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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Geometric Whitney problem: reconstruction of a man
ifold from a point cloud - Matti Lassas\, Helsinki
DTSTART;TZID=Europe/London:20160217T160000
DTEND;TZID=Europe/London:20160217T170000
UID:TALK61646AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/61646
DESCRIPTION:We study the geometric Whitney problem on how a Ri
emannian manifold (M\,g) can be constructed to app
roximate a metric space (X\,d_X). This problem i
s closely related to manifold interpolation (or m
anifold learning) where a smooth n-dimensional sur
face S in Euclidean m-space\, m>n\, needs to be co
nstructed to approximate a point cloud in m-space.
These questions are encountered in differential
geometry\, machine learning\, and in many inverse
problems encountered in applications. The determin
ation of a Riemannian manifold includes the constr
uction of its topology\, differentiable structure\
, and metric. \n\nWe give constructive solutions t
o the above problems. Moreover\, we characterize t
he metric spaces that can be approximated\, by Rie
mannian manifolds with bounded geometry: we give s
ufficient conditions to ensure that a metric space
can be approximated\, in the Gromov-Hausdorff or
quasi-isometric sense\, by a Riemannian manifold
of a fixed dimension and with bounded diameter\, s
ectional curvature\, and injectivity radius. Also\
, we show that similar conditions\, with modified
values of parameters\, are necessary.\n\nMoreover\
, we characterise the subsets of Euclidean spaces
that can be approximated in the Hausdorff metric b
y submanifolds of a fixed dimension\nand with boun
ded principal curvatures and normal injectivity ra
dius.\n\nThe above interpolation problems are al
so studied for unbounded metric sets and manifolds
. The results for Riemannian manifolds are based o
n a generalisation of the Whitney embedding constr
uction where approximative coordinate charts are e
mbedded in Euclidean m-space and interpolated to a
smooth surface.\nWe also give algorithms that sol
ve the problems for finite data.\n
LOCATION:MR13
CONTACT:Ivan Smith
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