Phase retrieval refers to the problem of reconstructing an unknown vector \(x_0 \in {\mathbb {C}}^n\) or \(x_0 \in {\mathbb {R}}^n \) from m measurements of the form \(y_i = \big \vert \langle \xi ^{\left( i\right) }, x_0 \rangle \big \vert ^2 \), where \( \left\{ \xi ^{\left( i\right) } \right\} ^m_{i=1} \subset {\mathbb {C}}^m \) are known measurement vectors. While Gaussian measurements allow for recovery of arbitrary signals provided the number of measurements scales at least linearly in the number of dimensions, it has been shown that ambiguities may arise for certain other classes of measurements \( \left\{ \xi ^{\left( i\right) } \right\} ^{m}_{i=1}\) such as Bernoulli measurements or Fourier measurements. In this paper, we will prove that even when a subgaussian vector \( \xi ^{\left( i\right) } \in {\mathbb {C}}^m \) does not fulfill a small-ball probability assumption, the PhaseLift method is still able to reconstruct a large class of signals \(x_0 \in {\mathbb {R}}^n\) from the measurements. This extends recent work by Krahmer and Liu from the real-valued to the complex-valued case. However, our proof strategy is quite different and we expect some of the new proof ideas to be useful in several other measurement scenarios as well. We then extend our results \(x_0 \in {\mathbb {C}}^n \) up to an additional assumption which, as we show, is necessary.

相位检索是指从m个形式的度量中重建未知向量\（x_0 \ in {\ mathbb {C}} ^ n \）或\（x_0 \ in {\ mathbb {R}} ^ n \）的问题\（y_i = \ big \ vert \ langle \ xi ^ {\ left（i \ right）}，x_0 \ rangle \ big \ vert ^ 2 \），其中\（\ left \ {\ xi ^ {\ left（i \ right）} \ right \} ^ m_ {i = 1} \ subset {\ mathbb {C}} ^ m \）是已知的测量向量。虽然高斯测量允许恢复任意信号，但条件是测量的数量至少在维度数量上成线性比例，但已经表明，某些其他类别的测量可能会产生歧义\（\ left \ {\ xi ^ {\ left （i \ right）} \ right \} ^ {m} _ {i = 1} \）例如伯努利测量或傅立叶测量。在本文中，我们将证明，即使在{\ mathbb {C}} ^ m \）中的高斯向量\（\ xi ^ {\ left（i \ right）} \ in不能满足小球概率假设，该PhaseLift方法仍然能够重构一大类的信号\（X_0 \在{\ mathbb {R}} ^ N \）从测量结果。这将Krahmer和Liu的最新工作从实值案例扩展到了复值案例。但是，我们的证明策略截然不同，我们希望一些新的证明思想也可以在其他几种测量方案中使用。然后，我们将结果\（x_0 \ in {\ mathbb {C}} ^ n \）扩展到一个额外的假设，如我们所示，这是必要的。