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Unformatted text preview: arn) a quadratic pseudo-metric d,
d(x, y ) = (x − y )T P (x − y ) 1/2 , with P ∈ Sn , which approximates the given distances, i.e., d(xi , yi ) ≈ di . (The pseudo-metric d is
a metric only when P ≻ 0; when P 0 is singular, it is a pseudo-metric.)
To do this, we will choose P ∈ Sn that minimizes the mean squared error objective
i=1 (di − d(xi , yi ))2 . (a) Explain how to ﬁnd P using convex or quasiconvex optimization. If you cannot ﬁnd an exact
formulation (i.e., one that is guaranteed to minimize the total squared error objective), give
a formulation that approximately minimizes the given objective, subject to the constraints.
(b) Carry out the method of part (a) with the data given in quad_metric_data.m. The columns
of the matrices X and Y are the points xi and yi ; the row vector d gives the distances di . Give
the optimal mean squared distance error.
We also provide a test set, with data X_test, Y_test, and d_test. Report the mean squared
distance error on the...
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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 University of California, Berkeley.
- Fall '13
- The Aeneid