We explore a spectral initialization method that plays a central role in contemporary research on signal estimation in nonconvex scenarios. In a noiseless phase retrieval framework, we precisely analyze the method's performance in the high-dimensional limit when sensing vectors follow a multivariate Gaussian distribution for two rotationally invariant models of the covariance matrix C. In the first model C is a projector on a lower dimensional space while in the second it is a Wishart matrix. Our analytical results extend the well-established case when C is the identity matrix. Our examination shows that the introduction of biased spatial directions leads to a substantial improvement in the spectral method's effectiveness, particularly when the number of measurements is less than the signal's dimension. This extension also consistently reveals a phase transition phenomenon dependent on the ratio between sample size and signal dimension. Surprisingly, both of these models share the same threshold value.

中文翻译：

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具有偏置空间方向的高维相位检索的谱初始化

我们探索了一种谱初始化方法，该方法在当代非凸场景中的信号估计研究中发挥着核心作用。在无噪声相位检索框架中，当感测向量遵循协方差矩阵 C 的两个旋转不变模型的多元高斯分布时，我们精确分析了该方法在高维限制下的性能。在第一个模型中，C 是较低维上的投影仪空间，而第二个是 Wishart 矩阵。当 C 是单位矩阵时，我们的分析结果扩展了已确立的情况。我们的研究表明，引入有偏差的空间方向可以显着提高谱方法的有效性，特别是当测量数量小于信号维度时。这种扩展还一致地揭示了依赖于样本大小和信号维度之间的比率的相变现象。令人惊讶的是，这两个模型具有相同的阈值。