RANDOM FIELDS AND GEOMETRY from the book of the same name by - PDF document

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

Mapping the Brain 2

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

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

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 1 ( t ) , . . . , f k ( t ) f ( t ) = F • Examples of F : √ k − 1 m � n k 1 x 2 � x 1 x 2 i i , i ) 1 / 2 , ( � k n � n + m 2 x 2 n +1 x 2 1 i • The result: � � L j ( A u ( f, M )) E � � dim M − j � � � j + l (2 π ) − j/ 2 L j + l ( M ) M ( k ) = D F,u l l l =0 where D F,u = F − 1 ([ u, ∞ )) 5

� We cannot avoid geometry • The result: � � dim M − j � � � j + l (2 π ) − j/ 2 L j + l ( M ) M ( k ) D F,u l l l =0 D F,u = F − 1 ([ u, ∞ )) Lipschitz-Killing curvatures L j , Whitney stratified manifolds M , Gaussian Minkowski functionals M ( k ) j • 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 ) remain! l 6

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, N � ω N ′ − j ρ N ′ − j L j ( M ) λ N ′ (Tube( M, ρ )) = j =0 • The L j can be defined via the tube formula • The L j are intrinsic 7

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

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 N � ψ ( M ) = c j L j ( M ) , j =0 where c 0 , . . . , c N are non-negative ( ψ -dependent) constants. 9

L 0 : The Euler characteristic • M ⊂ R N is nice and “triangulisable” • α k is the number of k -dimensional sim- plices 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 + · · · + ( − 1) d α N 10

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

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 L ( c ) c ∈ D 1 ([0 , 1]; M ) ( s,t ) � � g t ( c ′ , c ′ )( t ) dt L ( c ) = [0 , 1] and D 1 ([0 , 1]; M ) ( s,t ) contains all piece- wise 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

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 �� � ∆ = P ⊥ � S ( X, Y ) = ∇ X Y − ∇ X Y ∇ X Y TM • 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 ) = g ( S ( X, Y ) , ν ) � • Shape operator S : Defined by g ( S ν ( X ) , Y ) = S ν ( X, Y ) � for Y ∈ T ( M ), maps T ( M ) → T ( M ) . 13

� Lipschitz-Killing curvatures II 1: The general case of LK measures: L i ( M, A ) ⌊ j − i 2 ⌋ N � � C ( N − j, j − i − 2 m ) (2 π ) − ( j − i ) / 2 = ( − 1) m m ! ( j − i − 2 m )! j = i m =0 � � S ( T t ∂ j M ⊥ ) Tr T t ∂ j M � � � R m � S j − i − 2 m × ν N − j ∂ j M ∩ A × � 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 � (2 π ) − ( j − i ) / 2 C ( l − j, j − i ) = j = i � � 1 ( j − i )!Tr T t ∂ j M ( S j − i ) × η S ( R l − j ) ∂ j M ∩ A N t M ( − η ) H l − j − 1 ( dη ) H j ( dt ) × � 3: Lipschitz-Killing curvatures: L j ( M ) = L j ( M, M ) . 14

Tube formulae • On R l : M ⊂ R l a piecewise smooth manifold. For ρ < ρ c ( M, R l ) N � ρ l − i ω l − i L i ( M ) H l (Tube( M, ρ )) = i =0 • 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: N � ψ ( M ) = c j L j ( M ) , j =0 for additive, monotone, ‘invariant’ ψ 15

A Gaussian tube formula • Gauss measure γ k is the (product) mea- sure induced on R k by k i.i.d. standard Gaussian random variables. • Tube formula for WSM’s in R k : ∞ ρ j � j ! M ( k ) γ k (Tube( M, ρ )) = γ k ( M ) + ( M ) j j =1 • Example 1: M = [ u, ∞ ) ⊂ R 1 ([ u, ∞ )) = H j − 1 ( u )e − u 2 / 2 M (1) √ 2 π . j where ⌊ n/ 2 ⌋ ( − 1) j x n − 2 j � H n ( x ) = n ! j ! ( n − 2 j )! 2 j , n ≥ 0 j =0 √ 2 π Ψ( x ) e x 2 / 2 , H − 1 ( x ) = • Example 2: M = R k \ S u ( R k ) hinges on calculations involving the χ 2 k distri- bution, etc. 16

σ 2 c ( f ) • Recall the Gaussian excursion problem: � � � � u →∞ − u − 2 log � lim inf sup f t > u � P t ∈ M � dim M � � � − L j ( M ) ρ j ( u ) � j =0 � � � � u →∞ − u − 2 log � = lim inf sup f t > u � P t ∈ M � � � − E {L 0 ( A u ( M, f )) } � ≥ 1 1 2 + c ( f ) . 2 σ 2 17

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 σ 2 c = 0 and the error in the approximation is zero! • If M is convex, f is isotropic ( ⇒ no fi- nite expansion) and monotone then � � � � σ 2 f ′′ ( t ) c ( f ) = Var � f ( t ) 18

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 = ξ n ϕ n ( t ) n =1 • 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

Orthogonal expansions • Theorem: If { ϕ n } n ≥ 1 is an orthonor- mal basis for the reproducing kernel Hilbert space of C then f has the L 2 - representation ∞ � f t = ξ n ϕ n ( t ) n =1 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: ∞ � ϕ 2 1 = C ( t, t ) = Var f t = j ( t ) 1 • An observation: ∞ � X t f t = ξ n X t ϕ n ( t ) n =1 where X t ∈ T t M , assuming that M is a differentiable manifold and f t ∈ C 1 ( M ). 20