Almost everywhere injectivity conditions for the matrix recovery problem

TL;DR

This paper develops a framework for almost everywhere matrix recovery, determining the measurement conditions needed to recover nearly all matrices within certain algebraic varieties using tools from algebraic geometry.

Contribution

It introduces a new algebraic geometric framework for almost everywhere matrix recovery and analyzes measurement requirements under various algebraic settings.

Findings

Established measurement bounds for almost everywhere recovery

Applied algebraic geometry to characterize recovery conditions

Provided results for different algebraic variety settings

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 $Tr(A_jP), j=1,\ldots,N$ 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.