Phase Retrieval In The General Setting Of Continuous Frames For Banach Spaces

Abstract

Abstract. We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work [11], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame.We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over K = R or K = C) and verify that it is a necessary condition for uniqueness of the phase retrieval problem; when K = R the CP is also sufficient for uniqueness. In our general setting, we also prove a conjecture posed by Bandeira et al. [7], which was originally formulated for finite-dimensional spaces: for the case K = C the strong complement property (SCP) is a necessary condition for stability. To prove our main result, we show that the SCP can never hold for frames of infinite-dimensional Banach spaces.

Stable Gabor Phase Retrieval and Spectral Clustering

“…For instance, it turns out that the Gabor 3 If f is a regular tempered distribution, i.e., abusing notation, For instance, it turns out that the Gabor 3 If f is a regular tempered distribution, i.e., abusing notation,…”

“…A well-known source of instability (e.g., a very large constant c.f /), coined "multicomponent-type instability" in [1] arises whenever the measurementsˆ.f / are separated in the sense that f D u C v withˆ.u/ andˆ.v/ concentrated in disjoint subsets of . If B is a finite-dimensional Hilbert space over R (i.e., the realvalued case where only a sign and not the full phase needs to be determined), the correctness of this intuition has been proved in [8] and generalized in [3] to the setting of 1-dimensional real or complex Banach spaces: If B is a finite-dimensional Hilbert space over R (i.e., the realvalued case where only a sign and not the full phase needs to be determined), the correctness of this intuition has been proved in [8] and generalized in [3] to the setting of 1-dimensional real or complex Banach spaces:…”

“…More precisely, one can characterize c.f /, via the so-called -strong complement property (SCP), which indeed provides a measure for the disconnectedness of the measurements; see [3,8]. More precisely, one can characterize c.f /, via the so-called -strong complement property (SCP), which indeed provides a measure for the disconnectedness of the measurements; see [3,8].…”

“…Intiutively, in this case the function g D u v will produce measurements jˆ.g/j very close to the original measurements jˆ.f /j, while the distance d B .f; g/ is not small at all, resulting in an instability (see also Figure 1.1 for an illustration). If B is a finite-dimensional Hilbert space over R (i.e., the realvalued case where only a sign and not the full phase needs to be determined), the correctness of this intuition has been proved in [8] and generalized in [3] to the setting of 1-dimensional real or complex Banach spaces:…”

“…If B is a Banach space over R it is not very difficult to show that the "multicomponent-type instability" as just described is the only source of instability. More precisely, one can characterize c.f /, via the so-called -strong complement property (SCP), which indeed provides a measure for the disconnectedness of the measurements; see [3,8]. While these results provide a complete characterization of the stability of phase retrieval problems over R, we hasten to add that the verification of the -strong complement property is computationally intractable, which severely limits their applicability.…”