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2.7. GAUSSIAN PROCESSES 39
GTM is non-convex which means that we cannot be guaranteed to find the global
optima. Further, the GTM suffers from problems associated with mixture models
in high dimensional spaces [59].
2.7 Gaussian Processes
A D dimensional Gaussian distribution is defined by a D × 1 mean and a D × D
covariance matrix. A Gaussian process (GP) is the infinite dimensional general-
ization of the distribution where the mean and covariance is defined not by fixed
size matrices but a mean µ(x) and a covariance k(x,x ′ ) function, defined over
infinite index sets, x.
GP(µ(x), k(x,x ′ )). (2.43)
Evaluating a GP over a finite index set reduces the process to a distribution
with the dimensionality of the cardinality of the evaluation set. The covariance
function needs to specify a valid covariance matrix when evaluated for any finite
subset in its domain, this requires the covariance function to come from the same
family of functions as Mercer kernels [41, 45].
A GP generalizes the concept of a Gaussian distribution to infinite dimen-
sions, this has been exploited in machine learning by applying GPs to specify
distributions over infinite objects. One such application is when we are interested
in modeling relationships defined over continuous domains such as functions. If
we are interested in modeling a functional relationship f between input domain