  Dictionary learning is the task of determining a data-dependent transform
that yields a sparse representation of some observed data. The dictionary
learning problem is non-convex, and usually solved via computationally complex
iterative algorithms. Furthermore, the resulting transforms obtained generally
lack structure that permits their fast application to data. To address this
issue, this paper develops a framework for learning orthonormal dictionaries
which are built from products of a few Householder reflectors. Two algorithms
are proposed to learn the reflector coefficients: one that considers a
sequential update of the reflectors and one with a simultaneous update of all
reflectors that imposes an additional internal orthogonal constraint. The
proposed methods have low computational complexity and are shown to converge to
local minimum points which can be described in terms of the spectral properties
of the matrices involved. The resulting dictionaries balance between the
computational complexity and the quality of the sparse representations by
controlling the number of Householder reflectors in their product. Simulations
of the proposed algorithms are shown in the image processing setting where
well-known fast transforms are available for comparisons. The proposed
algorithms have favorable reconstruction error and the advantage of a fast
implementation relative to the classical, unstructured, dictionaries.
