RANDOM FIELDS AND GEOMETRY. Robert Adler and Jonathan Taylor

Transcription

1 RANDOM FIELDS AND GEOMETRY from the book of the same name by Robert Adler and Jonathan Taylor IE&M, Technion, Israel, Statistics, Stanford, US. ie.technion.ac.il/adler.phtml www-stat.stanford.edu/ jtaylor 1

2 Mapping the Brain 2

3 A result 70 years in the making: Under the model f(t) : M R is smooth, Gaussian E{f(t)} 0. E{f 2 (t)} σ 2 = 1. C(s, t) = E{f(s)f(t)} is known. P { sup t M f t > u } e u2 /2 n j=0 C j u α j + o ( ) e u2 (1+η)/2 THEOREM: For piecewise C 2 Whitney stratified manifolds M embedded in C 3 ambient manifolds M, and with convex support cones lim inf u u 2 log dim P M L j (M)ρ j (u) j= σ 2 c (f). 3

4 The main Gaussian result Excursion sets A u (f, M) = {t M : f(t) u} The result: dim M j=0 L j (M)ρ j (u) = E {L 0 (A u (f, M))} where ρ j (u) = (2π) (j+1)/2 H j 1 (u)e u2 2 H j is the j-th Hermite polynomial H n (x) = n! n/2 j=0 H 1 (x) = e x2 /2 x ( 1) j x n 2j j! (n 2j)! 2 j e x2 /2 dx The L j (M) are the Lipschitz-Killing curvatures of M 4

5 Non-Gaussian processes f 1 (t),..., f k (t) i.i.d. Gaussian satisfying all the assumptions in force until now F : R k R twice differentiable defines f(t) = F ( f 1 (t),..., f k (t) ) Examples of F : k 1 x 2 i, x 1 k 1 m ( k 2 x 2 i )1/2, The result: E { L j (A u (f, M)) } n1 x 2 i n n+m n+1 x2 i = dim M j l=0 [ j + l l ] (2π) j/2 L j+l (M) M (k) l ( DF,u ) where D F,u = F 1 ([u, )) 5

6 We cannot avoid geometry The result: dim M j [ j + l l l=0 ] (2π) j/2 L j+l (M) M (k) l D F,u = F 1 ([u, )) Lipschitz-Killing curvatures L j, Whitney stratified manifolds M, Gaussian Minkowski functionals M (k) j ( DF,u ) In dimension 1, with f stationary: M = [0, T ] L 0 (A u ) = f(0)>u + # { t [0, T ] : f(t) = u, f (t) > 0 } L 1 (A u ) = λ 1 {t [0, T ] : f(t) u} L 0 (M) = 1 L 1 (M) = T But the M (k) l remain! 6

7 Lipschitz-Killing curvatures: I The tube in R N of radius ρ around an N dimensional M, (N N ) is Tube(M, ρ) = {t M : d(t, M) ρ} For nice (e.g. convex) M, the volume of Tube(M, ρ) is, for ρ < r c(m), given by Weyl s tube formula, λ N (Tube(M, ρ)) = N j=0 ω N j ρn j L j (M) The L j can be defined via the tube formula The L j are intrinsic 7

8 Two examples: The solid ball B N (T ): so that λ N ( Tube ( B N (T ), ρ )) = (T + ρ) N ω N N = ( N ) T j ρ N j ω N j = j=0 N j=0 L j ( B N (T ) ) ω N j ρ N j( N ) T j ω N. j ω N j ( N ) = T j ω N. j ω N j The sphere S N 1 (T ) S T (R N ): Tube ( S N 1 (T ), ρ ) = B N (T + ρ) B N (T ρ) yields L j ( S N 1 (T ) ) = 2 ( N ) ω N T j j ω N j if N 1 j is even, and 0 otherwise. Note the scaling in T! 8

