Probability and its Applications, 16:264-280, 1971. Google Scholar
  34. Vladimir N. Vapnik. Statistical Learning Theory. John Wiley &Sons, 1998. Google Scholar
  35. Martin Werner and Dev Oliver. ACM SIGSPATIAL GIS Cup 2017: Range queries under Fréchet distance. SIGSPATIAL Special, 10(1):24-27, June 2018. URL: http://dx.doi.org/10.1145/3231541.3231549.
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