Reproducing kernel Hilbert space
View article: Schrodinger equation and heat equation with translation invariance on the infinite-dimensional vector space ; Self-adjoint Laplace operator with translation invariance on infinite-dimensional space
Schrodinger equation and heat equation with translation invariance on the infinite-dimensional vector space ; Self-adjoint Laplace operator with translation invariance on infinite-dimensional space Open
The standard Laplacian $-\triangle_{\mathbb R^n}$ in $L^2(\mathbb R^n)$ is self-adjoint and translation invariant on the finite-dimensional linear space $\mathbb R^n$. In this paper, we define a translation invariant operator $-\triangle_{…
View article: Nonparametric Learning of Covariate-based Markov Jump Processes Using RKHS Techniques
Nonparametric Learning of Covariate-based Markov Jump Processes Using RKHS Techniques Open
We propose a novel nonparametric approach for linking covariates to Continuous Time Markov Chains (CTMCs) using the mathematical framework of Reproducing Kernel Hilbert Spaces (RKHS). CTMCs provide a robust framework for modeling transitio…
View article: Distributional Random Forests for Complex Survey Designs on Reproducing Kernel Hilbert Spaces
Distributional Random Forests for Complex Survey Designs on Reproducing Kernel Hilbert Spaces Open
We study estimation of the conditional law $P(Y|X=\mathbf{x})$ and continuous functionals $Ψ(P(Y|X=\mathbf{x}))$ when $Y$ takes values in a locally compact Polish space, $X \in \mathbb{R}^p$, and the observations arise from a complex surve…
View article: Distributional Random Forests for Complex Survey Designs on Reproducing Kernel Hilbert Spaces
Distributional Random Forests for Complex Survey Designs on Reproducing Kernel Hilbert Spaces Open
We study estimation of the conditional law $P(Y|X=\mathbf{x})$ and continuous functionals $Ψ(P(Y|X=\mathbf{x}))$ when $Y$ takes values in a locally compact Polish space, $X \in \mathbb{R}^p$, and the observations arise from a complex surve…
View article: Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams
Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams Open
A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Persistence diagrams do not carry intrinsic spectral or kernel structures, so applications typically us…
View article: Recovery of the optimal control value function in reproducing kernel Hilbert spaces from verification conditions
Recovery of the optimal control value function in reproducing kernel Hilbert spaces from verification conditions Open
Approximating the optimal value function $v^*$ for infinite-horizon, nonlinear, autonomous optimal control problems is both challenging and essential for synthesizing real-time optimal feedback. We develop an abstract optimal recovery fram…
View article: Recovery of the optimal control value function in reproducing kernel Hilbert spaces from verification conditions
Recovery of the optimal control value function in reproducing kernel Hilbert spaces from verification conditions Open
Approximating the optimal value function $v^*$ for infinite-horizon, nonlinear, autonomous optimal control problems is both challenging and essential for synthesizing real-time optimal feedback. We develop an abstract optimal recovery fram…
View article: Categorical Hilbert theory
Categorical Hilbert theory Open
Categorical Hilbert theory is the study of Hilbert spaces and similar kinds of mathematical objects from the perspective of category theory. Similar to the goals of other category-theoretic reformulations of mathematical theories (e.g., ab…
View article: Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams
Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams Open
A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Persistence diagrams do not carry intrinsic spectral or kernel structures, so applications typically us…
View article: Escaping the native space of Sobolev kernels by interpolation
Escaping the native space of Sobolev kernels by interpolation Open
Classical convergence analysis for kernel interpolation typically assumes that the target function $f$ lies in the reproducing kernel Hilbert space $\mathcal{H}_k\!\left(Ω\right)$ induced by a kernel on a domain $Ω\subset\mathbb{R}^N$. For…
