Radial basis interpolation on homogeneous manifolds: Convergence rates^{∗}

J. Levesley^{†}and D. L. Ragozin^{‡}

January 25, 2001

Department of Mathematics Department of Mathematics
and Computer Science  Univ. of Washington
University of Leicester  Box 354350, Seattle
Leicester LE1 7RH, UK.  WA 98195

jl1@mcs.le.ac.uk         rag@math.washington.edu

Abstract

Pointwise error estimates for approximation on compact homogeneous manifolds using dadial kernels are presented. Tangent space techniques are used to lift the problem from the manifold to Euclidean space, where methods for proving such error estimates are well established.

1 Introduction

There is currently significant interest in approximation on spheres, related to many interesting geophysical problems. There are a number of different approximation methods currently available on spheres, including wavelets [5], piecewise polynomial splines [1], and the subject of this paper, radial functions (sometimes called zonal splines or variational splines) [5, 8]. Error estimates and convergence rates for radial approximation on spheres, of an optimal nature, are recent in vintage [7, 6], and rely on some technically demanding mathematics. In this paper we build on an idea of Bos and de Marchi [3] in order to provide convergence rates for radial interpolation on a much wider class of manifolds: the reflexive, compact homogeneous spaces.

∗Technical Report No. 2001/04, Department of Mathematics & Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England.

†Supported by EPSRC Grant GR/L36222 ‡Partially supported by NSF Grant DMS-9972004

Let M^{d} ⊂ IR^{d+k} be a d-dimensional embedded compact homogeneous manifold in the sense of Definition 1.1 in [10]. In particular there is a compact group G of isometries of IR^{d+k} such that for some η ∈ M^{d} (often referred to as the pole) M = {gη : g ∈ G}. The reflexive condition means that for all x, y ∈ M there is a g ∈ G with gx = y and gy = x. A kernel k : M ×M → IR is termed zonal (or G-invariant) if k(x, y) = k(gx, gy) for all g ∈ G and x, y ∈ M. Since the maps in G are isometries of Euclidean space, they preserve both Euclidean distance and the (arc-length) metric d(·, ·) induced on the components of M by the Euclidean metric. Thus the distance kernel d(x, y) is zonal, as are all the radial functions, ϕ(d(x, y)), which are kernels which depend only on the distance between x, y.

The manifold carries a unique normalised G-invariant measure which we call μ (μ can be viewed either as the component of Lebesgue measure tangential to M or as the measure induced by Haar measure on G). Then, we can define the inner product of real functions

(f, g) = ∫_{M} fgdμ.

An important part of analysing the interpolation process (and many other processes) is the construction of polynomials on the manifold. If Π_{j} is the space of all polynomials of total degree j in the ambient space IR^{d+k} then P_{j} := Π_{j}|_{M} is the space of degree j polynomials on M. We can also construct sets of harmonic polynomials H_{j} := P_{j} ⋂ P_{j−1}^{⊥}, where the orthogonality is with respect to the inner product (·, ·).

It is straightforward to show that H_{j} is G-invariant, in the sense that g·p_{j}(·) := p_{j}(g^{−1}·) ∈ H_{j} for all p_{j} ∈ H_{j}, and has an orthogonal decomposition into irreducible G-invariant subspaces (i.e., subspaces with no proper Ginvariant subspace)

H_{j} = ⊕_{l=1}^{h_{j}}Ξ_{jl}.

Now, if {Y_{jl}^{1}, …, Y_{jl}^{d_{jl}}} is an orthonormal basis for Ξ_{jl}, then

P_{jl}(x, y) = ∑_{m=1}^{d_{jl}} Y_{jl}^{m}(x)Y_{jl}^{m}(y), x, y ∈ M,

is the unique G-invariant reproducing kernel for Ξ_{jl}. In other words, if T_{jl} is the orthogonal projection onto Ξ_{jl},

T_{jl}f(x) = ∫_{M} P_{jl}(x, y)f(y)dμ(y), x ∈ M,  (1)
or T_{jl}f(x) = (f, P_{jl}(·, x)), x ∈ M,

for all f ∈ L_{2}(M). In particular,

T_{jl}f(x) = (T_{jl}f, P_{jl}(·, x)), x ∈ M.  (2)

