Nonlinear determination and phase retrieval under unimodular constraints

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.