Approximation Theory on Manifolds

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1 ATHEATICAL and COPUTATIONAL ETHODS Approximation Theory on anifolds JOSE ARTINEZ-ORALES Universidad Nacional Autónoma de éxico Instituto de atemáticas A.P. 273, Admon. de correos #3C.P Cuernavaca, or. EXICO Abstract: Let be a compact Riemannian manifold and let {x i } p be vectors in a Hermitian vector bundle over. The approximation problem here considered is to find a section f on which approximates the data {x i } p in the penalized least squares sense, that is, by minimizing a functional defined as the sum of the mean squared distance to the data plus a term that measures the energy of the section f. It is demonstrated that the approximation problem has a unique solution and a formula for the solution is obtained. Key Words: anifold, vector bundle, penalized least squares. 1 Introduction Let be a compact Riemannian manifold, and suppose that {x i } p is a set of scattered points lying on. In this paper we are interested in the following problem: Problem 1. Given vectors {x i } p, find a (smooth vector-valued function f defined on which approximates the data in the sense that f(x i x i, i 1,...,p. Data fitting problems where the underlying domain is a manifold arise in many areas, including e.g. geophysics and meteorology where a sphere is taken as a model of the earth. The question of whether interpolation or approximation should be carried out depends on the setting, although in practice measured data are almost always noisy, in which case approximation is probably more appropriate. One approach to Problem 1 is to create a least squares fit: find a function f which minimizes L(f p f(x i x i 2. When the data are especially noisy, it may be useful to replace this discrete least squares problem by a penalized least squares problem. The idea is to minimize a combination I(f L(f + λ E(f, where E(f is a measure of energy defined by Af(x 2 dx, where A is an operator acting on f. The parameter λ controls the trade-off between these two quantities, and is typically chosen to be a small positive number, see [4]. One way to define the energy functional is to take A as an appropriate differential operator, such as A ( + P m/2, where m is an even integer, is the Laplace-Beltrami operator on, and P is the projection on the constants. The definition of A for the case where m is odd is more complicated and can be found in [12]. The functionals 1 are similar to the functionals which are minimized in defining spherical thin plate splines, see [14] and [15]. The penalized least squares approach to Problem 1 is a generalization of the smoothing spline, ridge regression, and generalized cross-validation problems ([5], which have appeared in the statistical literature. References to the work done in this area are [3], [4], [10] and [13]. This paper analyses the problem from the approximation theory point of view. A general penalized least squares problem can be formulated as follows: Let X be a linear space. On X there are given two semi-definite inner products, and (, with associated semi-norms and, respectively. For given F X and λ > 0 find f X so that F f 2 + λ f 2 inf φ X { F φ 2 + λ φ 2 }. We consider a special case of 1. To mention results about the general problem 1, it is easy to prove the uniqueness of the minimal f in 1 ISSN: ISBN:

