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 representations of operator-valued reproducing kernels. Concrete examples of refining translation invariant and finite Hilbert-Schmidt operator-valued reproducing kernels are provided. Other examples include refinement of Hessian of scalar-valued translation-invariant kernels and transformation kernels. Existence and properties of operator-valued reproducing kernels preserved during the refinement process are also investigated.
A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces
Carl R. Woese Institute for Genomic Biology and the Coordinated Science Laboratory, University of Illinois Urbana-Champaign（卡尔·R·沃塞基因组生物学研究所和协调科学实验室，伊利诺伊大学厄巴纳-香槟分校）；Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois Urbana-Champaign（电气与计算机工程系和协调科学实验室，伊利诺伊大学厄巴纳-香槟分校）
Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks
Dept. of Computer Science and Artificial Intelligence, Andalusian Institute on Data Science and Computational Intelligence (DaSCI), University of Granada（计算机科学与人工智能系，数据科学与计算智能安达卢西亚研究所，格拉纳达大学）；ADIA Lab, Abu Dhabi, United Arab Emirates（阿布扎比分实验室，阿联酋）；Dpto. de Análisis Matemático, Facultad de Ciencias, Universidad de Granada（数学系，科学学院，格拉纳达大学）
Learning Reconstructive Embeddings in Reproducing Kernel Hilbert Spaces via the Representer Theorem
Universidad Rey Juan Carlos（胡安卡洛斯国王大学）；Centro de Investigación for Data, Complex Networks and Cybersecurity Sciences, Universidad Rey Juan Carlos（胡安卡洛斯国王大学数据、复杂网络与网络安全科学研究中心）
Kernel Mean Embeddings of Von Neumann-Algebra-Valued Measures
NTT Network Technology Laboratories, NTT Corporation（NTT 公司网络技术实验室）；Graduate School of Science and Technology, Keio University（庆应义塾大学科学与技术研究生院）；Faculty of Science, Ehime University（爱媛大学理学部）；Center for Advanced Intelligence Project, RIKEN（理化学研究所革新智能综合研究中心）；Faculty of Science and Technology, Keio University（庆应义塾大学理工学部）；Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）
Random Fourier Features for Operator-Valued Kernels
IBISC；Université d’Évry val d’Essonne（埃松省埃夫里大学）；LTCI；Télécom ParisTech（巴黎电信学院）；Université Paris-Saclay（巴黎萨克雷大学）；Department of Information and Computer Science（信息与计算机科学系）；Aalto University（阿尔托大学）
Reproducing kernel Hilbert C*-module and kernel mean embeddings
NTT Network Service Systems Laboratories, NTT Corporation（NTT网络服务系统实验室，NTT公司）；Graduate School of Science and Technology, Keio University（庆应义塾大学科学与技术研究生院）；Center for Data Science, Ehime University（爱媛大学数据科学中心）；Center for Advanced Intelligence Project, RIKEN（理化学研究所高级智能项目中心）；Faculty of Science and Technology, Keio University（庆应义塾大学科学与技术学部）
Operator-valued Kernels for Learning from Functional Response Data
Aix-Marseille Université（艾克斯-马赛大学）；Ecole Centrale de Lille（里尔中央理工学院）；Université de Lille（里尔大学）；INSA de Rouen（鲁昂国立应用科学学院）；Université de Rouen（鲁昂大学）；ENS Cachan（卡尚高等师范学院）
Single Image Inpainting and Super-Resolution with Simultaneous Uncertainty Guarantees by Universal Reproducing Kernels
Institute for Computer Science and Control (SZTAKI)（计算机科学与控制研究所）；Institute of Mathematics, Budapest University of Technology and Economics (BME)（布达佩斯技术大学数学研究所）
Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels
School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Fudan University（复旦大学）；Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学当代应用数学重点实验室）
Forecasting and Granger Modelling with Non-linear Dynamical Dependencies
Geneva School of Business Administration, HES-SO University of Applied Sciences of Western Switzerland（日内瓦商学院，瑞士西部应用科学与艺术大学）；University of Geneva（日内瓦大学）
Quantum Policy Gradient in Reproducing Kernel Hilbert Space
Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR)（高性能计算研究所（IHPC），科技研究局（A*STAR））；Centre for Frontier AI Research (CFAR), Agency for Science, Technology and Research (A*STAR)（前沿人工智能研究中心（CFAR），科技研究局（A*STAR））；Centre for Quantum Engineering, Research and Education, TCG CREST（量子工程、研究与教育中心（TCG CREST））
