The problem of estimating a random vector x from noisy linear measurements y=Ax+w with unknown parameters on the distributions of x and w, which must also be learned, arises in a wide range of statistical learning and linear inverse problems. We show that a computationally simple iterative message-passing algorithm can provably obtain asymptotically consistent estimates in a certain high-dimensional large-system limit (LSL) under very general parameterizations. Previous message passing techniques have required i.i.d.

Tweedie's formula, we show that we can perform gradient steps to sequentially optimize these objectives.

By now Bayesian methods are routinely used in practice for solving inverse problems.

Neural Information Processing Systems http://nips.cc/

We demonstrate the abilities of the proposed paradigm for image deblurring, super-resolution, and compressive sensing.

Sparse linear inverse problems are well studied in the literature of optimization. For example, it can be formulated into LASSO [29] and solved by many optimization algorithms [9, 3].

BA YESIAN PINNS FOR UNCERT AINTY-A W ARE INVERSE PROBLEMS (BPINN-IP) Ali MOHAMMAD-DJAF ARI ISCT, Bures-sur-Y vette, France Institute of Digital T win (IDT), EIT, Ningbo, China Dept. of Statistics, Central South University, Changcha, China ABSTRACT The main contribution of this paper is to develop a hierarchical Bayesian formulation of PINNs for linear inverse problems, which is called BPINN-IP . The proposed methodology extends PINN to account for prior knowledge on the nature of the expected NN output, as well as its weights. Also, as we can have access to the posterior probability distributions, naturally uncertainties can be quantified. Also, variational inference and Monte Carlo dropout are employed to provide predictive means and variances for reconstructed images. Un example of applications to deconvolution and super-resolution is considered, details of the different steps of implementations are given, and some preliminary results are presented.