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 ∈ ℂⁿ is measured as r = Bx and y = |Cx |, where B ∈ ℂ^{m1 ×n} and C ∈ 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.