Compressive Sensing with Sparse Measurement Matrices

This paper discusses compressive sensing with sparse measurement matrices. Sparse matrices have several attractive properties, like low computational complexity in both encoding and recovery, easy incremental updates to signals, and low storage requirement, etc. Typical examples of existing algorithms for sparse signal recovery with sparse measurement matrices include convex relaxation, matching pursuit, and Bayesian framework based approaches. In this paper, we propose an alternative technique to this problem. The proposed technique has a linear recovery complexity and relatively good empirical behavior. In this technique, we employ a permutation-based multi-dimensional measurement matrix, which is composed of several sub-matrices, each consisting of a block-diagonal matrix and a random permutation matrix. The measurement symbols generated from the same permutation matrix are referred to as a dimension. Such a measurement matrix brings some useful features to the measurement symbols. Fully exploiting these features enables us to design a simple and effective recovery algorithm. The proposed recovery algorithm employs an iterative process. In each iteration, the algorithm looks for measurement symbols with certain features, and uses such features to recover the source symbols related to these measurements. An iterative process is then applied, along with an interference cancellation operation, to reconstruct all source symbols gradually. The complexity of the proposed algorithm is relatively low, which grows linearly with the source signal length. Numerical results show that the proposed technique empirically offers a much lower sketch length than ℓ₁-minimization-based convex relaxation and Bayesian framework based algorithms. It achieves the empirical lower bound of sketch length and linear recovery complexity at the same time.