A Lower Bound Theorem. Lin Hu.

Transcription

1 American J. of Mathematics and Sciences Vol. 3, No -1,(January 014) Copyright Mind Reader Publications ISSN No: A Lower Bound Theorem Department of Applied Mathematics, Beijing University of Technology, Beijing 10014, P. R. China Department of Basic Courses, Beijing Union University, Beijing , P. R. China hulin9803@yeah.net Abstract Motivated by Candes and Donoho s work (Candés, E J, Donoho, D L, Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Stat. 30, (00)), this paper is devoted to giving a lower bound of minimax mean square errors for Riesz fractional integration transforms and Bessel transforms. Keywords Bessel transform; Riesz transform; error; noise. AMS(000) subject classifications 4C15, 4C40 1. Introduction The linear inverse problem for a statistical model with additive noise plays important roles in scientific settings ranging from medical imaging to physical chemistry ([1]). More precisely, consider the problem of recovering an image f from the noisy data Y = Kf + εw. (1.1) Here, f belongs to ε (A), which is the function space consisting of compactly supported and twice continuously differentiable away from a smooth edge; W denotes a Wiener sheet; ε is a noisy level; K stands for a linear operator from L (R ) to another Hilbert space. We use sup E ˆf f to denote the mean square V error on the function space V for the L risk and M(ε, V ) := inf ˆf sup E ˆf f to V 73

2 represent the minimax mean square error on the function space V for the L risk. In 010, Colonna and Easley use shearlets to deal with the inverse problem (1.1), when K is the Radon transform([3]). They give an upper bound of the mean square error ε 4 5 (log(ε 1 )) on the space ε (A) for the L risk. Moreover, they show a lower bound to the minimax mean square error ε 4 5 (log(ε 1 )) 5 for that class of functions, which means their upper bound essentially optimal, ignoring log factor. Note that Riesz fractional integration transforms and Bessel transforms play important roles in both theoretical analysis and practical applications. Hu and Liu ([5])apply shearlets to the inverse problem (1.1) for a family of linear operators including Riesz fractional integration transforms and Bessel transforms. Based on a shearlet shrinkage method, they obtain an upper bound of the mean square error ε 3/+α (log(ε 1 )) on the space ε (A) for the L risk. The goal of this paper is to give a lower bound of the minimax mean square error for Riesz fractional integration transforms and Bessel transforms. It turns out that the above mentioned upper bounds are optimal, ignoring log factor.. Main Theorem To introduce our main theorem, we begin with the definitions of Riesz fractional integration transforms and Bessel transforms ([6]). Definition 1 Riesz fractional integration transform I α is defined by f(y) I α (f)(x) = C α x y dy (0 < α < 1) α with some normalizing constant C α ; the Bessel operator B α by R with G α (x) = A α K 1 α ( x ) x 1 α B α f = G α f (0 < α < ) L 1 (R), where A α is a normalizing constant and 74

3 A Lower Bound Theorem 3 K v (z) represents McDonald function defined as K v (z) = 1 (z ) v 0 t v 1 z t e 4t dt. The following two lemmas play an important role in our discussion. Lemma 1 ([7]) Let f L p (R ) and I α be the Riesz fractional integration transform. If 1 < p < α and 1 q = 1 p α, then there exist a constant C > 0 such that I α f q C f p. Lemma ([, 4]) For N 1, let ξ {0, 1} N and X N(ξ, V ) be a multivariate Gaussian vector. Assume that V is invertible such that τi = V ar(x i X k, k 1 i) = 1 for all 1 i N. Then there is an absolute constant B such (V 1 ) ii that Theorem inf ˆξ sup ξ {0,1} N E ˆξ ξ BN. Now, we are ready to state the main theorem of this paper: C > 0 such that If the operator K in (1.1) take I α or B α, then there exits a constant M(ε, ε (A)) Cε 3/+α (ε 0). Proof. Firstly, we consider Riesz fractional integration transform I α : Let h be a smooth function of variable t with compact support contained in [0, π] and h m,j (t) = m α+ α+1 h(mt πj), j = 0, 1,, m 1 for m 1. We introduce a polar coordinates (r, θ) with origin at ( 1, 1 ). Set r 0 = 1 4, f 0 := 1 {r r0 }, where 1 A denotes the indicator function on a set A. Then the functions ψ m,j := 1 {r<hm,j +r 0 } f 0, j = 0, 1,, m 1 are disjointly supported and ψ m,j = h m,j 1 = m 4α+3 α+1 h 1 m 4α+3 α+1. Here and after, A B denotes A CB and A CB for some constant C > 0). 75

