  We consider the data-driven dictionary learning problem. The goal is to seek
an over-complete dictionary from which every training signal can be best
approximated by a linear combination of only a few codewords. This task is
often achieved by iteratively executing two operations: sparse coding and
dictionary update. In the literature, there are two benchmark mechanisms to
update a dictionary. The first approach, such as the MOD algorithm, is
characterized by searching for the optimal codewords while fixing the sparse
coefficients. In the second approach, represented by the K-SVD method, one
codeword and the related sparse coefficients are simultaneously updated while
all other codewords and coefficients remain unchanged. We propose a novel
framework that generalizes the aforementioned two methods. The unique feature
of our approach is that one can update an arbitrary set of codewords and the
corresponding sparse coefficients simultaneously: when sparse coefficients are
fixed, the underlying optimization problem is similar to that in the MOD
algorithm; when only one codeword is selected for update, it can be proved that
the proposed algorithm is equivalent to the K-SVD method; and more importantly,
our method allows us to update all codewords and all sparse coefficients
simultaneously, hence the term simultaneous codeword optimization (SimCO).
Under the proposed framework, we design two algorithms, namely, primitive and
regularized SimCO. We implement these two algorithms based on a simple gradient
descent mechanism. Simulations are provided to demonstrate the performance of
the proposed algorithms, as compared with two baseline algorithms MOD and
K-SVD. Results show that regularized SimCO is particularly appealing in terms
of both learning performance and running speed.
