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A Field-Theoretic View of Unlabeled Sensing

Hao Liang, Jingyu Lu, Manolis C. Tsakiris, Lihong Zhi

TL;DR

It is established 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.

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.

A Field-Theoretic View of Unlabeled Sensing

TL;DR

It is established 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.

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.
Paper Structure (8 sections, 18 theorems, 14 equations)

This paper contains 8 sections, 18 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Unlabeled Sensing
  3. Motivation
  4. Contributions
  5. Acknowledgments
  6. The Field $\mathscr{F}_m$
  7. The Field Extension $\mathscr{F}_n \subseteq \mathscr{F}_m$
  8. The Field Extension $\mathscr{F}_{n+1} \subseteq \mathscr{F}_m$

Key Result

Theorem 1.1

\newlabelthm:main0 Suppose that $A^* \in \mathbb{R}^{m \times n}$ and $\xi^* \in \mathbb{R}^{n \times 1}$ are generic, let $\pi$ be any permutation of the coordinates of $\mathbb{R}^{m \times 1}$, and set $y^* = \pi(A^* \xi^*) \in \mathbb{R}^{m \times 1}$. Then $\xi^*$ is the unique complex soluti

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Proof 1
  • Remark 2.3
  • Lemma 2.4: Exercises 12 & 13, Chapter 5, Atiyah-MacDonald
  • Lemma 2.5
  • Proof 2
  • Lemma 2.6
  • ...and 26 more