  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.
