mlsanityData to 7 Oct 2026

On Representer Theorems and Convex Regularization

arXiv 1806.09810 ↗First posted 26 Jun 2018math.OC cs.IT math.ITcited by 24

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Abstract (arXiv)

We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. An extension to a broader class of quasi-convex regularizers is also discussed. As a side result, we characterize the minimizers of the total gradient variation, which was still an unresolved problem.

Key facts. Cited by 24 papers in the map (data through 7 Oct 2026). First posted on arXiv 26 Jun 2018.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 6; 2023: 5; 2024: 6; 2025: 2; 2026: 2.

24 papers.

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  1. 2021Understanding neural networks with reproducing kernel Banach spacescited by 50
  2. 2021Two-layer neural networks with values in a Banach spacecited by 26
  3. 2022Learning sparse features can lead to overfitting in neural networkscited by 23
  4. 2023Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compressioncited by 17
  5. 2023Sparse Representer Theorems for Learning in Reproducing Kernel Banach Spacescited by 10
  6. 2022Stability of Image-Reconstruction Algorithmscited by 7
  7. 2023Weighted variation spaces and approximation by shallow ReLU networkscited by 7
  8. 2023Function-Space Optimality of Neural Architectures with Multivariate Nonlinearitiescited by 5
  9. 2022From Kernel Methods to Neural Networks: A Unifying Variational Formulationcited by 4
  10. 2024Hypothesis Spaces for Deep Learningcited by 4
  11. 2023FastPart: Over-Parameterized Stochastic Gradient Descent for Sparse optimisation on Measurescited by 3
  12. 2025Solving Nonlinear PDEs with Sparse Radial Basis Function Networkscited by 3
12 more
  1. 2022Off-the-grid learning of mixtures from a continuous dictionarycited by 2
  2. 2024Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Modelscited by 2
  3. 2024Nonuniform random feature models using derivative informationcited by 2
  4. 2024A Lipschitz spaces view of infinitely wide shallow neural networkscited by 2
  5. 2024Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numericscited by 2
  6. 2026Fenchel-Young Duality Gaps: Certified Early Stopping for Regularized Inverse Problemscited by 1
  7. 2021Analysis of Regularized Learning in Banach Spaces for Linear-functional Datacited by 0
  8. 2022Accelerating Non-Negative and Bounded-Variable Linear Regression Algorithms with Safe Screeningcited by 0
  9. 2022Simultaneous off-the-grid learning of mixtures issued from a continuous dictionarycited by 0
  10. 2024Decomposition of one-layer neural networks via the infinite sum of reproducing kernel Banach spacescited by 0
  11. 2025Effective regions and kernels in continuous sparse regularisation, with application to sketched mixturescited by 0
  12. 2026A Dual Certificate Approach to Sparsity in Infinite-Width Shallow Neural Networkscited by 0