Gabor phase retrieval is severely ill-posed

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