review6_2009_10_30

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

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Unformatted text preview: 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 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 ) . 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 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 5 EE263 Review session 6 Jong-Han Kim 2009.10.30....
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This note was uploaded on 08/23/2010 for the course EE 263 taught by Professor Boyd,s during the Fall '08 term at Stanford.

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

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