A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems
Authors: Floor van Maarschalkerwaart, Subhadip Mukherjee, Christoph Brune, Marcello Carioni
Abstract
Learned reconstruction operators for inverse problems are typically trained under a fixed noise model, and generalize poorly when the distribution during testing differs from the one assumed during training. Distributionally robust optimization (DRO) addresses this by optimizing against the worst-case distribution within a prescribed ambiguity set, but standard Wasserstein DRO perturbs the full joint distribution uniformly, which can be overly conservative and ignores the physics of the measurement process. We develop a structured DRO framework in which the ambiguity set is restricted to structured perturbations aligned with the data-acquisition process. This allows us to learn data-driven reconstruction operators that remain robust to distributional shifts. By constraining perturbations to subsets such as P(Y∣X), our framework models uncertainty in the forward operator and noise model more faithfully, accommodating any noise model expressible as a stochastic forward operator. We establish strong duality for this general formulation and derive explicit finite-dimensional dual representations for perturbations in the joint, marginal, and conditional distributions. A central result is an explicit worst-case risk bound that induces Tikhonov regularization on the Lipschitz constant of the reconstruction operator, and is less conservative relative to standard DRO for well-posed problems. Numerical experiments on deblurring and sinogram-to-CT reconstruction demonstrate improved robustness, stability, and interpretability over standard DRO and MSE baselines. In the linear setting, the learned operator becomes effectively low-rank, truncating at the intrinsic dimension of the data and recovering a data-driven analogue of truncated-SVD regularization.
Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying algorithm is usually formulated in the functional analytic setting, learned approaches are often viewed as purely discrete. In this survey we present a unified operator view for learned iterative networks. Specifically, we formulate a learned reconstruction operator, defining how to compute, and separately the learning problem, which defines what to compute. In this setting we present common approaches and show that many approaches are closely related in their core. We review linear as well as non-linear inverse problems in this framework and present a short numerical study to conclude.
We study distributionally robust optimization (DRO) for robust inference when the worst-case distribution is continuous, leading to significant computational challenges due to the infinite-dimensional nature of the optimization problem. Unlike traditional discrete DRO approaches, which often suffer from scalability issues, limited generalization, and costly worst-case inference, our framework exploits Brenier's theorem to characterize the least favorable distribution as the pushforward of a transport map from a continuous reference measure. This characterization motivates our study of the minimax problem in Wasserstein space. We propose an iterative algorithmic framework with multiple variants and establish global convergence guarantees under mild assumptions, deriving complexity bounds in terms of subgradient evaluations and inexact Jordan-Kinderlehrer-Otto updates. Numerical results with neural network-based transport maps demonstrate that the proposed method enables both stable training of robust classifiers and effective worst-case inference for classification tasks.
We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO⋆ surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.