Some new method of approximation and estimation on sphere

Transcription

1 Some new method of approximation and estimation on sphere N. Jarzebkowska B. Cmiel K. Dziedziul September 17-23, 2017, in Bedlewo (Poland)

2 [BD] Bownik, M.; D.K. Smooth orthogonal projections on sphere. Constr. Approx. 41 (2015) Challenges: To give a simple construction of Parseval (tight) frame on the sphere based on [BD], To assure additional properties - to create a multivariate approximation on L 2 (S d ) i. e. a sequence of nite dimension spaces such that V 0 (S d ) V 1 (S d ) L 2 (S d ), To use the kernels K j (x, y), x, y S d, j 0 of orthog. projections K j : L 2 (S d ) V j (S d ), K j (f )(x) = f (y)k j (x, y)dσ(y), S d to an estimation of density on sphere where dσ is the surface measure on S d, To visualize results.

3 Ausher-Weiss-Wickerhouser (AWW) operator on IR Dene a smooth real-valued function in by s(t) = [ exp ( ) t δ / ( exp t + δ 2 t δ t + δ ) ( + exp where 1I is a characteristic function. Note that 2 t δ t + δ s 2 (t) + s 2 ( t) = 1, t IR. A AWW operator for all t IR and g : IR IR g(t) t > δ, E(g)(t) = s 2 (t)g(t) + s(t)s( t) g( t) t [ δ, δ], 0 t < δ. AWW operator is an example of Hestenes operator. ) ] 1I( δ,δ)(t)+1i[δ, )(t),

4 A natural parametrization (coordinates) on S d Consider a natural parametrization of the sphere Φ d : [0, π] S d 1 S d, Φ d (θ, ξ) = (ξ sin θ, cos θ), (θ, ξ) [0, π] S d 1. The function Φ d : (0, π) S d 1 S d \ {1 d, 1 d } is dieomorphism, where 1 d = (0,..., 0, 1) S d is the orth Pole"

5 Ausher-Weiss-Wickerhouser (AWW) operator on S d Now we can dene Ausher-Weiss-Wickerhouser (AWW) operator E = E δ,s, pointwise for every g : S d IR in natural parametrization by E(g)(θ, ξ) = = g(θ, ξ), θ > π/2 + δ ( ) d 1 sin(π θ) s 2 2 (θ π/2)g(θ, ξ) + s(θ π/2)s(π/2 θ) sin θ g(π θ, ξ) 0, θ < π/2 δ, where ξ S d 1. For an indentity operator Id E + = Id E and E = E.

6 Two paches Let us decompose the sphere onto two patches A and A +, where 0 < δ < π/2 is xed A = {x S d : x d+1 sin δ}, A + = {x S d : x d+1 sin δ}.

7 Properties of AWW operator Lemma E : L 2 (S d, dσ d ) L 2 (S d, dσ d ) is an orthogonal projection. Denition We say that an operator P is localized on an set U, i.e., for any f : S d IR we have Pf (x) = 0 for x S d \ U and supp f U = = P U f = 0. Theorem The operator E + is localized on A + and the operator E on A,

8 Now we are ready to reformulate [BD Theorem 1.1] Denition Let F(S d ) be either the space B s p,q(s d ), s 0, 1 p, q, or W r p (S d ), r N, 1 p. Dene the spaces F(A ± ) := E ± (F(S d )) for Q Q d. Theorem 1 The both operators E ± (i.e. E +, E ) E ± : L 2 (S d ) L 2 (S d ) are orthogonal projection and E + + E = I, 2 Moreover, each F(Q) is a closed subspace of F(S d ) and we have a direct sum decomposition with the equivalence of norms F(S d ) = F(A + ) F(A ) f F(S d ) E + f F(S d ) E f F(S d ).

9 Daubechies multivariate wavelets For a xed N 2, let N φ be a univariate, compactly supported scaling function with support supp N φ = [0, 2N 1] associated with the compactly supported, orthogonal univariate Daubechies wavelet N ψ. Note that we take N ψ such that supp N ψ = [0, 2N 1]. For convenience, let ψ 0 = N φ and ψ 1 = N ψ. Let E = {0, 1} d be the vertices of the unit cube and let E = E \ {0} be the set of nonzero vertices. For each e = (e 1,..., e d ) E, dene ψ e (x) = ψ e 1 (x 1) ψ e d (xd ), x = (x 1,..., x d ) IR d. Observe that supp ψ e = [0, 2N 1] d.

10 Properties of Daubechies multivariate wavelets Let D be the set of dyadic cubes in IR d of the form I = 2 j (k + [0, 1] d ), j ZZ, k ZZ d. Denote the side length of I by l(i ) = 2 j. For any e E dene scaled wavelet, related to I by ψ e I (x) = 2 jd/2 ψ e (2 j x k), x IR d. It is well-known that {ψ e I (x) : I D, e E} is an orthonormal basis of L 2 (IR d ). Moreover, it is unconditional basis of the Sobolev space Wp r (S d ), r = 0, 1,... and 1 p < for suciently large choice of N N(r) depending on r. For our purposes it is convenient to consider a localized wavelet systems on a cube.

11 Localized wavelet system, related to the cube Denition Suppose that J = [ 1, 1] d is a cube in IR d and ε > 0. Dene its ε enlargement by J ε = [ 1 ε, 1 + ε] d. Let j 0 ZZ be the smallest integer such that (2N 1)2 j 0 ε/2. For any j j 0, consider families of dyadic cubes D j (e) = {I D : l(i ) = 2 j and supp ψ e I J ε } and D + j 0 (e) = D j (e). j=j 0 Dene a localized wavelet system, related to the cube J and ε > 0 by S(J, ε) := {ψ e I : e E, I D + j 0 (e)} {ψ 0 I : I D j0 (0)}.

