[论文解读] Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks

Isabel de la Higuera, Francisco Herrera+1 more|arXiv (Cornell University)|Feb 6, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

论文介绍了 featured reproducing kernel Banach spaces (RKBSs)，识别了 Banach 空间中 feature-map 表示和 representer-type 结果的结构条件，并展示了固定架构的神经网络如何诱导特殊的向量值 featured RKBSs。

ABSTRACT

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. Many modern learning models, however -- including fixed-architecture neural networks equipped with non-quadratic norms -- naturally give rise to non-Hilbertian geometries that fall outside this setting. In Banach spaces, continuity of point-evaluation functionals alone is insufficient to guarantee feature representations or kernel-based learning formulations. In this work, we develop a functional-analytic framework for learning in Banach spaces based on the notion of featured reproducing kernel Banach spaces. We identify the precise structural conditions under which feature maps, kernel constructions, and representer-type results can be recovered beyond the Hilbertian regime. Within this framework, supervised learning is formulated as a minimal-norm interpolation or regularization problem, and existence results together with conditional representer theorems are established. We further extend the theory to vector-valued featured reproducing kernel Banach spaces and show that fixed-architecture neural networks naturally induce special instances of such spaces. This provides a unified function-space perspective on kernel methods and neural networks and clarifies when kernel-based learning principles extend beyond reproducing kernel Hilbert spaces.

研究动机与目标

• Motivate learning in Banach spaces where non-quadratic norms arise in modern models.
• Characterize when a Banach space of functions admits a feature-map representation and a kernel.
• Establish existence and form of minimal-norm interpolation/regularization solutions in featured RKBSs.
• Extend the framework to vector-valued spaces and connect to fixed-architecture neural networks.

提出的方法

• Define and distinguish featured RKBSs from general RKBSs.
• Derive necessary and sufficient conditions for feature-map representations in RKBSs.
• Formulate supervised learning as minimal-norm interpolation and regularization in featured RKBSs and prove existence results.
• Develop conditional representer-type theorems for featured RKBSs.
• Extend to vector-valued featured RKBSs and construct associated kernels.
• Show that fixed-architecture neural networks induce special vector-valued featured RKBSs and provide training interpretations.

实验结果

研究问题

• RQ1 Under what structural conditions does a Banach space of functions admit a feature-map representation (featured RKBS)?
• RQ2 What are the necessary and sufficient conditions for representer-type results to hold in RKBSs?
• RQ3 How can learning problems be formulated and solved in featured RKBSs via minimal-norm interpolation and regularization?
• RQ4 How can the framework be extended to vector-valued settings and linked to neural network architectures?

主要发现

• A precise framework (featured RKBS) is developed that yields feature-map representations for Banach spaces.
• Representer-type results in Banach spaces require additional structural conditions beyond continuity of evaluation functionals.
• Learning in featured RKBSs can be posed as minimal-norm interpolation or regularization with existence guaranteed by functional-analytic arguments.
• The framework is extended to vector-valued featured RKBSs with corresponding kernel constructions.
• Fixed-architecture neural networks are shown to induce special vector-valued featured RKBSs, linking kernel methods with neural networks.