# On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions

arXiv: https://arxiv.org/abs/1901.01002 · HTML: https://mlsanity.com/paper/1901.01002/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2019.

## Built on by (most cited first, 29 of 29)

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- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [Duality for Neural Networks through Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2211.05020/) (2022)
- [Learning with Asymmetric Kernels: Least Squares and Feature Interpretation](https://mlsanity.com/paper/2202.01397/) (2022)
- [Sparse Representer Theorems for Learning in Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2305.12584/) (2023)
- [Neural reproducing kernel Banach spaces and representer theorems for deep networks](https://mlsanity.com/paper/2403.08750/) (2024)
- [Random Fourier Features for Asymmetric Kernels](https://mlsanity.com/paper/2209.08461/) (2022)
- [Nonlinear SVD with Asymmetric Kernels: feature learning and asymmetric Nystr"om method](https://mlsanity.com/paper/2306.07040/) (2023)
- [Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities](https://mlsanity.com/paper/2310.03696/) (2023)
- [Solving Nonlinear PDEs with Sparse Radial Basis Function Networks](https://mlsanity.com/paper/2505.07765/) (2025)
- [Learning Functional Transduction](https://mlsanity.com/paper/2302.00328/) (2023)
- [Learning Analysis of Kernel Ridgeless Regression with Asymmetric Kernel Learning](https://mlsanity.com/paper/2406.01435/) (2024)
- [A Lipschitz spaces view of infinitely wide shallow neural networks](https://mlsanity.com/paper/2410.14591/) (2024)
- [Mirror Descent on Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2411.11242/) (2024)
- [Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators](https://mlsanity.com/paper/2509.26371/) (2025)
- [Gradient Descent in Neural Networks as Sequential Learning in RKBS](https://mlsanity.com/paper/2302.00205/) (2023)
- [Enhancing Kernel Flexibility via Learning Asymmetric Locally-Adaptive Kernels](https://mlsanity.com/paper/2310.05236/) (2023)
- [Barron Space for Graph Convolution Neural Networks](https://mlsanity.com/paper/2311.02838/) (2023)
- [Novel Kernel Models and Exact Representor Theory for Neural Networks Beyond the Over-Parameterized Regime](https://mlsanity.com/paper/2405.15254/) (2024)
- [Decomposition of one-layer neural networks via the infinite sum of reproducing kernel Banach spaces](https://mlsanity.com/paper/2409.18132/) (2024)
- [Sobolev norm inconsistency of kernel interpolation](https://mlsanity.com/paper/2504.20617/) (2025)
- [Quantum-inspired probability metrics define a complete, universal space for statistical learning](https://mlsanity.com/paper/2508.21086/) (2025)
- [Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks](https://mlsanity.com/paper/2602.07141/) (2026)
- [Self-Regularized Learning Methods](https://mlsanity.com/paper/2603.17160/) (2026)
- [Understanding Deep Learning via Entropy Space Theory](https://mlsanity.com/paper/2608.29279/) (2026)

## Abstract (arXiv)

Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with ℓ1 norm, the p-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction. Furthermore, we propose a framework of constructing RKBSs that unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps. A new class of Orlicz RKBSs is proposed. Finally, we develop representer theorems for machine learning in RKBSs constructed in our framework, which also unifies representer theorems in existing RKBSs.