Curvature-Based Characterization of Radial Basis Functions: Application to Interpolation

. Choosing the scale or shape parameter of radial basis functions (RBFs) is a well-documented but still an open problem in kernel-based methods. It is common to tune it according to the applications, and it plays a crucial role both for the accuracy and stability of the method. In this paper, we first devise a direct relation between the shape parameter of RBFs and their curvature at each point. This leads to characterizing RBFs to scalable and unscalable ones. We prove that all scalable RBFs lie in the 1 c 2 -class which means that their curvature at the point x j is proportional to 1 c 2 j , where c j is the corresponding spatially variable shape parameter at x j . Some of the most commonly used RBFs are then characterized and classified accordingly to their curvature. Then, the fundamental theory of plane curves helps us recover univariate functions from scattered data, by enforcing the exact and approximate solutions have the same curvature at the point where they meet. This leads to introducing curvature-based scaled RBFs with shape parameters depending on the function values and approximate curvature values of the function to be approximated. Several numerical experiments are devoted to show that the method performs better than the standard fixed-scale basis and some other shape parameter selection methods


Introduction
Given a set of n distinct points {x j } n j=1 ⊂ R d and corresponding data values {f j } n j=1 , the RBF interpolant is given by where ϕ(r), r ≥ 0, is some radial function, and ∥.∥ is the Euclidean norm throughout the paper. The expansion coefficients λ j are determined from the interpolation conditions s(x j ) = f j for j = 1, . . . , n, which leads to a symmetric linear system Aλ = f , where A = [ϕ(∥x i − x j ∥)] 1≤i,j≤n . The existence of a solution is assured for positive definite RBFs and also for conditionally positive definite RBFs by adding a lower degree polynomial to (1.1). We can introduce a shape parameter as ϕ r c allowing to scale the basis function ϕ making it flatter as c → ∞ and spiky as c → 0. In order for the system matrix to be well-conditioned, the shape parameter c must not be too large, but large shape parameters are required to obtain better accuracy for the interpolation with RBF (cf. e.g., [16]).
In spatially scaled RBFs the shape parameter varies with the centers getting a more flexible approximation. This means working with functions ϕ ∥x−xj ∥ cj for 1 ≤ j ≤ n. For more details see [2,5,9]. In [1], the scale parameter is considered as a scale function and treating it as an additional coordinate. This approach can be fully understood as the standard fixed-scale method applied to a certain sub-manifold of R d+1 . For a scale function c : R d → (0, ∞), a variably scaled kernel (VSK) can be written as ϕ(∥x − x j ∥ 2 + (c(x) − c(x j )) 2 ), in which the tuning of the shape parameter is replaced by the choice of a suitable scale function depending on the data. A question arises: how to find the scale function? Recently, Rossini [14] considered the problem of interpolating functions with gradient faults and tried to provide a strategy to choose a proper scaling function. But it needs the knowledge of the discontinuity curve.
In this paper, we propose an explicit formula for the shape parameter of RBFs based on the value of the curvature in each point. This leads to characterizing RBFs to scalable and unscalable ones. Some of the most commonly used RBFs are then characterized. Then we use the fundamental theory of plane curves for recovering univariate functions from scattered data by enforcing the exact and approximate solutions have the same curvature at the point where they meet. This leads to introducing curvature-based scaled RBFs.
The paper is organized as follows. In the next section, we provide some basic definitions and theorems from differential geometry [11]. The characterization of RBFs based on the relation between their shape parameter and curvature at each point is outlined in Section 3. In Section 4, curvature-based scaled RBFs are introduced in order to interpolate univariate functions. Some numerical examples are presented in Section 5. The last section is devoted to a brief conclusion and description of future works.

