In this paper, we propose a novel GPU-based method for highly parallel compressed sensing of n-dimensional (nD) signals based on the smoothed l(0) (SL0) algorithm. We demonstrate the efficiency of our approach by showing several examples of nD tensor reconstructions. Moreover, we also consider the traditional 1D compressed sensing, and compare the results. We show that the multidimensional SL0 algorithm is computationally superior compared to the 1D variant due to the small dictionary sizes per dimension. This allows us to fully utilize the GPU and perform massive batch-wise computations, which is not possible for the 1D compressed sensing using SL0. For our evaluations, we use light field and light field video data sets. We show that we gain more than an order of magnitude speedup for both one-dimensional as well as multidimensional data points compared to a parallel CPU implementation. Finally, we present a theoretical analysis of the SL0 algorithm for nD signals, which generalizes previous work for 1D signals.