2 if [, ] λ :, + λ(, is an inner product on X, and the existence of f if X is a Hilbert space for [, ] λ. Since the existence and uniqueness of such a minimal f are our main results, it would suffice to prove that our (X, [, ] λ is a Hilbert space. Instead, we give a constructive proof of the existence and uniqueness of the minimizing function. As a byproduct of this proof, we obtain an explicit formula of the minimizing function. This explicit formula of the minimizing function can be used in applications of the type 1, and an estimation of the approximation rate of the corresponding numerical scheme can be obtained. Such estimation of the approximation rate will be the subject of a future paper. A number of numerical examples are discussed in [1] where both figures and tables related to storage, exactness, and computational time are included. In the present paper an application of penalized least squares fitting is developed by letting the manifold to be the n-dimensional sphere. In particular we consider the case of the two-sphere, and a figure is included to visually show the quality of the fitting. The paper is organized as follows: in Section 2 some basic definitions and preliminary results are given. In Section 3, the existence and uniqueness of a global minimum of the functional I is demonstrated, and an explicit formula for such minimum obtained. Finally, in Section 4 we apply our method to the n- dimensional sphere case and explicit formulas are obtained. 2 Definitions of Function Spaces and Notation This section is intended to give a theoretical formulation of the penalized least squares scheme. We begin by defining the space in which the minimization process will be carried out, and also introduce the notation that will be used throughout the paper. The description of our setting in this section might be too short. References on various definitions that here are used are [2] and [7]. Let be a compact n-dimensional C Riemannian manifold such that if, then the boundary is a closed submanifold and near its boundary is a direct product [0,δ,δ > 0. To avoid technical difficulties we use also a manifold ((,0]. For a finite dimensional hermitian vector bundle V over, let Γ(V be the space of all C sections of V. Denote by, the Hermitian inner product and set,. For f Γ(V, we define ATHEATICAL and COPUTATIONAL ETHODS norms f k by f 2 k l k (D l f(x 2 dx, where dx is the volume element defined by the Riemannian metric, and Df denotes covariant differentiation with respect to the Riemannian connection. Let Γ k (V be the completion of Γ(V with respect to the norm f k. To simplify the notation we write instead of 0. Denote by π : V the projection of the bundle. Suppose that {x i } p is a set of scattered points lying on, and consider vectors {x i } p V such that πx i x i. For k [ n 2 ] + 1 consider an isomorphism A : Γ k (V Γ 0 (V, and define the functional I(f x i f(x i 2 + λ Af 2. By the Sobolev lemma (see [11], p. 116, every section in Γ k (V is continuous. Therefore, the domain of the functional I is set to be Γ k (V. 2.1 The map K In the following section the existence and uniqueness of a global minimum of the functional I is demonstrated. Such a minimum is given in terms of the inverse of a matrix which contains all of the information supplied by the vectors {x i } p. We call this matrix the system matrix. Next we develop some technical machinery. We start by defining the system matrix and other fiber maps. Let E be the fiber of V and let Hom(E denote the space of linear homomorphisms of E into itself. Let V be the C vector bundle over with fiber Hom(E. By Theorem 4 in [9], the operator A 1 : Γ 0 (V Γ k (V is associated with a kernel K(, Γ 0 (V. For x,y and H Hom(E x,e y, denote by H Hom(E y,e x the adjoint of H defined by e x,h e y He x,e y, e x E x, e y E y. Definition 2 For i,j 1,...,p, let K ij Hom(E xj,e xi, (1 P i Hom(E x1 E xp,e xi, (2 K Hom(E x1 E xp, (3 be the maps defined by ISSN: ISBN:

