C. Carmeli, E. De Vito, A. Toigo, V. Umanità
This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem.
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This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on reproducing kernels in vector-valued reproducing kernel Hilbert spaces. In particular we extend reproducing kernels to relative reproducing kernels and prove some theorems in this subject.
Vector-valued Reproducing Kernel Banach Spaces with Group Lasso Norms
该研究为稀疏多任务学习的核方法建立数学基础，构建了带$\ell_{p,1}$群Lasso范数的向量值再生核巴拿赫空间，证明其满足线性表示定理并提出适配的再生核，可用于稀疏多任务学习。
This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representat…
Targeting at sparse multi-task learning, we consider regularization models with an $\ell^1$ penalty on the coefficients of kernel functions. In order to provide a kernel method for this model, we construct a class of vector-valued reproducing kernel Banach spaces with the $\ell^1$ norm. The notion of multi-task admissible kernels is proposed so that the constructed spaces could have desirable properties including the crucial linear representer theorem. Such kernels are related to bounded Lebesgue constants of a kernel interpolation question. We study the Lebesgue constant of multi-task kernels…
Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBS. The theory is then applied to multi-task machine learning. Especially, the representer theorem and characterization equations for the minimizer of regularized learning schemes in vector-valued RKBS are established.
Motivated by the need of processing functional-valued data, or more general, operatorvalued data, we introduce the notion of the operator reproducing kernel Hilbert space (ORKHS). This space admits a unique operator reproducing kernel which reproduces a family of continuous linear operators on the space. The theory of ORKHSs and the associated operator reproducing kernels are established. A special class of ORKHSs, known as the perfect ORKHSs, are studied, which reproduce the family of the standard point-evaluation operators and at the same time another different family of continuous linear op…
We give necessary and sufficient conditions under which the reproducing kernel of a Pontryagin space of $d \times 1$ vector polynomials is determined by a generalized Nevanlinna pair of $d \times d$ matrix polynomials.
The aim of this paper is to present a unified framework in the setting of Hilbert $C^*$-modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and $C^*$-valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the $C^*$-algebra of adjointable operators acting on a Hilbert $C^*$-module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert $C^*$-modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and …
Mercer's Theorem for Vector-Valued Reproducing Kernel Hilbert Spaces in Kaplansky-Hilbert Modules over $L_{\infty}(Ω)$
本文在$L_{\infty}(Ω)$上Kaplansky-希尔伯特模块上扩展了Mercer定理，通过构造等距同构和向量值提升公理化体系，建立了偏积分算子的三个基本性质的等价性。
Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$-tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $Ω$, where $E$ is a separable Hilbert space and $Ω$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the c…
We present a characterization for a positive definite operator valued kernel to be universal or $C_{0}$-universal, and apply these characterizations to a family of operator valued kernels that are shown to be well behaved. Later, we obtain a characterization for an operator valued differentiable kernel to be $C^{q}$-universal and $C_{0}^{q}$-universal. In order to obtain such characterization and examples we generalize some well known results concerning the structure of differentiable kernels to the operator valued context. On the examples is given an emphasis on the radial kernels on Euclidea…
Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators
研究神经网络背后函数空间，提出向量值再生核巴拿赫空间伴随对概念，扩展核定义，证明浅度\(\mathbb{R}^d\)值神经网络及神经算子模型属于积分vv - RKBS，并建立表示定理，揭示函数空间优化与神经架构的关系。
A Survey on Reproducing Kernel Krein Spaces
本文综述了再生核 Krein 空间及其与算子值 Hermitian 核的相互作用，利用线性化与 Kolmogorov 分解等抽象概念探讨存在性与唯一性，并强调其算子值域特征及 Szego 核的普遍性。
Bounded Composition Operators on Hilbert Space of Complex-Valued Harmonic Functions
本文研究复值调和函数希尔伯特空间上的复合算子，刻画了其有界性、等距性与生成自映射类型，并建立其与再生核的关系。
Vector valued de Branges spaces, CNU contractions and functional models
本文通过Hilbert空间直和分解构造向量值再生核Hilbert空间，证明其为de Branges空间，并用于建立完全非酉压缩算子的函数模型，同时给出Fredholm型判据并建立特征函数与投影算子值函数的关系。
A short note on similarity of operator-valued multishifts
本文研究带可逆算子权重的算子值多重移位的相似性，给出完全由算子权重表述的刻画，并将其用于判定对角核再生核 Hilbert 空间上坐标乘法算子元组的相似性。
Banach Spaces Generated by Finite-Valued Functions: Superreflexive Rigidity, Hankel Operators, and Universality
该研究建立有限值函数生成巴拿赫空间的超自反性等几何性质与语言正则性的关联，构造出含所有可分实巴拿赫空间等距拷贝的二元语言空间，揭示正则语言的几何刚性与非正则情形的普适性差异。
Complementing earlier results on dynamics of unilateral weighted shifts, we obtain a sufficient (but not necessary, with supporting examples) condition for hypercyclicity, mixing and chaos for $M_z^*$, the adjoint of $M_z$, on vector-valued analytic reproducing kernel Hilbert spaces $\mathcal{H}$ in terms of the derivatives of kernel functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. Here $M_z$ denotes the multiplication operator by the coordinate function $z$, that is \[ (M_z f) (w) = w f(w), \] for all $f \in \mathcal{H}$ and $w \in \mathbb{D}$. We analyze the special case of quas…
Some aspects of vector valued de Branges spaces of entire functions
本文研究向量值de Branges空间中整个函数的某些方面，扩展了标量理论中的因子分解与等距嵌入结果，探讨了Fredholm算子值整个函数的全局因子分解及再生核的解析等价性，并研究了与这些空间相关的算子值整个函数及其与算子节点的联系。
This paper considers universal Hilbert space operators in the sense of Rota, and gives criteria for universality of semigroups in the context of uniformly continuous semigroups and contraction semigroups. Specific examples are given. Universal semigroups provide models for these classes of semigroups: following a line of research initiated by Shimorin, models for concave semigroups are developed, in terms of shifts on reproducing kernel Hilbert spaces.
