Recovering data permutations from noisy observations: The linear regime

Minoh Jeong, Alex Dytso, Martina Cardone, H. Vincent Poor

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


This article considers a noisy data structure recovery problem. The goal is to investigate the following question: Given a noisy observation of a permuted data set, according to which permutation was the original data sorted? The focus is on scenarios where data is generated according to an isotropic Gaussian distribution, and the noise is additive Gaussian with an arbitrary covariance matrix. This problem is posed within a hypothesis testing framework. The objective is to study the linear regime in which the optimal decoder has a polynomial complexity in the data size, and it declares the permutation by simply computing a permutation-independent linear function of the noisy observations. The main result of this article is a complete characterization of the linear regime in terms of the noise covariance matrix. Specifically, it is shown that this matrix must have a very flat spectrum with at most three distinct eigenvalues to induce the linear regime. Several practically relevant implications of this result are discussed, and the error probability incurred by the decision criterion in the linear regime is also characterized. A core technical component consists of using linear algebraic and geometric tools, such as Steiner symmetrization.

Original languageEnglish (US)
Article number3041697
Pages (from-to)854-869
Number of pages16
JournalIEEE Journal on Selected Areas in Information Theory
Issue number3
StatePublished - Nov 2020

All Science Journal Classification (ASJC) codes

  • Computer Networks and Communications
  • Media Technology
  • Artificial Intelligence
  • Applied Mathematics


  • Gaussian noise
  • Linear estimation
  • M-ary hypothesis testing
  • Permutation recovery
  • Steiner symmetrization


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