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research
Finite Volume Spaces and Sparsification
Authors
Ilan Newman
Yuri Rabinovich
Publication date
1 January 2010
Publisher
View
on
arXiv
Abstract
We introduce and study finite
d
d
d
-volumes - the high dimensional generalization of finite metric spaces. Having developed a suitable combinatorial machinery, we define
β
1
\ell_1
β
1
β
-volumes and show that they contain Euclidean volumes and hypertree volumes. We show that they can approximate any
d
d
d
-volume with
O
(
n
d
)
O(n^d)
O
(
n
d
)
multiplicative distortion. On the other hand, contrary to Bourgain's theorem for
d
=
1
d=1
d
=
1
, there exists a
2
2
2
-volume that on
n
n
n
vertices that cannot be approximated by any
β
1
\ell_1
β
1
β
-volume with distortion smaller than
Ξ©
~
(
n
1
/
5
)
\tilde{\Omega}(n^{1/5})
Ξ©
~
(
n
1/5
)
. We further address the problem of
β
1
\ell_1
β
1
β
-dimension reduction in the context of
β
1
\ell_1
β
1
β
volumes, and show that this phenomenon does occur, although not to the same striking degree as it does for Euclidean metrics and volumes. In particular, we show that any
β
1
\ell_1
β
1
β
metric on
n
n
n
points can be
(
1
+
Ο΅
)
(1+ \epsilon)
(
1
+
Ο΅
)
-approximated by a sum of
O
(
n
/
Ο΅
2
)
O(n/\epsilon^2)
O
(
n
/
Ο΅
2
)
cut metrics, improving over the best previously known bound of
O
(
n
log
β‘
n
)
O(n \log n)
O
(
n
lo
g
n
)
due to Schechtman. In order to deal with dimension reduction, we extend the techniques and ideas introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of graph Sparsification, and develop general methods with a wide range of applications.Comment: previous revision was the wrong file: the new revision: changed (extended considerably) the treatment of finite volumes (see revised abstract). Inserted new applications for the sparsification technique
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Last time updated on 30/10/2017