# On the reduction of a random basis

Ali Akhavi; Jean-François Marckert; Alain Rouault

ESAIM: Probability and Statistics (2009)

- Volume: 13, page 437-458
- ISSN: 1292-8100

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topAkhavi, Ali, Marckert, Jean-François, and Rouault, Alain. "On the reduction of a random basis." ESAIM: Probability and Statistics 13 (2009): 437-458. <http://eudml.org/doc/250640>.

@article{Akhavi2009,

abstract = {For p ≤ n, let b1(n),...,bp(n) be independent random vectors in $\mathbb\{R\}^n$ with the same distribution invariant by
rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of
this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If $\widehat b_\{1\}^\{(n)\},\ldots,
\widehat b_p^\{(n)\}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends
upon the sequence of ratios of squared lengths of consecutive vectors $r_j^\{(n)\} = \Vert \widehat b^\{(n)\}_\{n-j+1\}\Vert^2 / \Vert
\widehat b^\{(n)\}_\{n-j\} \Vert^2$, j = 1,...,p - 1. We show that as n → +∡ the process $(r_j^\{(n)\}-1,j\geq 1)$ tends in distribution
in some sense to an explicit process $(\{\mathcal R\}_j -1,j\geq 1)$; some properties of the latter are provided. The probability that a random
random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞.
},

author = {Akhavi, Ali, Marckert, Jean-François, Rouault, Alain},

journal = {ESAIM: Probability and Statistics},

keywords = {Random matrices; random basis; orthogonality index; determinant; lattice reduction.; random matrices; lattice reduction; beta distribution; Gram-Schmidt orthogonalization},

language = {eng},

month = {9},

pages = {437-458},

publisher = {EDP Sciences},

title = {On the reduction of a random basis},

url = {http://eudml.org/doc/250640},

volume = {13},

year = {2009},

}

TY - JOUR

AU - Akhavi, Ali

AU - Marckert, Jean-François

AU - Rouault, Alain

TI - On the reduction of a random basis

JO - ESAIM: Probability and Statistics

DA - 2009/9//

PB - EDP Sciences

VL - 13

SP - 437

EP - 458

AB - For p ≤ n, let b1(n),...,bp(n) be independent random vectors in $\mathbb{R}^n$ with the same distribution invariant by
rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of
this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If $\widehat b_{1}^{(n)},\ldots,
\widehat b_p^{(n)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends
upon the sequence of ratios of squared lengths of consecutive vectors $r_j^{(n)} = \Vert \widehat b^{(n)}_{n-j+1}\Vert^2 / \Vert
\widehat b^{(n)}_{n-j} \Vert^2$, j = 1,...,p - 1. We show that as n → +∡ the process $(r_j^{(n)}-1,j\geq 1)$ tends in distribution
in some sense to an explicit process $({\mathcal R}_j -1,j\geq 1)$; some properties of the latter are provided. The probability that a random
random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞.

LA - eng

KW - Random matrices; random basis; orthogonality index; determinant; lattice reduction.; random matrices; lattice reduction; beta distribution; Gram-Schmidt orthogonalization

UR - http://eudml.org/doc/250640

ER -

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