Title: Optimal Frames for Phase Retrieval from Edge Vectors of Optimal Polygons Show Chinese title

Title: 相位重构的最优框架来自最优多边形的边向量

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. Show Chinese abstract

Abstract: 本文旨在表征相位恢复的最优框架，定义为使相位恢复条件数达到最小值的框架。 在二维实数情况下，我们揭示了相位恢复的最优框架与最初由Reinhardt于1922年提出的周长最大化等直径问题之间的联系。 我们的工作表明，周长最大化等直径问题的每个最优解都会自然地导致${\mathbb R}^2$中的一个最优框架。 通过将最优多边形问题重新表述为关于单位根差异的问题，我们表征了所有最优多边形。 基于这一联系，我们随后表征了当$m \geq 3$具有奇数因子时，在${\mathbb R}^2$中具有$m$个向量的相位恢复的最优框架。 作为关键推论，我们证明了调和框架$E_m$对于任何偶数整数$m \geq 4$都是{\em 不是}最优的。 这一发现反驳了Xia、Xu和Xu（Math. Comp., 94(356): 2931-2960）提出的一个猜想。 先前的工作已经证明，当$m$是奇数时，谐波框架$E_m \subset {\mathbb R}^2$确实是最优的。 探索相位恢复与离散几何之间的联系，本文旨在阐明相位恢复的进展，并为周长最大化的等直径问题提供新的视角。