The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous [2]. On the other hand, in finite-dimensional settings, unique solvability of the problem implies uniform stability.

A prominent example of such a phase retrieval problem is the recovery of a signal from the modulus of its Gabor transform. In this paper, we study Gabor phase retrieval and ask how the stability degrades on a natural family of finite-dimensional subspaces of the signal domain $L^2(\mathbb{R})$. We prove that the stability constant scales at least quadratically exponentially in the dimension of the subspaces. Our construction also shows that typical priors such as sparsity or smoothness promoting penalties do not constitute regularization terms for phase retrieval.

最近已经证明，从函数的帧系数的大小重建函数的问题在无限维空间中永远不会统一稳定[5]。对于可能连续的帧，此结果也适用[2]。另一方面，在有限维设置中，问题的独特可解性意味着稳定的稳定性。

这种相位恢复问题的一个突出示例是从其Gabor变换的模数中恢复信号。在本文中，我们研究Gabor相位检索，并询问稳定性如何在信号域的有限维子空间的自然族上退化${\mathrm{大号}}^2（\mathbb{[R}）$。我们证明，稳定常数在子空间的维度上至少平方成指数比例。我们的构造还表明，诸如稀疏性或平滑度提升惩罚之类的典型先验不构成用于相位检索的正则化项。