Sparse signal recovery from a mixture of linear and magnitude-only measurements

We consider the problem of exact sparse signal recovery from a combination of linear and magnitude-only (phaseless) measurements. A k-sparse signal x in ∈ ℂⁿ is measured as r = Bx and y = |Cx|, where B in ∈ ℂ^{m1 × n} and ∈ ℂ^{m2 × n} are measurement matrices and |·| is the element-wise absolute value. We show that if max(2m₁,1) + m₂ ≥ 4k - 1, then a set of generic measurements are sufficient to recover every k-sparse x exactly, establishing the trade-off between the number of linear and magnitude-only measurements.