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Expanding the regression function in a basis of
eigenfunctions of

(333) 
yields for functional (247)

(334) 
Under the assumption of output noise for training data
the data terms may for example be replaced by the logarithm
of a mixture of Gaussians.
Such mixture functions with varying mean can develop flat regions
where the error is insensitive (robust) to changes of .
Analogously, Gaussians with varying mean can be added
to obtain errors which are flat compared to Gaussians for large
absolute errors.
Similarly to such Gaussian mixtures
the meansquare error data term
may be replaced
by an insensitive error
,
which is zero for absolute errors smaller and linear
for larger absolute errors (see Fig.5).
This results in a quadratic programming problem
and is equivalent to Vapnik's support vector machine
[225,74,226,214,215,49].
For a more detailed discussion
of the relation between support vector machines
and Gaussian processes see
[229,208].
Figure 5:
Three
robust error functions which are
insensitive to small errors.
Left: Logarithm of mixture with two Gaussians with equal variance
and different means.
Middle: Logarithm of mixture with 11 Gaussians with equal variance
and different means.
Right: insensitive error.

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