Factor Graphs with NUV Priors and Iteratively Reweighted Descent for Sparse Least Squares and More

Normal priors with unknown variance (NUV) are well known to include a large class of sparsity promoting priors and to blend well with Gaussian message passing. Essentially equivalently, sparsifying norms (including the L1 norm) as well as the Huber cost function from robust statistics have variational representations that lead to algorithms based on iteratively reweighted L2-regularization. In this paper, we rephrase these well-known facts in terms of factor graphs. In particular, we propose a smoothed-NUV representation of the Huber function and of a related nonconvex cost function, and we illustrate their use for sparse least-squares with outliers and in a natural (piecewise smooth) prior for imaging. We also point out pertinent iterative algorithms including variations of gradient descent and coordinate descent.