Trust your source: quantifying source condition elements for variational regularisation methods

TL;DR

The paper tackles the practical estimation of source condition elements for variational regularisation in linear inverse problems. It reformulates source conditions as convex minimisation problems using proximal/Bregman tools, enabling iterative computation of the condition element $v$ and the associated range data, with guarantees when the forward operator $K$ is injective. Through 1D polynomial LASSO regression and 2D Fourier sub-sampling with total variation regularisation, the method yields quantitative error bounds and informs optimal sampling strategies in the Fourier domain, showing potential benefits for MRI-like applications. The work further outlines extensions to composite functionals and data-driven, regularised designs, pointing to operator correction and learning-based regularisers as promising future directions for improved convergence and reconstruction quality.

Abstract

Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.

Figures (14)

• Figure 1: Noisy sampled data \ref{['eq:results-fx-additive-noise']}, for the ground truth $\varphi$ defined by in \ref{['eq:results-fx-1d-polynomial-regression_deg5']} (left) and \ref{['eq:results-fx-1d-polynomial-regression-deg20']} (right).
• Figure 2: Top row: source condition element $v$ computed for examples \ref{['eq:results-fx-1d-polynomial-regression_deg5']} (left) and \ref{['eq:results-fx-1d-polynomial-regression-deg20']} (right). Note the difference in the ordinate scales: the norm of the source condition element for the higher order polynomial \ref{['eq:results-fx-1d-polynomial-regression-deg20']} is much higher it and has larger oscillations than the ones for the lower order polynomial \ref{['eq:results-fx-1d-polynomial-regression_deg5']}. Bottom row: comparison between $\Phi^\top v$ and the sign of the true coefficients $w^\dagger$. For both plots, every time $\text{sign}(w^\dagger)=\pm 1$, then also $\Phi^\top v = \pm 1$. This indicates that the estimated source condition $v$ is correct.
• Figure 3: Comparison between noisy samples $f^\delta$, data $f$ and range condition element $g_\alpha$ for $\alpha = \delta / \| v \|$, computed for examples \ref{['eq:results-fx-1d-polynomial-regression_deg5']} (left) and \ref{['eq:results-fx-1d-polynomial-regression-deg20']} (right).
• Figure 4: Figure \ref{['subfig:slp_full_sc']} depicts the source condition element $v^K$ that satisfies $v^K \in \partial \text{TV}(u^\dagger)$ for the Shepp-Logan phantom $u^\dagger$. Figure \ref{['subfig:slp_full_sg']} shows the corresponding vector $q^K$ such that $A^\top q^K = v^K$. As one would expect this for an element in the subgradient, the Euclidean norm of $q^K$ with respect to the vector components is bounded by one.
• Figure 5: The Fourier transform of the Shepp-Logan phantom depicted in Figure \ref{['subfig:slp-sanity1']}, and the sub-sampled Fourier transform that emulates a simple low-pass filter. Note that we depict the logarithm of the absolute value of the Fourier transformed data plus the constant $1/4000$ for better visualisation.

Theorems & Definitions (8)

• Remark 3.1
• Proposition 3.1
• proof
• Proposition 3.2
• Remark 3.2
• Proposition 3.3
• Remark 3.3
• Remark 3.4