How many entries of a typical orthogonal matrix can be approximated by independent normals

By Tiefeng Jiang. 2006

136 citations recorded by the source corpus.

We solve an open problem of Diaconis that asks what are the largest orders of p n and q n such that Z n , the p n x q n upper left block of a random matrix r n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the variation distance between the joint distribution of entries of Z n and that of p n q n independent standard normals goes to zero provided p n = o(√n) and q n = o(/n). We also show that the above variation distance does not go to zero if p n = [x√n] and q n = [y√n] for any positive numbers x and y. This says that the largest orders of p n and q n are o(n 1/2 ) in the sense of the above approximation. Second, suppose Γ n = (γ ij ) n×n is generated by performing the Gram-Schmidt algorithm on the columns of Y n = (y ij ) n×n , where {y ij ; 1 ≤ i, j ≤ n} are i.i.d. standard normals. We show that e n (m)::= max 1≤i≤n 1≤j≤m √n. γ ij -y ij | goes to zero in probability as long as m = m n = o(n/log n). We also prove that e n (m n ) → 2√α in probability when m n = [nα/logn] for any a > 0. This says that m n = o(n/log n) is the largest order such that the entries of the first m n columns of Γ n can be approximated simultaneously by independent standard normals.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2fe77985fbcff0ee0662b406bf9c735d5ca57422

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Citations in the corpus

A Random Rotation Perturbation Approach to Privacy Preserving Data Classification (2005, methodology): “When the random rotation matrix generated following the Haar distribution, a considerable number of matrix entries are approximately independent normal N(0, 1/d) [13].”
The computational complexity of linear optics (2010, background): “1, results of Jiang [38] (which we learned about only recently) strongly suggest that an improved analysis would yield m = O ( n2 ) .”
Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles (2009, methodology): “First, this is true for the orthogonal case by Theorem 3 from [42].”
The Random Matrix Theory of the Classical Compact Groups (2019, methodology): “[60] Tiefeng Jiang.”
Random matrices: Universal properties of eigenvectors (2011, methodology): “The distribution of coefficients of a matrix distributed using the Haar measure on the unitary or orthogonal groups has been studied by many authors [2], [16], [11], [23], [5], [4], [3], [12], [13].”
Maxima of entries of Haar distributed matrices (2005, background): “tance goes to zero is an open problem, see section 6.3 from [10]; it has been solved recently by Jiang [ 24 ]: the largest order is o(n 1/2 )).”
MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS (2007, unknown): “aussian distribution, with covariance matrix 1 n (bij) k i,j=1. Then dL∗(W,Y) ≤ 4 √ λk3/2 n− 1 for n ≥ 4, where λ is the largest eigenvalue of 1 n (bij) k i,j=1. Remark. In the remarkable recent work [22], Tiefeng Jiang has shown that the entries of any pn × qn submatrix of an n × n random orthogona”
A Fourier view on the R-transform and related asymptotics of spherical integrals (2005, background): “Jiang [17] where he shows that the entries of the first O(N/ logN) columns of an Haar distributed unitary matrix can be simultaneously approximated by independent standard normal variables.”