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In this paper, we consider a compressed sensing problem of reconstructing asparse signal from an undersampled set of noisy linear measurements. Theregularized least squares or least absolute shrinkage and selection operator(LASSO) formulation is used for signal estimation. The measurement matrix isassumed to be constructed by concatenating several randomly orthogonal bases,referred to as structurally orthogonal matrices. Such measurement matrix ishighly relevant to large-scale compressive sensing applications because itfacilitates fast computation and also supports parallel processing. Using thereplica method from statistical physics, we derive the mean-squared-error (MSE)formula of reconstruction over the structurally orthogonal matrix in thelarge-system regime. Extensive numerical experiments are provided to verify theanalytical result. We then use the analytical result to study the MSE behaviorsof LASSO over the structurally orthogonal matrix, with a particular focus onperformance comparisons to matrices with independent and identicallydistributed (i.i.d.) Gaussian entries. We demonstrate that the structurallyorthogonal matrices are at least as well performed as their i.i.d. Gaussiancounterparts, and therefore the use of structurally orthogonal matrices ishighly motivated in practical applications.

On Sparse Vector Recovery Performance in Structurally Orthogonal Matrices via LASSO