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2.8. GP-LVM 49
prior over the parameters leads to the following objective,
L = Lr +
lnθi + 1
2 ||xi|| 2 . (2.58)
i
For a covariance function specifying a distribution over linear functions a
closed form solution for Eq 2.56 exists [33]. However for general covariance
functions the solution is found through gradient based optimization .
As previously discussed, infinitely many solutions to the latent variable for-
mulation of dimensionality reduction exists, to proceed the solution needs to be
constrained by prior information. The GP-LVM solution is constrained by the
GP marginal likelihoods trade-off between smooth solutions and a good data-fit
Eq.2.50. By fixing the dimensionality of the latent representation a solution can
be found.
2.8.1 Latent Constraints
The GP-LVM objective seeks the locations of the latent coordinates X that max-
imize the marginal likelihood of the data. One advantage of directly optimizing
the latent locations is that additional constraints on X can easily be incorporated
in the GP-LVM framework. In the following section we will review some of the
extensions in terms of latent constraints that has been applied to the GP-LVM.
The fundamental difference between spectral and generative dimensionality
reduction is the assumption made by the spectral algorithms that the latent coordi-
nates can be found as a smooth mapping from the observed data. This means that
we are interested in finding latent locations such that the locality in the observed
data is preserved. Further this assumption implies that a smooth inverse to the
i