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CORR
2008
Springer

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

8 years 12 months ago
Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications
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...
David L. Donoho, Jared Tanner
Added 09 Dec 2010
Updated 09 Dec 2010
Type Journal
Year 2008
Where CORR
Authors David L. Donoho, Jared Tanner
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