Analytic Kramer sampling and quasi Lagrange-type interpolation in vector valued RKHS
本文研究了向量值RKHS中的抽象Kramer采样定理，推广了拟Lagrange型插值概念，并探讨了其与广义后向平移算子不变条件及de Branges空间的联系。
Localised Operator Valued Kernels Invariant under Actions of $*$-Semigroupoids
本文研究了在$*-$半群oid作用下保持不变的局部化算子值核，证明了其广义$*-$表示的存在性，并探讨了Hermitian核的再生核Krein空间生成条件。
A contraction $T$ on a Hilbert space $\mathcal{H}$ is said to be pure if the sequence $\lbrace T^{*n} \rbrace_{n}$ converges to $0$ in the strong operator topology. In this article, we prove that for contractions $T$, which commute with certain tractable tuples of commuting operators $X = (X_1,\ldots,X_n)$ on $\mathcal{H}$, the following statements are equivalent: (i) $T$ is a pure contraction on $\mathcal{H}$, (ii) the compression $P_{\mathcal{W}(X)}T|_{\mathcal{W}(X)}$ is a pure contraction, where $\mathcal{W}(X)$ is the wandering subspace corresponding to the tuple $X$. An operator-valued m…
We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$into $\mathbb C^n$. Given a finite measure $μ$ on $X$, we represent the reproducing kernel $K$ as convergent series in terms of the eigenfunctions of a suitable compact operator depending on $K$ and $μ$. Our result holds under the mild assumption that $K$ is measurable and the associated Hilbert space is separable. Furthermore, we show that $X$ has a natural second countable topology with respect to which the eigenfunctions are continuous and the series representi…
Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks
本文提出广义再生核巴拿赫空间框架，用于构建抽象神经网络，扩展了传统RKBS在处理巴拿赫值余域问题中的应用，推动了神经架构设计与数学建模的结合。
Vector valued de Branges spaces of entire functions based on pairs of Fredholm operator valued functions and functional model
本文构造基于 Fredholm 算子值整函数对的 de Branges 算子及相应向量值整函数 RKHS，刻画空间结构、自伴扩张、抽样公式及其与整算子和压缩算子特征函数的联系。
This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.
A Schur-class function in $d$ variables is defined to be an analytic contractive-operator valued function on the unit polydisk. Such a function is said to be in the Schur--Agler class if it is contractive when evaluated on any commutative $d$-tuple of strict contractions on a Hilbert space. It is known that the Schur--Agler class is a strictly proper subclass of the Schur class if the number of variables $d$ is more than two. The Schur--Agler class is also characterized as those functions arising as the transfer function of a certain type (Givone--Roesser) of conservative multidimensional line…
Explicit Universal and Approximate-Universal Kernels on Compact Metric Spaces
本文提出紧度量空间上通用核的显式构造方法，并引入近似通用性概念，构造可处理的近似通用核。
The main purpose of this paper is providing a systematic study and classification of non-scalar kernels for Reproducing Kernel Hilbert Spaces (RKHS), to be used in the analysis of deformation in shape spaces endowed with metrics induced by the action of groups of diffeomorphisms. After providing an introduction to matrix-valued kernels and their relevant differential properties, we explore extensively those, that we call TRI kernels, that induce a metric on the corresponding Hilbert spaces of vector fields that is both translation- and rotation-invariant. These are analyzed in an effective man…
The techniques developed by Popescu, Muhly-Solel and Good for the study of algebras generated by weighted shifts are applied to generalize results of Sarkar and of Bhattacharjee-Eschmeier-Keshari-Sarkar concerning dilations and invariant subspaces for commuting tuples of operators. In that paper the authors prove Beurling-Lax-Halmos type results for commuting tuples $T=(T_1,\ldots,T_d)$ operators that are contractive and pure; that is $Φ_T(I)\leq I$ and $Φ_T^n(I)\searrow 0$ where $$Φ_T(a)=Σ_i T_iaT_i^*.$$ Here we generalize some of their results to commuting tuples $T$ satisfying similar condi…
We consider positive semidefinite kernels valued in the $*$-algebra of adjointable operators on a VE-space (Vector Euclidean space) and that are invariant under actions of $*$-semigroups. A rather general dilation theorem is stated and proved: for these kind of kernels, representations of the $*$-semigroup on either the VE-spaces of linearisation of the kernels or on their reproducing kernel VE-spaces are obtainable. We point out the reproducing kernel fabric of dilation theory and we show that the general theorem unifies many dilation results at the non topological level.
