Related papers: An Operator Learning Approach via Function-valued …
In this paper, we discuss the convergence analysis of the conjugate gradient-based algorithm for the functional linear model in the reproducing kernel Hilbert space framework, utilizing early stopping results in regularization against…
Deep kernel learning aims at designing nonlinear combinations of multiple standard elementary kernels by training deep networks. This scheme has proven to be effective, but intractable when handling large-scale datasets especially when the…
Simultaneously solving multiple related learning tasks is beneficial under a variety of circumstances, but the prior knowledge necessary to correctly model task relationships is rarely available in practice. In this paper, we develop a…
In this paper, we introduce a new kernel function which differs from previous functions, and play an important role for generating a new design of primal-dual interior point algorithms for semidefinite linear complementarity problem. Its…
We introduce a nonparametric algorithm to learn interaction kernels of mean-field equations for 1st-order systems of interacting particles. The data consist of discrete space-time observations of the solution. By least squares with…
We consider the problem of operator-valued kernel learning and investigate the possibility of going beyond the well-known separable kernels. Borrowing tools and concepts from the field of quantum computing, such as partial trace and…
This is a tutorial and survey paper on kernels, kernel methods, and related fields. We start with reviewing the history of kernels in functional analysis and machine learning. Then, Mercer kernel, Hilbert and Banach spaces, Reproducing…
The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and…
The split common fixed-point problem is an inverse problem that consists in finding an element in a fixed-point set such that its image under a bounded linear operator belongs to another fixed-point set. Recently Censor and Segal proposed…
We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves…
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…
Eigenvalue problems for elliptic operators play an important role in science and engineering applications, where efficient and accurate numerical computation is essential. In this work, we propose a novel operator inference approach for…
The Koopman operator, as a linear representation of a nonlinear dynamical system, has been attracting attention in many fields of science. Recently, Koopman operator theory has been combined with another concept that is popular in data…
We present a Representer Theorem result for a large class of weak formulation problems. We provide examples of applications of our formulation both in traditional machine learning and numerical methods as well as in new and emerging…
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…
This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator,…
We propose a novel online learning paradigm for nonlinear-function estimation tasks based on the iterative projections in the L2 space with probability measure reflecting the stochastic property of input signals. The proposed learning…
In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape…
In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of…
This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as…