Locality and stability for phase retrieval

A frame \((x_j)_{j\in J}\) ${(x_j)}_{j\in J}$ for a Hilbert space H is said to do phase retrieval if for all distinct vectors \(x,y\in H\) $x,y\in H$ the magnitude of the frame coefficients \((|\langle x, x_j\rangle |)_{j\in J}\) $(|⟨x,x_j⟩{|)}_{j\in J}$ and \((|\langle y, x_j\rangle |)_{j\in J}\) $(|⟨y,x_j⟩{|)}_{j\in J}$ distinguish x from y (up to a unimodular scalar). We consider the weaker condition where the magnitude of the frame coefficients distinguishes x from every vector y in a small neighborhood of x (up to a unimodular scalar). We prove that some of the important theorems for phase retrieval hold for this local condition, whereas some theorems are completely different. We prove as well that when considering stability of phase retrieval, the worst stability inequality is always witnessed at orthogonal vectors. This allows for much simpler calculations when considering optimization problems for phase retrieval.