Nonlinear determination and phase retrieval under unimodular constraints

TL;DR

This paper investigates nonlinear phase retrieval problems in Hilbert spaces with a focus on unimodular constraints, providing comprehensive characterizations, geometric insights, and thresholds for successful determination under various conditions.

Contribution

It introduces a unified framework for phase and sign retrieval, characterizes $ heta$-PR for countable and uncountable sets, and establishes sharp thresholds and invariance properties.

Findings

Complete characterization of $ heta$-PR for countable sets

Equivalence of $ heta$-PR to recurrence relations for cyclic sets

Sharp thresholds and genericity results in $ extbf{C}^d$

Abstract

We study nonlinear determination problems in Hilbert spaces in which inner products are observed up to prescribed rotations in the complex plane. Given a Hilbert space $H$ and a subset $\Theta$ of the unit circle $\mathbb{T}$, we say that a system $\mathbf{G}\subseteq H$ does $\Theta$-phase retrieval ($\Theta$-PR) if for all $f,h\in H$ the condition that for every $g\in\mathbf{G}$ there exists $\theta_g\in\Theta$ with $\langle f,g\rangle=\theta_g\langle h,g\rangle$ forces $f=\theta h$ for some $\theta\in\Theta$. This framework unifies classical phase retrieval ($\Theta=\mathbb{T}$) and sign retrieval ($\Theta=\{1,-1\}$). For every countable $\Theta$ we give a complete characterization of $\Theta$-PR in terms of covers of $\mathbf{G}$ and geometric relations among vectors in the corresponding orthogonal complements, extending the complement-property characterization of Cahill, Casazza, and Daubechies. For cyclic phase sets we show that $\Theta$-PR is equivalent to the existence of specific second-order recurrence relations. We apply this to obtain a sharp lattice density criterion for $\Theta$-PR of exponential systems. For uncountable $\Theta$ we obtain a topological dichotomy in the Fourier determination setting, showing that $\Theta$-PR is characterized in terms of connectedness of $\Theta$. We further develop a M\"obius-invariant framework, proving that $\Theta$-PR is preserved under circle automorphisms and is governed by projective invariants such as the cross ratio. Finally, in $\mathbb{C}^d$ we determine sharp impossibility thresholds and prove that for countable $\Theta$ the property is generic once one passes the failure regime, yielding the minimal number of vectors required for $\Theta$-PR.