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 language | English (US) |
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Pages (from-to) | 24-39 |
Number of pages | 16 |
Journal | Journal of Multivariate Analysis |
Volume | 107 |
DOIs | |
State | Published - May 2012 |
Keywords
- Chen-Stein method
- Coherence
- Correlation coefficient
- Limiting distribution
- Maximum
- Phase transition
- Random matrix
- Sample correlation matrix