仿射相位恢复的强凸性
Strong convexity of affine phase retrieval
摘要：
从具有预先已知的一些条目的强度测量中恢复信号被称为{ \em仿射相位恢复}。本文证明了在一定的条件下，当测量值为复高斯随机向量且测量数为$m \gtrsim d \log d$时，仿射相位恢复的自然最小二乘公式在整个空间上是强凸的，其中$d$是信号的维数.在此基础上，我们证明了简单的梯度下降法从任意初始点以高概率线性收敛到目标解。这些结果显示了仿射相位恢复和经典相位恢复之间的本质区别，其中经典相位恢复的最小二乘公式是非凸的。
英文摘要：
The recovery of a signal from the intensity measurements with some entries being known in advance is termed as {\em affine phase retrieval}. In this paper, we prove that a natural least squares formulation for the affine phase retrieval is strongly convex on the entire space under some mild conditions, provided the measurements are complex Gaussian random vecotrs and the measurement number $m \gtrsim d \log d$ where $d$ is the dimension of signals. Based on the result, we prove that the simple gradient descent method for the affine phase retrieval converges linearly to the target solution with high probability from an arbitrary initial point. These results show an essential difference between the affine phase retrieval and the classical phase retrieval, where the least squares formulations for the classical phase retrieval are non-convex.
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