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High-Dimensional Menger-Type Curvatures—Part II: d-Separation and a Menagerie of Curvatures

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Abstract

We estimate d-dimensional least squares approximations of an arbitrary d-regular measure μ via discrete curvatures of d+2 variables. The main result bounds the least squares error of approximating μ (or its restrictions to balls) with a d-plane by an average of the discrete Menger-type curvature over a restricted set of simplices. Its proof is constructive and even suggests an algorithm for an approximate least squares d-plane. A consequent result bounds a multiscale error term (used for quantifying the approximation of μ with a sufficiently regular surface) by an integral of the discrete Menger-type curvature over all simplices. The preceding paper (part I) provided the opposite inequalities of these two results. This paper also demonstrates the use of a few other discrete curvatures which are different from the Menger-type curvature. Furthermore, it shows that a curvature suggested by Léger (Ann. Math. 149(3), pp. 831–869, 1999) does not fit within our framework.

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Authors and Affiliations

  1. 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN, 55455, USA

    Gilad Lerman

  2. 1326 Stevenson Center, Vanderbilt University, Nashville, TN, 37240, USA

    J. Tyler Whitehouse

Authors
  1. Gilad Lerman
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  2. J. Tyler Whitehouse
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Correspondence to Gilad Lerman.

Additional information

Communicated by Mauro Maggioni.

This work has been supported by NSF grant #0612608

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Lerman, G., Whitehouse, J.T. High-Dimensional Menger-Type Curvatures—Part II: d-Separation and a Menagerie of Curvatures. Constr Approx 30, 325–360 (2009). https://doi.org/10.1007/s00365-009-9073-z

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  • DOI: https://doi.org/10.1007/s00365-009-9073-z

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