21 research outputs found
The generic crystallographic phase retrieval problem
In this paper we consider the problem of recovering a signal from its power spectrum assuming that the signal is sparse with
respect to a generic basis for . Our main result is that if the
sparsity level is at most in this basis then the generic sparse
vector is uniquely determined up to sign from its power spectrum. We also prove
that if the sparsity level is then every sparse vector is
determined up to sign from its power spectrum. Analogous results are also
obtained for the power spectrum of a vector in which extend
earlier results of Wang and Xu \cite{arXiv:1310.0873}.Comment: 20 page
Phase retrieval with semi-algebraic and ReLU neural network priors
The key ingredient to retrieving a signal from its Fourier magnitudes,
namely, to solve the phase retrieval problem, is an effective prior on the
sought signal. In this paper, we study the phase retrieval problem under the
prior that the signal lies in a semi-algebraic set. This is a very general
prior as semi-algebraic sets include linear models, sparse models, and ReLU
neural network generative models. The latter is the main motivation of this
paper, due to the remarkable success of deep generative models in a variety of
imaging tasks, including phase retrieval. We prove that almost all signals in
R^N can be determined from their Fourier magnitudes, up to a sign, if they lie
in a (generic) semi-algebraic set of dimension N/2. The same is true for all
signals if the semi-algebraic set is of dimension N/4. We also generalize these
results to the problem of signal recovery from the second moment in
multi-reference alignment models with multiplicity free representations of
compact groups. This general result is then used to derive improved sample
complexity bounds for recovering band-limited functions on the sphere from
their noisy copies, each acted upon by a random element of SO(3)