Signal Recovery on Algebraic Varieties Using Linear Samples

TL;DR

This paper explores the minimum linear measurements needed to recover signals on algebraic varieties, using algebraic geometry tools, with applications to phase retrieval and low-rank matrix recovery.

Contribution

It introduces an algebraic geometry-based method to determine measurement bounds for signal recovery on algebraic varieties, applied to phase retrieval and low-rank matrix problems.

Findings

Provides bounds on measurements for unique recovery

Applies method to phase retrieval and low-rank matrices

Highlights open problems for future research

Abstract

The recovery of an unknown signal from its linear measurements is a fundamental problem spanning numerous scientific and engineering disciplines. Commonly, prior knowledge suggests that the underlying signal resides within a known algebraic variety. This context naturally leads to a question: what is the minimum number of measurements required to uniquely recover any signal belonging to such an algebraic variety? In this survey paper, we introduce a method that leverages tools from algebraic geometry to address this question. We then demonstrate the utility of this approach by applying it to two problems: phase retrieval and low-rank matrix recovery. We also highlight several open problems, which could serve as a basis for future investigations in this field.