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 Open problem (Grötschel, Lovász, Schrijver [GLS90].) In randomized polynomial time (Dyer, Gritzmann, Hufnagel [DGH98].)

3 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)

4 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].)

5 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.

6 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.

7 Computing the Surface Area of a Convex Body Previous approach : Choose appropriate Consider the convex body, its - neighbourhood and their difference.

8 Computing the Surface Area of a Convex Body Previous approach: Compute Surface area by interpolation

9 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.

10 Heat Flow t = 0 t = 0.05 t = t = 0.075

11 Heat diffusing out of in time Terminology Motivation

12 Fact For small, Surface Area Terminology Heat diffusing out of in time

13 Heat diffusing out in time Fact For small, Surface Area Terminology Algorithm Choose random points in

14 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.

15 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

16 Finding : for some unit vector T 2-isotropic : For all unit vectors Set smallest eigenvalue of

17 If samples were generated uniformly at random, Output {Heat Flow} Algorithm’s relation to Heat Flow

18 Complexity of rounding the body (and finding) - Complexity of estimating volume – Complexity of generating random points - Algorithm’s Complexity

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

20 Analysis: Upper bound on Terminology: Heat flow

21 Analysis: Upper bound on Terminology: Heat flow : Let Then,

22 Analysis: Upper bound on Terminology: Heat flow : Plot of for t = 1/4

23 Analysis: Upper bound on Terminology: S = Surface Area, V = Volume Heat flow : The “Alexandrov-Fenchel inequalities”imply that which leads to,

24 Analysis: Lower bound on Terminology: Heat flow

25 Analysis: Lower bound on Terminology: Heat flow : Let Then,

26 Analysis: Lower bound on Terminology: Heat flow : Plot of for t = 1/4

27 Analysis: Lower bound on Terminology: Heat flow : For the upper bound we had ?

28 Analysis: Lower bound on Proof: Surface Area is monotonic, that is, Lemma:

29 Analysis: Lower bound on Terminology: Heat flow : implies that

30 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.)

31 Other Considerations: Need to upper bound by We show Need to lower bound by We show

32 Upper bound for : We show Infinitesimally,

33 Lower bound for : We show Prove that Method : Consider

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