# Sparse Representer Theorems for Learning in Reproducing Kernel Banach Spaces

arXiv: https://arxiv.org/abs/2305.12584 · HTML: https://mlsanity.com/paper/2305.12584/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 21 May 2023.

## Builds on (most cited first, 9 of 9)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions](https://mlsanity.com/paper/1901.01002/) (2019)
- [Sparsity of solutions for variational inverse problems with finite-dimensional data](https://mlsanity.com/paper/1809.05045/) (2018)
- [On Representer Theorems and Convex Regularization](https://mlsanity.com/paper/1806.09810/) (2018)
- [A unifying representer theorem for inverse problems and machine learning](https://mlsanity.com/paper/1903.00687/) (2019)
- [Duality for Neural Networks through Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2211.05020/) (2022)
- [Convex Optimization in Sums of Banach Spaces](https://mlsanity.com/paper/2104.13127/) (2021)
- [Parameter Choices for Sparse Regularization with the ℓ1 Norm](https://mlsanity.com/paper/2205.07170/) (2022)

## Built on by (most cited first, 10 of 10)

- [Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression](https://mlsanity.com/paper/2305.16534/) (2023)
- [Hypothesis Spaces for Deep Learning](https://mlsanity.com/paper/2403.03353/) (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)
- [Transformers Can Implement Preconditioned Richardson Iteration for In-Context Gaussian Kernel Regression](https://mlsanity.com/paper/2605.08475/) (2026)
- [Convergence Analysis of Statistical Inverse Problems on Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2608.16404/) (2026)
- [Featured Reproducing Kernel Banach Spaces for Learning and Neural Networks](https://mlsanity.com/paper/2602.07141/) (2026)
- [Beyond NNGP: Large Deviations and Feature Learning in Bayesian Neural Networks](https://mlsanity.com/paper/2602.22925/) (2026)
- [Regularization of Statistical Inverse Problems on Non-Reflexive Banach Spaces](https://mlsanity.com/paper/2608.17533/) (2026)
- [Layerwise Decoupling for Stable Structured Sparsification of Fully Connected Layers](https://mlsanity.com/paper/2609.21126/) (2026)

## Abstract (arXiv)

Sparsity of a learning solution is a desirable feature in machine learning. Certain reproducing kernel Banach spaces (RKBSs) are appropriate hypothesis spaces for sparse learning methods. The goal of this paper is to understand what kind of RKBSs can promote sparsity for learning solutions. We consider two typical learning models in an RKBS: the minimum norm interpolation (MNI) problem and the regularization problem. We first establish an explicit representer theorem for solutions of these problems, which represents the extreme points of the solution set by a linear combination of the extreme points of the subdifferential set, of the norm function, which is data-dependent. We then propose sufficient conditions on the RKBS that can transform the explicit representation of the solutions to a sparse kernel representation having fewer terms than the number of the observed data. Under the proposed sufficient conditions, we investigate the role of the regularization parameter on sparsity of the regularized solutions. We further show that two specific RKBSs: the sequence space ℓ1(mathbbN) and the measure space can have sparse representer theorems for both MNI and regularization models.