Nonlinear determination and phase retrieval under unimodular constraints

TL;DR

This work develops a unified framework for nonlinear determination in Hilbert spaces under unimodular constraints, formalized as $Θ$-PR. It provides a countable-Θ characterization via covers and orthogonality relations, plus a simple second-order recurrence description when $Θ$ is cyclic, and a density-criterion for exponential systems in the finite- or Paley–Wiener setting. A Möbius-invariant viewpoint links circle automorphisms and cross ratios to $Θ$-PR, enabling transfer of PR properties across equivalent phase-sets. In finite dimensions, the authors establish sharp impossibility thresholds and show that $Θ$-PR is generic for countable $Θ$, with a clear dichotomy for uncountable $Θ$ (arc-containing sets reduce to standard PR). The results yield new insights into the geometry of phase retrieval, Fourier-analytic uniqueness, and the minimal redundancy required for nonlinear determination, with implications for lattice-density design and cross-field applications.

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 $Θ$ of the unit circle $\mathbb{T}$, we say that a system $\mathbf{G}\subseteq H$ does $Θ$-phase retrieval ($Θ$-PR) if for all $f,h\in H$ the condition that for every $g\in\mathbf{G}$ there exists $θ_g\inΘ$ with $\langle f,g\rangle=θ_g\langle h,g\rangle$ forces $f=θh$ for some $θ\inΘ$. This framework unifies classical phase retrieval ($Θ=\mathbb{T}$) and sign retrieval ($Θ=\{1,-1\}$). For every countable $Θ$ we give a complete characterization of $Θ$-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 $Θ$-PR is equivalent to the existence of specific second-order recurrence relations. We apply this to obtain a sharp lattice density criterion for $Θ$-PR of exponential systems. For uncountable $Θ$ we obtain a topological dichotomy in the Fourier determination setting, showing that $Θ$-PR is characterized in terms of connectedness of $Θ$. We further develop a Möbius-invariant framework, proving that $Θ$-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 $Θ$ the property is generic once one passes the failure regime, yielding the minimal number of vectors required for $Θ$-PR.

Theorems & Definitions (59)

• Definition 1.1
• Proposition 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Corollary 1.10