Nonconvex image reconstruction via expectation propagation.

The problem of efficiently reconstructing tomographic images can be mapped into a Bayesian inference problem over the space of pixels densities. Solutions to this problem are given by pixels assignments that are compatible with tomographic measurements and maximize a posterior probability density. This maximization can be performed with standard local optimization tools when the log-posterior is a convex function, but it is generally intractable when introducing realistic nonconcave priors that reflect typical images features such as smoothness or sharpness. We introduce a new method to reconstruct images obtained from Radon projections by using expectation propagation, which allows us to approximate the intractable posterior. We show, by means of extensive simulations, that, compared to state-of-the-art algorithms for this task, expectation propagation paired with very simple but non-log-concave priors is often able to reconstruct images up to a smaller error while using a lower amount of information per pixel. We provide estimates for the critical rate of information per pixel above which recovery is error-free by means of simulations on ensembles of phantom and real images.

Universal behavior of the full particle statistics of one-dimensional Coulomb gases with an arbitrary external potential.

We present a complete theory for the full particle statistics of the positions of bulk and extremal particles in a one-dimensional Coulomb gas (CG) with an arbitrary potential, in the typical and large deviations regimes. Typical fluctuations are described by a universal function which depends solely on the general properties of the external potential. The rate function controlling large deviations is, rather unexpectedly, not strictly convex and has a discontinuous third derivative around its minimum for both extremal and bulk particles. This implies, in turn, that the rate function cannot predict the anomalous scaling of the typical fluctuations with the system size for bulk particles, and it may indicate the existence of an intermediate phase in this case. Moreover, its asymptotic behavior for extremal particles differs from the predictions of the Tracy-Widom distribution. Thus many of the paradigmatic properties of the full particle statistics of Dyson log gases do not carry over into their one-dimensional counterparts, hence proving that one-dimensional CG belongs to a different universality class. Our analytical expressions are thoroughly compared with Monte Carlo simulations, showing excellent agreement.