相关论文: Eignets for function approximation on manifolds
In this paper we analyze a greedy procedure to approximate a linear functional defined in a Reproducing Kernel Hilbert Space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals, including…
In this paper we first identify a basic limitation in gradient descent-based optimization methods when used in conjunctions with smooth kernels. An analysis based on the spectral properties of the kernel demonstrates that only a vanishingly…
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower…
Neural operators aim to learn mappings between infinite-dimensional function spaces, but their performance often degrades on complex or irregular geometries due to the lack of geometry-aware representations. We propose the Finite Element…
Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…
An invex function generalizes a convex function in the sense that every stationary point is a global minimizer. Recently, invex functions and their subclasses have attracted attention in signal processing and machine learning. However,…
We present simple, user-friendly bounds for the expected operator norm of a random kernel matrix under general conditions on the kernel function $k(\cdot,\cdot)$. Our approach uses decoupling results for U-statistics and the non-commutative…
We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space. To this end, we use an additive family to map the input…
The family of Mat\'ern kernels are often used in spatial statistics, function approximation and Gaussian process methods in machine learning. One reason for their popularity is the presence of a smoothness parameter that controls, for…
We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal…
The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the…
The aim of this work is to introduced the concept of the best one-sided approximation of unbounded functions in weighted space by using algebraic operators in terms the average modulus of smoothness. We also show an estimate of the degree…
In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…
This paper considers the problem of minimizing the summation of a differentiable function and a nonsmooth function on a Riemannian manifold. In recent years, proximal gradient method and its invariants have been generalized to the…
Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…
A central problem in machine learning is often formulated as follows: Given a dataset $\{(x_j, y_j)\}_{j=1}^M$, which is a sample drawn from an unknown probability distribution, the goal is to construct a functional model $f$ such that…
We study approximation error bounds of isogeometric function spaces on a specific type of singularly parameterized domains. In this context an isogeometric function is the composition of a piecewise rational function with the inverse of a…
The purpose of this paper is to establish that for any compact, connected C^{\infty} Riemannian manifold there exists a robust family of kernels of increasing smoothness that are well suited for interpolation. They generate Lagrange…
This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are…
We consider optimal distributed computation of a given function of distributed data. The input (data) nodes and the sink node that receives the function form a connected network that is described by an undirected weighted network graph. The…