It is clear that P_{jl} is symmetric for all j = 0, 1, …, and l = 1, …, h_{j}, i.e., P_{jl}(x, y) = P_{jl}(y, x). The self-adjoint projection operators T_{jl} associated with the G-invariant kernels P_{jl} are special cases of the general kernel operator T_{k} associated with a continuous zonal kernel k(x, y) by

T_{k}f(x) = ∫_{M} k(x, y)f(y)dμ(y), x ∈ M.  (3)

All these operators are G-equivariant in the sense that g · T_{k}(f) = T_{k}(g · f) and, at least on reflexive spaces, commute with each of the projectors T_{jl} (Proposition 3.7 in [10]). Since Ξ_{jl} is an irreducible G-invariant subspace, the commuting properties of the self-adjoint T_{k} imply T_{k}T_{jl} = a_{jl}T_{jl} for some scalar a_{jl}. Thus on reflexive spaces, by the density of polynomials, each zonal kernel k has an expansion (convergent as operators on L_{2}(M))

k(x, y) = ∑_{j=0}^{∞} ∑_{m=1}^{h_{j}} a_{jl}P_{jl}(x, y).

In [10] the fundamentality of the set {k(·, y) : y ∈ M}, of translates of a zonal kernel k in the continuous functions on M is explored. The crucial result is that a zonal kernel has dense translates if and only if a_{jl} ≠ 0 for all j, l, i.e. the associated integral operator, T_{k} does not behave like the null operator on any of the G-invariant polynomial subspaces. As is shown in [11] the positive definiteness of a kernel k is equivalent to k being positive definite on each of these G-invariant polynomial subspaces. Here, k is positive definite if the quadratic form

∑_{i,j=1}^{N} c_{i}c_{j}k(x_{i}, x_{j})

is non-negative for any choice of points x_{1}, …, x_{N} ∈ M and coefficients c_{1}, …, c_{N} ∈ IR.

A fixed positive definite kernel k gives rise to a reproducing kernel Hilbert space W, often called the native space, whose elements have smoothness of the same order as k. Given the values of f at distinct x_{1}, …, x_{N} ∈ M, we approximate f ∈ W by the k-spline interpolant

s_{k}(x) = ∑_{j=1}^{N} c_{j}k(x, x_{j}),

where the coefficients c_{1}, …, c_{N} are determined by satisfying the interpolation equations s_{k}(x_{i}) = f(x_{i}), i = 1, …, x_{N}. This interpolation problem has a unique solution if k is strictly positive definite, i.e., the above quadratic form is always positive when x_{i} are distinct.. It is such kernels that we shall be interested in.

The reader should view the following work as tracking closely the corresponding analysis on the sphere, but we depart from the rationale of Light and v. Golitschek [6] as we try to convert error bounds to convergence rates. They describe a scaling operation on the sphere under which the Lagrange polynomials remain bounded, this being a key element of the analysis. The novelty of our approach is that we transplant the scaling, and in fact the entire error estimation onto the tangent space, and use well known properties of the scaling of Lagrange polynomials in Euclidean space. This is possible because distances on the manifold and its tangent space are comparable.

In Section 2 we develop the Hilbert space theory and show how to produce an error estimate for interpolation. In Section 3 we use this error estimate to produce convergence rates for positive definite kernels which are also radial (functions of geodesic distance on the manifold) or zonal.

2 Hilbert Space, Reproducing Kernel, and Error Estimates

In this section we describe the construction of a Hilbert space from a positive definite zonal kernel k, and see that k is the reproducing kernel for this Hilbert space. This construction is very standard (see e.g. Atteia [2]), and is only included for completeness.

As indicated above, when M is reflexive, every positive definite zonal kernel k whose translates are dense in C(M) has an expansion in terms of the reproducing kernels for the minimal G-invariant polynomial subspaces of L_{2} (see Theorems 12 and 16 in [11]):

k(x, y) = ∑_{j=0}^{∞} ∑_{l=1}^{h_{j}} a_{kj}P_{jl}(x, y), a_{jl} > 0.

For the rest of this paper we assume

∑_{j=0}^{∞} ∑_{l=1}^{h_{j}} d_{jl}a_{jl} < ∞.  (4)

Since d_{jl} = P_{jl}(x, x)^{1/2}P_{jl}(y, y)^{1/2} > |P_{jl}(x, y)|, this condition ensures that the positive definite zonal kernel defined by the sum above is jointly continuous, so that point evaluation makes sense.

