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GP-EIGEN:  Find eigenvalues/vectors of a Gaussian process's covariance.

GP-eigen prints the eigenvalues and/or eigenvectors of the covariance
matrix for a Gaussian process, as constructed at a specified set of
input points.  The ratio of the largest to the smallest eigenvalue is
a measure of the "conditioning" of the matrix; a large ratio indicates
that computational problems may arise.  The eigenvalues and
eigenvectors are also of theoretical interest.

Usage:

gp-eigen [ -values ] [ -vectors ] log-file index [ extra ]
[ / test-inputs ]

Computes the eigenvalues and/or eigenvectors for the Gaussian process
stored at the specified iteration in the log file.  If no extra
argument is given, the covariance matrix for latent values is used,
without the noise variance being added to the diagonal.  If the extra
argument is the string "+noise", the noise is added.  This is valid
only for a regression model, and only if the noise variances do not
vary on a case-by-case basis.  If there is more than one target
variable, the noise added is that for the first target.  If the extra
argument is a positive number, that number is added to the diagonal.

By default, the covariance matrix is for the set of training inputs,
but this can be overridden by specifying a file of inputs at the end
of the command line.

By default, or if only the "-values" option is specified, the
eigenvalues are printed on per line, in descending order of magnitude.
If both "-values" and "-vectors" are specified, the eigenvalues are
followed, on the same line, by their eigenvectors.  If only "-vectors"
is specified, only the eigenvectors are printed, but they are still
ordered according to decreasing eigenvalue.

The eigenvalues and eigenvectors are computed by Jacobi iteration,
stopping when all off-diagonal elements are less than 1e-10.  If the
matrix is poorly conditioned, it is possible for some of the
eigenvalues found to be negative, even though these cannot be correct,
since the covariance matrix is supposed to be positive semi-definite.

Copyright (c) 1995-2004 by Radford M. Neal
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