Interpolation of scattered data: Distance matrices and conditionally positive definite functions

By Charles A. Micchelli. 1986

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Fastfood: Approximate Kernel Expansions in Loglinear Time (2014, background): “(RKHS) (Aronszajn, 1944; Wahba, 1990; Micchelli, 1986) and one may show that estimators using norms in feature space as penalty are equivalent to estimators using smoothness in an RKHS (Girosi, 1998; Smola et al.”
On the Schoenberg Transformations in Data Analysis: Theory and Illustrations (2010, background): “The class Φ1 is central to Brownian and fractional Brownian motion (see e.g. Alpay et al. 2009), while lower-order classes Φp≤3 are fundamental in Geostatistics (see e.g. Christakos 1984) and spatial interpolation (see e.g. Micchelli 1986; Stein 1999).”
Deep Sets (2017, background): “This is a well established property (see e.g. (Micchelli, 1986) for details in the case of radial basis function networks).”
Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations (2000, methodology): “The procedure rst requires the construction of a matrix W = [B; 2H]T (25) 7”
Preventing Over-Fitting during Model Selection via Bayesian Regularisation of the Hyper-Parameters (2007, background): “The Gram matrix for the radial basis function kernel is thus of full rank (Micchelli, 1986), and so the kernel model is able to form an arbitrary shattering of the data.”
The Nature of Statistical Learning Theory (2000, background): “For the theory of RBF machines see (Micchelli, 1986), (Powell, 1992).”
On the Approximation Order and Numerical Stability of Local Lagrange Interpolation by Polyharmonic Splines (2003, background): “This linear system has always a solution, which is unique, provided that the point set X is Pd m-unisolvent [19].”
A Study on SMO-Type Decomposition Methods for Support Vector Machines (2006, unknown): “definite [ 22 ] and we have the result of Corollary 1.”
Concurrent learning for convergence in adaptive control without persistency of excitation (2010, background): “Function NN is reduced to selecting distinct points for storage [65], [36].”
Use of radial basis functions for solving the second‐order parabolic equation with nonlocal boundary conditions (2008, background): “In the case of GA, MQ, IMQ, and IQ radial basis functions, it is shown that interpolation matrix is nonsingular [42, 43].”
Limit problems for interpolation by analytic radial basis functions (2008, background): “The first nonzero entry of U in row j, counted from left to right, corresponds to a unique column multi-index α, thus defining the sequence (5).”