Define the inner product

[f, g] := ∑_{j=0}^{∞} ∑_{l=1}^{h_{j}} a_{jl}^{−1}(T_{jl}f, T_{jl}g),

where we recall that (,) is the usual L_{2} inner product. Let ∥f∥ = [f, f]^{1/2}. Then, the Hilbert space W is the subspace of L_{2} satisfying

W := {f ∈ L_{2} : ∥f∥ < ∞}.

Condition (4) also ensures that k(·, x) ∈ W for every x ∈ M.

It is easy to show that point evaluation at x ∈ M in this space is realised by inner product with k(·, x), since

[f, k(·, x)] = ∑_{j=0}^{∞} ∑_{l=1}^{h_{j}} a_{jl}^{−1}(T_{jl}f, T_{jl}k(·, x))

= ∑_{j=0}^{∞} ∑_{l=1}^{h_{j}} a_{jl}^{−1}(T_{jl}f, a_{jl}P_{jl}(·, x))

= ∑_{j=0}^{∞} ∑_{l=1}^{h_{j}} T_{jl}f(x)
= f(x),  (5)

where, in the penultimate line we have used Equation (2).

We can now immediately show Proposition 1 Let s_{k} be the k-spline interpolant to f ∈ W. Then s_{k} is the interpolant to f from W with minimum norm.

Proof The following holds for any g ∈ H:

[g, g] = [g − s_{k} + s_{k}, g − s_{k} + s_{k}]
= [g − s_{k}, g − s_{k}] + [s_{k}, s_{k}] + 2[s_{k}, g − s_{k}].

Since k is the reproducing kernel for W,

[s_{k}, g − s_{k}] = [∑_{n=1}^{N} α_{n}k(·, x_{n}), g − s_{k}]

= ∑_{n=1}^{N} α_{n}g(x_{n}) − s_{k}(x_{n})
= 0,

if g is also an interpolant to f. Thus, if g is an interpolant to f,

[g, g] = [g − s_{k}, g − s_{k}] + [s_{k}, s_{k}] ≥ [s_{k}, s_{k}].  (6)

Remarks 1 In fact, one way of characterising the k-spline interpolant is via a minimum norm property (see e.g. Light and Wayne [9] for details of such an approach).

We can use (6) in a different way to give an error estimate for k-spline interpolation in W.

Theorem 1 Let f ∈ W and s_{k} be the k-spline interpolant to f at x_{1}, …, x_{N}. Then, for x ∈ M,

|f(x) − s_{k}(x)| ≤ ∥f∥P (x, x_{1}, …, x_{N}),

where

P (x, x_{1}, …, x_{N})

= min_{α_{1},...,α_{N} ∈IR} {k(x, x) − 2 ∑_{n=1}^{N} α_{n}k(x, x_{n}) + ∑_{n,m=1}^{N} α_{n}α_{m}k(x_{n}, x_{m})}^{1/2} .

Proof Using the reproducing kernel property we have

|f(x) − s_{k}(x)| = [f(x) − s_{k}(x), k(·, x)]
= [f(x) − s_{k}(x), k(·, x) − ∑_{n=1}^{N} α_{n}k(·, x_{n})],

since f and s_{k} are equal at x_{n}, n = 1, …, N. The coefficients α_{1}, …, α_{N} are arbitrary real numbers. Using the Cauchy-Schwarz inequality we get

|f(x) − s_{k}(x)| ≤ ∥f − s_{k}∥ ∥k(·, x) − ∑_{n=1}^{N} α_{n}k(·, x_{n})∥

≤ ∥f∥ ∥k(·, x) − ∑_{n=1}^{N} α_{n}k(·, x_{n})∥,

using (6). Now,

∥k(·, x) − ∑_{n=1}^{N} α_{n}k(·, x_{n})∥

= [k(·, x) − ∑_{n=1}^{N} α_{n}k(·, x_{n}), k(·, x) − ∑_{m=1}^{N} α_{m}k(·, x_{m})]

= {k(x, x) − 2 ∑_{n=1}^{N} α_{n}k(x, x_{n}) + ∑_{n,m=1}^{N} α_{n}α_{m}k(x_{n}, x_{m})}^{1/2}

using the reproducing kernel property and the symmetry of k. The required result follows by minimisation of the final expression over all values of α_{1}, …, α_{N} ∈ IR.

