基于交替投影的随机高斯感知矢量相位恢复
Phase retrieval with random Gaussian sensing vectors by alternating projections
摘要：
我们考虑一个相位恢复问题，我们要重建一个$n$维矢量从其无相位标量积与$m$感测矢量。我们假设感测向量是从复正态分布独立采样的。我们建议用经典的交替投影的非凸方法来解决这个问题。我们表明，当$m\geq Cn$为$C$足够大，交替投影成功的概率很高，只要他们仔细初始化。我们还表明，有一个政权中的交替投影方法的停滞点消失，初始化程序变得无用。然而，在该方案中，$m$必须为$n^2$的量级。最后，我们推测，从我们的数值实验，在制度$m=O(n)$，有停滞点，但其吸引盆地的大小是小的，如果$m/n$是足够大的，所以交替的投影可以成功的概率接近$1$，即使没有特殊的初始化。
英文摘要：
We consider a phase retrieval problem, where we want to reconstruct a $n$-dimensional vector from its phaseless scalar products with $m$ sensing vectors. We assume the sensing vectors to be independently sampled from complex normal distributions. We propose to solve this problem with the classical non-convex method of alternating projections. We show that, when $m\geq Cn$ for $C$ large enough, alternating projections succeed with high probability, provided that they are carefully initialized. We also show that there is a regime in which the stagnation points of the alternating projections method disappear, and the initialization procedure becomes useless. However, in this regime, $m$ has to be of the order of $n^2$. Finally, we conjecture from our numerical experiments that, in the regime $m=O(n)$, there are stagnation points, but the size of their attraction basin is small if $m/n$ is large enough, so alternating projections can succeed with probability close to $1$ even with no special initialization.
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