Characterize low-rank intersections for generic Toeplitz subspaces under permutations

Characterize all permutation matrices P for which a generic n-by-d Toeplitz matrix V, with n at least 2d, satisfies rank([V, PV])<2d, and prove that every such permutation is covered by the rank formulas of Theorem 1.1, equivalently Theorem \ref{thm:MainTechnical}.

Background

The paper reduces the Toeplitz unlabeled sensing problem to understanding the rank of the concatenated matrix [V, PV] as the permutation matrix P varies. Theorem \ref{thm:MainTechnical} gives rank formulas for permutations satisfying specific conditions involving the ranks r_t of truncated matrices derived from P and Jordan blocks.

The authors state that a complete answer to this rank-classification question is not yet available. They formulate a conjecture asserting that every permutation producing rank less than 2d is already covered by Theorem \ref{thm:MainTechnical}; this has been verified computationally only for n=2d and d≤5.

References

While a complete answer to Question \ref{que:rank-P} remains elusive, we pose the following conjecture, which we have verified by exhaustive computation for $n=2d$ and $d \le 5$: For $V$ generic Toeplitz, all permutations for which $\rank \left[V, \, PV \right] < 2d$ are covered by Theorem \ref{thm:MainTechnical}.

— Toeplitz Unlabeled Sensing  (2502.12778 - Hong et al., 18 Feb 2025) in Section 2, immediately preceding Conjecture \ref{conj}