保存相位：相位恢复的注入性和稳定性
Saving phase: Injectivity and stability for phase retrieval
摘要：
凸优化的最新进展使有限维向量空间上的相位恢复问题取得了新的进展。然而，某些基本问题仍然存在：什么样的测量向量唯一地确定每个信号的全局相位因子，需要多少个？此外，哪些测量组合提供稳定性？本文提出了解决这些问题的几个结果。我们开始由特征的内射性，我们确定的补充性质确实是一个必要条件在复杂的情况下。然后，我们提出了一个猜想，4 M-4通用测量向量是必要的和充分的内射在M维，我们证明了这一猜想的特殊情况下，M= 2，3。接下来，我们将注意力转移到稳定性，无论是在最坏的情况下还是在一般情况下。在这里，我们通过引入一个数值版本的补充属性，在实际情况下的最坏情况下的稳定性。这个新的属性与压缩感知的受限等距属性有一些相似之处，并且可以用于导出强度测量映射上的陡峭的Lipschitz下界。本地化的帧缺乏这种属性（建议不稳定），而高斯随机测量显示，以满足此属性的高概率。最后，我们提出的结果，使用随机噪声模型在真实和复杂的情况下，我们利用Cramer-Rao下界，以确定稳定性与更强的版本的注入特性。
英文摘要：
Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every signal up to a global phase factor, and how many are needed to do so? Furthermore, which measurement ensembles lend stability? This paper presents several results that address each of these questions. We begin by characterizing injectivity, and we identify that the complement property is indeed a necessary condition in the complex case. We then pose a conjecture that 4M-4 generic measurement vectors are both necessary and sufficient for injectivity in M dimensions, and we prove this conjecture in the special cases where M=2,3. Next, we shift our attention to stability, both in the worst and average cases. Here, we characterize worst-case stability in the real case by introducing a numerical version of the complement property. This new property bears some resemblance to the restricted isometry property of compressed sensing and can be used to derive a sharp lower Lipschitz bound on the intensity measurement mapping. Localized frames are shown to lack this property (suggesting instability), whereas Gaussian random measurements are shown to satisfy this property with high probability. We conclude by presenting results that use a stochastic noise model in both the real and complex cases, and we leverage Cramer-Rao lower bounds to identify stability with stronger versions of the injectivity characterizations.
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