View article: Escaping the native space of Sobolev kernels by interpolation
Escaping the native space of Sobolev kernels by interpolation Open
Classical convergence analysis for kernel interpolation typically assumes that the target function $f$ lies in the reproducing kernel Hilbert space $\mathcal{H}_k\!\left(Ω\right)$ induced by a kernel on a domain $Ω\subset\mathbb{R}^N$. For…
View article: 图灵机空间的概率测度构造与复杂度函数Hilbert空间 A Probability Measure on Turing Machine Space and the Hilbert Space of Complexity Functions
图灵机空间的概率测度构造与复杂度函数Hilbert空间 A Probability Measure on Turing Machine Space and the Hilbert Space of Complexity Functions Open
本文在图灵机空间上构造自然的概率测度,并为复杂度函数建立严格的Hilbert空间理论框架
View article: 图灵机空间的概率测度构造与复杂度函数Hilbert空间 A Probability Measure on Turing Machine Space and the Hilbert Space of Complexity Functions
图灵机空间的概率测度构造与复杂度函数Hilbert空间 A Probability Measure on Turing Machine Space and the Hilbert Space of Complexity Functions Open
本文在图灵机空间上构造自然的概率测度,并为复杂度函数建立严格的Hilbert空间理论框架
View article: The Joint Range of Quadratic Mapping on Hilbert Space
The Joint Range of Quadratic Mapping on Hilbert Space Open
We present a novel technical method for analyzing the hidden convex structure embedded in the joint range of a quadratic mapping defined on a Hilbert space. Our approach stands out by relying exclusively on elementary mathematical principl…
View article: The Joint Range of Quadratic Mapping on Hilbert Space
The Joint Range of Quadratic Mapping on Hilbert Space Open
We present a novel technical method for analyzing the hidden convex structure embedded in the joint range of a quadratic mapping defined on a Hilbert space. Our approach stands out by relying exclusively on elementary mathematical principl…
View article: An Operator Theoretical Approach to Mercer's Theorem
An Operator Theoretical Approach to Mercer's Theorem Open
This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of th…
View article: An Operator Theoretical Approach to Mercer's Theorem
An Operator Theoretical Approach to Mercer's Theorem Open
This is a survey article on Mercer's Theorem in its most general form and its relations with the theory of reproducing kernel Hilbert spaces and the spectral theory of compact operators. We provide a modern introduction to the basics of th…
View article: Bayesian-Frequentist Convergence for High-Dimensional Nonparametric Inference
Bayesian-Frequentist Convergence for High-Dimensional Nonparametric Inference Open
This paper provides a comprehensive analysis of Bayesian-Frequentist convergence in the context of high-dimensional nonparametric inference. In an era characterized by increasingly complex and large datasets, both Bayesian and Frequentist …
View article: Bayesian-Frequentist Convergence for High-Dimensional Nonparametric Inference
Bayesian-Frequentist Convergence for High-Dimensional Nonparametric Inference Open
This paper provides a comprehensive analysis of Bayesian-Frequentist convergence in the context of high-dimensional nonparametric inference. In an era characterized by increasingly complex and large datasets, both Bayesian and Frequentist …
View article: Conservative Motion Theory – BHUO V: Fredholm Evolution of Black Holes from UO–Tank Formation to Giant Exchange Kernels
Conservative Motion Theory – BHUO V: Fredholm Evolution of Black Holes from UO–Tank Formation to Giant Exchange Kernels Open
Within the Conservative Motion Theory (CMT), black holes are modeled not as geometricsingularities but as phase transitions of the gravitational kernel in Hilbert space. In earlierparts of the BHUO (Black Hole Universe–Outside) series, we …
View article: Conservative Motion Theory (CMT) – BHUO VI Observational and Physical Signatures of the Universe–Outside Tank and Giant Exchange Kernel
Conservative Motion Theory (CMT) – BHUO VI Observational and Physical Signatures of the Universe–Outside Tank and Giant Exchange Kernel Open
Recent developments in the Conservative Motion Theory (CMT) show that classicalblack–hole singularities correspond not to geometric divergences but to operator–theoreticphase transitions. When the self–adjoint reflection–positive gravitati…
View article: Conservative Motion Theory (CMT) – BHUO VI Observational and Physical Signatures of the Universe–Outside Tank and Giant Exchange Kernel
