Which Spaces can be Embedded in $L_p$-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy
In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces. Metric entropy characterizes the information complexity of function spaces and has implications for their approximability and learnability. Classical results show that embedding a function space into a reproducing kernel Hilbert space (RKHS) implies a bound on its metric entropy growth. Surprisingly, we prove a \textbf{converse}: a bound on the metric entropy growth of a function space allows its embedding to a $L_p-$type Reproducing Kernel Banach Space (RKBS). This shows that the ${L}_p-$type RKBS provides a broad modeling framework for learnable function classes with controlled metric entropies. Our results shed new light on the power and limitations of kernel methods for learning complex function spaces.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions
In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using t…
Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks
Reproducing kernel Hilbert spaces provide a foundational framework for kernel-based learning, where regularization and interpolation problems admit finite-dimensional solutions through classical representer theorems. Man…
Representation of Reinforcement Learning Policies in Reproducing Kernel Hilbert Spaces
We propose a general framework for policy representation for reinforcement learning tasks. This framework involves finding a low-dimensional embedding of the policy on a reproducing kernel Hilbert space (RKHS). The usage…
reinforcement-learningReinforcement LearningReinforcement Learning (RL)Mirror Descent on Reproducing Kernel Banach Spaces
Recent advances in machine learning have led to increased interest in reproducing kernel Banach spaces (RKBS) as a more general framework that extends beyond reproducing kernel Hilbert spaces (RKHS). These works have res…
Upper Bounds for Learning in Reproducing Kernel Hilbert Spaces for Non IID Samples
In this paper, we study a Markov chain-based stochastic gradient algorithm in general Hilbert spaces, aiming to approximate the optimal solution of a quadratic loss function. We establish probabilistic upper bounds on it…
Learning Theory