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

• Result
• An O(n4) randomized algorithm to approximate
  the surface area of a convex body in n dimensions
  given by a membership oracle.
• Previous best O(n8.5)

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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.
• dimension n

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Computing the Volume of Convex bodies

• Basic question how to estimate volume?
• Volume cannot be approximated in deterministic
  poly time within (n/log n)n (Bárány,
  F?redi BF88 )
• Volume can be approximated in randomized poly
  time (Dyer, Freize, Kannan DFK89.)
• Numerous improvements in complexity --
• most recent O(n4) ( Lovász, Vempala
  LV04.)

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What about surface area?

• Surface area is hard. Volume reduces to surface
  area.
• 2 volume K surface area of thin
  cylinder C(K)
• Surface area cannot be computed faster than
  volume.

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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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Computing the Surface Area of a convex body
previous approach

• (Dyer, Gritzmann, Hufnagel DGH98.)
• V Vd Vd - V
• Vd V Sd andn (Brunn-Minkowski)
• S surface area.

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Computing the Surface Area of a convex body
previous approach

• (Dyer, Gritzmann, Hufnagel DGH98.)
• Construct an oracle for the inflated body
• Each call costs O(n4.5). ( Lovász, Vempala
  LV06).
• Estimate surface area as (Vd V)/d.
• Need O(n4) oracle calls to estimate Vd.
• ( Lovász, Vempala LV04).
• Bound (Vd V)/d S using Alexandrov-Fenchel
  inequalities.
• The cost with best present technology is
  O(n8.5).

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Present approach Heat Flow

• Intuition
• Heat flows out of a body through its boundary.
• In a short interval of time, the amount of heat
  flowing out of the body is proportional to its
  surface area.

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How to compute heat flow
u0(x) 1, for x in K, initial
heat distribution. Heat at time t is
given by Heat kernel
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Heat equation

• Let
• Then, u satisfies the heat equation
  
  

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Approximating surface area

Lemma For sufficiently small ,
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Algorithm to approximate

• Choose random points
• in

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Algorithm to approximate

• Choose random points
• in
• Perturb each point independently
• by a random vector from a spherical
• Gaussian G(0, 2nt)
• Count number of perturbed points
• landing outside
• Obtain Volume estimate
• Output

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Choice of t

• Small t high accuracy but many oracle
  calls
• Large t few oracle calls but low
  accuracy
• Strike a balance
• 
  desired accuracy
  radius of a large ball
• 
  in K

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Algorithms Complexity

• Complexity of finding t -
• Complexity of estimating volume
• Complexity of generating
• random points -
• Final complexity
• for a precision .
• Same dependence on n as best volume
• Algorithm.

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Main Theorem
The output of the given algorithm is an
approximation of the surface area with
probability
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Main Theorem
The output of the given algorithm is an
approximation of the surface area with
probability (The probability of obtaining
precision can be boosted to by
repeating the algorithm times and
taking the median of the outputs.)
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Clustering and Surface Area of Cuts

• Semi-supervised Classification - Labelled and
  unlabelled data
• Low Density Separation (Chapelle, Zien 05.)

Quality of cut
(N, Belkin,Niyogi 06)
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Other Connections

• Manifold Learning Learning invariants of
  Manifolds from data
• (Zomorodian-Carlsson 04, Nadler et al 06,
  Belkin-Niyogi 05)

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Other Connections

• Manifold Learning Learning invariants of
  Manifolds from data
• (Zomorodian-Carlsson 04, Nadler et al 06,
  Belkin-Niyogi 05)
• Numerical Integration Example of integration on
  a manifold

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Thank you !
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Computing Cheeger ratio for smooth non-convex
bodies
Given membership oracle and sufficiently
many random samples from the body, fraction of
perturbed points landing outside
for the same algorithm

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Analysis Upper bound on

Terminology
Heat flow

Let
Then,
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Analysis Upper bound on

Terminology
Heat flow

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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 !
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Computing the Surface Area of a Convex Body

• Previous approach involves computing the
• Volume of cost given
  membership oracle for
  (with present Technology)
• Answering
• each oracle query to
• takes time .
• Computing volume takes
• time.

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Thank you !