19 research outputs found
Locality and stability for phase retrieval
A frame for a Hilbert space is said to do phase
retrieval if for all distinct vectors the magnitude of the frame
coefficients and distinguish from (up to a unimodular scalar). We
consider the weaker condition where the magnitude of the frame coefficients
distinguishes from every vector in a small neighborhood of (up to a
unimodular scalar). We prove that some of the important theorems for phase
retrieval hold for this local condition, where as 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.Comment: 14 page
Stable phase retrieval and perturbations of frames
A frame for a Hilbert space is said to do phase
retrieval if for all distinct vectors the magnitude of the frame
coefficients and distinguish from (up to a unimodular scalar). A
frame which does phase retrieval is said to do -stable phase retrieval if
the recovery of any vector from the magnitude of the frame
coefficients is -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