4 4 Let Then H m ε (A), which implies H m := {f = f 0 + M(ε, ε (A)) M(ε, H m ) := inf ˆf ξ j ψ m,j }. sup E ˆf f. (.1) H m So, the problem reduces to estimate f H m. Furthermore, we can restrict the estimator of the form: ˆf = f 0 + ˆξ j ψ m,j. In fact, let P m denote the L projection on the smallest affine subspace containing H m. Then for f H m, P m ˆf f = P m ˆf Pm f ˆf f. This implies the risk of a general estimator ˆf greater than or equal to that of a corresponding estimator P m ˆf. Moreover, ˆf f = (ˆξ j ξ j )ψ m,j ˆξ ξ ψ m,j m 4α+3 α+1 ˆξ ξ (.) due to the orthogonality of ψ m,j and the fact that ψ m,j m 4α+3 α+1. Hence, it is sufficient to estimate ξ {0, 1} M. Let g j := I α ψ m,j. Applying Lemma 1 to g j with q =, p =, one has 1 + α g j = I α ψ m,j ψ m,j. This with the fact ψ m,j = h m,j 1+α 1 1+α 1+α m (4α+3)(α+1) α+1 leads to g j m (4α+3)(α+1) α+1. Because a Riesz fractional integration transform I α is invertible, the functions g j are linearly independent. Let V m stand for the smallest affine space containing I α f 0 + θ j g j for arbitrary {θ j } {0, 1} m. Note that, for each function v(x) L (R ) orthogonal to V m, the law of R v(x)y dx is N(0, v ) independently of ξ. So, the projection of the Riesz fractional integration data on the span V m is sufficient for ξ. 76

5 A Lower Bound Theorem 5 Since {g j } is linear independent, the linear functions g j, f Rf 0 give a nondegenerate set of affine coordinates for f V m. For each j, define Y j := Y, g j I α f 0, g j = g j, g i ξ i + Then the vector Y := (Y j ) i=0 i=0 ε W, g i. gives a nondegenerate set of affine coordinates for the projection of the Riesz fractional integration data on the space V m. Hence, Y = (Y j ) m is a sufficient statistic for ξ and Y N(Gξ, ε G), where G is the matrix with the i, j element G j,i = g j, g i. Because g j is linearly independent, the matrix G is invertible. Define X := = G 1 Y, then X N(ξ, ε G 1 ). Note that Y is a sufficient statistic for ξ, so is X. We may restrict our attention to estimator X. Let V := ε G 1 be the covariance matrix of X and τ j := V ar(x j X k, k j) be the conditional variance of X i given the other coordinates. Take m such that m ε α+1 ( 3 +α)(α+1). Then g j m (4α+3)(α+1) α+1 ε for all 1 j m 1 and τ j = V ar(x j X k, k j) = inf ˆξ This with (.1) and (.) shows 1 (V 1 ) jj = M(ε, ε (A)) inf ˆf 1 ε (G) jj = ε g j 1. By Lemma, sup ξ {0,1} m E ˆξ j ξ j Bm. sup E ˆf f Bm α+ α+1. H m Using m ε α+1 ( 3 +α)(α+1). one receives M(ε, ε (A)) Cε 3/+α. This completes the proof for K = I α. It remains to conclude the theorem for K = B α. By Definition 1, one has G α (x) = A α 1 t α α x α e t x 4t dt C. Hence, 0 x α B α ψ m,j = B α ψ m,j I α ψ m,j m (4α+3)(α+1) α+1 due to ψ m,j > 0. Note that the Bessel operator B α is invertible. Then the exactly 77

6 6 same arguments as above show M(ε, ε (A)) Cε 3/+α for Bessel transform Bα. This completes the proof of our main theorem. Acknowledgments This work is supported by the National Natural Science Foundation of China (No ) and Natural Science Foundation of Beijing (No ). The author would like to thank her adviser, Prof. suggestions and comments. Youming Liu for his important References [1] Bertero, M., 1989, Linear inverse and ill-posed problems, In Advances in Electronics and Electron Physics (P.W. Hawkes, ed.), Academic Press, New York. [] Candés, E.J, Donoho, D.L., 00, Recovering edges in ill-posed inverse problems: optimality of curvelet frames, Ann. Stat., 30, pp [3] Colonna, F., Easley, G., Guo, K., Labate, D., 010, Radon transform inversion using the shearlet representation, Applied and Computational Harmonic Analysis, 9, pp [4] Guo, K., Labate, D., 01, Optimal Recovery of 3D X-Ray Tomographic Data using the Shearlet Representation, to appear in Advances Comput. Math. [5] Lin, H., Youming, L., Shearlet Approximations to the Inverse of A Family of Linear Operators, to appear in Inequalities and applications. [6] Stefan, G., Samko., 00, Hypersingular Integrals and Their Applications, London: Taylor Francis, New York. [7] Zhou, M. Q., 1999, Harmonic Analysis, Beijing University Press, Beijing. 78