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Unformatted text preview: ances pi − pj 2 , so
the origin can be chosen arbitrarily. For example, it can be assumed without loss of generality
that p1 = 0. With this assumption there is a unique Gram matrix Y for a given Euclidean
distance matrix X . Find Y from (23), and relate it to the lefthand side of the inequality (24).
(d) Show that X is a Euclidean distance matrix if and only if
diag(X ) = 0, (I − 1T
11 )X (I − 11T )
n 0. (25) Hint. Use the same argument as in part (c), but take the mean of the vectors pk at the origin,
i.e., impose the condition that p1 + p2 + · · · + pn = 0.
(e) Suppose X is a Euclidean distance matrix. Show that the matrix W ∈ Sn with elements
wij = e−xij , i, j = 1, . . . , n, is positive semideﬁnite.
Hint. Use the following identity from probability theory. Deﬁne z ∼ N (0, I ). Then
E eiz Tx 1 = e− 2 x 2
for all x, where i = −1 and E denotes expectation with respect to z . (This is the characteristic function of a multivariate normal distribution.)
7.3 Minimum total covering ball volume. We consider a collection of n points with locations x1 , . . . , xn...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.
- Fall '13
- The Aeneid