Relationships between the Phase Retrieval Problem and Permutation Invariant Embeddings

TL;DR

This paper explores the link between phase retrieval and permutation invariant embeddings, revealing their equivalence and providing new inversion algorithms for permutation orbits.

Contribution

It establishes a theoretical connection between phase retrieval and permutation invariant embeddings, enabling novel inversion algorithms for permutation orbits.

Findings

Real phase retrieval is equivalent to Euclidean embeddings of quotient spaces.

Provides algorithms for inverting orbits induced by permutation groups.

Links phase retrieval problems to permutation invariant embedding techniques.

Abstract

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.