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).

翻译：从傅里叶幅度恢复信号（即求解相位恢复问题）的关键要素在于对目标信号施加有效的先验知识。本文研究信号位于半代数集合这一先验条件下的相位恢复问题。由于半代数集合包含线性模型、稀疏模型以及ReLU神经网络生成模型，该先验具有很强的通用性。鉴于深度生成模型在包括相位恢复在内的多种成像任务中取得的显著成功，后者成为本文的主要研究动机。我们证明：若信号位于维数为N/2的（一般性）半代数集合中，则几乎所有的R^N空间信号均可通过其傅里叶幅度确定（至多相差一个符号）。当半代数集合维数降至N/4时，该结论对所有信号成立。我们还将这些结果推广至紧群重数自由表示的多参考对齐模型中从二阶矩恢复信号的问题。该通用结论随后被用于推导球面上带限函数从含噪声副本（每个副本受到SO(3)随机元素作用）中恢复的样本复杂度改进界。