Sparse Signal Recovery from a Mixture of Measurements

We consider the problem of exact sparse signal recovery from a combination of linear and magnitude-only (phaseless) measurements. A$k$-sparse signal ${bf x} in {BBC^n}$ is measured as ${bf r} = {bf Bx}$and ${bf y} = vert {bf Cx} vert$, where ${bf B} in {BBC^{{m_1} times n}}$and ${bf C} in {BBC^{{m_2} times n}}$ are measurement matrices and$vert cdot vert$ is the element-wise absolute value.

We show that if $max (2{m_1},1) + {m_2} geq 4k – 1$, then a set of generic measurements are sufficient to recover every $k$-sparse ${bf x}$ exactly, establishing the trade-off between the number of linear and magnitude-only measurements.