# Sparsity of solutions for variational inverse problems with finite-dimensional data

arXiv: https://arxiv.org/abs/1809.05045 · HTML: https://mlsanity.com/paper/1809.05045/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2018.

## Built on by (most cited first, 25 of 25)

- [What Kinds of Functions do Deep Neural Networks Learn? Insights from Variational Spline Theory](https://mlsanity.com/paper/2105.03361/) (2021)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [Two-layer neural networks with values in a Banach space](https://mlsanity.com/paper/2105.02095/) (2021)
- [Ridges, Neural Networks, and the Radon Transform](https://mlsanity.com/paper/2203.02543/) (2022)
- [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)
- [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)
- [Stability of Image-Reconstruction Algorithms](https://mlsanity.com/paper/2206.07128/) (2022)
- [Weighted variation spaces and approximation by shallow ReLU networks](https://mlsanity.com/paper/2307.15772/) (2023)
- [Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities](https://mlsanity.com/paper/2310.03696/) (2023)
- [From Kernel Methods to Neural Networks: A Unifying Variational Formulation](https://mlsanity.com/paper/2206.14625/) (2022)
- [Hypothesis Spaces for Deep Learning](https://mlsanity.com/paper/2403.03353/) (2024)
- [Solving Nonlinear PDEs with Sparse Radial Basis Function Networks](https://mlsanity.com/paper/2505.07765/) (2025)
- [Sparsest Univariate Learning Models Under Lipschitz Constraint](https://mlsanity.com/paper/2112.13542/) (2021)
- [Solving the inverse problem of microscopy deconvolution with a residual Beylkin-Coifman-Rokhlin neural network](https://mlsanity.com/paper/2407.03239/) (2024)
- [Nonuniform random feature models using derivative information](https://mlsanity.com/paper/2410.02132/) (2024)
- [A Lipschitz spaces view of infinitely wide shallow neural networks](https://mlsanity.com/paper/2410.14591/) (2024)
- [Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics](https://mlsanity.com/paper/2412.01619/) (2024)
- [Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators](https://mlsanity.com/paper/2509.26371/) (2025)
- [Towards a Sampling Theory for Implicit Neural Representations](https://mlsanity.com/paper/2405.18410/) (2024)
- [Manifold limit for the training of shallow graph convolutional neural networks](https://mlsanity.com/paper/2601.06025/) (2026)
- [A Dual Certificate Approach to Sparsity in Infinite-Width Shallow Neural Networks](https://mlsanity.com/paper/2603.17785/) (2026)
- [Decomposition of one-layer neural networks via the infinite sum of reproducing kernel Banach spaces](https://mlsanity.com/paper/2409.18132/) (2024)
- [Physics-Modeled Neural Networks](https://mlsanity.com/paper/2605.08176/) (2026)

## Abstract (arXiv)

In this paper we characterize sparse solutions for variational problems of the form minu∈ X φ(u) + F(A u), where X is a locally convex space, A is a linear continuous operator that maps into a finite dimensional Hilbert space and φ is a seminorm. More precisely, we prove that there exists a minimizer that is `sparse' in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer φ (possibly translated by an element in the null space of φ). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear differential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in [31] under weaker hypotheses.