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Dictionary learning is a branch of signal processing and machine learningthat aims at finding a frame (called dictionary) in which some training dataadmits a sparse representation. The sparser the representation, the better thedictionary. The resulting dictionary is in general a dense matrix, and itsmanipulation can be computationally costly both at the learning stage and laterin the usage of this dictionary, for tasks such as sparse coding. Dictionarylearning is thus limited to relatively small-scale problems. In this paper,inspired by usual fast transforms, we consider a general dictionary structurethat allows cheaper manipulation, and propose an algorithm to learn suchdictionaries --and their fast implementation-- over training data. The approachis demonstrated experimentally with the factorization of the Hadamard matrixand with synthetic dictionary learning experiments.

Learning computationally efficient dictionaries and their implementation as fast transforms