On Sparse Vector Recovery Performance in Structurally Orthogonal Matrices via LASSO

In this paper, we consider the compressed sensing problem of reconstructing a sparse signal from an undersampled set of noisy linear measurements. The regularized least squares or least absolute shrinkage and selection operator (LASSO) formulation is used for signal estimation. The measurement matrix is assumed to be constructed by concatenating several randomly orthogonal bases, which we refer to as structurally orthogonal matrices. Such measurement matrix is highly relevant to large-scale compressive sensing applications because it facilitates rapid computation and parallel processing. Using the replica method in statistical physics, we derive the mean-squared-error (MSE) formula of reconstruction over the structurally orthogonal matrix in the large-system regime. Extensive numerical experiments are provided to verify the analytical result. We then consider the analytical result to investigate the MSE behaviors of the LASSO over the structurally orthogonal matrix, with an emphasis on performance comparisons with matrices with independent and identically distributed (i.i.d.) Gaussian entries. We find that structurally orthogonal matrices are at least as good as their i.i.d. Gaussian counterparts. Thus, the use of structurally orthogonal matrices is attractive in practical applications.