Matrix recovery from bilinear and quadratic measurements

TL;DR

This paper investigates conditions under which matrix recovery from bilinear and quadratic measurements can be effectively linearized, reducing complexity and enabling solutions in applications like Time Encoding Machines and Localisation.

Contribution

It provides theoretical conditions for solvability of bilinear and quadratic matrix recovery problems and demonstrates linearization with minimal additional measurements.

Findings

Bilinear problems are solvable under specific conditions.

Quadratic problems can be linearized with a linear number of extra measurements.

Applications include Time Encoding and Continuous Localisation.

Abstract

Matrix (or operator) recovery from linear measurements is a well-studied problem. However, there are situations where only bilinear or quadratic measurements are available. A bilinear or quadratic problem can easily be transformed into a linear one, but it raises questions when the linearized problem is solvable and what is the cost of linearization. In this work, we study a few specific cases of this general problem and show when the bilinear problem is solvable. Using this result and certain properties of polynomial rings, we present a scenario when the quadratic problem can be linearized at the cost of just a linear number of additional measurements. Finally, we link our results back to two applications that inspired it: Time Encoding Machines and Continuous Localisation.