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...

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: David L. Donoho, Jared Tanner
• Keywords: CORR 2008, Education, Expected Number, Matrix Ensembles, Random Matrix