Conjugate phase retrieval on graphs and with applications in shift-invariant spaces

In this paper, we study the conjugate phase retrieval for complex-valued \mbox{signals} residing on graphs, and explore its applications to shift-invariant spaces. Given a complex-valued graph signal $\bf f$ residing on the graph $\mathcal G$, we introduce a graph ${\mathcal G}_{\bf f}$ and show that its connectivity is sufficient to determine $\bf f$ up to a global unimodular constant and conjugation. We then construct two explicit graph models and show that graph signals residing on them can be recovered, up to a unimodular constant and conjugation, from its absolute values on the vertices and the relative magnitudes between neighboring vertices. Building on this graph-based framework, we apply our results to shift-invariant spaces generated by real-valued functions. For signals in the Paley-Wiener space, we show that any complex-valued function can be recovered, up to a unimodular constant and conjugation, from structured phaseless samples taken at three times the Nyquist rate. For more general shift invariant spaces, we establish the conjugate phase retrievability of signals from phaseless samples collected on a discrete sampling set, in conjunction with relative magnitude measurements between neighboring sample points. Two numerical reconstruction algorithms are introduced to recover the signals in the Paley-Wiener space and general shift-invariant spaces, up to a unimodular constant and conjugation, from the given phaseless measurements.