distinguishing random orthogonal matrix from Gaussian random matrix

2012-12-02T16:23:36Z

Jiang's paper (http://projecteuclid.org/euclid.aop/1158673325) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ orthogonal matrix (uniform w.r.t. Haar measure). Let $D_1$ be the distribution of topleft $m \times m$ block of $G$, and $D_2$ be the distribution of topleft $m\times m$ block of $Z$. Then the total variance distance between $D_1$ and $D_2$ goes to 0 if $m=o(\sqrt n)$ and the t.v.d. is at least a constant if $m$ is at the order of $\sqrt n$.

This suggests that, given a random matrix which comes from the distribution of $G$ or the distribution of $Z$, we can tell which case it is with probability, say, $\geq 3/4$, by reading its top-left $\sqrt{n}\times\sqrt{n}$ block (dimension up to a constant factor).

My question is, What property should we examine on the top-left $\sqrt{n}\times\sqrt{n}$ block? Operator norm? Have there been any results on this?

Answer by Omer for distinguishing random orthogonal matrix from Gaussian random matrix

2012-12-03T19:47:12Z

The answer is implicit in Jiang's paper. There is a formula for the Radon-Nykodym derivative of the two measures, and you just check whether this is more or less than 1. This would precisely achieve the total variation distance which is at least some constant.

distinguishing random orthogonal matrix from Gaussian random matrix - MathOverflow

distinguishing random orthogonal matrix from Gaussian random matrix

Answer by Omer for distinguishing random orthogonal matrix from Gaussian random matrix