Exact solutions of infinite dimensional total-variation regularized problems

Abstract

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 function. The first contribution describes the solution's structure: we show that under mild assumptions, there always exists 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 exact solutions of the infinite dimensional problem can be obtained by solving one or two consecutive finite dimensional convex programs depending on the measurement functions structures. These results extend recent advances in the understanding of total-variation regularized inverse problems.

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

“…We can furthermore characterize the extremal points of the ball associated to φ N according to the following theorem. Note that a similar result was also obtained by [31] and [23] in different settings and more restrictive hypotheses. For this purpose, for x ∈ R d , denote by G x the fundamental solution G translated by x, i.e., such that LG x = δ x .…”

“…Hence, {A N v i } i is contained in a dim H N − 1 dimensional Hilbert space (obviously, w N = 0 in this case). Then, applying Carathéodory theorem again, we deduce that p ≤ dim H N as a consequence of the minimality of p. Define then (26) (23) and the linearity of A N we infer that…”

“…In this section we consider the case where φ(u) = Lu M , namely the Radon norm of a linear, translation-invariant scalar differential operator L. This was already treated in [31] and in [23] in different settings. Our goal is to show that our theory applies straightforwardly to this case.…”

“…Finally, we apply our main result to the setting considered in [31] and [23], i.e., where the regularizer is given by φ(u) = Lu M for a scalar linear differential operator L. We remove the hypotheses concerning the structure of the null-space of L and we work in the space of finite-order distributions equipped with the weak* topology. This allows us to have a general framework for these inverse problems that does not require additional assumptions on the Banach structure of the minimization domain (see [5] and [23] for comparison). It also justifies once more the usage of locally convex spaces in the abstract theory.…”

On Representer Theorems and Convex Regularization

“…Fisher and Jerome [23] proposed an interesting result, which can be seen as an extension of (20). This result was recently revisited in [48] and [25]. Below, we follow the presentation in [25].…”