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The Hessians ,
We now calculate the Hessian of the functional .
For fixed one finds the Hessian
by differentiating again the gradient (166)
of
|
(181) |
i.e.,
|
(182) |
Here the diagonal matrix
is non-zero only at data points.
Including the dependence of on
one obtains for the Hessian of in (175)
by calculating the derivative of the gradient in (180)
|
(183) |
i.e.,
Here we used
= 0.
It follows from the normalization
that any -independent function
is right eigenvector of
with zero eigenvalue.
Because =
this factor or its transpose is
also contained in the second line of
Eq. (184),
which means that has a zero mode.
Indeed, functional is invariant under multiplication
of with a -independent factor. The zero modes can be projected out
or removed by including additional conditions, e.g. by
fixing one value of for every .
Next: General Gaussian prior factors
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Joerg_Lemm
2001-01-21