Performance analysis of L1-norm minimization for compressed sensing with non-zero-mean matrix elements

We study performance of the L-1-norm minimization for compressed sensing with noiseless linear measurements when the elements of the measurement matrix are independent and identically-distributed Gaussian with non-zero mean. Using replica method in statistical mechanics, we derive in the large-system limit the condition for perfect estimation of sparse vectors in terms of the four parameters: the ratio of the number of measurements to the dimension of the sparse vector, the ratio of the number of non-zeros in the sparse vector to the dimension of the vector, the bias of the matrix elements, and the imbalance of the distribution of non-zeros of the sparse vector. We find that when the distribution of non-zeros is balanced the bias of the matrix elements does not affect the condition for perfect estimation. When the distribution of non-zeros is not balanced, on the other hand, the L-1-norm minimization will be successful with a smaller number of linear measurements if one uses a biased measurement matrix. Numerical experiments are also conducted to confirm the derived condition.