Perturbation-Aware Distributionally Robust Optimization for Inverse Problems

TL;DR

This work advances inverse problems by embedding perturbation-aware distributionally robust optimization (DRO) within an entropy-regularized Wasserstein framework. By constraining perturbations to a class $K$ and using a Wasserstein ball around the empirical distribution $\mu^*$, it yields reconstructions robust to X-, Y-, and conditional ($Y|X$) perturbations, including Gaussian and anisotropic noise. A weak duality result enables a computationally tractable algorithm based on a biased stochastic mirror descent with a bisection over the dual variable, demonstrated on matrix inversion and image deconvolution tasks. The approach provides a flexible, model-based mechanism for robust inverse problems and offers a path toward strong duality under further assumptions, with practical benefits in handling diverse noise models in data acquisition.

Abstract

This paper builds on classical distributionally robust optimization techniques to construct a comprehensive framework that can be used for solving inverse problems. Given an estimated distribution of inputs in $X$ and outputs in $Y$, an ambiguity set is constructed by collecting all the perturbations that belong to a prescribed set $K$ and are inside an entropy-regularized Wasserstein ball. By finding the worst-case reconstruction within $K$ one can produce reconstructions that are robust with respect to various types of perturbations: $X$-robustness, $Y|X$-robustness and, more general, targeted robustness depending on noise type, imperfect forward operators and noise anisotropies. After defining the general robust optimization problem, we derive its (weak) dual formulation and we use it to design an efficient algorithm. Finally, we demonstrate the effectiveness of our general framework to solve matrix inversion and deconvolution problems defining $K$ as the set of multivariate Gaussian perturbations in $Y|X$.

Figures (4)

• Figure 1: Left: entries of the reconstructed $g$ through the iterations. Right: values of the standard deviation $\sigma$ through the iterations.
• Figure 2: Left: entries of the matrix $g$ through the iterations. Right: entries of the covariance matrix $\Sigma$ through the iterations. Note that the lines for "01" and "10" in $\Sigma$ overlap.
• Figure 3: Samples from a Gaussian with average $Hx$ and optimal covariance $\Sigma_{opt}$ together with the ellipse representing the of eigenvectors of $H$.
• Figure 4: Deconvolution result for a given MNIST image with different noise types and levels, compared to a Tikhonov regularized inverse.

Theorems & Definitions (4)

• Definition 3.1
• Proposition 4.1
• proof
• Remark 4.2