The choice of an appropriate frame, or dictionary, is a crucial step in the sparse representation of a given class of signals. Traditional dictionary learning techniques generally lead to unstructured dictionaries which are costly to deploy and train, and do not scale well to higher dimensional signals. In order to overcome such limitation, we propose a learning algorithm that constrains the dictionary to be a sum of Kronecker products of smaller sub-dictionaries. This approach, named SuKro, is demonstrated experimentally on an image denoising application.
The purpose of this paper is manifold. In a first part, we present a new alternating least squares (ALS)-based method for estimating the matrix factors of a Kronecker product, the so-called Kronecker ALS (KALS) method. Four other methods are also briefly described. In a second part, we consider the design of multiple-input multiple-output (MIMO) wireless communication systems using tensor modelling. Eight systems are presented in a unified way, and their theoretical performance is compared in terms of maximal diversity gain. Exploiting a Kronecker product of symbol and channel matrices, and applying the algorithms introduced in the first part, we propose three semi-blind and two supervised receivers, called Kronecker receivers, for jointly estimating the channel and the transmitted symbols. Necessary identifiability conditions are established. Finally, extensive Monte Carlo simulation results are provided to compare the performance of three tensor-based systems, on the one hand, and of the five proposed Kronecker receivers for the tensor space-time-frequency (TSTF) coding system, on the other hand.
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