Unlabeled Sensing Using Rank-One Moment Matrix Completion

TL;DR

This paper introduces a novel approach to unlabeled sensing by formulating it as a rank-one matrix completion problem, enabling the recovery of solutions to linear systems with unknown permutations.

Contribution

It presents a new polynomial system formulation and algorithms for solving unlabeled sensing problems via rank-one moment matrix completion.

Findings

Algorithms demonstrate efficiency in numerical experiments.

Proposed methods are robust to noise and variations.

Unique solutions can be reliably recovered in generic cases.

Abstract

We study the unlabeled sensing problem that aims to solve a linear system of equations $A x =\pi(y) $ for an unknown permutation $\pi$. For a generic matrix $A$ and a generic vector $y$, we construct a system of polynomial equations whose unique solution satisfies $ A\xi^*=\pi(y)$. In particular, $\xi^*$ can be recovered by solving the rank-one moment matrix completion problem. We propose symbolic and numeric algorithms to compute the unique solution. Some numerical experiments are conducted to show the efficiency and robustness of the proposed algorithms.