Given a dataset, the task of learning a transform that allows sparse representations of the data bears the name of dictionary learning. In many applications, these learned dictionaries represent the data much better than the static well-known transforms (Fourier, Hadamard etc.). The main downside of learned transforms is that they lack structure and, therefore, they are not computationally efficient, unlike their classical counterparts. These posse several difficulties especially when using power limited hardware such as mobile devices, therefore, discouraging the application of sparsity techniques in such scenarios. In this paper, we construct orthogonal and nonorthogonal dictionaries that are factorized as a product of a few basic transformations. In the orthogonal case, we solve exactly the dictionary update problem for one basic transformation, which can be viewed as a generalized Givens rotation, and then propose to construct orthogonal dictionaries that are a product of these transformations, guaranteeing their fast manipulation. We also propose a method to construct fast square but nonorthogonal dictionaries that are factorized as a product of few transforms that can be viewed as a further generalization of Givens rotations to the nonorthogonal setting. We show how the proposed transforms can balance very well data representation performance and computational complexity. We also compare with classical fast and learned general and orthogonal transforms.
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