9 Fundamental nature of L j Let ψ be a real valued function on nice sets in R N which is Invariant under rigid motions. Additive, in that ψ (M 1 M 2 ) = ψ (M 1 ) + ψ (M 2 ) ψ (M 1 M 2 ) Monotone, in that M 1 M 2 ψ (M 1 ) ψ (M 2 ). Then ψ (M) = N j=0 c j L j (M), where c 0,..., c N are non-negative (ψ-dependent) constants. 9

10 L 0 : The Euler characteristic M R N is nice and triangulisable α k is the number of k-dimensional simplices in the triangulation α 0 = number of vertices α 1 = number of lines.... α k = number of full simplices L 0 (M) Euler characteristic of M is ϕ(a) = α 0 α ( 1) d α N 10

11 Whitney stratified manifolds WSM s can be written as M = dim M k=0 k M with rules about glueing strata. Piecewise smooth manifolds are WSM s which have convex support cones 11

12 Riemannian manifolds Riemannian metrics. For each t T, g t : T t M T t M R is linear, positive definite, symmetric. g t (X t, X t ) = 0 X t = 0 (M, g) is called a Riemannian manifold g is NOT a metric, but τ g is: τ g (s, t) = inf c D 1 ([0,1];M) (s,t) L(c) L(c) = [0,1] g t (c, c )(t) dt and D 1 ([0, 1]; M) (s,t) contains all piecewise C 1 maps c : [0, 1] M with c(0) = s, c(1) = t. The canonical Gaussian metric: g t (X t, Y t ) = E {X t f, Y t f(t)} 12

13 Riemannian curvature Curvature operator: R(X, Y ) = X Y Y X [X,Y ]. Curvature tensor:, R(X, Y, Z, W ) = g(r(x, Y )Z, W ) Second fundamental form S S(X, Y ) = X Y X Y = P T M ( X Y ) Scalar second fundamental form S ν If ν is a unit normal vector field on M, The scalar second fundamental form of M in M for ν is S ν (X, Y ) = ĝ (S(X, Y ), ν) Shape operator S: Defined by ĝ(s ν (X), Y ) = S ν (X, Y ). for Y T (M), maps T (M) T (M) 13

14 Lipschitz-Killing curvatures II 1: The general case of LK measures: L i (M, A) = j i 2 N (2π) (j i)/2 j=i m=0 j M A C(N j, j i 2m) ( 1) m m! (j i 2m)! ( S(T t j M ) TrT t j M R mŝj i 2m ) ν N j N t M( ν N j ) H N j 1 (dν N j )H j (dt) 2: M embedded in R l with Euclidean metric: L i (M, A) = N j=i (2π) (j i)/2 C(l j, j i) j M A S(R l j ) 1 (j i)! TrT t j M (S j i η ) Nt M ( η)h l j 1(dη)H j (dt) 3: Lipschitz-Killing curvatures: L j (M) = L j (M, M). 14

15 Tube formulae On R l : M R l a piecewise smooth manifold. For ρ < ρ c (M, R l ) H l (Tube(M, ρ)) = N i=0 ρ l i ω l i L i (M) On the sphere S λ (R l ): Similar: But the constants are different and we need a one-parameter family L λ of Lipschitz-Killing curvatures. On general manifolds: Assuming piecewise smooth basic form remains, but constants change. General representation: ψ (M) = N j=0 c j L j (M), for additive, monotone, invariant ψ 15

16 A Gaussian tube formula Gauss measure γ k is the (product) measure induced on R k by k i.i.d. standard Gaussian random variables. Tube formula for WSM s in R k : γ k (Tube(M, ρ)) = γ k (M) + j=1 ρ j j! M(k) j (M) Example 1: M = [u, ) R 1 M (1) j ([u, )) = H j 1 (u) e u2 /2 2π. where H n (x) = n! n/2 j=0 ( 1) j x n 2j j! (n 2j)! 2 j, n 0 H 1 (x) = 2π Ψ(x) e x2 /2, Example 2: M = R k \ S u (R k ) hinges on calculations involving the χ 2 k distribution, etc. 16

