Optimal Frames for Phase Retrieval from Edge Vectors of Optimal Polygons
Zhiqiang Xu, Zili Xu, Xinyue Zhang
TL;DR
The paper investigates stability and optimality in phase retrieval by linking the minimal phaseless-measurement condition number to Reinhardt's perimeter-maximizing isodiametric problem. It develops a bijection between optimal tight frames and edge-sets of optimal polygons in $\mathbb{R}^2$, yielding the exact relation $\beta_{\boldsymbol{A}}=\frac{1}{\sqrt{1-2r(P)}}$ and enabling explicit minimizers when $m$ has an odd factor. It provides a full characterization of all optimal polygons via a discrepancy problem on roots of unity and consequently a complete description of all optimal frames in ${\mathbb R}^2$ for such $m$, including a counterexample showing the harmonic frame $E_m$ is not optimal for even $m$ (e.g., $m=4$). These results fuse phase retrieval stability with discrete geometry and discrepancy theory, offering new insights and potential extensions to higher dimensions and complex settings.
Abstract
This paper aims to characterize the optimal frame for phase retrieval, defined as the frame whose condition number for phase retrieval attains its minimal value. In the context of the two-dimensional real case, we reveal the connection between optimal frames for phase retrieval and the perimeter-maximizing isodiametric problem, originally proposed by Reinhardt in 1922. Our work establishes that every optimal solution to the perimeter-maximizing isodiametric problem inherently leads to an optimal frame in ${\mathbb R}^2$. By recasting the optimal polygons problem as one concerning the discrepancy of roots of unity, we characterize all optimal polygons. Building upon this connection, we then characterize all optimal frames with $m$ vectors in ${\mathbb R}^2$ for phase retrieval when $m \geq 3$ has an odd factor. As a key corollary, we show that the harmonic frame $E_m$ is {\em not} optimal for any even integer $m \geq 4$. This finding disproves a conjecture proposed by Xia, Xu, and Xu (Math. Comp., 94(356): 2931-2960). Previous work has established that the harmonic frame $E_m \subset {\mathbb R}^2$ is indeed optimal when $m$ is an odd integer. Exploring the connection between phase retrieval and discrete geometry, this paper aims to illuminate advancements in phase retrieval and offer new perspectives on the perimeter-maximizing isodiametric problem.