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Toeplitz Unlabeled Sensing

Xin Hong, Manolis C. Tsakiris

TL;DR

The paper addresses the identifiability of a vector in a Toeplitz-structured subspace from a permuted observation, formalized as Unlabeled Sensing Property (USP). It introduces a rank-based framework, proving a MainTechnical theorem that expresses the rank of [V, PV] for generic Toeplitz V in terms of d, t, and r_t, and derives Toeplitz-specific USP conditions and a complete cyclic-permutation result. The work yields concrete criteria under which unlabeled recovery is possible and provides a suite of proofs and examples to illustrate the sharpness of these criteria, with potential applications to signal processing where LTI filter outputs are observed after a permutation. The results advance the theory by embedding the combinatorial Toeplitz structure into the rank analysis, offering exact thresholds and a clear path for cyclic-case conclusions.

Abstract

Unlabeled sensing is the problem of recovering an element of a vector subspace of R^n, from its image under an unknown permutation of the coordinates and knowledge of the subspace. Here we study this problem for the special class of subspaces that admit a Toeplitz basis.

Toeplitz Unlabeled Sensing

TL;DR

The paper addresses the identifiability of a vector in a Toeplitz-structured subspace from a permuted observation, formalized as Unlabeled Sensing Property (USP). It introduces a rank-based framework, proving a MainTechnical theorem that expresses the rank of [V, PV] for generic Toeplitz V in terms of d, t, and r_t, and derives Toeplitz-specific USP conditions and a complete cyclic-permutation result. The work yields concrete criteria under which unlabeled recovery is possible and provides a suite of proofs and examples to illustrate the sharpness of these criteria, with potential applications to signal processing where LTI filter outputs are observed after a permutation. The results advance the theory by embedding the combinatorial Toeplitz structure into the rank analysis, offering exact thresholds and a clear path for cyclic-case conclusions.

Abstract

Unlabeled sensing is the problem of recovering an element of a vector subspace of R^n, from its image under an unknown permutation of the coordinates and knowledge of the subspace. Here we study this problem for the special class of subspaces that admit a Toeplitz basis.

Paper Structure

This paper contains 7 sections, 11 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Proof of Theorem \ref{['thm:MainTechnical']}
  4. Remaining Proofs
  5. Proof of Theorem \ref{['thm:US']}
  6. Proof of Proposition \ref{['prp:cyclic-full-rk']}
  7. Examples

Key Result

Theorem 1

If there exists an integer $0\leq t \leq \frac{d}{2}$ such that or if there exists an integer $-\frac{d}{2} \le t < 0$ such that then $\mathop{\mathrm{rank}}\nolimits \left[V,PV\right] = d + r_{t} + 2|t|$, for $V$ generic Toeplitz.

Theorems & Definitions (23)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Proposition 1
  • Lemma 1: MarsagliaStyan1972
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 13 more