3 Problem 3 Definition 1. For i,j 1,...,p, let K ij K(x i,xk(x j,x dx,(4 P i (e x1,...,e xp e xi, (5 P i K K ij P j. (6 j1 Definition 4. Let, be the inner product in defined by E x1 E xp (e x1,...,e xp,(e x 1 e xi,e x i, (7 e xi, e x i E xi. The map K has the following properties: (i it is selfadjoint with respect to the inner product 7, (ii it is positive semidefinite and, if we denote by I the identity map in E x1 E xp, then (iii the map λ I +K is invertible. These assertions are consequence of the following theorem. Theorem 5 For (e x1,...,e xp,(e x 1 E x1 E xp, we have (i (ii (iii (e x1,...,e xp, K(e x 1 K(e x1,...,e xp,(e x 1. (e x1,...,e xp, K(e x1,...,e xp K(x i, e xi 2. (e x1,...,e xp,(λ I + K(e x1,...,e xp λ (e x1,...,e xp,(e x1,...,e xp. In order to prove this theorem a technical lemma is needed. Let (f,g f(x,g(x dx, be the inner product of two sections f,g Γ 0 (V. ATHEATICAL and COPUTATIONAL ETHODS Lemma 6 Let 1 i p and (e x1,...,e xp E x1 E xp. Then, for e x i E xi, e x i,p i K(e x1,...,e xp (K(x i, e x i,k(x j, e xj. j1 Proof: This is a computation. P i K(e x1,...,e xp (8 K ij P j (e x1,...,e xp (by 6 (9 j1 j1 j1 K ij e xj (by 5 (10 K(x i,xk(x j,x dxe xj (by 4.(11 Taking the inner product with e x i gives e x i,p i K(e x1,...,e xp (12 e x i,k(x i,xk(x j,x e xj dx (13 j1 j1 K(x i,x e x i,k(x j,x e xj dx(14 (by definition of K(x i,x (15 (K(x i, e x i,k(x j, e xj (by 2.1.(16 j1 In the sequel, purely formal calculations such as the preceding one will sometimes be omitted. Proof of Theorem 1. (i This is a computation. (e x1,...,e xp, K(e x 1 (17 e xi,p i K(e x 1 (18 (by Definition 4 (19 (K(x i, e xi,k(x j, e x j (20 j1 (by Lemma 1 (21 e x j,p j K(e x1,...,e xp (22 j1 (again by Lemma 1, where the upper line denotes complex conjugation P j K(e x1,...,e xp,e x j (23 j1 K(e x1,...,e xp,(e x 1. (24 ISSN: ISBN:

4 ATHEATICAL and COPUTATIONAL ETHODS For part (ii we have K(πx, xaf(x, x dx (e x1,...,e xp, K(e x1,...,e xp (25 Af(x,K(πx,x x dx (K(x i, e xi,k(x j, e xj (26 (Af,K(πx, x (f,a K(πx, x. j1 (by 20 (27 ( K(x i, e xi, K(x j, e xj (28 Lemma 10 The section f min satisfies λaf min + K(x i, f min (x i K(x i, x i. j1 K(x i, Proof: Denote by e xi 2. (29 x (x 1,...,x p. Finally part (iii follows from (ii. 3 The minimum of the functional In this section we show that there exist a unique global minimum of the functional I in Γ k (V. We start by defining our candidate for this. Definition 7 Let f min be the section defined by f min A 1 K(x i, P i (λ I + K 1 (x 1,...,x p. Notice that for any e xi E xi, K(x i, e xi Γ 0 (V since K(, Γ 0 (V. Therefore A 1 K(x i, e xi Γ k (V. In particular f min Γ k (V. As a preliminary result, we compute the value of the functional I at f min. Theorem 8 I(f min x i f min (x i,x i. An obvious consequence of this theorem is that if f min interpolates the data {x i } p, then I(f min is zero. To prove Theorem 2 we need two technical lemmas. Lemma 9 For x V and f Γ 0 (V, we have f(πx,x (f,a K(πx, x, We first compute f min (x i (30 A 1 K(x j,x i P j (λ I + K 1 x(by 7(31 j1 K(x i,x K(x j,x P j (λ I + K 1 xdx (32 j1 K(x i,xk(x j,x dxp j (λ I + K 1 x (33 j1 K ij P j (λ I + K 1 x(by 4 (34 j1 P i K(λ I + K 1 x, (35 by 6. Then K(x i, f min (x i K(x i, P i K(λ I + K 1 x. (36 On the other hand, λaf min λ K(x i, P i (λ I + K 1 x, by 7. Adding this to 36 gives λaf min + K(x i, f min (x i where A denotes the adjoint of A. Proof: This is a computation. f(πx,x (A 1 Af(πx,x K(πx, xaf(xdx, x K(x i, P i (λ I + K(λ I + K 1 x(37 K(x i, P i x (38 K(x i, x i. (39 ISSN: ISBN:

