Learning multiplication-free linear transformations

Cristian Rusu

Abstract: In this paper, we propose several dictionary learning algorithms for sparse representations that also impose specific structures on the learned dictionaries such that they have low coding complexity and are numerically efficient to use: reduced number of addition/multiplications and even avoid multiplications altogether. We base our work on factorizations of the dictionaries 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 compare against well-known numerically efficient transforms such as fast Fourier, fast discrete cosine transforms, and fast structured dictionaries.