Tensor-based algorithms for learning multidimensional separable dictionaries

Compressive Sensing (CS) allows to acquire signals at sampling rates significantly lower than the Nyquist rate, provided that the signals possess a sparse representation in an appropriate basis. However, in some applications of CS, the dictionary providing the sparse description is partially or entirely unknown. It has been shown that dictionary learning algorithms are able to estimate the basis vectors from a set of training samples. In some applications the dictionary is multidimensional, e.g., when estimating jointly azimuth and elevation in a 2-D direction of arrival (DOA) estimation context. In this paper we show that existing dictionary learning algorithms can be extended to exploit this structure, thereby providing a more accurate estimate of the dictionary. As examples we choose two prominent dictionary learning algorithms, the method of optimal directions (MOD) and the KSVD algorithm. We propose tensor-based multidimensional extensions for both algorithms and show their improved performances numerically.