Optimal Centers’ Allocation in Smoothing or Interpolating with Radial Basis Functions

This work was supported by FEDER/Junta de Andalucía-Consejería de Transformación Económica, Industria, Conocimiento y Universidades (Research Project A-FQM-76-UGR20, University of Granada) and by the Junta de Andalucía (Research Group FQM191).Function interpolation and approximation are classical problems of vital importance in many science/engineering areas and communities. In this paper, we propose a powerful methodology for the optimal placement of centers, when approximating or interpolating a curve or surface to a data set, using a base of functions of radial type. In fact, we chose a radial basis function under tension (RBFT), depending on a positive parameter, that also provides a convenient way to control the behavior of the corresponding interpolation or approximation method. We, therefore, propose a new technique, based on multi-objective genetic algorithms, to optimize both the number of centers of the base of radial functions and their optimal placement. To achieve this goal, we use a methodology based on an appropriate modification of a non-dominated genetic classification algorithm (of type NSGA-II). In our approach, the additional goal of maintaining the number of centers as small as possible was also taken into consideration. The good behavior and efficiency of the algorithm presented were tested using different experimental results, at least for functions of one independent variable.Junta de Andalucía-Consejería de Transformación Econímica, Industria, Conocimiento y Universidades A-FQM-76-UGR20Universidad de GranadaEuropean Regional Development FundJunta de Andalucía FQM19

LR B-splines to approximate bathymetry datasets: An improved statistical criterion to judge the goodness of fit

The task of representing remotely sensed scattered point clouds with mathematical surfaces is ubiquitous to reduce a high number of observations to a compact description with as few coefficients as possible. To reach that goal, locally refined B-splines provide a simple framework to perform surface approximation by allowing an iterative local refinement. Different setups exist (bidegree of the splines, tolerance, refinement strategies) and the choice is often made heuristically, depending on the applications and observations at hand. In this article, we introduce a statistical information criterion based on the t-distribution to judge the goodness of fit of the surface approximation for remote sensing data with outliers. We use a real bathymetry dataset and illustrate how concepts from model selection can be used to select the most adequate refinement strategy of the LR B-splines.publishedVersio

Nonlinear Matrix Approximation with Radial Basis Function Components

We introduce and investigate matrix approximation by decomposition into a sum of radial basis function (RBF) components. An RBF component is a generalization of the outer product between a pair of vectors, where an RBF function replaces the scalar multiplication between individual vector elements. Even though the RBF functions are positive definite, the summation across components is not restricted to convex combinations and allows us to compute the decomposition for any real matrix that is not necessarily symmetric or positive definite. We formulate the problem of seeking such a decomposition as an optimization problem with a nonlinear and non-convex loss function. Several modern versions of the gradient descent method, including their scalable stochastic counterparts, are used to solve this problem. We provide extensive empirical evidence of the effectiveness of the RBF decomposition and that of the gradient-based fitting algorithm. While being conceptually motivated by singular value decomposition (SVD), our proposed nonlinear counterpart outperforms SVD by drastically reducing the memory required to approximate a data matrix with the same L2 error for a wide range of matrix types. For example, it leads to 2 to 6 times memory save for Gaussian noise, graph adjacency matrices, and kernel matrices. Moreover, this proximity-based decomposition can offer additional interpretability in applications that involve, e.g., capturing the inner low-dimensional structure of the data, retaining graph connectivity structure, and preserving the acutance of images