相关论文: $H$-Sets for Kernel-Based Spaces

The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 were computed by Feigin and Veselov [Int. Math. Res. Not. 2002…

We consider the following open questions. Fix a Hilbert function, $h$, that occurs for a reduced zero-dimensional subscheme of $\mathbb P^2$. Among all subschemes, $X$, with Hilbert function $h$, what are the possible Hilbert functions and…

We present a systematic study of the family of positive definite (p.d.) kernels with the use of their associated feature maps and feature spaces. For a fixed set $X$, generalizing Loewner, we make precise the corresponding partially ordered…

We show that if a reproducing kernel Hilbert space $H_K,$ consisting of functions defined on ${\bf E},$ enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number $n,$ and…

In this paper, a quantum computational framework for algebraic topology based on simplicial set theory is presented. This extends previous work, which was limited to simplicial complexes and aimed mostly to topological data analysis. The…

Given a hypergraph $H = (V,E)$, what is the smallest subset $X \subseteq V$ such that $e \cap X \neq \emptyset$ holds for all $e \in E$? This problem, known as the hitting set problem, is a basic problem in parameterized complexity theory.…

The performance of reproducing kernel Hilbert space-based methods is known to be sensitive to the choice of the reproducing kernel. Choosing an adequate reproducing kernel can be challenging and computationally demanding, especially in…

The Computation of discrete Contractive semigroups becomes necessary when we deal with several types of evolution equations in Discretizable Hilbert spaces, in this work we study some properties of the discrete forms of the contractive…

The vanishing ideal I of a subspace arrangement is an intersection of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of a product J of the…

This work presents a generalized notion of multiset mapping thus resolving a long standing obstacle in structural study of multiset processing. It has been shown that the mapping defined herein can model a vast array of notions as special…

This paper explores the use of "generated sets" $\{ \{ k \boldsymbol{\zeta} \} : k = 1, \ldots, n \}$ for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent…

We study scattered piecewise interpretable Hilbert spaces from a model theoretic point of view. We establish strong connections between the Hilbert space structure theorems of [Chevalier Hrushovski 2021] and the model theoretic notions of…

A subset $M$ of a separable Hilbert space $H$ is $\ell^1$-bounded if there exists a Riesz basis $\mathcal{F} = \{e_n\}_{n \in \mathbb{N}}$ for $H$ such that $\sup_{x \in M} \sum_{n \in \mathbb{N}} |\langle x, e_n\rangle| < \infty.$ A…

We present several generative and predictive algorithms based on the RKHS (reproducing kernel Hilbert spaces) methodology, which, most importantly, are scale up efficiently with large datasets or high-dimensional data. It is well recognized…

The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in…

This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt's theorem…

This article gives a new insight of kernel-based (approximation) methods to solve the high-dimensional stochastic partial differential equations. We will combine the techniques of meshfree approximation and kriging interpolation to extend…

A proof that minimum uncertainty states of the simplest periodic quantum system exist in a state space that is represented by a Colombeau algebra of generalised functions but not in Hilbert space or in the space of Schwartz distributions is…

This paper studies an intriguing phenomenon related to the good generalization performance of estimators obtained by using large learning rates within gradient descent algorithms. First observed in the deep learning literature, we show that…

H-measures and semiclassical (Wigner) measures were introduced in earlyn 1990s and since then they have found numerous applications in problems involving $\mathrm{L}^2$ weakly converging sequences. Although they are similar objects, neither…