  In this paper, we consider a 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,
referred to as structurally orthogonal matrices. Such measurement matrix is
highly relevant to large-scale compressive sensing applications because it
facilitates fast computation and also supports parallel processing. Using the
replica method from 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 use the analytical result to study the MSE behaviors
of LASSO over the structurally orthogonal matrix, with a particular focus on
performance comparisons to matrices with independent and identically
distributed (i.i.d.) Gaussian entries. We demonstrate that the structurally
orthogonal matrices are at least as well performed as their i.i.d. Gaussian
counterparts, and therefore the use of structurally orthogonal matrices is
highly motivated in practical applications.
