bv_cvxbook_extra_exercises

# You will nd that the optimal weight vector v has some

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Unformatted text preview: areroot of the sum of the squares of the Euclidean distances to the mean of the coordinates of the nodes in S : fS ( X ) = j ∈S 1/2 xj − x 2 ¯2 where xi = ¯ 1 xik , | S | k ∈S i = 1, . . . , p. (iv) fS (X ) is the sum of the ℓ1 -distances to the (coordinate-wise) median of the coordinates of the nodes in S : fS ( X ) = j ∈S xj − x ˆ where 1 xi = median({xik | k ∈ S }), ˆ i = 1, . . . , p. 15.2 Let W ∈ Sn be a symmetric matrix with nonnegative elements wij and zero diagonal. We can interpret W as the representation of a weighted undirected graph with n nodes. If wij = wji &gt; 0, there is an edge between nodes i and j , with weight wij . If wij = wji = 0 then nodes i and j are not connected. The Laplacian of the weighted graph is deﬁned as L(W ) = −W + diag(W 1). This is a symmetric matrix with elements Lij (W ) = n k=1 wik −wij i=j i = j. The Laplacian has the useful property that y T L(W )y = i≤ j wij (yi − yj )2 for all vectors y ∈ Rn . (a) Show that the function f : Sn → R, f (...
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