Differential geometry basics
Let us consider parametric vector-valued curves α : [a, b] ⊂ R → R n , n ≥ 1. Definition 1. Let α : [a, b] → R n be a regular curve. Then are the unit tangent vector of α(t) and its arc length from a to t, respectively.
Theorem 1. Let C be a regular plane curve given by α(t). Then the curvature κ of C at t is given by [11] κ[α](t) = dT ds , which measures the rate of change of the unit tangent vector with respect to arc length. It is a measure of how much a curve deviates from a straight line.
The following definition can be simply resulted from the above theorem. (2.1)

Curvature-based characterization of RBFs
In this section, we present the relation between the spatially variable shape parameter and the curvature of RBFs. Then, we can classify RBFs to scalable and unscalable functions. Hereafter, we assume that ϕ(r) is at least in C 2 [0, ∞).
In the sequel, we discuss how scalable RBFs are described using Theorem 2.
Definition 3. The radial basis function ϕ is called scalable if it has nonzero curvature at the origin. Otherwise, we call it unscalable.
Remark 2. Scalability means that we can change the curvature of the basis functions (3.1) at the center point x j by changing the corresponding shape parameter c j .
Remark 3. It can be noted from Definition 3 that the RBFs which are not at least C 2 at the origin as well as RBFs with zero curvature at the origin are not scalable. They are shape parameter-free.

Classification of RBFs by curvature
Now, we are going to classify some commonly used RBFs into scalable and unscalable functions. According to Theorem 2, for RBF ϕ(r) we consider ψ(r) = ϕ r c and compute its curvature at the origin.
Since for n = 1, the singularities of the function and first derivative at the origin are removable, but the singularity of the second derivative at the origin is not. TPS are not C 2 at the origin for n = 1. Moreover, by using (2.1), we conclude that κ ψ (r) r=0 = 0, for n ≥ 2. As it happens with radial powers, TPS are unscalable and the use of a shape parameter c with TPS is pointless.
Hyperbolic Secant: ϕ(r) = sech (r) . This function is C ∞ at the origin. Since RTH: It is a new transcendental RBF of the form ϕ(r) = r tanh (r) introduced for the first time by Heidari et al. [7]. It can be scaled as r tanh r c which is a smooth approximant to r with higher accuracy and better convergence properties than the MQ RBF. But, according to Theorem 2, we scale it as ψ(r) = r c tanh r c . This function is C ∞ at the origin. Since using (2.1) we obtain κ ψ (r) r=0 = 2 c 2 , which leads to η = 2 in (3.5).
Bump function: This function is where J ν is the Bessel function of the first kind of order ν. While these functions are not defined at the origin, they can be extended to be C ∞ (R d ) [3]. Since Again, using (2.1), we conclude that κ ψ (r) Matérn: ϕ(r) = r ν K ν (r), ν > 0, where K ν is the modified Bessel function of the second kind of order ν, that can be defined as a function of the Bessel function of the first kind as follows Then, where M = π/(2 sin(πν)). By (2.1) we then have indicates the Laguerre polynomial of degree n and order 1/2, that is Then, We summarize our findings in Table 1.

Interpolating univariate functions via curvature-based scaled radial basis functions
In this section, we intend to recover functions f : At first, the fundamental theorem of curve theory [11] is given, where τ denotes torsion. Gen.
Then, the Corollary 1 helps us recover univariate functions from scattered data, by enforcing the exact and approximate solutions have the same curvature at the point where they meet.
where g is a slowly increasing function. Then, for the interpolant η the positive constant appeared in Theorem 2 and κ j = κ f (x)| x=xj , we get By using (4.1) for k = 0 we get

