# Probabilistic Non-linear Component Analysis through Gaussian Process Latent Variable Models

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**University of California, Berkeley, U.S.A.**on May 6, 2004

#### Abstract

It is known that Principal Component Analysis has an underlying
probabilistic representation based on a latent variable model. PCA
is recovered when the latent variables are integrated out and the
parameters of the model are optimised by maximum likelihood. It is
less well known that the dual approach of integrating out the
parameters and optimising with respect to the latent variables also
leads to PCA. The marginalised likelihood in this case takes the
form of Gaussian process mappings, with linear Covariance functions,
from a latent space to an observed space, which we refer to as a
Gaussian Process Latent Variable Model (GPLVM) [@Lawrence:gplvm03].
It is straightforward to *non-linearise* this model by
substituting the linear covariance function for a non-linear
one. The result is a non-linear probabilistic PCA model. In this
talk we will present a practical algorithm for optimising the latent
variables in a non-linear GPLVM and discuss some relations with
other models. Finally we will present results from a SIGGRAPH paper
which uses the GPLVM to learn styles in an inverse kinematics
problem [@Grochow:styleik04].