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CORR

2008

Springer

2008

Springer

Let A be an n by N real valued random matrix, and HN denote the N-dimensional hypercube. For numerous random matrix ensembles, the expected number of k-dimensional faces of the random n-dimensional zonotope AHN obeys the formula Efk(AHN )/fk(HN ) = 1 - PN-n,N-k, where PN-n,N-k is a fair-coin-tossing probability: PN-n,N-k Prob{N - k - 1 or fewer successes in N - n - 1 tosses }. The formula applies, for example, where the columns of A are drawn i.i.d. from an absolutely continuous symmetric distribution. The formula exploits Wendel's Theorem[20]. Let RN + denote the positive orthant; the expected number of k-faces of the random cone ARN + obeys Efk(ARN + )/fk(RN + ) = 1 - PN-n,N-k. The formula applies to numerous matrix ensembles, including those with iid random columns from an absolutely continuous, centrally symmetric distribution. The probabilities PN-n,N-k change rapidly from nearly 0 to nearly 1 near k 2n - N. Consequently, there is an asymptotically sharp threshold in the be...

Added |
09 Dec 2010 |

Updated |
09 Dec 2010 |

Type |
Journal |

Year |
2008 |

Where |
CORR |

Authors |
David L. Donoho, Jared Tanner |

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