Search

We investigate the use of the so-called variably scaled kernels (VSKs) for learning tasks, with a particular focus on support vector machine (SVM) classifiers and kernel regression networks (KRNs). Concerning the kernels used to train the models, under appropriate assumptions, the VSKs turn out to be more expressive and more stable than the standard ones. Numerical experiments and applications to breast cancer and coronavirus disease 2019 (COVID-19) data support our claims. For the practical implementation of the VSK setting, we need to select a suitable scaling function. To this aim, we propose different choices, including for SVMs a probabilistic approach based on the naive Bayes (NB) classifier. For the classification task, we also numerically show that the VSKs inspire an alternative scheme to the sometimes computationally demanding feature extraction procedures. [ABSTRACT FROM AUTHOR]

The data driven extrapolation requires the definition of a functional model depending on the available data and has the application scope of providing reliable predictions on the unknown dynamics. Since data might be scattered, we drive our attention towards kernel models that have the advantage of being meshfree. Precisely, the proposed numerical method makes use of the so-called Variably Scaled Kernels (VSKs), which are introduced to implement a feature augmentation-like strategy based on discrete data. Due to the possible uncertainty on the data and since we are interested in modelling the behaviour of the target functions, we seek for a regularized solution by ridge regression. Focusing on polyharmonic splines, we investigate their implementation in the VSK setting and we provide error bounds in Beppo–Levi spaces. The performances of the method are then tested on functions showing exponential or rational decay. Comparisons with Support Vector Regression (SVR) are also carried out and highlight that the proposed approach is effective, particularly since it does not require to train complex architecture constructions. [ABSTRACT FROM AUTHOR]

We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure. [ABSTRACT FROM AUTHOR]

Abstract Edge detection is a widely used tool in signal/image processing with the aim of identifying abrupt changes or discontinuities in a signal/digital image. For the detection of jump discontinuities in 1D problems, we present an iterative method based on interpolation with Variably Scaled Kernels (VSKs). This is shown to outperform an existing iterative edge detection method based on multiquadric (MQ) radial basis function interpolation. To extend our purely one-dimensional edge detector to any dimension, we then introduce an innovative non iterative technique that detects jumps/edges by identifying the local maxima of the normalized absolute values of the RBF interpolant coefficients. The RBF interpolant is built-upon the compactly supported C 2 Wendland function and exploits its advantageous properties to provide a robust and low-cost method. Numerical examples in 1D and 2D are included to illustrate its effectiveness and efficiency. In the context of edge detection for digital images, comparisons with Canny method are also presented. [ABSTRACT FROM AUTHOR]