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Local angles and dimension estimation from data on manifolds

Author

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  • Díaz, Mateo
  • Quiroz, Adolfo J.
  • Velasco, Mauricio

Abstract

For data living in a manifold M⊆Rm and a point p∈M, we consider a statistic Uk,n which estimates the variance of the angle between pairs (Xi−p,Xj−p) of vectors, for data points Xi, Xj, near p, and we evaluate this statistic as a tool for estimation of the intrinsic dimension of M at p. Consistency of the local dimension estimator is established and the asymptotic distribution of Uk,n is found under minimal regularity assumptions. Performance of the proposed methodology is compared against state-of-the-art methods on simulated data and real datasets.

Suggested Citation

  • Díaz, Mateo & Quiroz, Adolfo J. & Velasco, Mauricio, 2019. "Local angles and dimension estimation from data on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 229-247.
    • Handle: RePEc:eee:jmvana:v:173:y:2019:i:c:p:229-247
    DOI: 10.1016/j.jmva.2019.02.014
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    References listed on IDEAS

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    1. Kaufmann, E. & Reiss, R. -D., 1992. "On conditional distributions of nearest neighbors," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 67-76, July.
    2. Brito, M.R. & Quiroz, A.J. & Yukich, J.E., 2013. "Intrinsic dimension identification via graph-theoretic methods," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 263-277.
    3. Brito, María R. & Quiroz, Adolfo J. & Yukich, J. E., 2002. "Graph-Theoretic Procedures for Dimension Identification," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 67-84, April.
    4. Jonathan M. Borwein & Marc Chamberland, 2007. "Integer Powers of Arcsin," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2007, pages 1-10, May.
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