5 Proof of Theorem 2. We start by computing x i f min (x i 2 x i 2 x i,f min (x i f min (x i,x i + + f min (x i,f min (x i x i 2 x i,f min (x i f min (x i,x i + +(A K(x i, f min (x i,f min, by Lemma 2, where in this case xf min (x i and ff min. Substituting this and into 2, gives Af min 2 (A Af min,f min I(f min ( x i 2 x i,f min (x i f min (x i,x i +(A K(x i, f min (x i,f min +λ(a Af min,f min ( x i 2 x i,f min (x i f min (x i,x i ( ( + A K(x i, f min (x i + +λaf min,f min ATHEATICAL and COPUTATIONAL ETHODS Corollary 12 The section given by 7 is the unique minimum of the functional I. Proof: This is a consequence of Theorem 3. If I(f min I(f then A(f f min 0, by 40. Hence f f min, since A is injective. Proof of Theorem 3. We begin by computing the right hand side of 40. We start with (f f min (x i 2 f(x i 2 f(x i,f min (x i f min (x i,f(x i + f min (x i,f min (x i f(x i 2 (f,a K(x i, f min (x i (A K(x i, f min (x i,f +(f min,a K(x i, f min (x i, by Lemma 2. On the other hand, A(f f min 2 Af 2 (Af,Af min (Af min,af + +(Af min,af min Af 2 (f,a Af min (A Af min,f +(f min,a Af min. (by factoring f min and A ( x i 2 x i,f min (x i f min (x i,x i ( + A K(x i, x i,f min (by 10 ( x i 2 x i,f min (x i f min (x i,x i + x i,f min (x i (by Lemma 2 x i f min (x i,x i Theorem 11 For all f Γ k (V, I(f I(f min (f f min (x i 2 + λ A(f f min 2. (40 By 41 and 41, (f f min (x i 2 + λ A(f f min 2 ( f(x i 2 (f,a K(x i, f min (x i (A K(x i, f min (x i,f +(f min,a K(x i, f min (x i + +λ( Af 2 (f,a Af min (A Af min,f + (f min,a Af min f(x i 2 ( ( f,a K(x i, f min (x i + λaf min + ( ( A ( K(x i, f min (x i + λaf min,f ( f min,a K(x i, f min (x i + ISSN: ISBN:

6 +λaf min + λ Af 2 ( ( f(x i 2 f,a K(x i, x i ( ( A K(x i, x i,f ( ( + f min,a K(x i, x i + λ Af 2 ( f(x i 2 (f,a K(x i, x i (A K(x i, x i,f +(f min,a K(x i, x i + λ Af 2 ( f(x i 2 f(x i,x i x i,f(x i + + f min (x i,x i + λ Af 2 (by Lemma 2 ( f(x i 2 f(x i,x i x i,f(x i + + f min (x i,x i + λ Af 2 + x i 2 x i 2 f(x i x i 2 + λ Af 2 x i f min (x i,x i I(f I(f min, by 2 and 8. 4 Conclusion The penalized least squares on varieties allows one to solve a problem of approximation of vector data in non-euclidean topological spaces, and gives an exact solution to the minimization problem in a functional space of infinite dimension. Such approaches are beginning to take on much importance today, as they exploit today s computational potential to solve problems with a high degree of complexity quickly and efficiently. References: ATHEATICAL and COPUTATIONAL ETHODS [1] Alfeld, P., Neamtu,., and Schumaker, L. L., Fitting scattered data on sphere-like surfaces using spherical splines, J. Comput. Appl. ath. 73 (1996, [2] Clemens, H., Introduction to Hodge Theory, Appunti dei Corsi Tenuti da Docenti della Scuola, Scuola Normale Superiore, Pisa, [3] Eubank, R. L. and Gunst, R. F., Diagnostics for penalized least-squares estimators, Statist. Probab. Lett. 4 (1986, [4] von Golitschek,., and Schumaker, L. L., Data fitting by penalized least squares, in Algorithms for Approximation II, J. C. ason and. G. Cox (eds., Chapman& Hall, London, 1990, [5] Good, P. I., Resampling ethods. A Practical Guide to Data Analysis, Birkhuser, Boston, [6] Groemer, H., Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, New York, [7] Lee, J.., Riemannian anifolds. An Introduction to Curvature, Springer Verlag, New York, [8] artínez-orales, J. L., Generalized Legendre series and the fundamental solution of the Laplacian on the n-sphere, preprint. [9] [10] Nowak, R. D., Penalized least squares estimation of Volterra filters and higher order statistics, IEEE Trans. Signal Process 46 (1998, [11] Omori, H., Infinite-Dimensional Lie Groups, American athematical Society, Providence, [12] Seeley, R. T., Complex powers of an elliptic operator, A..S. Proc. Symp. Pure ath. 10 (1967, Corrections in: The resolvent of an elliptic boundary problem. Am. J. ath. 91, (1969. [13] Sun, W., A penalized least squares method for image reconstruction and its stability analysis, Information 3 1 (2000, [14] Wahba, G., Spline Interpolation and smoothing on the sphere, SIA J. Sci. Statist. Comput. 2 (1981, [15] Wahba, G., Errata: Spline interpolation and smoothing on the sphere, SIA J. Sci. Statist. Comput. 3 (1982, ISSN: ISBN: