Bayesian sparsity and class sparsity priors for dictionary learning and
  coding
Alberto Bocchinfuso Daniela Calvetti Erkki Somersalo
Dictionary learning methods continue to gain popularity for the solution of
challenging inverse problems. In the dictionary learning approach, the
computational forward model is replaced by a large dictionary of possible
outcomes, and the problem is to identify the dictionary entries that best match
the data, akin to traditional query matching in search engines. Sparse coding
techniques are used to guarantee that the dictionary matching identifies only
few of the dictionary entries, and dictionary compression methods are used to
reduce the complexity of the matching problem. In this article, we propose a
work flow to facilitate the dictionary matching process. First, the full
dictionary is divided into subdictionaries that are separately compressed. The
error introduced by the dictionary compression is handled in the Bayesian
framework as a modeling error. Furthermore, we propose a new Bayesian
data-driven group sparsity coding method to help identify subdictionaries that
are not relevant for the dictionary matching. After discarding irrelevant
subdictionaries, the dictionary matching is addressed as a deflated problem
using sparse coding. The compression and deflation steps can lead to
substantial decreases of the computational complexity. The effectiveness of
compensating for the dictionary compression error and using the novel group
sparsity promotion to deflate the original dictionary are illustrated by
applying the methodology to real world problems, the glitch detection in the
LIGO experiment and hyperspectral remote sensing.