Related papers: On the optimal shape parameter for kernel methods:…
A function approximation method is developed that aims to approximate a function in a small neighborhood of a state that travels within a compact set. The development is based on the theory of universal reproducing kernel Hilbert spaces…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
Kernel methods are powerful and flexible approach to solve many problems in machine learning. Due to the pairwise evaluations in kernel methods, the complexity of kernel computation grows as the data size increases; thus the applicability…
The radial basis function (RBF) approach is applied in predicting nuclear masses for 8 widely used nuclear mass models, ranging from macroscopic-microscopic to microscopic types. A significantly improved accuracy in computing nuclear masses…
The primary hyperparameter in kernel regression (KR) is the choice of kernel. In most theoretical studies of KR, one assumes the kernel is fixed before seeing the training data. Under this assumption, it is known that the optimal kernel is…
Kernel methods give powerful, flexible, and theoretically grounded approaches to solving many problems in machine learning. The standard approach, however, requires pairwise evaluations of a kernel function, which can lead to scalability…
A causal rate distortion function with a general fidelity criterion is formulated on abstract alphabets and a coding theorem is derived. Existence of the minimizing kernel is shown using the topology of weak convergence of probability…
Let $\Omega\subset \mathbb{R}^d$ be a bounded domain. We consider the problem of how efficiently shallow neural networks with the ReLU$^k$ activation function can approximate functions from Sobolev spaces $W^s(L_p(\Omega))$ with error…
In this paper the infinite horizon optimal regulation problem is solved online for a deterministic control-affine nonlinear dynamical system using the state following (StaF) kernel method to approximate the value function. Unlike…
We propose a kernel-based nonparametric framework for mean-variance optimization that enables inference on economically motivated shape constraints in finance, including positivity, monotonicity, and convexity. Many central hypotheses in…
We show how the essential spectral radius of a bounded positive kernel, acting on bounded functions, is linked to its lower approximation by certain absolutely continuous kernels. The standart Doeblin's condition can be interpreted in this…
Kernel methods, particularly kernel ridge regression (KRR), are time-proven, powerful nonparametric regression techniques known for their rich capacity, analytical simplicity, and computational tractability. The analysis of their predictive…
Kernel learning methods are among the most effective learning methods and have been vigorously studied in the past decades. However, when tackling with complicated tasks, classical kernel methods are not flexible or "rich" enough to…
The objective of this study is to investigate the efficient determination of $C$ and $\gamma$ for Support Vector Regression with RBF or mahalanobis kernel based on numerical and statistician considerations, which indicates the connection…
These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing…
This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections,…
There is a constant c contained in the famous radial basis function shifted surface spline. It's called shape parameter. RBF people only know that this constant is very influential, while its optimal choice is unknown. This paper presents…
We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to…
This paper investigates the approximation properties of shallow neural networks with activation functions that are powers of exponential functions. It focuses on the dependence of the approximation rate on the dimension and the smoothness…
Despite their many appealing properties, kernel methods are heavily affected by the curse of dimensionality. For instance, in the case of inner product kernels in $\mathbb{R}^d$, the Reproducing Kernel Hilbert Space (RKHS) norm is often…