17 σ 2 c (f) Recall the Gaussian excursion problem: lim inf u u 2 log = lim inf u u 2 log P {sup f t > u t M dim M j=0 P {sup t M } L j (M)ρ j (u) f t > u } E {L 0 (A u (M, f))} σ 2 c (f). 17

18 σ 2 c (f): cont The critical radius, r c, of a set M is the radius of the largest ball that can be rolled around M so that, at each point, it touches M only once. If ϕ(m) has no boundary, then σ 2 c (f) = (cot (r c )) 2 If r c = π/2, then σc 2 = 0 and the error in the approximation is zero! If M is convex, f is isotropic ( no finite expansion) and monotone then σc 2 (f) = Var ( f (t) f(t) ) 18

19 σ 2 c (f): In general Reproducing kernel Hilbert space H f is the space of functions on M satisfying f(s), C(t, s) H = f(t), where the inner product is determined by the covariance function C via C(s, ), C(t, ) H = C(s, t), An orthonormal basis {ϕ n } for H will always give a orthonormal expansion for f. i.e. f t = n=1 ξ n ϕ n (t) THEOREM: In general, σ 2 c (f) can be defined in terms of the critical radius of S(H), the unit ball of the RKHS. 19

20 Orthogonal expansions Theorem: If {ϕ n } n 1 is an orthonormal basis for the reproducing kernel Hilbert space of C then f has the L 2 - representation f t = n=1 ξ n ϕ n (t) where the {ξ n } n 1 are i.i.d. N(0, 1). Convergence: Sum converges uniformly f is continuous (w.p. 1). A crucial identity: 1 = C(t, t) = Varf t = 1 ϕ 2 j (t) An observation: X t f t = n=1 ξ n X t ϕ n (t) where X t T t M, assuming that M is a differentiable manifold and f t C 1 (M). 20

21 The rôle of the sphere An astounding consequence: If C is smooth enough every Gaussian process with expansion of order K, dim(m) K < can be rewritten on a subset of S K 1 via and t (ϕ 1 (t),..., ϕ K (t)) f(t) f (ϕ 1 (t),..., ϕ K (t)) f (u) = u, ξ Change of parameter space: to ϕ(m) S K 1 From M Covariance structure of f : E { f (u)f (v) } = E { u, ξ v, ξ } = E { } u i ξ i v j ξ j i = u, v Consequence: If the expansion is finite, one case covers all. j 21

22 Suprema and tubes Rewriting the canonical process: f u = = K ξ n u n n=1 1/2 K ξn 2 K n=1 n=1 χ 2 K K n=1 U n u n ξ n ( Kn=1 ξ 2 n) 1/2 u n for uniform U n, independent of χ 2 K. Consequently: P = = { sup f u λ } u M { P sup 0 u M { P 0 } f (u) > λ χ 2 K = x φ ξ 2 (x) dx K sup U, u > λ/ x u M } φ ξ 2 K (x) dx BUT, P {sup u M U, u > y} is a tube volume!! 22

23 Back to non-canonical f For the canonical process on the sphere we now know that excursion probabilities are related to tube volumes By Weyl s tube formula these are related to Lipschitz-Killing curvatures The supremum of a non-canonical process over M is the same as the canonical process over ϕ(m). To get an answer in terms of M, we need to carry back the Riemannian (Euclidean) structure of ϕ(m) (i.e. of S K 1 ) to M and so need smooth ϕ. Computation: The induced Riemannian metric that f induces on M under the map ϕ 1 is given by g(x t, Y t ) = n X t ϕ n (t) Y t ϕ n (t) = E {X t f t Y t f t } 23

24 ie.technion.ac.il/adler.phtml. www-stat.stanford.edu/ jtaylor. keith 24