  This paper studies the problem of power allocation in compressed sensing when
different components in the unknown sparse signal have different probability to
be non-zero. Given the prior information of the non-uniform sparsity and the
total power budget, we are interested in how to optimally allocate the power
across the columns of a Gaussian random measurement matrix so that the mean
squared reconstruction error is minimized. Based on the state evolution
technique originated from the work by Donoho, Maleki, and Montanari, we revise
the so called approximate message passing (AMP) algorithm for the
reconstruction and quantify the MSE performance in the asymptotic regime. Then
the closed form of the optimal power allocation is obtained. The results show
that in the presence of measurement noise, uniform power allocation, which
results in the commonly used Gaussian random matrix with i.i.d. entries, is not
optimal for non-uniformly sparse signals. Empirical results are presented to
demonstrate the performance gain.
