Heat flow and a faster Algorithm to Compute the Surface Area of a Convex Body

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Heat flow and a faster Algorithm to Compute the
Surface Area of a Convex Body
Hariharan 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 measure of the quality of the cut ( is
the prob. density and 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, F?redi 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 Model

• Given
• 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)
• Answering
• each oracle query to
• takes time .
• Computing volume takes
• time.

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Heat Flow
t 0
t 0.025

t 0.05
t 0.075
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Terminology

• Heat diffusing out of in
  time

Motivation
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Terminology

• Heat diffusing out of in
  time

Fact

• For small , Surface Area

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Terminology

• Heat diffusing out in time

Fact

• For small , Surface Area

Algorithm

• Choose
• random points in

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Algorithm

• Choose random points
• in
• 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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Algorithms relation to Heat Flow

• If samples were generated uniformly at random,
• Output Heat Flow

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Algorithms Complexity

• Complexity of rounding the body (and finding )
  -
• Complexity of estimating volume
• Complexity of generating
• random points -

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Cheeger ratio for smooth non-convex bodies
Given membership oracle and sufficiently
many random samples from the body, fraction
escaping

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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

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 inequalitiesimply
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

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

Lemma

Proof Surface Area is monotonic, that is,
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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
  outside
• has 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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Thank you !