Almost everywhere injectivity conditions for the matrix recovery problem

Abstract

Matrix recovery is raised in many areas. In this paper, we build up a framework for almost everywhere matrix recovery which means to recover almost all the $P\in \mathcal{M}\subset {\mathbb{H}}^{p\times q}$ from $Tr(A_{j}P),j=1,\dots ,N$ where $A_j\in V_j\subset {\mathbb{H}}^{p\times q}$. We mainly focus on the following question: how many measurements are needed to recover almost all the matrices in $\mathcal{M}$? For the case where both $\mathcal{M}$ and $V_j$ are algebraic varieties, we use the tools from algebraic geometry to study the question and present some results to address it under many different settings.

22 pages

References in corpus (2)

• An algebraic characterization of injectivity in phase retrieval
• On The 2D Phase Retrieval Problem