SIAM Journal on Scientific and Statistical Computing
The fast Gauss transform with variable scales
SIAM Journal on Scientific and Statistical Computing
Fast Fourier transforms for nonequispaced data
SIAM Journal on Scientific Computing
Norm estimates for inverses of Toeplitz distance matrices
Journal of Approximation Theory
On shifted cardinal interpolation by Gaussians and multiquadrics
Journal of Approximation Theory
A New Error Estimate of the Fast Gauss Transform
SIAM Journal on Scientific Computing
Improved Fast Gauss Transform and Efficient Kernel Density Estimation
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
Fast convolution with radial kernels at nonequispaced knots
Numerische Mathematik
Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation
SIAM Journal on Numerical Analysis
Rapid evaluation of radial basis functions
Journal of Computational and Applied Mathematics
A sharp error estimate for the fast Gauss transform
Journal of Computational Physics
Fast Radial Basis Function Interpolation via Preconditioned Krylov Iteration
SIAM Journal on Scientific Computing
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Fast Gauss Transforms based on a High Order Singular Value Decomposition for Nonlinear Filtering
SSP '07 Proceedings of the 2007 IEEE/SP 14th Workshop on Statistical Signal Processing
Fast radial basis function interpolation with Gaussians by localization and iteration
Journal of Computational Physics
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
Applied Numerical Mathematics
Applied Numerical Mathematics
IEEE Transactions on Signal Processing
Preconditioning for radial basis functions with domain decomposition methods
Mathematical and Computer Modelling: An International Journal
Computers & Mathematics with Applications
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The Fast Gauss Transform is an algorithm for summing a series of Gaussians which is sometimes much faster than direct summation. Gaussian series in d dimensions are of the form @?"j"="1^N@l"jexp(-[@a/h]^2@?x-x"j@?^2) where the x"j are the centers, h is the average separation between centers and @a is the relative inverse width parameter. We show that the speed-up of the Fast Gauss Transform is bounded by a factor @W(@a). When @a@?1, @W can be large. However, when applied to Gaussian radial basis function interpolation, it is difficult to apply the Gaussian basis in this parameter range because the interpolation matrix is exponentially ill-conditioned: the condition number @k~(1/2)exp@p^24@a^2 for a uniform, one-dimensional grid, and larger still in two dimensions or when the grid is irregular. Furthermore, the Gaussian RBF interpolant is ill-conditioned for most series in the sense that the interpolant is the small difference of terms with exponentially large coefficients. Fornberg and Piret developed a ''QR-basis'' that ameliorates this difficulty for approximations on the surface of a sphere, but because the recombined basis functions are perturbed spherical harmonics, not Gaussians, the Fast Gauss Transform is no longer applicable. The solution of the interpolation matrix system by a preconditioned iteration is less sensitive to condition numbers than direct methods because iterations are self-correcting and also because the preconditioning reduces the spread of the eigenvalues. However, each iteration requires a matrix-vector multiply which is fast only if this operation can be performed by some species of Fast Summation. When @a~O(1), alas, we show that @W is not large and the Fast Gauss Transform is not accelerative. Gaussian RBFs are unusual among RBF species through the absence of long-range interactions due to the exponential decay of the Gaussians with distance from their centers; many other RBF species do have long-range interactions, and it is well-established that these can be accelerated by fast multipole and treecode algorithms. We offer a less rigorous scale analysis argument to explain why the underlying difficulty in accelerating short-range interactions is not peculiar to the Gaussian RBF basis or to the Fast Gauss Transform, but rather is likely to be a generic difficulty in accelerating the short-range interactions of almost any RBF basis with almost any Fast Summation.