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COUNTING THE FACES OF RANDOMLY-PROJECTED HYPERCUBES AND ORTHANTS, WITH APPLICATIONS
RANDOMLY-PROJECTED ORTHANTS
2015/8/21
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 e...
COUNTING FACES OF RANDOMLY-PROJECTED POLYTOPES WHEN THE PROJECTION RADICALLY LOWERS DIMENSION
RANDOMLY-PROJECTED RADICALLY LOWERS DIMENSION
2015/8/21
The modern trend in statistics and probability is to consider the case where both
the number of dimensions d and the sample size n are large [19, 21]. In that case, the
intuition fostered by the cla...
Neighborliness of Randomly-Projected Simplices in High Dimensions
Neighborly Polytopes Convex Hull of Gaussian Sample
2015/8/21
Let A be a d by n matrix, d < n. Let T = T
n−1 be the standard regular simplex in R
n
.
We count the faces of the projected simplex AT in the case where the projection is random,
the dimens...