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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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Correspondence to Gilad Lerman.

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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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