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The key ingredient to retrieving a signal from its Fourier magnitudes,namely, to solve the phase retrieval problem, is an effective prior on thesought signal. In this paper, we study the phase retrieval problem under theprior that the signal lies in a semi-algebraic set. This is a very generalprior as semi-algebraic sets include linear models, sparse models, and ReLUneural network generative models. The latter is the main motivation of thispaper, due to the remarkable success of deep generative models in a variety ofimaging tasks, including phase retrieval. We prove that almost all signals inR^N can be determined from their Fourier magnitudes, up to a sign, if they liein a (generic) semi-algebraic set of dimension N/2. The same is true for allsignals if the semi-algebraic set is of dimension N/4. We also generalize theseresults to the problem of signal recovery from the second moment inmulti-reference alignment models with multiplicity free representations ofcompact groups. This general result is then used to derive improved samplecomplexity bounds for recovering band-limited functions on the sphere fromtheir noisy copies, each acted upon by a random element of SO(3).

Phase retrieval with semi-algebraic and ReLU neural network priors