# Exact solutions of infinite dimensional total-variation regularized problems

arXiv: https://arxiv.org/abs/1708.02157 · HTML: https://mlsanity.com/paper/1708.02157/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Aug 2017.

## Built on by (most cited first, 2 of 2)

- [Stability of Image-Reconstruction Algorithms](https://mlsanity.com/paper/2206.07128/) (2022)
- [Sparsest Univariate Learning Models Under Lipschitz Constraint](https://mlsanity.com/paper/2112.13542/) (2021)

## Abstract (arXiv)

We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary convex function. The first contribution is about the solu-tion's structure: we show that under suitable assumptions, there always exist an m-sparse solution, where m is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exacts solutions of the infinite dimensional problem can be obtained by solving two consecutive finite dimensional convex programs. These results extend recent advances in the understanding of total-variation reg-ularized problems.