3 Pointwise Convergence Rates for Zonal Kernels

In this section we consider zonal kernels and in particular radial kernels of the form

k(x, y) = ϕ(d(x, y)),

where d(x, y) is the geodesic distance between x and y, and ϕ : IR → IR is a univariate, even, real-valued function. As we have already noted, the G-invariance of d(x, y) shows radial kernels are clearly zonal. For the analysis in this section we will assume that k is positive definite and 2r-times continuously differentiable in each variable near the diagonal in M × M.

We are interested in establishing (pointwise) convergence rates for the interpolants as the as the interpolation points x_{1}, …, x_{N} become dense about the point of interest x. Since we will not be concerned with global constants, we are able to exploit the compactness of M and simply give local estimates. By passing to the tangent space T_{x} = IR^{d} at x it becomes possible to reduce all the critical work to the Euclidean case.

Here are two simple results which lie at the heart of the Euclidean case. The proof of the first is elementary.

Lemma 1 Let Z := {z_{1}, …, z_{t}} be a Π_{2r} unisolvent set in IR^{d} with t = dim(Π_{2r}). Let p_{1}, …, p_{t} be the Lagrange polynomials for Z, i.e. p_{i}(z_{j}) = δ_{ij}. Then:

i. λZ := {λz_{1}, …, λz_{t}} is unisolvent.

ii. The Lagrange polynomials for λZ are p_{i}^{λ} = p_{i}(·/λ), i = 1, …, t. In particular p_{i}^{λ}(0) = p_{i}(0).

Lemma 2 Fix an integer r ≥ 0 and a Π_{2r} unisolvent set {z_{1}, …, z_{t}} in IR^{d}. Then there exist d_{0} > 0 and C_{0} such that:

i. Any set {v_{1}, …, v_{t}} with ∥z_{i} − v_{i}∥ ≤ d_{0} is Π_{2r} unisolvent,

ii. If {v_{1}, …, v_{t}} are as in (i) and {p_{i}, i = 1, …, t} is a Lagrange basis for Π_{2r} centered on v_{i} then |p_{i}(0)| ≤ C_{0}.

Proof If {q_{1}, …, q_{t}} is any basis for Π_{2r}, then {v_{1}, …, v_{t}} is unisolvent if and only if det(q_{j}(v_{i})) ≠ 0. Hence by the continuity of (v_{1}, …, v_{t}) ↦ det(q_{j}(v_{i})), and the unisolvency of {z_{1}, …, z_{t}} there is a d_{0} for which (i) holds. Since the map from unisolvent sets of distinct centers to Lagrange polynomials is continuous, so is (v_{1}, …, v_{t}) ↦ (p_{1}(0), …, p_{t}(0)). But {(v_{1}, …, v_{t}) : ∥v_{i} − z_{i}∥ ≤ d_{0}} is compact, so (ii) holds with

C_{0} = sup_{i,v_{j}}{|p_{i}(0)|} < ∞

These two Lemmas provide the basis for us to estimate the size of the analog of P (x, x_{1}, …, x_{N}) for the Euclidean space case.

Proposition 2 Suppose K(x, y) is a strictly positive definite symmetric function on some compact neighborhood \overline{U} × \overline{U} of (0, 0) in IR^{d} × IR^{d}. Further suppose K is in C^{2r}(\overline{U} × \overline{U}). Then there exist constants C > 0 and ρ_{0} such that for any finite set u_{1}, …, u_{N} in U with

sup_{v∈\overline{U}} min_{j} ∥v − u_{j}∥ ≤ ρ < ρ_{0},  (7)

the following estimate holds:

min_{α_{1},...,α_{N} ∈IR} {K(0, 0) − 2 ∑_{n=1}^{N} α_{n}K(0, u_{n}) + ∑_{n,m=1}^{N} α_{n}α_{m}K(u_{n}, u_{m})} ≤ Cρ^{2r}. (8)

