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Quantifying curvelike structures of measures by using \(L _{2}\) Jones quantities. (English) Zbl 1076.28005

This interesting paper applies geometric measure theory to analysis of data in a non-trivial way. It is motivated by the following general question: given a data set \(X\) in the Euclidean space \({\mathbb R}^n\), can it be approximated by a single manifold with sufficiently good smoothness properties, or else by a collection of such manifolds? One of the pioneers of this approach was P. Jones [Invent. Math. 102, No. 1, 1–15 (1990; Zbl 0731.30018)].
The setting in the paper under review is as follows: the distribution of data points is modelled by a locally finite Borel measure \(\mu\) in \({\mathbb R}^n\), which is being approximated by a rectifiable (sometimes in a weakened sense) curve \(\Gamma\) with regard to a suitable distance between \(\mu\) and the arc-length measure on \(\Gamma\). The measures \(\mu\) belong to a special class, and the approximation errors by lines can be quantitatively measured at various scales using the so-called \(\beta_2\) numbers, which are analogues of \(\beta_\infty\) numbers introduced by Jones. The main result of the paper (Th. 4.10) gives upper bounds on approximation errors of a given measure \(\mu\), expressed in terms of a function, \(J(x)\), associated to \(\mu\). The technique used is multiscale analysis and grids.
Reviewer’s remark: Another significant contribution to approximating data sets with manifolds has been made by Gorban’ and his school in Krasnoyarsk, Russia, who have developed the so-called “elastic maps” method, see e.g. [A. N. Gorban, A. Y. Zinovyev and D. C. Wunsch, “Application of the method of elastic maps in analysis of genetic texts”, Proceedings of IJCNN2003, available from http://mystic.math.neu.edu/gorban/GZW2003.pdf].

MSC:

28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
68P20 Information storage and retrieval of data
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
30C85 Capacity and harmonic measure in the complex plane

Citations:

Zbl 0731.30018
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References:

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