A Field-Theoretic View of Unlabeled Sensing

TL;DR

This paper introduces a novel algebraic approach to unlabeled sensing, framing the problem through field theory and symmetric polynomials, leading to polynomial systems that can solve high-dimensional instances.

Contribution

It provides a new theoretical framework connecting unlabeled sensing with algebraic geometry, enabling polynomial system formulations for generic data.

Findings

Solution reduces to a system of n+1 polynomial equations

The approach scales to high-dimensional problems

Offers new algebraic algorithms for unlabeled sensing

Abstract

Unlabeled sensing is the problem of solving a linear system of equations, where the right-hand-side vector is known only up to a permutation. In this work, we study fields of rational functions related to symmetric polynomials and their images under a linear projection of the variables; as a consequence, we establish that the solution to an n-dimensional unlabeled sensing problem with generic data can be obtained as the unique solution to a system of n + 1 polynomial equations of degrees 1, 2, . . . , n + 1 in n unknowns. Besides the new theoretical insights, this development offers the potential for scaling up algebraic unlabeled sensing algorithms.