Phase retrieval in infinite-dimensional Hilbert spaces (Q2826799)

A family \(\Phi=\{\varphi_n\}\) in a separable Hilbert space \(\mathcal{H}\) does phase retrieval if the inequalities \(|\langle f,\, \varphi_n\rangle|\leq |\langle g,\, \varphi_n\rangle|\) and \(\|f\|=\|g\|\) imply that \(f=\alpha g\) with \(|\alpha|=1\). In other words, the nonlinear operator \(\mathcal{A}_\Phi: f\mapsto \{|\langle f, \varphi_n\rangle|\}\) is injective in \(\mathcal{H}/{\sim}\), where \(f\sim g\) means that \(f=\alpha g\), \(|\alpha|=1\). In finite dimensions, frames almost always do phase retrieval if they are redundant enough. The first result, Prop.~1, states that in an infinite-dimensional separable Hilbert space \(\mathcal{H}\), to any frame \(\{\varphi_n\}\) that does phase retrieval and any \(\epsilon>0\), there is another frame \(\{\psi_n\}\) that does not do phase retrieval, even though \(\sum \|\varphi_n-\psi_n\|^2<\epsilon\). The main result of the paper, Theorem 2.2, proves that phase retrieval is inherently unstable: given a frame \(\{\varphi_n\}\) with \(\|\varphi_n\|\geq c>0\) for all \(n\), for every \(\delta>0\) there exist \(f,g\in\mathcal{H}\) with \(\inf_\alpha \|f-\alpha g\|\geq 1\) but \(\|\mathcal{A}_\Phi(f)-\mathcal{A}_\Phi(g)\|<\delta\). The authors show (Theorem 2.7) that, however, phase retrieval is in a sense somewhat stable over elements that are well-approximated by finite frame expansions: Let \(V_m\) be a family of finite-dimensional subspaces of \(\mathcal{H}\) of strictly increasing dimension and denote by \(P_m\) the orthogonal projection onto \(V_m\). Suppose that there is a strictly increasing \(G(m)\) such that on \(V_m\), NEWLINE\[NEWLINE\inf_\alpha \|f-\alpha g\|\leq G(m) \|\mathcal{A}_\Phi(f)-\mathcal{A}_\Phi(g)\|.NEWLINE\]NEWLINE Define \(B_\gamma(R)=\{f\in\mathcal{H}: \|f-P_mf\|\leq G(m+1)^{-\gamma}R\|f\|\}\) uniformly in \(m\). Then, for \(f,g\in B_\gamma(R)\), one has NEWLINE\[NEWLINE\inf_\alpha \|f-\alpha g\|\leq C(1+\|f\|+\|g\|) \|\mathcal{A}_\Phi(f)-\mathcal{A}_\Phi(g)\|^{(\gamma-1)/\gamma}.NEWLINE\]NEWLINENEWLINENEWLINEExample 2.11 considers the frame of quarter-integer shifted sinc functions for the real-valued Paley-Wiener space, defines \(V_n\) to be the span of the sinc functions shifted by integers between \(-n\) and \(n\), and proves that there exist \(f,g\in V_{2m}\) with suitable polynomial decay such that \(\min_\pm\|f\pm\alpha g\|> C(m+1)^{-1}2^{3m} \|\mathcal{A}_\Phi(f)-\mathcal{A}_\Phi(g)\|\) (\(C\) independent of \(m\)). This shows that \(G\) can be a fast-growing function.