论文标题
$ L_2（\ Mathbb {r}）$的无限尺寸子空间的稳定阶段检索
Stable phase retrieval for infinite dimensional subspaces of $L_2(\mathbb{R})$
论文作者
论文摘要
当使用框架或连续的框架作为无限尺寸希尔伯特空间时，相近的相位检索始终是不稳定的。我们考虑对$ L_2 $的子空间设置的相位检索的概括，该子空间与使用连续帧进行相位检索时，当子空间是连续帧的分析操作员的范围时。然后，我们证明确实存在$ l_2 $的无限尺寸子空间，而相位检索是稳定的。也就是说，我们提供了一种构建无限尺寸子空间$ y \ subseteq l_2 $的方法\ big \ | | f | - | g | \ big \ | _ {l_2} \ qquad \ ququad \ textrm {for All} f，g \ iny。$$ 这种结构还导致了有限维度中相位检索统一的稳定性的新结果。我们的构造具有确定性的组件和一个随机组件。当使用次高斯随机变量时，我们在$ n $随机向量的订单上使用$ m $时实现具有高概率和稳定性的相位检索。没有次高斯或任何其他更高的时刻假设，我们就能在$ n \ log（n）$随机向量的订单上使用$ m $时，具有高概率和稳定性的相位检索，而稳定性则独立于尺寸$ n $。
Phase retrieval is known to always be unstable when using a frame or continuous frame for an infinite dimensional Hilbert space. We consider a generalization of phase retrieval to the setting of subspaces of $L_2$ which coincides with using a continuous frame for phase retrieval when the subspace is the range of the analysis operator of a continuous frame. We then prove that there do exist infinite dimensional subspaces of $L_2$ where phase retrieval is stable. That is, we give a method for constructing an infinite dimensional subspace $Y\subseteq L_2$ such that there exists $C\geq 1$ so that $$\min\big(\big\|f-g\big\|_{L_2},\big\|f+g\big\|_{L_2}\big)\leq C \big\| |f|-|g| \big\|_{L_2} \qquad\textrm{ for all }f,g\in Y. $$ This construction also leads to new results on uniform stability of phase retrieval in finite dimensions. Our construction has a deterministic component and a random component. When using sub-Gaussian random variables we achieve phase retrieval with high probability and stability constant independent of the dimension $n$ when using $m$ on the order of $n$ random vectors. Without sub-Gaussian or any other higher moment assumptions, we are able to achieve phase retrieval with high probability and stability constant independent of the dimension $n$ when using $m$ on the order of $n\log(n)$ random vectors.
