A kd-tree algorithm to discover the boundary of a black box hypervolume

Or how to peel potatoes by recursively cutting them in halves
  • Jean-Baptiste Rouquier
  • Isabelle Alvarez
  • Romain Reuillon
  • Pierre-Henri Wuillemin
Article

Abstract

In the framework of Decision Support Systems, mathematical viability theory can be used to classify the states and the trajectories of a dynamical system evolving in a set of desirable states. Since obtaining this viability theory output is a complex and computationally intensive task, we propose in this article to consider a compact representation of this set and its approximations using kd-trees. Given a subset of \(\mathbb {R}^{n}\) of non null measure, defined through a black box function (an oracle), and assuming some regularity properties on this set, we build a kd-tree based data structure representing this set, which can be used as an input to the viability algorithm. This data structure has a complexity close to gaining one dimension, both in terms of space and in number of calls to the oracle, compared to the exhaustive computation on a regular grid. This data structure produces a characteristic function (i.e. a function that can be used in lieu of the oracle), allows to measure the volume of the set, and to compute the distance to the boundary of the set for any point. It offers distance guarantee between the points of the original set and its kd-tree approximation.

Keywords

Kd-tree Approximation of hyper-volume Viability Decision support system 

Mathematics Subject Classifications (2010)

68T05 68Q32 68W25 68W40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alvarez, I., de Aldama, R., Martin, S., Reuillon, R.: Assessing the Resilience of Socio-Ecosystems: Coupling Viability Theory and Active Learning with kd-Trees. Application to Bilingual Societies. In: proceedings of IJCAI, pp 2776–2782 (2013). http://ijcai.org/papers13/Papers/IJCAI13-409.pdf
  2. 2.
    Aubin, J.P., Bayen, A., Saint-Pierre, P.: Viability Theory: New Directions. Springer (2011). doi:10.1007/978-3-642-16684-6. http://hal.inria.fr/inria-00636570
  3. 3.
    Bentley, J.L.: Multidimensional binary search trees used for associative searching. Commun. ACM 18(9), 509–517 (1975). doi:10.1145/361002.361007 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bentley, J.L.: Multidimensional binary search trees in database applications. IEEE Trans. Softw. Eng. SE-5(4), 333–340 (1979). doi:10.1109/TSE.1979.234200 CrossRefGoogle Scholar
  5. 5.
    Coquelin, P.A., Martin, S., Munos, R.: A dynamic programming approach to viability problems. In: IEEE International Symposium on Approximate Dynamic Programming and Reinforcement Learning, 2007. ADPRL 2007, pp 178–184 (2007)Google Scholar
  6. 6.
    Deffuant, G., Chapel, L., Martin, S.: Approximating Viability Kernels With Support Vector Machines. IEEE Trans. Autom. Control 52(5), 933–937 (2007). doi:10.1109/TAC.2007.895881 [ http://hal.archives-ouvertes.fr/hal-00616841]CrossRefMathSciNetGoogle Scholar
  7. 7.
    Jain, A.K.: Data clustering: 50 years beyond k-means. Pattern Recogn. Lett. 31 (8), 651–666 (2010). doi:10.1016/j.patrec.2009.09.011 CrossRefGoogle Scholar
  8. 8.
    Lindenbaum, M., Markovitch, S., Rusakov, D.: Selective sampling for nearest neighbor classifiers. Mach. Learn. 54(2), 125–152 (2004)MATHCrossRefGoogle Scholar
  9. 9.
    Moore, A.W.: An intoductory tutorial on kd-trees (1991). http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.10.818
  10. 10.
    Rosenthal, P., Linsen, L.: Direct isosurface extraction from scattered volume data. In: Ertl, T., Joy, K., Santos, B. (eds.) EUROVIS - Eurographics /IEEE VGTC Symposium on Visualization, Eurographics Association, pp 99–106 (2006), 10.2312/VisSym/EuroVis06/099-106. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.89.2313
  11. 11.
    Saint-Pierre, P.: Approximation of the viability kernel. Appl. Math. Optim. 29, 187–209 (1994)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Settles, B.: Active learning literature survey. Computer Sciences Technical Report 1648. University of Wisconsin–Madison (2009)Google Scholar
  13. 13.
    Sicard, M., Perrot, N., Reuillon, R., Mesmoudi, S., Alvarez, I., Martin, S.: A viability approach to control food processes: Application to a camembert cheese ripening process. Food Control 23(2), 312–319 (2012)CrossRefGoogle Scholar
  14. 14.
  15. 15.
    Wald, I., Mark, W.R., Günther, J., Boulos, S., Ize, T., Hunt, W., Parker, S.G., Shirley, P.: State of the art in ray tracing animated scenes. Comput. Graphics Forum 28(6), 1691–1722 (2009). doi:10.1111/j.1467-8659.2008.01313.x

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jean-Baptiste Rouquier
    • 1
    • 2
  • Isabelle Alvarez
    • 3
    • 4
  • Romain Reuillon
    • 1
  • Pierre-Henri Wuillemin
    • 4
    • 5
  1. 1.Institut des Systèmes ComplexesParisFrance
  2. 2.Dataiku.comParisFrance
  3. 3.Irstea, LISCAubièreFrance
  4. 4.Sorbonne Universites, UPMC, Univ Paris 06, UMR 7606, LIP6ParisFrance
  5. 5.CNRSParisFrance

Personalised recommendations