论文标题
尖峰，根和调制：有限支持的复合措施的相位检索
Spikes, Roots, and Modulations: Phase Retrieval for Finitely-Supported Complex Measures
论文作者
论文摘要
我们研究了有限支持的分布的恢复，即从强度测量中恢复狄拉克测量的复杂线性组合。分布$μ= \ sum_ {j = 1}^{s} c_ {j}δ_{t_ {t_ {j}} $由系数向量$ c \ in \ in \ mathbb {c}^s $ in \ in \ mathbb {c}^s $及其支持$ \ \ \ \ \ { $λ> 0 $。强度测量值评估（平方）对应用于$μ$的一组线性功能的幅度，通过对$ \ hatμ$，$μ$的傅立叶变换或评估调制样品之间的差异而获得。遵循Alexeev等人的策略，将线性函数的结构以及非线性幅度测量的结构编码为图，其中顶点表示$ \hatμ$ $ $ \ hat $ \ \ \ hat $ \ f_1，v_2，v_2，v_2，\ dots，v_n \} \ a a（v_n \} \ sectents Indertents Indistions y的点评估） 它。我们表明，ramanujan图表$ d \ ge 3 $和$ n> \ frac {6（1 + 6/\ ln（s/λΩ））s}} {1-2 \ sqrt {d-1}/d} $ vertices提供$ m =（d + 1）n $ agnitudes，可用于确定一个整体的速度，以确定一个整体的速度。以包括额外的过采样步骤为代价，并且还要求$ n-1 $是主要的，我们构建了一种基于Prony方法的显式恢复算法。
We study the recovery of a finitely supported distribution, a complex linear combination of Dirac measures, from intensity measurements. The distribution $μ=\sum_{j=1}^{s}c_{j}δ_{t_{j}}$ is given by a coefficient vector $c\in\mathbb{C}^s$ and its support $\{t_1, t_2, \dots, t_s\}$ is contained in $ [0,Λ]$ for some $Λ>0$. The intensity measurements evaluate (squared) magnitudes of a set of linear functionals applied to $μ$, obtained by sampling $\hat μ$, the Fourier transform of $μ$, or by evaluating differences between modulated samples. Following a strategy by Alexeev et al., the structure of the linear functionals, and hence of the non-linear magnitude measurement, is encoded with a graph, where the vertices represent point evaluations of $\hat μ$ at $\{v_1, v_2, \dots, v_n\} \subset [-Ω,Ω]$ and each edge represents a (modulated) difference between vertices incident with it. We show that a Ramanujan graph with degree $d \ge 3$ and $n>\frac{6(1 + 6 /\ln(s/ΛΩ)) s}{1-2\sqrt{d-1}/d}$ vertices provides $M=(d+1)n$ magnitudes that are sufficient for identifying the complex measure up to an overall unimodular multiplicative constant. At the cost of including an additional oversampling step and with an additional requirement that $n-1$ is prime, we construct an explicit recovery algorithm that is based on the Prony method.