This note finds a new characterization of complete Nevanlinna-Pick kernels on the Euclidean unit ball. The classical theory of Sz.-Nagy and Foias about the characteristic function is extended in this note to a commuting tuple $\bfT$ of bounded operators satisfying the natural positivity condition of $1/k$-contractivity for an irreducible unitarily invariant complete Nevanlinna-Pick kernel. The characteristic function is a multiplier from $H_k \otimes \cE$ to $H_k \otimes \cF$, {\em factoring} a certain positive operator, for suitable Hilbert spaces $\cE$ and $\cF$ depending on $\bfT$. There is…
Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations
本文探讨复值再现核希尔伯特空间中的代表定理，应用于福克空间、哈代空间及超振荡问题，揭示超振荡与机器学习核方法的联系。
In this work we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a r…
Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space
本文探讨了并行磁共振成像在再生核希尔伯特空间中的近似方法，通过空间灵敏度矩阵构建核函数，将图像重建问题转化为函数近似问题，并扩展了采样影响的分析。
Time-Frequency Analysis and Coorbit Spaces of Operators
本文提出带算子窗的算子值短时傅里叶变换，建立算子共轨空间理论，完全分类生成等价范数的算子，并利用原子分解与局部化算子刻画相应空间。
On the extension of positive definite kernels to topological algebras
本文将算子值正定函数扩展至拓扑代数，描述其再生核空间，并为整函数情形下解析函数的幂级数展开赋予精确含义。
We describe those reproducing kernel Hilbert spaces of holomorphic functions on domains in ${\Bbb C}^d$ for which an analogue of the Nevanlinna-Pick theorem holds, in other words when the existence of a (possibly matrix-valued) function in the unit ball of the multiplier algebra with specified values on a finite set of points is equivalent to the positvity of a related matrix. Our description is in terms of a certain localization property of the kernel.
Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks
本文提出了一种基于特色再生核巴拿赫空间的框架，用于在巴拿赫空间中进行学习，扩展了核方法和神经网络的应用范围。
In this paper we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications, and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example Krein spaces, in the study of stability questions. We focus on the non-rational case, and establish new connections with the theory of generalized Schur functions and their associate…
Toeplitz, Hankel, de Branges and two truncated matrix moment problems
本文借助de Branges型向量值整函数再生核希尔伯特空间及对应开单位圆盘的同类空间，分别研究截断矩阵Hamburger矩与三角矩问题，用再生核计算投影以简化推导、得到简洁公式。
Universality of shallow and deep neural networks on non-Euclidean spaces
研究浅层和深层神经网络在一般拓扑空间上的输入范围，探讨其通用逼近性质，建立在任意拓扑空间上连续向量值函数空间中的稠密条件，并讨论深度窄设置下的通用逼近保持条件。
Octonionic monogenic and slice monogenic Hardy and Bergman spaces
本文研究八元数单演与切片单演 Hardy 和 Bergman 空间，通过引入平行内积克服非结合性困难，并给出单位球、半空间及条形域上切片再生核的显式公式与序列刻画。
Factorization of positive definite kernels. Correspondences: $C^{*}$-algebraic and operator valued context vs scalar valued kernels
本文引入一类广义正定核，通过标量值核和RKHS方法，给出其Stinespring型分解，并刻画核支配关系，恢复纯态与不可约表示的经典对应。
Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels
本文研究了从波兰空间到可分希尔伯特空间的回归算子估计问题，通过正则化随机梯度下降算法在在线和有限时间 horizon 设置下分析 ill-posed 性问题，建立与维度无关的预测和估计误差界，并提供结构预测和参数 PDEs 的应用示例。
A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning
本文提出了一种基于核的随机近似框架，用于非线性算子学习，通过通用Mercer算子值核实现无维度多项式收敛，适用于广泛的操作学习任务。
Stochastic Gradient Descent for Operator Learning in Hilbert Spaces: Convergence Rates and Minimax Lower Bounds
本文提出在一般希尔伯特空间中利用SGD学习算子的正则性条件和收敛分析，揭示了非线性算子学习的收敛特性。
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