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2.8. GP-LVM 47
forcing the model to explain the data using few kernel functions leading to a
sparse model.
As noted in [64, 45, 14] the RVM is a special case of a GP with covariance
function,
k(xi,xj) =
N
l=1
1
c(xi,xk)c(xj,xl), (2.55)
αl
where c is the kernel basis function as in Eq. 2.51. The covariance function is
different in form as it depends on the training data xl. Further, it correspond
to a degenerate covariance matrix having at most rank N as it is an expansion
around the training data. Training the RVM is the same as optimizing a
GP regression model i.e. finding the hyper-parameters that maximizes the
marginal likelihood of the model. However, as noted in [45] the covariance
function of the RVM has some undesirable effects. Using a standard RBF
kernel for the GP the predictive variance associated with a point far away
from the training data will be large, i.e. the model will be uncertain in regions
where it has not previously seen data. Rather the opposite is true using the
covariance function specified by the RVM as a both terms in the predictive
variance Eq. 2.47 will be close to zero while for a standard RBF kernel the
first term will be large.
2.8 GP-LVM
Lawrence [33] suggested an alternative Gaussian latent variable model capable
of handling non-linear generative mappings while at the same time avoiding the
problems associated with the GTM. Both the PPCA and the GTM specifies a