12 Lemma The localized wavelet system S (J, ε) has following properties: S(J, ε) is an orthonormal sequence in L 2 (J ε ), for every f L 2 (J ε ) with supp f J ε/2 we have f 2 = L f, ψ e 2 I 2 + f, ψ 0 I 2. e E I D + j 0 (e) I D j0 (0) magnitudes of coecients { f, g } g S(J,ε) characterize functions f F(IR d ) satisfying supp f J ε/2, where F is either the Sobolev space Wp s or the Besov space B p,q s, 0 < s < r, 1 < p, q.

13 By appropriate choice of ε, ε = k or ε = 2 k, k IN (we x ε) we get a sequence of nite dimensional spaces V j0 V j L 2 (J ε ), where Recall V j = span L2 (J ε){ψ 0 I : I D j (0)}. D j (0) = {I D : l(i ) = 2 j and supp ψ 0 I J ε }.

14 Localized wavelet system S(J, ɛ) is transformed to S d by two stereographic projections ( S ± : S d \ {1 d } IR d x 1, S ± (x 1,..., x d+1 ) =,..., 1 ± x d+1 We dene variable change operators T ± d : L2 ([ 1 ε, 1 + ε] d ) L 2 (S d ) x d 1 ± x d+1 ). given by T ± (ψ)(u) = ψ(s ±(u)) Jd (S ± (u)), u Sd, where J d the Jacobian of S 1 ± ( ) d+1 2 J d (x 1,..., x d+1 ) = 1 + x x. 2 1 d+1 Both operators T ± d are isometric isomorphism.

15 This leads us to two a local wavelet system on S d. Namely S ± = S ± (A ± ) = T ± d (S(J, ɛ)) = = {T ± d (ψe I ) : e E, I D + j 0 (e)} {T ± d (ψ0 I ) : I D j0 (0)}, Lemma The system S ± has following properties: 1 S ± is an orthogonal system in L 2 (S d ), ( cos δ 2 If ɛ 2 1 sin δ ), 1 then E ± (S ± ) is Parseval frame for E ± (L 2 (S d )), a.e. for all f E ± (L 2 (S d )) f 2 = f, E ± (g) 2. 2 g S ± 3 the coecients { f, g } g E ± (S ±) characterize functions f E ± (F(S d )), where F is a Sobolev space Wp s, r = 0, 1,..., or Besov space Bp,q s, 0 < s < r, 1 < p, q.

16 We dene a wavelet system corresponding to A + and A S := E + (S + ) E (S ), where E ± (S ± ) = {E ± T ± d (ψe I ) : e E, I D + j 0 (e)} {E ± T ± d (ψ0 I ) : I D j0 (0)}. We can dene a sequence of nite dimensional spaces j 0 V j (S d ) = E + T + (V d j+j 0 ) E T (V d j+j 0 ). and { W j (S d ) = span L2 (S d ) E ± T ± d (ψe I ) : e E, I D j+j0 (e) }.

17 Theorem ( cos δ If ɛ 2 1 sin δ ), 1 the wavelet system S is a Parseval frame in L 2 (S d ). The sequence of {V j (S d )} j 0 has following properties V 0 (S d ) V 1 (S d ) L 2 (S d ), (1) V j+1 (S d ) = V j (S d ) W j (S d ), (2) and the set V j 0 j(s d ) is dense in L 2 (S d ). Moreover, the magnitude of the coecients { f, g } g S characterize f F(S d ), where F is Sobolev space Wp r, r = 0, 1,..., or Besov space Bp,q s, 0 < s < r, 1 < p, q.

18 Denition of a wavelet estimator Let n be a sample size. Denition Let {K j : j 0} be a family of kernels K j : S d S d IR corresponding of the orthogonal projection on V j (S d ). Let X 1,... X n be iid with density function f on S d with respect to Lebesgue'a measure. For j 0 we dene an estimator of f f n (j)(x) = 1 n n K j (x, X i ). i=1 We denote the balls of density functions in Besov spaces on sphere Σ(s, B) = {f B2, (S s d ) : f (x)dσ d (x) = 1, f 0, f s,2 B}. S d

19 Adaptive estimation Theorem Let d/2 < r < R and let X 1,... X n be iid with density function f B s 2, (Sd ), where s is unknown and r s R. Then for any U > 0 there are constants c = c(r, R, U) and C = C(U) such that for all s, n and B > 1 we have sup E f n (j n ) f 2 cb 2d/(2s+d) n 2s/(2s+d), 2 f Σ(s,B), f U where j n = min } 2ld {j [j min, j max ] : l, j<l jmax f n (j) f n (l) C 22 n (3) and j min = log2 n log2 n, j max =. 2R + d 2r + d

20 Proof Main tool in the proof: Talagrand's inequality. Task: to nd analytic condition on the kernels following a result of A. Bull, R. Nickl, Adaptive condence sets in L 2, Probability Theory and Related Fields, vol. 156 (3-4), , 2013 on IR and an interval. Explanation: Note that Ef (j) = K j (f ). i.e. E f (j) f 2 2 = E f (j) Ef (j) K j (f ) f 2 2 ERROR = Variance + Square of bias(approximation problem) Problem of an optimization: for a given sample size nd a best level j 0.

21 Rysunek: From top: estimator of function f1 for n function f1 on the bottom. N. JarzebkowskaB. Cmiel K. Dziedziul Sphere = 100, n = and true