Phase transition in limiting distributions of coherence of high-dimensional random matrices

T. Tony Cai, Tiefeng Jiang

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

The coherence of a random matrix, which is defined to be the largest magnitude of the Pearson correlation coefficients between the columns of the random matrix, is an important quantity for a wide range of applications including high-dimensional statistics and signal processing. Inspired by these applications, this paper studies the limiting laws of the coherence of n× p random matrices for a full range of the dimension p with a special focus on the ultra high-dimensional setting. Assuming the columns of the random matrix are independent random vectors with a common spherical distribution, we give a complete characterization of the behavior of the limiting distributions of the coherence. More specifically, the limiting distributions of the coherence are derived separately for three regimes: 1nlogp→0, 1nlogp→β∈(0,∞), and 1nlogp→∞. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension p grows as a function of n. Applications to statistics and compressed sensing in the ultra high-dimensional setting are also discussed.

Original languageEnglish (US)
Pages (from-to)24-39
Number of pages16
JournalJournal of Multivariate Analysis
Volume107
DOIs
StatePublished - May 2012

Keywords

  • Chen-Stein method
  • Coherence
  • Correlation coefficient
  • Limiting distribution
  • Maximum
  • Phase transition
  • Random matrix
  • Sample correlation matrix

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