The interpolation condition gives
for sufficiently large n. Now, by considering (2.1), we get Then, according to (4.1) and (4.2), the first and second derivatives appeared in the above equation approaches to 0 for j ̸ = i. So, Therefore, (4.3) results the proof. ⊓ ⊔ Remark 4. Our theoretical results of Section 3 justify the well-known fact that the classical RBF approximation has difficulty in accurately approximating flat functions [8]. Actually, the curvature of flat functions is almost small and (3.5) leads to large value of the shape parameter c which leads to severe ill-conditioning problem. But, the factor g(n) in (4.2) controls the shape parameter c j does not become too large. So, in application, we work with the following new scaled RBFs which use a more simple and stable version of (4.2).
Definition 4. Let ϕ be a scalable RBF with ϕ(0) ̸ = 0 satisfying (4.1). The curvature-based scaled radial basis functions (CBS RBFs) are defined as where c j = 1/g(n) η |f j |/(κ j |ϕ(0)|). . . , f n ] T . According to Theorem 4, using (4.5) makes the exact and approximate solutions have the same curvature at the point where they meet, for sufficiently large number of data points and avoids from numerical instability according to Remark 4.
In Table 2, we listed those RBFs having conditions of Definition 4.
Hyperbolic Secant sech (r) 1 Remark 5. With a spatially variable c j , the standard proofs for non-singularity of the RBF interpolation matrix A no longer apply. But for small values of c j (c j → 0), the A-matrices corresponding to the RBFs in Table 2, will tend to the identity matrix, and non-singularity is obviously assured. Moreover, for large values of c j (c j → ∞) the singularities are unlikely to occur [6]. On the other hand, the factor g(n) in (4.6) controls the shape parameter c j does not become too large.
Remark 6. The relation (4.6) shows that the shape parameter selection in the univariate RBF interpolation method with scalable RBFs relates to the function values and curvature values of the function to be approximated. Moreover, we can observe the following.
The scalable RBFs are not good candidates for approximating functions having zero curvature everywhere, like lines. So, one should use the classical fixed-scale RBF interpolation method with an unscalable RBF. Clearly, the power RBF ϕ(r) = r is the most appropriate one.
The classical RBF interpolation method with scalable RBFs may lead to inaccurate solutions for functions having very small values of κ j .
We discuss these situations in the next section devoted to numerical experiments. Now, consider the two variable function f : R 2 → R. Then, it corresponds with a parametric surface F : R 2 → R 3 called the graph of f by F (x, y) = (x, y, f (x, y)). The fundamental theorem of surfaces states that the congruent parametric surfaces in R 3 have the same first and second fundamental forms and conversely two parametric surfaces in R 3 with the same first and second fundamental forms are congruent [12]. Discussion about the concepts of the first and second fundamental forms and the methodology of recovering multivariate functions in differential geometry points of view, needs a detailed study, and we leave it to our further work.

Numerical results
We now provide some examples that show different roles of the curvature-based scaled RBFs. We take different number of equidistant and Chebyshev-Gauss-Lobatto (CGL) center points in [a, b]. We use the maximum absolute error norm L ∞ = max 1≤i≤m |f i −f i |, where f andf represent the exact and approximate solutions, respectively. In order to compute interior curvature values κ j , j = 2, . . . , n − 1, we use the formula for the curvature of the circle which circumscribes the triangle formed by three two-dimensional points (x j−1 , f j−1 ), (x j , f j ), and (x j+1 , f j+1 ) [13]. That would be an approximation for the curvature at the middle point x j . The computation goes as follows.
where B, C, and D are the lengths of three sides, S is the area of the triangle, and κ j is the curvature of circumscribing circle. The values of κ 1 and κ n can be obtained by (2.1) using the following numerical differentiation formulas which are of second-order accuracy [17].
Remark 7. Since division by very small numbers leads to magnification of the absolute error, we use a tolerance σ that one avoids using f j in (4.6) for each center point with |f j | < σ. We also set κ j = ϵ M for each center point with κ j < ϵ M , with ϵ M ≈ 2.204e − 16 the machine epsilon.
The optimal shape parameter c for the standard fixed-scale RBF interpolation method is founded by trial end error. In all tests, we took σ = 0.3 and m = 125 equidistant evaluation points. All experiments have been done using an Intel(R) Core(TM) i5-10210U CPU @ 1.60GHz, 2.11 GHz.

