bv_cvxbook_extra_exercises

Set ci contains ni examples of feature vectors in

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Unformatted text preview: ibution, with φ(u) = (u)+ + β , where α and β are normalizing constants, and (a)+ = max{a, 0}. Now let x be the random variable x = Az + b, where A ∈ Rn×n is nonsingular. The distribution of x is parametrized by A and b. Suppose x1 , . . . , xN are independent samples from the distribution of x. Explain how to find a maximum likelihood estimate of A and b using convex optimization. If you make any further assumptions about A and b (beyond invertiblility of A), you must justify it. Hint. The density of x = Az + b is given by px ( v ) = 1 pz (A−1 (v − b)). | det A| 58 7 Geometry 7.1 Efficiency of maximum volume inscribed ellipsoid. In this problem we prove the following geometrical result. Suppose C is a polyhedron in Rn , symmetric about the origin, and described as C = {x | − 1 ≤ aT x ≤ 1, i = 1, . . . , p}. i Let E = {x | xT Q−1 x ≤ 1}, with Q ∈ Sn , be the maximum volume ellipsoid with center at the origin, inscribed in C . Then ++ the ellipsoid √ nE = {x | xT Q−1 x ≤ n} √ (i.e., the ellipsoid E , scaled by a factor n abou...
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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.

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