Stable Phase Retrieval in Infinite Dimensions

Rima Alaifari, Ingrid Daubechies, Philipp Grohs, Rujie Yin

Published 2016-08-31Version 1

The problem of phase retrieval is to determine a signal $f\in\mathcal{H}$, with $\mathcal{H}$ a Hilbert space, from intensity measurements $|F(\omega)|:=|\langle f,\varphi_\omega\rangle|$ associated with a measurement system $(\varphi_\omega)_{\omega\in \Omega}\subset\mathcal{H}$. Such problems can be seen in a wide variety of applications, ranging from X-ray crystallography, microscopy to audio processing and deep learning algorithms and accordingly, a large body of literature treating the mathematical and algorithmic solution of phase retrieval problems has emerged in recent years. Recent work [9,3] has shown that, whenever $\mathcal{H}$ is infinite-dimensional, phase retrieval is never uniformly stable, and that, although it is always stable in the finite dimensional setting, the stability deteriorates severely in the dimension of the problem [9]. On the other hand, all observed instabilities are of a certain type: they occur whenever the function $|F|$ of intensity measurements is concentrated on disjoint sets $D_j\subset\Omega$, i.e., when $F=\sum_{j=1}^k F_j$ where $F_j$ is concentrated on $D_j$ (and $k \geq 2$). Indeed, it is easy to see that intensity measurements of any function $\sum_{j=1}^k e^{i\alpha_j} F_j$ will be close to those of $F$ while the functions themselves need not be close at all. 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 $D_j$, i.e., recovering $F$ up to the equivalence $$F \sim \sum_{j=1}^k e^{i\alpha_j}F_j.$$ We present concrete applications where this new notion of stability is natural and meaningful and show that in this setting stable phase retrieval can actually be achieved, e.g. if the measurement system is a Gabor frame or a frame of Cauchy wavelets.