Distribution-free uncertainty quantification for inverse problems: Application to weak lensing mass mapping

Aims. In inverse problems, the aim of distribution-free uncertainty quantification (UQ) is to obtain error bars in the reconstruction with coverage guarantees that are independent of any prior assumptions about the data distribution. This allows for a better understanding of how intermediate errors introduced during the process affect subsequent stages and ultimately influence the final reconstruction. In the context of mass mapping, uncertainties could lead to errors that affect how the underlying mass distribution is understood or that propagate to cosmological parameter estimation, thereby impacting the precision and reliability of cosmological models. Current surveys, such as Euclid or Rubin, will provide new weak lensing datasets of very high quality. Accurately quantifying uncertainties in mass maps is therefore critical to fully exploit their scientific potential and to perform reliable cosmological parameter inference. Methods. In this paper, we extend the conformalized quantile regression (CQR) algorithm, initially proposed for scalar regression, to inverse problems. We compared our approach with another distribution-free approach based on risk-controlling prediction sets (RCPS). Both methods are based on a calibration dataset, and they offer finite-sample coverage guarantees that are independent of the data distribution. Furthermore, they are applicable to any mass mapping method, including black box predictors. In our experiments, we applied UQ to three mass-mapping methods: the Kaiser-Squires inversion, iterative Wiener filtering, and the MCALens algorithm. Results. Our experiments reveal that RCPS tends to produce overconservative confidence bounds with small calibration sets, whereas CQR is designed to avoid this issue. Although the expected miscoverage rate is guaranteed to stay below a user-prescribed threshold regardless of the mass mapping method, selecting an appropriate reconstruction algorithm remains crucial for obtaining accurate estimates, especially around peak-like structures, which are particularly important for inferring cosmological parameters. Additionally, the choice of mass mapping method influences the size of the error bars.