Related papers: Minimum complexity interpolation in random feature…
This monograph develops a unified, application-driven framework for kernel methods grounded in reproducing kernel Hilbert spaces (RKHS) and optimal transport (OT). Part I lays the theoretical and numerical foundations on positive-definite…
Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of…
Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproducing kernel Hilbert space. Beyond this classical setting,…
Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically…
Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures $\mu$ from some set $M$ to functions in a reproducing kernel Hilbert space (RKHS) with kernel $k$. The RKHS distance of two…
Meshfree methods, including the reproducing kernel particle method (RKPM), have been widely used within the computational mechanics community to model physical phenomena in materials undergoing large deformations or extreme topology…
This paper studies the generalization properties of a recently proposed kernel method, the Random Feature models with Learnable Activation Functions (RFLAF). By applying a data-dependent sampling scheme for generating features, we provide…
The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their…
This paper addresses the problem of approximating an unknown function from point evaluations. When obtaining these point evaluations is costly, minimising the required sample size becomes crucial, and it is unreasonable to reserve a…
We study algorithms to estimate geometric properties of raw point cloud data through implicit surface representations. Given that any level-set function with a constant level set corresponding to the surface can be used for such…
Traditionally, kernel methods rely on the representer theorem which states that the solution to a learning problem is obtained as a linear combination of the data mapped into the reproducing kernel Hilbert space (RKHS). While elegant from…
We present a novel diffusion scheme for online kernel-based learning over networks. So far, a major drawback of any online learning algorithm, operating in a reproducing kernel Hilbert space (RKHS), is the need for updating a growing number…
We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive…
Kernel methods form a powerful, versatile, and theoretically-grounded unifying framework to solve nonlinear problems in signal processing and machine learning. The standard approach relies on the kernel trick to perform pairwise evaluations…
Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we…
A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing…
We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a…
In this paper, we give a new generalization error bound of Multiple Kernel Learning (MKL) for a general class of regularizations, and discuss what kind of regularization gives a favorable predictive accuracy. Our main target in this paper…
We investigate how to train kernel approximation methods that generalize well under a memory budget. Building on recent theoretical work, we define a measure of kernel approximation error which we find to be more predictive of the empirical…
Recent empirical work has shown that hierarchical convolutional kernels inspired by convolutional neural networks (CNNs) significantly improve the performance of kernel methods in image classification tasks. A widely accepted explanation…