Proof Let Z = {z_{1}, …, z_{t}} be the fixed unisolvent set from from Lemma 2, such that keeping within d_{0} of the z_{i} preserves unisolvency. Choose r > 0 such that the ball \overline{B_{r}(0)} ⊂ \overline{U}. Then set ρ_{0} = rd_{0}/ max_{i} ∥z_{i}∥. If ρ < ρ_{0}, then \frac{ρ}{d_{0}} Z ⊂ B_{r}(0) ⊂ \overline{U}. Hence for each \frac{ρ}{d_{0}} z_{j} ∈ \overline{U}, the density assumption (7) yields a u_{i_{j}} with ∥\frac{ρ}{d_{0}} z_{j} − u_{i_{j}}∥ ≤ ρ. Hence v_{j} = \frac{d_{0}}{ρ} u_{i_{j}} is Π_{2r} unisolvent by the same Lemma and the corresponding Lagrange polynomials satisfy |p_{j}(0)| ≤ C_{0}. If the indexing on the u_{i} is changed so that i_{j} = j, then scaling the v_{j} by \frac{ρ}{d_{0}} produces, by Lemma 1, a Π_{2r}–unisolvent subset {u_{1}, …, u_{t}}. Furthermore, if {p_{1}, …, p_{t}} are the Lagrange polynomials for u_{1}, …, u_{t}, then

max{|p_{1}(0)|, …, |p_{t}(0)|} ≤ C_{0}.  (9)

Moreover, for j = 1, …, t,

∥u_{j}∥ ≤ ρ(∥z_{j}∥/d_{0} + 1) ≤ ρC_{u}, where C_{u} = 1 + max_{j} ∥z_{j}∥/d_{0}  (10)

Now we proceed to prove the bound (8) by the coefficients α_{1}, …, α_{N} as follows. Let α_{j} = p_{j}(0), j = 1, …, t and α_{j} = 0 otherwise. This choice of coefficients is made since

∑_{j=1}^{t} p_{j}(0)q(u_{j}) = q(0),  (11)

for all q ∈ Π_{2r}. Then the left hand side (LHS) of (8) may be bounded as follows:

LHS ≤ {K(0, 0) − 2 ∑_{j=1}^{t} p_{j}(0)K(0, u_{j}) + ∑_{i,j=1}^{t} p_{i}(0)p_{j}(0)K(u_{i}, u_{j})}(12)

Since K(·, ·) is 2r times continuously differentiable in each variable, for fixed w ∈ U we may expand

K(w, z) = ∑_{|α|<2r} c_{α}^{w}(z − w)^{α} + R_{2r}^{w}(z), z ∈ U,  (13)

where R_{2r}^{w} is a Taylor series remainder satisfying

R_{2r}^{w} ≤ C_{R}∥z − w∥_{2}^{2r},  (14)

for some constant C_{R} independent of z and w. Putting z = w in the above expansion we see that

c_{0}^{w} = K(w, w).  (15)

Plugging (13) into (12) gives

LHS ≤ {K(0, 0) − 2 ∑_{j=1}^{t} p_{j}(0) { ∑_{|α|<2r} c_{α}^{0}(u_{j})^{α} + R_{2r}^{0}(u_{j})}

+ ∑_{i,j=1}^{t} p_{i}(0)p_{j}(0) { ∑_{|α|<2r} c_{α}^{u_{i}}(u_{j} − u_{i})^{α} + R_{2r}^{u_{i}}(u_{j})}} .

Using the polynomial reproduction (11) we get

LHS ≤ {K(0, 0) − 2K(0, 0) − 2 ∑_{j=1}^{t} p_{j}(0)R_{2r}^{0}(u_{j})

+ ∑_{i=1}^{t} p_{i}(0) { ∑_{|α|<2r} c_{α}^{u_{i}}(−u_{i})^{α}} + ∑_{i,j=1}^{t} p_{i}(0)p_{j}(0)R_{2r}^{u_{i}}(u_{j})}

= {−K(0, 0) − 2 ∑_{j=1}^{t} p_{j}(0)R_{2r}^{0}(u_{j})

+ ∑_{i=1}^{t} p_{i}(0) {K(u_{i}, 0)) − R_{2r}^{u_{i}}(0)} + ∑_{i,j=1}^{t} p_{i}(0)p_{j}(0)R_{2r}^{u_{i}}(u_{j})} ,

where we have used (12) and (15) in the above argument.

