# Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks

arXiv: https://arxiv.org/abs/2602.07141 · HTML: https://mlsanity.com/paper/2602.07141/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 6 Feb 2026.

## Builds on (most cited first, 11 of 11)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions](https://mlsanity.com/paper/1901.01002/) (2019)
- [A representer theorem for deep kernel learning](https://mlsanity.com/paper/1709.10441/) (2017)
- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [A Unified Approach to Controlling Implicit Regularization via Mirror Descent](https://mlsanity.com/paper/2306.13853/) (2023)
- [Sparse Representer Theorems for Learning in Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2305.12584/) (2023)
- [Deep Networks are Reproducing Kernel Chains](https://mlsanity.com/paper/2501.03697/) (2025)
- [Enhanced Feature Learning via Regularisation: Integrating Neural Networks and Kernel Methods](https://mlsanity.com/paper/2407.17280/) (2024)
- [Hypothesis Spaces for Deep Learning](https://mlsanity.com/paper/2403.03353/) (2024)

## Abstract (arXiv)

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.