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Minimal unimodal decomposition on trees

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Abstract

The decomposition of a density function on a domain into a minimal sum of unimodal components is a fundamental problem in statistics, leading to the topological invariant of unimodal category of a density. This paper gives an efficient algorithm for the construction of a minimal unimodal decomposition of a tame density function on a finite metric tree.

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Notes

  1. That is, using the language of this paper.

References

  • Baryshnikov, Y., Ghrist, R.: Unimodal category and topological statistics. In: Proceedings on NOLTA (2011)

  • Behboodian, J.: On the modes of a mixture of two normal distributions. Technometrics 12, 131–139 (1970)

    Article  Google Scholar 

  • Carlsson, G., Mémoli, F.: Classifying clustering schemes. Found. Comput. Math. 13(2), 221–252 (2013)

    Article  MathSciNet  Google Scholar 

  • Carreira-Perpinan, M., Williams, C.: On the number of modes of a Gaussian mixture. In: Griffin, L.D., Lillholm, M. (eds.) Scale-Space Methods in Computer Vision. Lecture Notes in Computer Science, vol. 2695, pp. 625–640. Springer, Berlin (2003)

    Chapter  Google Scholar 

  • Cornea, O., Lupton, G., Oprea, J., Tanré, D.: Lusternik–Schnirelmann category. Am. Math. Soc. (2003)

  • Eisenberger, I.: Genesis of bimodal distributions. Technometrics 6, 357–363 (1964)

    Article  MathSciNet  Google Scholar 

  • Farber, M.: Topological complexity of motion planning. Discrete Comput. Geom. 29(2), 211–221 (2003)

    Article  MathSciNet  Google Scholar 

  • Ghrist, R.: Elementary Applied Topology. Createspace, Scotts Valley (2014)

    MATH  Google Scholar 

  • Govc, D.: Unimodal category and the monotonicity conjecture. arXiv:1709.06547 [math.AT] (2017)

  • Huntsman, S.: Topological mixture estimation. Proc. Mach. Learn. Res. 80, 2088–2097 (2018)

    Google Scholar 

  • Kakiuchi, I.: Unimodality conditions of the distribution of a mixture of two distributions. Math. Semin. Notes Kobe Univ. 9, 315–325 (1981)

    MathSciNet  MATH  Google Scholar 

  • Kemperman, J.: Mixtures with a limited number of modal intervals. Ann. Stat. 19, 2120–2144 (1991)

    Article  MathSciNet  Google Scholar 

  • Kleinberg, J.: An impossibility theorem for clustering. In: Proceedings on NIPS, pp. 446–453 (2002)

  • Robertson, C., Fryer, J.: Some descriptive properties of normal mixtures. Skand. Aktuarietidskr. 1969, 137–146 (1969)

    MathSciNet  MATH  Google Scholar 

  • Singh, G., Mémoli, F., Carlsson, G.: Topological methods for the analysis of high dimensional data sets and 3d object recognition. In: SPBG, pp. 91–100 (2007)

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Acknowledgements

This research was done while YB was visiting the Departments of Mathematics and ESE of the University of Pennsylvania—the hospitality of both departments is warmly appreciated.

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Correspondence to Yuliy Baryshnikov.

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Work supported by the National Science Foundation via DMS-1622370. Work supported by the Office of the Assistant Secretary of Defense Research and Engineering through ONR N00014-16-1-2010.

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Baryshnikov, Y., Ghrist, R. Minimal unimodal decomposition on trees. J Appl. and Comput. Topology 4, 199–209 (2020). https://doi.org/10.1007/s41468-019-00046-7

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  • DOI: https://doi.org/10.1007/s41468-019-00046-7

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