Permutations Unlabeled beyond Sampling Unknown

Research output: Contribution to journalArticle


A recent unlabeled sampling result by Unnikrishnan, Haghighatshoar, and Vetterli states that with probability one over Gaussian random matrices A with iid entries, any x can be uniquely recovered from an unknown permutation of y = A x as soon as A has at least twice as many rows as columns. We show that this condition on A implies something much stronger: that an unknown vector x can be recovered from measurements y = T A x, when the unknown T belongs to an arbitrary set of invertible, diagonalizable linear transformations \mathcal {T}. The set \mathcal {T} can be finite or countably infinite. When it is the set of m \times m permutation matrices, we have the classical unlabeled sampling problem. We show that for almost all A with at least twice as many rows as columns, all x can be recovered either uniquely, or up to a scale depending on \mathcal {T}, and that the condition on the size of A is necessary. Our proof is based on vector space geometry. Specializing to permutations, we obtain a simplified proof of the uniqueness result of Unnikrishnan, Haghighatshoar, and Vetterli. In this letter, we are only concerned with uniqueness; stability and algorithms are left for future work.

Original languageEnglish (US)
Article number8678393
Pages (from-to)823-827
Number of pages5
JournalIEEE Signal Processing Letters
Issue number6
StatePublished - Jun 2019


  • Sampling
  • shuffled regression
  • unknown permutation
  • unknown transformation
  • unlabeled sampling

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering
  • Applied Mathematics

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