Stable phase retrieval and perturbations of frames

A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ is said to do phase retrieval if for all distinct vectors $x,y\in H$ the magnitude of the frame coefficients $(|\langle x, x_j\rangle|)_{j\in J}$ and $(|\langle y, x_j\rangle|)_{j\in J}$ distinguish $x$ from $y$ (up to a unimodular scalar). A frame which does phase retrieval is said to do $C$-stable phase retrieval if the recovery of any vector $x\in H$ from the magnitude of the frame coefficients is $C$-Lipschitz. It is known that if a frame does stable phase retrieval then any sufficiently small perturbation of the frame vectors will do stable phase retrieval, though with a slightly worse stability constant. We provide new quantitative bounds on how the stability constant for phase retrieval is affected by a small perturbation of the frame vectors. These bounds are significant in that they are independent of the dimension of the Hilbert space and the number of vectors in the frame.Comment: 14 page

Complex phase retrieval from subgaussian measurements

Phase retrieval refers to the problem of reconstructing an unknown vector x0∈Cn or x0∈Rn from m measurements of the form yi=∣∣⟨ξ(i),x0⟩∣∣2, where {ξ(i)}mi=1⊂Cm are known measurement vectors. While Gaussian measurements allow for recovery of arbitrary signals provided the number of measurements scales at least linearly in the number of dimensions, it has been shown that ambiguities may arise for certain other classes of measurements {ξ(i)}mi=1 such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector ξ(i)∈Cm does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals x0∈Rn from the measurements. This extends recent work by Krahmer and Liu from the real-valued to the complex-valued case. However, our proof strategy is quite different and we expect some of the new proof ideas to be useful in several other measurement scenarios as well. We then extend our results x0∈Cn up to an additional assumption which, as we show, is necessary