  In this paper, we propose several dictionary learning algorithms for sparse
representations that also impose specific structures on the learned
dictionaries such that they are numerically efficient to use: reduced number of
addition/multiplications and even avoiding multiplications altogether. We base
our work on factorizations of the dictionary in highly structured basic
building blocks (binary orthonormal, scaling and shear transformations) for
which we can write closed-form solutions to the optimization problems that we
consider. We show the effectiveness of our methods on image data where we can
compare against well-known numerically efficient transforms such as the fast
Fourier and the fast discrete cosine transforms.
