Tensor-Based Sparsity Order Estimation for Big Data Applications

In Big Data Processing we typically face very large data sets that are highly structured. To save the computation and storage cost, it is desirable to extract the essence of the data from a reduced number of observations. One example of such a structural constraint is sparsity. If the data possesses a sparse representation in a suitable domain, it can be recovered from a small number of linear projections into a low-dimensional space. In this case, the degree of sparsity, referred to as sparsity order, is of high interest. It has recently been shown that if the measurement matrix obey certain structural constraints, one can estimate the sparsity order directly from the compressed data. The rich structure of the measurement matrix allows to rearrange the multiple-snapshot measurement vectors into a fourth-order tensor with rank equal to the desired sparsity order. In this paper, we exploit the multilinear structure of the data for accurate sparsity order estimation with improved identifiability. We discuss the choice of the parameters, i.e., the block size, block offset, and number of blocks, to maximize the sparsity order that can be inferred from a certain number of observations, and compare state-of-the-art order selection algorithms for sparsity order estimation under the chosen parameter settings. By performing an extensive campaign of simulations, we show that the discriminant function based method and the random matrix theory algorithm outperform other approaches in small and large snapshot-number scenarios, respectively.