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

T. Tony Cai and Tiefeng Jiang

Journal of Multivariate Analysis, 2012, vol. 107, issue C, 24-39

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.

Keywords: Coherence; Correlation coefficient; Limiting distribution; Maximum; Phase transition; Random matrix; Sample correlation matrix; Chen–Stein method (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (9)

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DOI: 10.1016/j.jmva.2011.11.008

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