Approximating the surface volume of convex bodies

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1 Approximating the surface volume of convex bodies Hariharan Narayanan Advisor: Partha Niyogi Department of Computer Science, University of Chicago Approximating the surface volume of convex bodies p.

2 Surface volumes A natural measure of the quality of a cut in binary classification is its volume. If data points are from an open set in R d, this cut is the boundary of an open set, and the computation of its volume is of interest. Approximating the surface volume of convex bodies p.

3 Complexity of surface volumes The following is known about the hardness of approximating the volume of a convex set deterministically. Theorem 1 (Bárány-Füredi) There is no polynomial time deterministic algorithm that would compute a lower bound vol(k) and an upper bound vol(k) so that vol(k) vol(k) ( c n log n ) n. Approximating the surface volume of convex bodies p.

4 Complexity of surface volumes Let K be a convex body and C(K) be the cylinder over it of height h. Then vol C(K) hvol K vol K = 2. For small h, this approximates the volume of K. So approximating the surface volume is at least as hard as approximating the volume. K C(K) Figure 1: A cylinder of height h over K Approximating the surface volume of convex bodies p.

5 Randomized algorithms for volume The volume of a convex body can be computed in randomized polynomial-time as shown by Dyer, Frieze and Kannan [2]. Their algorithm took O(d 23 ) steps. In a series of papers, this was brought down to the current best - O (d 4 ) by Lovász and Vempala. Approximating the surface volume of convex bodies p.

6 Random algorithms for surface volum Grötschel, Lovász and Schrijver (1987) mention computing the surface volume of a convex body to be an open problem. The first and to our knowledge only work on surface volumes of convex bodies is by Dyer, Gritzmann and Hufnagel (1998), who gave a randomized polynomial time algorithm for this task. Their complexity analysis is sketchy. Approximating the surface volume of convex bodies p.

7 Random algorithms for surface volum The time complexity is {time for volume} {time for a quadratic program}. This appears to be O(d 9 ) (d = dimension) with present technology. On the other hand our running time is (in a restricted setting) O ( d 4 ɛ + d3.5 R 3 2 r 2 τɛ 3 where 1/τ is a condition number. ). Approximating the surface volume of convex bodies p.

8 Diffusion and Surface Volume The heat equation u = u t. u(x, 0) = f(x). has the solution u(, t) = f(x) K t (x, ), where K t (x, y), the heat kernel is (4πt) d/2 e x y 2 /4t. Approximating the surface volume of convex bodies p.

9 Diffusion and Surface Volume Let f be the function that takes the value 1 on points in M and 0 outside, i. e.u = 1 M. Then, u(x, t) = 1 M K t (, x). Define F t (M) = π/t R d M u(y, t)dy. Approximating the surface volume of convex bodies p.

10 Diffusion and Surface Volume In other words, we are assuming that the initial heat content per unit volume of M is 1. The quantity of heat that diffuses across the boundary from time 0 to t is K t (x, y)dxdy = t/πf t (M). R d M M Approximating the surface volume of convex bodies p. 1

11 Diffusion and Surface Volume We prove that lim F t (M) = vol M. t 0 If the condition number of M is τ = 1 Theorem ( 2 ( vol M = 1 + O d 3/2 )) t ln(1/t) F t (M) + (et ln(1/t))d/2 O( t )vol M Approximating the surface volume of convex bodies p. 1

12 Condition Number The Condition Number of a submanifold X of R d is defined to be 1/τ where τ is the largest number satisfying the following property: The open normal bundle about X of radius r is imbedded in R d for every r < τ. Approximating the surface volume of convex bodies p. 1

13 Condition Number Figure 2: M and two tangent spheres of radius τ Approximating the surface volume of convex bodies p. 1

14 Condition Number Alternate Definition when X is the boundary M of an open set M: Defined to be 1/τ where τ is the largest number satisfying the following property: For any r < τ, to every point p on M it is possible to draw two tangent spheres S 1 (p, r) and S 2 (p, r) such that S 1 (p, r) M and S 2 (p, r) M =. Approximating the surface volume of convex bodies p. 1

15 vol M and F t (M) Proposition 1 Let the dimension d 3, and let t < e 1 satisfy Then, t ln( 1 t ) < ɛ 40 2d 3/2. (1 2ɛ 5 )F t(m) < M < (1 + ɛ 2 )F t(m). Approximating the surface volume of convex bodies p. 1

16 Computing vol M when M is convex Let M be a convex body in R d. Let O be a point inside M with the property that a ball B(r ) of radius r and centre O is contained entirely inside M and a concentric ball B(R ) of radius R contains M. O Approximating the surface volume of convex bodies p. 1

17 Computing vol M when M is convex Consider the random variable z defined according to the following process. Definition 1 1. Choose a random point x out of the convex body M, from the uniform distribution. 2. Add to x a Gaussian random variable n having density function K t (0, n) = e n 2 /4t (4πt) d/2. 3. If x + n is outside M, set z to 1, else set z to 0. Approximating the surface volume of convex bodies p. 1

18 Computing vol M when M is convex Lemma 1 (vol M) πe[z ] t = F t (M). Proof: E[z ] = R d M M 1 K t (x, y)( vol M )dxdy. The lemma follows, since π F t (M) := t R d M M K t (x, y)dxdy. Approximating the surface volume of convex bodies p. 1

19 Computing vol M when M is convex Unfortunately, we cannot sample exactly from the uniform distribution. So, consider the random variable z defined according to the following process. Choose a random point x out of the convex body M, out of some fixed distribution with dɛ density ρ f that is within t 10R (1+ɛ /2) π of the uniform distribution on M in variation distance. Approximating the surface volume of convex bodies p. 1

