Stable Phase Retrieval in Infinite Dimensions

The problem of phase retrieval is to determine a signal $$f\in \mathcal {H}$$f∈H, with $$ \mathcal {H}$$H a Hilbert space, from intensity measurements $$|F(\omega )|$$|F(ω)|, where $$F(\omega ):=\langle f, \varphi _\omega \rangle $$F(ω):=⟨f,φω⟩ are measurements of f with respect to a measurement system $$(\varphi _\omega )_{\omega \in \Omega }\subset \mathcal {H}$$(φω)ω∈Ω⊂H. Although phase retrieval is always stable in the finite-dimensional setting whenever it is possible (i.e. injectivity implies stability for the inverse problem), the situation is drastically different if $$\mathcal {H}$$H is infinite-dimensional: in that case phase retrieval is never uniformly stable (Alaifari and Grohs in SIAM J Math Anal 49(3):1895–1911, 2017; Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016); moreover, the stability deteriorates severely in the dimension of the problem (Cahill et al. 2016). On the other hand, all empirically observed instabilities are of a certain type: they occur whenever the function |F| of intensity measurements is concentrated on disjoint sets $$D_j\subset \Omega $$Dj⊂Ω, i.e. when $$F= \sum _{j=1}^k F_j$$F=∑j=1kFj where each $$F_j$$Fj is concentrated on $$D_j$$Dj (and $$k \ge 2$$k≥2). Motivated by these considerations, we propose a new paradigm for stable phase retrieval by considering the problem of reconstructing F up to a phase factor that is not global, but that can be different for each of the subsets $$D_j$$Dj, i.e. recovering F up to the equivalence $$\begin{aligned} F \sim \sum _{j=1}^k e^{\mathrm {i}\alpha _j} F_j. \end{aligned}$$F∼∑j=1keiαjFj.We present concrete applications (for example in audio processing) where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, for instance, if the measurement system is a Gabor frame or a frame of Cauchy wavelets.