Hierarchical Infinite Divisibility for Multiscale Shrinkage

A new shrinkage-based construction is developed for a compressible vector
${\mmb x}\in\BBR^{n}$
, for cases in which the components of
${\mmb x}$
are naturally associated with a tree structure. Important examples are when
${\mmb x}$
corresponds to the coefficients of a wavelet or block-DCT representation of data. The method we consider in detail, and for which numerical results are presented, is based on the gamma distribution. The gamma distribution is a heavy-tailed distribution that is infinitely divisible, and these characteristics are leveraged within the model. We further demonstrate that the general framework is appropriate for many other types of infinitely divisible heavy-tailed distributions. Bayesian inference is carried out by approximating the posterior with samples from an MCMC algorithm, as well as by constructing a variational approximation to the posterior. We also consider expectation-maximization (EM) for a MAP (point) solution. State-of-the-art results are manifested for compressive sensing and denoising applications, the latter with spiky (non-Gaussian) noise.