# Phaseless Reconstruction from Space-Time Samples

## TL;DR

This paper investigates the problem of reconstructing a function in a Hilbert space from the magnitudes of its space-time measurements, providing conditions for solvability with applications to crystallography, scattering, and deep learning.

## Contribution

It offers new necessary and sufficient conditions for phaseless reconstruction from space-time samples, connecting the problem to various applied fields.

## Key findings

- Established conditions for reconstructability from modulus measurements.
- Linked the problem to applications in crystallography and deep learning.
- Provided theoretical insights into nonlinear inverse problems.

## 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 $\{\lvert \langle x, \phi_i\rangle \rvert$, $ \ldots$, $\lvert \langle A^{L_i}x, \phi_i \rangle \rvert : i \in\mathscr I\}$, where $\{\phi_i; i \in\mathscr I\}\subset \mathcal{H}$ 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.