distinguishing random orthogonal matrix from Gaussian random matrix - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:25:27Z http://mathoverflow.net/feeds/question/115185 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115185/distinguishing-random-orthogonal-matrix-from-gaussian-random-matrix distinguishing random orthogonal matrix from Gaussian random matrix unknown (google) 2012-12-02T16:23:36Z 2012-12-03T19:47:12Z <p>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 $\sqrt n$.</p> <p>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).</p> <p><strong>My question is</strong>, What property should we examine on the top-left $\sqrt{n}\times\sqrt{n}$ block? Operator norm? Have there been any results on this?</p> http://mathoverflow.net/questions/115185/distinguishing-random-orthogonal-matrix-from-gaussian-random-matrix/115332#115332 Answer by Omer for distinguishing random orthogonal matrix from Gaussian random matrix Omer 2012-12-03T19:47:12Z 2012-12-03T19:47:12Z <p>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.</p>