# 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].