1 heat flow and a faster algorithm to compute the surface area of a convex body hariharan narayanan,...
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Heat flow and a faster Algorithm to Compute the Surface Area of a
Convex BodyHariharan Narayanan, University of Chicago
Joint work with
Mikhail Belkin, Ohio state University
Partha Niyogi, University of Chicago
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Computing the Surface Area of a Convex Body
• Open problem (Grötschel, Lovász, Schrijver [GLS90].)
• In randomized polynomial time (Dyer, Gritzmann, Hufnagel [DGH98].)
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Clustering and Surface Area of Cuts
• Semi-supervised Classification - Labelled and unlabelled data
• Low Density Separation (Chapelle, Zien [CZ05].)
is a measureof the quality
of the cut ( is theprob. densityand is the surface area measure on the cut)
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Prior work on Computing the Volume of Convex bodies
n dimension, c fixed constant
• Volume cannot be approximated in deterministic poly time within
(Bárány, Fϋredi [BF88] )• Volume can be approximated in randomized
poly time within (Dyer, Freize, Kannan [DFK89].)
• Numerous improvements in complexity - Best known is ( Lovász, Vempala [LV04].)
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The ModelGiven:
• Membership oracle for convex body K.• The radius r and centre O of a ball contained in K.• Radius R of a ball with centre O containing K.
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Complexity of Computing the Surface Area
At least as hard as Volume:
Let Then the surface area of C(K) is an approximation of twice the volume of
K.
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Computing the Surface Area of a Convex Body
Previous approach : Choose appropriate• Consider the convex body , its -
neighbourhood and their difference.
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Computing the Surface Area of a Convex Body
Previous approach:
• Compute Surface area by interpolation
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Computing the Surface Area of a Convex Body
Previous approach involves computing the Volume of ; cost appears to be=
given membership oracle for (with present Technology) :
• Answeringeach oracle query to takes time .• Computing volume takes time.
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Heat Flow
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t = 0
t = 0.05
t = 0.025
t = 0.075
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• Heat diffusing out of in time
Terminology
Motivation
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Fact
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• For small , Surface Area
Terminology• Heat diffusing out of in time
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• Heat diffusing out in time
Fact• For small , Surface Area
Terminology
Algorithm
• Choose random points in
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Algorithm• Choose random pointsin
•Perturb each by a random vector from a multivariate Gaussian
•Set fraction of perturbed points landing outside
•Obtain estimate of the Volume.
•Output
as the estimate for Surface Area.
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Choice of t:
• Find radius of a ball in , large in the following sense – For chosen uniformly at random from
for some unit vector
• Set
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Finding :
for some unit vector
T
2-isotropic :For all unit vectors
Set smallest
eigenvalue of
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• If samples were generated uniformly at random, Output {Heat Flow}
Algorithm’s relation to Heat Flow
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• Complexity of rounding the body (and finding ) -
• Complexity of estimating volume –
• Complexity of generatingrandom points -
Algorithm’s Complexity
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Given membership oracle and sufficiently many random samples from the body, “fraction escaping”
Cheeger ratio for smooth non-convex bodies
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Analysis: Upper bound on
Terminology:
Heat flow
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Analysis: Upper bound on
Terminology:
Heat flow :
Let
Then,
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Analysis: Upper bound on
Terminology:
Heat flow :
0.5 1 1.5 2
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Plot of for t = 1/4
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Analysis: Upper bound on
Terminology: S = Surface Area, V = Volume
Heat flow :
The “Alexandrov-Fenchel inequalities”imply that
which leads to ,
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Analysis: Lower bound on
Terminology:
Heat flow
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Analysis: Lower bound on
Terminology:
Heat flow :
Let
Then,
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Analysis: Lower bound on
Terminology:
Heat flow :
0.5 1 1.5 2
0.1
0.2
0.3
0.4
0.5
Plot of for t = 1/4
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Analysis: Lower bound on
Terminology:
Heat flow :
For the upper bound we had
?
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Analysis: Lower bound on
Proof:
Surface Area is monotonic,that is,
Lemma:
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Analysis: Lower bound on
Terminology:
Heat flow :
implies that
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Other Considerations:
• We have the upper bound ; Need to upper bound by .
• The fraction of perturbed points that fall outsidehas Expectation ; Need to lower bound by to ensure that is close to its expectation(since we are using random samples.)
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Other Considerations:
• Need to upper bound by • We show
• Need to lower bound by• We show
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Upper bound for :
• We show
Infinitesimally ,
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Lower bound for :
• We show Prove that
Method : Consider
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