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res:
bibo_abstract:
- We introduce the adaptive neighborhood graph as a data structure for modeling
a smooth manifold M embedded in some Euclidean space ℝ d. We assume that M is
known to us only through a finite sample P ⊂ M, as is often the case in applications.
The adaptive neighborhood graph is a geometric graph on P. Its complexity is at
most min{2O(k)n, n2}, where n = P and k = dim M, as opposed to the n[d/2] complexity
of the Delaunay triangulation, which is often used to model manifolds. We prove
that we can correctly infer the connected components and the dimension of M from
the adaptive neighborhood graph provided a certain standard sampling condition
is fulfilled. The running time of the dimension detection algorithm is d20(k7
log k) for each connected component of M. If the dimension is considered constant,
this is a constant-time operation, and the adaptive neighborhood graph is of linear
size. Moreover, the exponential dependence of the constants is only on the intrinsic
dimension k, not on the ambient dimension d. This is of particular interest if
the co-dimension is high, i.e., if k is much smaller than d, as is the case in
many applications. The adaptive neighborhood graph also allows us to approximate
the geodesic distances between the points in P.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: Joachim
foaf_name: Giesen, Joachim
foaf_surname: Giesen
- foaf_Person:
foaf_givenName: Uli
foaf_name: Uli Wagner
foaf_surname: Wagner
foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1494-0568
bibo_doi: 10.1007/s00454-004-1120-8
bibo_issue: '2'
bibo_volume: 32
dct_date: 2004^xs_gYear
dct_publisher: Springer@
dct_title: Shape dimension and intrinsic metric from samples of manifolds@
...