Phaseless Reconstruction from Space-Time Samples

Abstract

Phaseless reconstruction from space-time samples is a nonlinear problem of recovering a function $x$ in a Hilbert space $\mathcal{H}$ from the modulus of linear measurements $\{|⟨x,{\varphi }_i⟩|$, $\dots$, $|⟨A^{L_i}x,{\varphi }_i⟩|:i\in \mathcal{I}\}$, where $\{{\varphi }_i;i\in \mathcal{I}\}\subset \mathcal{H}$ is a set of functionals on $\mathcal{H}$, and $A$ is a bounded operator on $\mathcal{H}$ that acts as an evolution operator. In this paper, we provide various sufficient or necessary conditions for solving this problem, which has connections to $X$-ray crystallography, the scattering transform, and deep learning.

23 pages, 4 figures