Tensor-Based Dictionary Learning for Multidimensional Sparse Recovery: the K-HOSVD

In many applications of compressive sensing, 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 an example we choose the well-known K-SVD algorithm [1], propose a tensor extension and show its improved performance numerically. Consider a sparse recovery problem given by