Now, we can use the symmetry of K and re-expand K(u_{i}, 0) = K(0, u_{i}) to give

LHS ≤ {−K(0, 0) − 2 ∑_{i=1}^{t} p_{i}(0)R_{2r}^{0}(u_{i})

+ ∑_{i=1}^{t} p_{i}(0) { ∑_{|α|<2r} c_{α}^{0}(u_{i})^{α} + R_{2r}^{0}(u_{i}) − R_{2r}^{u_{i}}(0)}

+ ∑_{i,j=1}^{t} p_{i}(0)p_{j}(0)R_{2r}^{u_{i}}(u_{j})}

= {−K(0, 0) − 2 ∑_{j=1}^{t} p_{j}(0)R_{2r}^{0}(u_{j}) + K(0, 0)

+ ∑_{i=1}^{t} p_{i}(0)(R_{2r}^{0}(u_{i}) − R_{2r}^{u_{i}}(0)) + ∑_{i,j=1}^{t} p_{i}(0)p_{j}(0)R_{2r}^{u_{i}}(u_{j})}

=  − ∑_{i=1}^{t} p_{i}(0)(R_{2r}^{0}(u_{i}) + R_{2r}^{u_{i}}(0)) + ∑_{i,j=1}^{t} p_{i}(0)p_{j}(0)R_{2r}^{u_{i}}(u_{j})

≤ C_{R} max_{i,j=1,t}{∥u_{i} − u_{j}∥_{2}^{2r}, ∥u_{i}∥_{2}^{2r}} { ∑_{i,j=1}^{t} |p_{i}(0)||2 + |p_{j}(0)||}
≤ C_{R}(2C_{u})^{2r}ρ^{2r}t^{2}C_{L}(2 + C_{L}),

using (14) and (10). Combining all the constants yields (8).

To lift this up to a local error bound on the manifold requires only the Lipschitz equivalence of the Euclidean metric on IR^{d+k} restricted to M and the (arc-length) metric d(·, ·). Thus, we fix x ∈ M and, for ρ > 0, let D_{ρ}(x) = {y ∈ M : d(x, y) ≤ ρ}. Let T_{x} = IR^{d} be the tangent space to M at x considered as a d-dimensional hyperplane in IR^{d+k}. Let N_{ρ} = {y ∈ IR^{d} : ∥y∥ ≤ ρ}. Since M is an embedded C^{∞} manifold the orthogonal projection, π_{x}, from IR^{d+k} to T_{x} has a derivative at x which is the identity map when restricted to T_{x}. Hence by the inverse function theorem there exists ρ_{0} > 0 and a C^{∞} map ϕ : N_{ρ_{0}} → M which is the inverse of the orthogonal projection restricted to a closed neighborhood of x in M. Since ϕ : N_{ρ_{0}} ⊂ IR^{d} → M is an invertible C^{∞} map on a compact neighborhood, whose inverse is orthogonal projection restricted to M, there exists a constant C_{1} such that

C_{1}d(ϕ(y), ϕ(z)) ≤ ∥y − z∥ ≤ ||ϕ(y) − ϕ(z)||
≤ d(ϕ(y), ϕ(z)), y, z ∈ N_{ρ_{0}}.  (16)

When we combine the purely local Proposition 2 with the metric equivalence the principal local error estimate for s_{k} spline interpolation is obtained.

Theorem 2 Let k(x, y) be a strictly positive definite zonal kernel on M. Assume k is 2r-times continuously differentiable in each variable. Let x ∈ S ⊂ M. Suppose that the interpolation points x_{1}, …, x_{N} satisfy

max_{y∈S} min_{i=1,...,N} d(y, x_{i}) ≤ ρ,

for some sufficiently small ρ. Then, for f ∈ H,

|f(x) − s_{k}(x)| ≤ C∥f∥ρ^{r},

where C is independent of ρ and f.

Proof Let π_{x} be the orthogonal projection onto the tangent space T_{x} to M at x, as in the preceding discussion. So we assume the x ∈ T_{x} is the origin. Also if ϕ is a local inverse to π_{x}, then K(w, z) = k(ϕ(w), ϕ(z)) defines a positive definite function on a compact neighborhood of 0, satisfying the hypotheses of Proposition 2. Now taking account of the metric stretching in (16), the result in (8) carries over to M. Then combining with Theorem 1 yields the result.

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