This paper discusses the connection between the phase retrieval problem and permutation invariant embeddings. We show that the real phase retrieval problem for $\mathbb{R}^d/O(1)$ is equivalent to Euclidean embeddings of the quotient space $\mathbb{R}^{2\times d}/S_2$ performed by the sorting encoder introduced in an earlier work. In addition, this relationship provides us with inversion algorithms of the orbits induced by the group of permutation matrices.

翻译：本文探讨了相位恢复问题与置换不变嵌入之间的关联。我们证明了$\mathbb{R}^d/O(1)$的实值相位恢复问题等价于通过先前工作中引入的排序编码器对商空间$\mathbb{R}^{2\times d}/S_2$进行的欧几里得嵌入。此外，这种关系为我们提供了由置换矩阵群诱导的轨道的反演算法。