# Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

@article{GonzlezGuilln2014EuclideanDB, title={Euclidean Distance Between Haar Orthogonal and Gaussian Matrices}, author={Carlos E. Gonz{\'a}lez-Guill{\'e}n and Carlos Palazuelos and Ignacio Villanueva}, journal={Journal of Theoretical Probability}, year={2014}, volume={31}, pages={93-118} }

In this work, we study a version of the general question of how well a Haar-distributed orthogonal matrix can be approximated by a random Gaussian matrix. Here, we consider a Gaussian random matrix $$Y_n$$Yn of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix $$U_n$$Un. If $$F_i^m$$Fim denotes the vector formed by the first m-coordinates of the ith row of $$Y_n-\sqrt{n}U_n$$Yn-nUn and $$\alpha \,=\,\frac{m}{n}$$α=mn…

## 6 Citations

### Sampling Quantum Nonlocal Correlations with High Probability

- Mathematics
- 2016

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form $${\gamma=(\langle u_i,v_j\rangle)_{i,j=1}^n}$$γ=(⟨ui,vj⟩)i,j=1n, where the vectors ui and vj…

### Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations

- Mathematics
- 2020

Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and…

### Random Matrices Generating Large Growth in LU Factorization with Pivoting

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2021

It is shown more generally that a rank-$1$ perturbation to an orthogonal matrix producing large growth for any form of pivoting also generates large growth under reasonable assumptions, and it is demonstrated that GMRES-based iterative refinement can provide stable solutions to $Ax = b$ when large growth occurs in low precision LU factors, even when standard iteratives cannot.

### Random Constructions in Bell Inequalities: A Survey

- Physics
- 2015

The aim of the present work is to review some of the recent results in this direction by focusing on the main ideas and removing most of the technical details, to make the previous study more accessible to a wide audience.

### Understanding Self-Predictive Learning for Reinforcement Learning

- Computer ScienceArXiv
- 2022

This work identifies that a faster paced optimization of the predictor and semi-gradient updates on the representation, are crucial to preventing the representation collapse and proposes bidirectional self-predictive learning, a novel self-Predictive algorithm that learns two representations simultaneously.

## References

SHOWING 1-10 OF 22 REFERENCES

### Sampling Quantum Nonlocal Correlations with High Probability

- Mathematics
- 2016

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form $${\gamma=(\langle u_i,v_j\rangle)_{i,j=1}^n}$$γ=(⟨ui,vj⟩)i,j=1n, where the vectors ui and vj…

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

- Mathematics
- 2006

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…

### Probability Inequalities for Sums of Bounded Random Variables

- Mathematics
- 1994

If S is a random variable with finite rnean and variance, the Bienayme-Chebyshev inequality states that for x > 0,
$$\Pr \left[ {\left| {S - ES} \right| \geqslant x{{{(\operatorname{var}…

### Maxima of entries of Haar distributed matrices

- Mathematics
- 2005

Abstract.Let Γn=(γij) be an n×n random matrix such that its distribution is the normalized Haar measure on the orthogonal group O(n). Let also Wn:=max1≤i,j≤n|γij|. We obtain the limiting distribution…

### Borel theorems for random matrices from the classical compact symmetric spaces.

- Mathematics
- 2008

We study random vectors of the form (Tr(A (1) V), ..., Tr(A (r) V)), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A (v) are…

### Brownian motion and the classical groups

- Mathematics
- 2002

Let G be chosen from the orthogonal group On according to Haar measure, and let A be an ? ? ? real matrix with non-random entries satisfying TrAA1 = n. We show that TrAT converges in distribution to…

### Random matrices: Universal properties of eigenvectors

- Mathematics
- 2011

The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing)…

### Asymptotic Theory Of Finite Dimensional Normed Spaces

- Mathematics
- 1986

The Concentration of Measure Phenomenon in the Theory of Normed Spaces.- Preliminaries.- The Isoperimetric Inequality on Sn?1 and Some Consequences.- Finite Dimensional Normed Spaces, Preliminaries.-…

### Adaptive estimation of a quadratic functional by model selection

- Mathematics
- 2000

We consider the problem of estimating ∥s∥ 2 when s belongs to some separable Hilbert space and one observes the Gaussian process Y(t) = (s, t) + σ L(t), for all t ∈ H, where L is some Gaussian…