Exact solutions of infinite dimensional total-variation regularized problems
Provides theoretical guarantees and a practical computational framework for infinite-dimensional TV-regularized problems, benefiting researchers in inverse problems and signal processing.
This paper proves that infinite-dimensional total-variation regularized inverse problems always have an m-sparse solution (where m is the number of measurements) and shows that exact solutions can be computed via two finite-dimensional convex programs, extending recent theoretical advances.
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.