20 Computing vol M when M is convex Add to x a Gaussian random variable n having density function K t (0, n) := e n 2 /4t (4πt) d/2. If x + n is outside M, set z to 1, else set z to 0. Approximating the surface volume of convex bodies p. 2

21 Computing vol M when M is convex Lemma 2 Let z be the random variable defined above. Let the dimension d 3, and let t < e 1 satisfy 1 t ln( t ) < ɛ 40 2d 3/2. Then, 1 ɛ /10 < (vol M) πe[z] tft (M) < 1 + ɛ /10. Approximating the surface volume of convex bodies p. 2

22 Computing vol M when M is convex Compute an estimate ˆv(M), of the volume of M to within a multiplicative error of ɛ/3 with confidence 7/8. Compute an estimate Ê[z] of E[z] with error ɛ/3, confidence 7/8. Call this estimate Ê[z]. Using a form of Hoeffding s ) inequality, we find (R that this takes O d ɛ samples. 3 Output π t Ê[z]ˆv(M). Approximating the surface volume of convex bodies p. 2

23 Computing vol M when M is convex Theorem 3 Let M be a convex body in R d. Let O be a point in M such that the ball of radius r centered at O is contained in M, and the concentric ball of radius R contains it. Let the condition number of M be 1/τ. Then, it is possible to find the surface area of M in time O ( d 4 ɛ + d3.5 R 3 2 r 2 τɛ 3 within an error of ɛ with probability greater than 3/4. ), Approximating the surface volume of convex bodies p. 2

24 Upper bound for M K(x, y)dx E_1 B_1 1 O_1 A H_1 F_1 R R R^2 D_1 G_1 M C Approximating the surface volume of convex bodies p. 2

25 Upper bound M K(x, y)dx < K(x, y)dx R d B 1 K(x, y)dx H 1 R d B 1 H 1 K(x, y)dx + B c 1 K(x, y)dx Approximating the surface volume of convex bodies p. 2

26 Upper bound Choose R = 2dt ln (1/t). Let the mass outside the ball of radius R that the gaussian with density 1 e x2 /4t be ɛ.then (4πt) d/2 ) ɛ = O ((et ln(1/t)) d/2. B 1 has radius > R and so K(x, y)dx < K(x, y)dx + ɛ. M H 1 Approximating the surface volume of convex bodies p. 2

27 Lower bound for M K(x, y)dx A M H_2 F_2 R C (R^2)/2 D_2 G_2 O_2 1 B_2 E_2 Approximating the surface volume of convex bodies p. 2

28 Lower bound M K(x, y)dx = > K(x, y)dx(since B 2 M) B 2 K(x, y)dx H 2 B 2 K(x, y)dx K(x, y)dx H 2 H 2 B2 c K(x, y)dx K(x, y)dx H 2 B c 2 Approximating the surface volume of convex bodies p. 2

29 Bounds for F t (M) y r H Let H K(x, y)dx =: h(r). Approximating the surface volume of convex bodies p. 2

30 Bounds for F t (M) Then, we have shown that 1. h(r + R 2 /2) ɛ < M K(x, y)dx 2. If r > R 2, h(r R 2 /2) + ɛ > M K(x, y)dx. Approximating the surface volume of convex bodies p. 3

31 Bounds for F t (M) Definition 2 Let [M] r denote the set of points at a distance of r to the manifold M. Let π r be map from [M] r to M that takes a point P on [M] r to the foot of the perpendicular from P to M. Lemma 3 Let y [M] r. Let the Jacobian of a map f be denoted by Df. (1 r) d 1 Dπ r (y) (1 + r) d 1. Approximating the surface volume of convex bodies p. 3

32 Bounds for F t (M) tau Q P Q P M Figure 4: M and two tangent spheres of radius τ Approximating the surface volume of convex bodies p. 3

33 Bounds for F t (M) Lemma 4 R d M R M K(x, y)dxdy ɛvol M. Lemma 5 (1 e α2 /4t ) π/t α 0 h(r)dr π/t. Approximating the surface volume of convex bodies p. 3

34 References [1] M. Belkin and P. Niyogi (2004). Semi-supervised Learning on Riemannian Manifolds. In Machine Learning 56, Special Issue on Clustering, [2] M. Dyer, A. Frieze and R. Kannan, A random polynomial time algorithm for approximating the volume of convex sets (1991) in Journal of the Association for Computing Machinary, 38:1-17, [3] M.Dyer, P Gritzmann and A. Hufnagel, On the complexity of computing Mixed Volumes, In SIAM J, Comput. volume 27, No 2, pp , April 1998 [4] M. R. Jerrum, L. G. Valiant and V. V. Vazirani (1986), Random generation of Combinatorial structures from a uniform distribution. Theoretical Computer Science, 43, [5] E. Levina and P.J. Bickel (2005). Maximum Likelihood estimation of intrinsic dimension. In Advances in NIPS 17, Eds. L. K. Saul, Y. Weiss, L. Bottou. [6] R. M. Karp and M. Luby, (1983). Monte-Carlo algorithms for enumeration and reliablility problems. Proc. of the 24th 33-1

35 IEEE Foundations of Computer Science (FOCS 83),56-64 [7] P.Niyogi, S. Weinberger, S. Smale (2004), Finding the Homology of Submanifolds with High Confidence from Random Samples. Technical Report TR , University of Chicago [8] L. Lovász and S. Vempala (2004), Hit-and-run from a corner Proc. of the 36th ACM Symposium on the Theory of Computing, Chicago [9] S. Vempala and L. Lovász, Simulated annealing in convex bodies and an O (n 4 ) volume algorithm Proc. of the 44th IEEE Foundations of Computer Science (FOCS 03), Boston,