High-Dimensional Menger-Type Curvatures—Part II: d-Separation and a Menagerie of Curvatures
By Gilad Lerman, J. Tyler Whitehouse. 2008
50 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/aab41a651d47a31f4be4d4a86f4bfb2538ff28f2
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In Applied Mathematics
- Pawel Strzelecki
- H. von der Mosel
- Gilad Lerman
- Guangliang Chen
- J. Tyler Whitehouse
- Michael Struwe
- Changyou Wang
- Armin Schikorra
In Numerical Analysis
Citations in the corpus
- High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities (2008, methodology): “This paper and [8] show that this Menger-type curvature measures a cumulative flatness or oscillation of μ around the ball B.”
- Regularizing and self-avoidance effects of integral Menger curvature (2010, unknown): “A generalization of total Menger curvature M2(·) to higher-dimensional sets was recently initiated in [19,20].”
- Integral Menger curvature for surfaces (2009, background): “constant! Because of that we do not work with the cMT curvature introduced by Lerman and Whitehouse in [16]: for sufficiently large p, the only surface with finite energy would be a plane.”
- Spectral Curvature Clustering (SCC) (2009, methodology): “It is shown in Lerman and Whitehouse (2008d) (following the methods of Lerman and Whitehouse 2008a, 2008b, 2008c) that under certain conditions the least squares error of approximating certain probability measures μ by d-flats is comparable to the average of c2p (with respect to μ d+2).”