Convex regularization in statistical inverse learning problems

Abstract

We consider a statistical inverse learning problem, where the task is to estimate a function f based on noisy point evaluations of Af , where A is a linear operator. The function Af is evaluated at i.i.d. random design points u n , n = 1, ..., N generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and p-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography.

Shearlet-based regularization in statistical inverse learning with an application to X-ray tomography

“…• In theorem 3.3, we derive convergence rates on the symmetric Bregman distance for the regularization term R(f ) = 1 p f p X + ι + (f ) where 1 < p < 2, X is the Besov space B s p and ι + is the indicator function of the non-negativity constraint. Under suitable assumptions, our rates coincide with those of the unconstrained case proved in [7], for both noise regimes.…”

“…Following the path paved by [7], we do not derive any lower bounds and we do not prove any minimaxoptimality of our results. As we briefly discuss in section 6, in line with [7,11] similar techniques can lead to minimax-optimal rates in inverse problems when considered against suitable source conditions. 1.3.…”

“…The semidiscrete Radon transform is introduced in subsection 2.3, and the discrete formulation is described in subsection 2.4. In section 3 and 4 we extend the convergence rates proven in [7] for 1 < p < 2 to the case of constrained shearlet-based regularization, for 1 < p < 2. In particular, in section 3 we prove that the concentrations rates derived in [7] for the unconstrained case hold true also when the solution is constrained to the non-negative orthant, in the case of Besov priors.…”

“…The goal of this paper is, therefore, to extend the results in [7] to work with a sparsity-enforcing model which is not limited to working with orthonormal bases for the sparsifying system, and applies in particular to shearlet-based regularization, thanks to powerful theory of shearlet coorbit spaces [22]. In addition, our model allows to work also with randomized tomographic data where generally a nonnegativity constraint is included in the minimized functional.…”

“…Our work lies at the intersection of convex regularization and statistical learning in the context of tomographic inverse problems. Apart from [7] which constitutes the foundation for our work (see also references therein), general convex regularization is not considered in statistical learning problems, which focuses more on deriving minimax optimal convergence rates, in a suitable metric, for Lasso-like problems (see, for example, [3,16,25] and references therein or [36] for an overview).…”