On the connection between uniqueness from samples and stability in Gabor phase retrieval

TL;DR

This work shows that uniqueness from samples and stability in Gabor phase retrieval are not fundamentally linked: one can have counterexamples that break uniqueness on lattices while preserving the stability properties of the continuous problem. Using the Bargmann–Fock framework, the authors construct dense families of counterexamples in $L^2(\mathbb{R})$ and relate instability directions to low Laplacian eigenvalues, yielding a finite-dimensional obstruction picture. They prove that Gaussian recovery on lattices is possible for sufficiently fine sampling, yet the counterexamples can be made arbitrarily close to the Gaussian, forcing the conclusion that restricting to a small function class is necessary for robust uniqueness. The results imply that stability and uniqueness require careful priors or structure beyond mere local Lipschitz bounds, and they articulate a spectral-geometry pathway (via Poincaré and Cheeger constants, and Laplacian eigenfunctions) to understand and potentially mitigate instability directions in Gabor phase retrieval.

Abstract

Gabor phase retrieval is the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of $\mathbb{R}^2$). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples are dense in $L^2(\mathbb{R})$. Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

Figures (6)

• Figure 1: The Gabor transform magnitudes $|\mathcal{G}h_a^+|$ for a=1/6. The two bumps move apart as the sample rate $a$ goes to zero. As a consequence, the local Lipschitz constant of $h_a^+$ increases as $a$ goes to zero.
• Figure 2: A plot of the Gabor transform magnitude of the counterexamples whose existence is postulated in Theorem \ref{['thm:main']}.
• Figure 3: The Gabor transform magnitude of the fifth Hermite function (Fig. \ref{['fig:Gabor_Hermite']}) and of a counterexample $g_\delta^+$ to sampled Gabor phase retrieval on $\mathbb{R} \times \tfrac{1}{4} \mathbb{Z}$ (Fig. \ref{['fig:counterexample']}).
• Figure 4: We consider $a = 1/2$ and $\gamma = \exp(-5\pi)$. The roots of $\mathcal{G} f^+$ are indicated by circles and the roots of $\mathcal{G} f^-$ are indicated by disks. We have also drawn the local maxima of $\mathcal{G} f^\pm$ as squares and indicated the region on which 99% of the $L^2$-mass of $\mathcal{G} f^\pm$ is concentrated in light gray. We highlight that we have chosen $\gamma < \gamma_0(1/2,R=3) = \exp(-4\pi)$ such that the roots of $\mathcal{G} f^\pm$ fall outside the open ball of radius $R=3$. We also note that there is no gray region around the local maximum at $(2,0)$ indicating that very little mass is concentrated on the small bump.
• Figure 5: A sketch of an example of a function $a(x)$ (left) corresponding to a difficult "toy phase retrieval problem". There is a function $\phi$ (right) for which $\int_0^1 a(x) \phi'(x)^2 \,\mathrm{d} x$ is small even though $\phi$ not close to a constant.

Theorems & Definitions (25)

• Remark 1
• Remark 2
• Definition 1
• Lemma 2
• proof
• Theorem 3
• proof
• Remark 3: Some explanations on the proof
• Theorem 4
• Example 1