1. 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

2. 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)

3. Given: The Model 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

4. 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, Fϋredi [BF88] )
Volume can be approximated in randomized poly time (Dyer, Freize, Kannan [DFK89].)
Numerous improvements in complexity --
most recent O*(n4) ( Lovász, Vempala [LV04].)

5. 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.

6. Computing the Surface Area of a Convex Body
Open problem (Grötschel, Lovász, Schrijver [GLS90].)
In randomized polynomial time (Dyer, Gritzmann, Hufnagel [DGH98].)

7. Computing the Surface Area of a convex body: previous approach
(Dyer, Gritzmann, Hufnagel [DGH98].)
V Vδ Vδ - V
Vδ – V = Sδ + … + anδn (Brunn-Minkowski)
S surface area.

8. 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 (Vδ – V)/δ.
Need O*(n4) oracle calls to estimate Vδ.
( Lovász, Vempala [LV04]).
Bound |(Vδ – V)/δ – S| using Alexandrov-Fenchel inequalities.
The cost with best present technology is O*(n8.5).

9. 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.

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

11. Heat equation
Let
Then, u satisfies the heat equation

12. Approximating surface area
Lemma:
For sufficiently small ,

13. Algorithm to approximate
Choose random points in

14. 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

15. Choice of t: Strike a balance :
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

16. Algorithm’s 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.

17. Main Theorem: The output of the given algorithm is an
approximation of the surface area with probability

18. 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.)

19. 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)

20. Other Connections
Manifold Learning: Learning invariants of Manifolds from data
(Zomorodian-Carlsson ’04, Nadler et al ’06, Belkin-Niyogi ’05)

21. 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

22. Thank you !

23. 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

24. Analysis: Upper bound on
Terminology:
Heat flow :
Let
Then,
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

25. Analysis: Upper bound on
Terminology:
Heat flow
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

26. Analysis: Upper bound on
Terminology:
Heat flow :
Plot of for t = 1/4
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

27. Analysis: Upper bound on
Terminology: S = Surface Area, V = Volume
Heat flow :
The “Alexandrov-Fenchel inequalities”imply that
which leads to ,
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

28. Analysis: Lower bound on
Terminology:
Heat flow
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

29. Analysis: Lower bound on
Terminology:
Heat flow :
Let
Then,
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

30. Analysis: Lower bound on
Terminology:
Heat flow :
Plot of for t = 1/4
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

31. Analysis: Lower bound on
Terminology:
Heat flow :
For the upper bound we had ?
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

32. Analysis: Lower bound on
Lemma:
Proof:
Surface Area is monotonic,
that is,
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

33. Analysis: Lower bound on
Terminology:
Heat flow :
implies that
You may want to add a slide on the formal definition of heat flow after or before the slide with 4 pictures.

34. 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.)

35. Other Considerations:
Need to upper bound by
We show
Need to lower bound by

36. Upper bound for :
We show
Infinitesimally ,

37. Lower bound for :
We show
Prove that
Method : Consider

38. Thank you !

39. 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.

40. Thank you !