Uncertainty Visualization via Low-Dimensional Posterior Projections

In ill-posed inverse problems it is commonly desirable to obtain insight into the full spectrum of plausible solutions rather than extracting only a single reconstruction. Information about the plausible solutions and their likelihoods is encoded in the posterior distribution. However for high-dimensional data this distribution is challenging to visualize. In this work we introduce a new approach for estimating and visualizing posteriors by employing energy-based models (EBMs) over low-dimensional subspaces. Specifically we train a conditional EBM that receives an input measurement and a set of directions that span some low-dimensional subspace of solutions and outputs the probability density function of the posterior within that space. We demonstrate the effectiveness of our method across a diverse range of datasets and image restoration problems showcasing its strength in uncertainty quantification and visualization. As we show our method outperforms a baseline that projects samples from a diffusion-based posterior sampler while being orders of magnitude faster. Furthermore it is more accurate than a baseline that assumes a Gaussian posterior.