review6_2009_10_30

review6_2009_10_30 - 1 EE263 Review session 6 Jong-Han Kim...

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1 EE263 Review session 6 Jong-Han Kim 2009.10.30. EE263 Review session 6 Least norm solutions Least norm vs. least squares General norm minimization Multiobjective least squares Regularized least squares
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2 EE263 Review session 6 Jong-Han Kim 2009.10.30. Least-norm solutions of undetermined equations consider y = Ax with a fat and full rank matrix A . set of all solutions has form x = x p + Qz where the columns of Q form the basis for null ( A ) .
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3 EE263 Review session 6 Jong-Han Kim 2009.10.30. one choice of x p is x ln = A T ( AA T ) 1 y in fact, x ln minimizes bardbl x bardbl among all solutions of y = Ax . i.e., it is the optimal solution of the following. minimize bardbl x bardbl subject to Ax = y
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4 EE263 Review session 6 Jong-Han Kim 2009.10.30. for any x satisfying Ax = y , ( x x ln ) T x ln = ( x x ln ) T A T ( AA T ) 1 y = ( Ax Ax ln ) T ( AA T ) 1 y = 0 therefore bardbl x bardbl 2 = bardbl x x ln + x ln bardbl 2 = bardbl x x ln bardbl 2 + bardbl x ln bardbl 2 ≥ bardbl x ln bardbl 2
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5 EE263 Review session 6 Jong-Han Kim 2009.10.30. the least norm problem is equivalent to minimize x T x subject to Ax = y introducing the Lagrangian L ( x, λ ) = x T x + λ T ( Ax y ) optimality conditions are, x L = 2 x + A T λ = 0 λ L = Ax y = 0
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