Test problem 1
In the first experiment, we approximate the function (cf. [10]) Absolute errors using the proposed CBS and classical fixed-scale RBFs interpolation methods with the Inverse Quadric RBF (β = −1), and Gaussian RBF, using g(n) = ln(n), and 200 equidistant center points are shown in Figures 1,  and 2, respectively. It should be noted that the results for the CGL center points are the same. Results show that the proposed method leads to more accurate results than the fixed-scale RBFs interpolation method. Furthermore, the numerical errors in Figures 1-2 which are of 10 −13 and 10 −15 are really smaller than those reported in [10, p. 904], which is of order 10 −6 with 1024 center points.
The exact and approximate solutions of f 1 (x) and κ(f 1 (x)) using the CBS RBFs interpolation method with the Gaussian RBF are given in Figures 3(a)  and 3(b), respectively. Now, let us change (5.1) into the highly oscillating function

Test problem 2
The next function we test is (cf. [15]) It is flat near the boundaries but has a steep gradient near x = 0. Absolute errors using the proposed CBS and classical fixed-scale RBFs interpolation methods with the Hyperbolic secant RBF, using g(n) = √ n, c = 0.09 and 200, 300 equidistant center points are shown in Figure 6.  It is flat near the boundaries but has a steep gradient near x = 0. Absolute errors using the proposed CBS and classical fixed-scale RBFs interpolation methods with the Hyperbolic secant RBF, using g(n) = √ n, c = 0.09 and 200, 300 equidistant center points are shown in Figure 6.
Results show that the proposed method leads to more accurate results than the fixed-scale RBFs interpolation method. Moreover, the numerical errors are smaller than those reported in [15] for the random, linearly, and exponentially varying shape parameters.
The exact and approximate solutions of f 3 (x) and κ(f 3 (x)) using the CBS RBFs interpolation method with the Gaussian RBF, using g(n) = √ n, and 200 CGL center points are given in Figures 7(a) and 7(b), respectively.

Test problem 3 (Runge function)
In the last example, we consider the Runge function on [−1, 1], that is The L ∞ error norms using the proposed CBS and classical fixed-scale RBFs interpolation methods with the Gaussian RBF, using g(n) = √ n, c = 0.08, and different number of equidistant and CGL center points are reported in Tables 3  and 4, respectively. It can be noted from Tables 3-4, that the proposed method leads to more accurate results than the fixed-scale RBFs interpolation method. The exact and approximate solutions of f 4 (x) and κ(f 4 (x)) using the CBS RBFs interpolation method for n = 200 equidistant center points with the Inverse Multiquadric RBF (β = − 1 2 ), g(n) = √ n are given in Figures 8(a) and 8(b), respectively. In Figure 8(a), we see that no Runge-type oscillations arise. In order to show the important role of the values κ j in the shape parameter formula (4.6), we multiply 25 by the large number 10 9 , and work with f 5 (x) = 1 1 + 25 × 10 9 × x 2 .   The exact and approximate solutions of f 5 (x) and κ(f 5 (x)) using the CBS RBFs interpolation method for n = 101 equidistant center points with the Inverse Multiquadric RBF and g(n) = √ n are given in Figures 9(a) and 9(b), respectively. As we can see in Figure 9(b), this change leads to the large value of curvature at the points x = 0. Now, we plot the absolute errors using the proposed CBS and classical fixed-scale RBFs interpolation methods with the Gaussian RBF, using the same previous parameters g(n) = √ n and c = 0.08 in Figure 10. This figure reveals superiority of the proposed method.

Conclusions
In this paper, we first introduce an explicit technique for the shape parameter selection based on the curvature characterization, at r = 0, for RBFs. This leads to characterizing RBFs to scalable and unscalable ones. Some of the most commonly used RBFs are characterized and discussed. Then we use the fundamental theory of plane curves for recovering univariate functions from scattered data by enforcing the exact and approximate solutions have the same curvature at the point where they meet. This leads to introducing curvaturebased scaled RBFs. Several numerical experiments are devoted to show that the method performs better than the standard fixed-scale basis and some other shape parameter selection methods. In order to generalize the proposed method to higher dimensions, we should work with parametric surfaces and some more tools in differential geometry, like first and second fundamental forms. We leave this to our further works.