Conservative Motion Theory (CMT) – BHUO VI Observational and Physical Signatures of the Universe–Outside Tank and Giant Exchange Kernel Open
Recent developments in the Conservative Motion Theory (CMT) show that classicalblack–hole singularities correspond not to geometric divergences but to operator–theoreticphase transitions. When the self–adjoint reflection–positive gravitati…
View article: Conservative Motion Theory – BHUO II Hilbert–Fredholm Structure of Black–Hole Collapse, UO–Tank Formation, and Giant Exchange Kernels
Conservative Motion Theory – BHUO II Hilbert–Fredholm Structure of Black–Hole Collapse, UO–Tank Formation, and Giant Exchange Kernels Open
This paper develops the second part of the BHUO (Black-Hole Universe-Outside) serieswithin the framework of the Conservative Motion Theory (CMT). We provide a fully analyticand operator-theoretic account of the transition from extreme grav…
View article: Conservative Motion Theory — BHUO III Fredholm Analysis of the Giant Exchange Kernel (GEX)
Conservative Motion Theory — BHUO III Fredholm Analysis of the Giant Exchange Kernel (GEX) Open
In the first two papers of the BHUO series we established the Fredholm mechanism behindthe gravitational → Universe–Outside transition (BHUO I) and the existence and uniquenessof the spherical Universe–Outside kernel (BHUO II). This third …
View article: Conservative Motion Theory — BHUO III Fredholm Analysis of the Giant Exchange Kernel (GEX)
Conservative Motion Theory — BHUO III Fredholm Analysis of the Giant Exchange Kernel (GEX) Open
In the first two papers of the BHUO series we established the Fredholm mechanism behindthe gravitational → Universe–Outside transition (BHUO I) and the existence and uniquenessof the spherical Universe–Outside kernel (BHUO II). This third …
View article: Conservative Motion Theory – BHUO II Hilbert–Fredholm Structure of Black–Hole Collapse, UO–Tank Formation, and Giant Exchange Kernels
Conservative Motion Theory – BHUO II Hilbert–Fredholm Structure of Black–Hole Collapse, UO–Tank Formation, and Giant Exchange Kernels Open
This paper develops the second part of the BHUO (Black-Hole Universe-Outside) serieswithin the framework of the Conservative Motion Theory (CMT). We provide a fully analyticand operator-theoretic account of the transition from extreme grav…
View article: Conservative Motion Theory – BHUO V: Fredholm Evolution of Black Holes from UO–Tank Formation to Giant Exchange Kernels
Conservative Motion Theory – BHUO V: Fredholm Evolution of Black Holes from UO–Tank Formation to Giant Exchange Kernels Open
Within the Conservative Motion Theory (CMT), black holes are modeled not as geometricsingularities but as phase transitions of the gravitational kernel in Hilbert space. In earlierparts of the BHUO (Black Hole Universe–Outside) series, we …
View article: Existence Manifold Theory: A Foundational Framework Expanding Internal Geometry and Interpretive Degrees of Freedom in AI
Existence Manifold Theory: A Foundational Framework Expanding Internal Geometry and Interpretive Degrees of Freedom in AI Open
Description Existence Manifold Theory (EMT) proposes a new geometric kernel for AI systems—one that does not merely interpret the internal states of large language models, but reconstructs the space in which reasoning itself takes place. C…
View article: Kernel Fusion: Bridging Explicit and Implicit Feature Spaces for Enhanced AI Generalization
Kernel Fusion: Bridging Explicit and Implicit Feature Spaces for Enhanced AI Generalization Open
This paper introduces Kernel Fusion, a novel framework designed to enhance the generalization capabilities of artificial intelligence models by effectively bridging explicit and implicit feature spaces. The framework leverages the strength…
View article: Kernel Fusion: Bridging Explicit and Implicit Feature Spaces for Enhanced AI Generalization
Kernel Fusion: Bridging Explicit and Implicit Feature Spaces for Enhanced AI Generalization Open
This paper introduces Kernel Fusion, a novel framework designed to enhance the generalization capabilities of artificial intelligence models by effectively bridging explicit and implicit feature spaces. The framework leverages the strength…