An algebraic characterization of injectivity in phase retrieval

A complex frame is a collection of vectors that span ${\mathbb{C}}^M$ and define measurements, called intensity measurements, on vectors in ${\mathbb{C}}^M$. In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from $4M-4$ generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.