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(d) Use CVX, with the overloaded max(), abs(), and sum() functions.
(e) Use the CVX function huber().
(f) The current version of CVX handles the logarithm using an iterative procedure, which is slow
and not entirely reliable. However, you can reformulate this problem as
maximize ( m
k=1 ((1 − (Ax − b)k )(1 + (Ax − b)k )))1/2m , and use the CVX function geo_mean().
5.5 ℓ1.5 optimization. Optimization and approximation methods that use both an ℓ2 -norm (or its
square) and an ℓ1 -norm are currently very popular in statistics, machine learning, and signal and
image processing. Examples include Huber estimation, LASSO, basis pursuit, SVM, various ℓ1 regularized classiﬁcation methods, total variation de-noising, etc. Very roughly, an ℓ2 -norm corresponds to Euclidean distance (squared), or the negative log-likelihood function for a Gaussian;
in contrast the ℓ1 -norm gives ‘robust’ approximation, i.e., reduced sensitivity to outliers, and also
tends to yield sparse solutions (of whatever the argument of the norm is). (All of this is just
background; you don’t need to know any of this to solve the problem...
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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