论文标题
阶段检索基本结果的基本证明
An elementary proof of a fundamental result in phase retrieval
论文作者
论文摘要
Edidin [3]在阶段检索中证明了一个基本结果：定理：正交投影的家族$ \ {p_i \} _ {i = 1}^m $在$ \ mathbb {r}^n $中进行阶段检索，并且只有每一个$ 0 \ not = x n $ = x \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in $ \ {p_ix \} _ {i = 1}^m $ spans $ \ mathbb {r}^n $。该结果的证明依赖于代数的几何形状，因此该领域的许多人无法访问。我们将在没有代数几何形状的情况下给出这个结果的基本证明。我们还将通过证明“如果”部分失败，并且仅在$ \ mathbb {c}^n $中保留“ If”部分失败，而“ If”部分失败，则可以解决此结果的复杂版本。最后，我们将证明这些技术可用于验证规范检索的两种分类。
Edidin [3] proved a fundamental result in phase retrieval: Theorem: A family of orthogonal projections $\{P_i\}_{i=1}^m$ does phase retrieval in $\mathbb{R}^n$ if and only if for every $0\not= x\in \mathbb{R}^n$, the family $\{P_ix\}_{i=1}^m$ spans $\mathbb{R}^n$. The proof of this result relies on Algebraic Geometry and so is inaccessible to many people in the field. We will give an elementary proof of this result without Algebraic Geometry. We will also solve the complex version of this result by showing that the "if" part fails and the "only if" part holds in $\mathbb{C}^n$. Finally, we will show that these techniques can be used to verify two classifications of norm retrieval.
