Phaseless Reconstruction from Space-Time Samples

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 ${lvert langle x, phi_irangle rvert$, $ ldots$, $lvert langle A^{L_i}x, phi_i rangle rvert : i inmathscr I}$, where ${phi_i; i inmathscr I}subset mathcal{H}$ 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.