'Kernel' principal component analysis (PCA) is an elegant non(cid:173) linear generalisation of the popular linear data analysis method, where a kernel function implicitly defines a nonlinear transforma(cid:173) tion into a feature space wherein standard PCA is performed. Un(cid:173) fortunately, the technique is not'sparse', since the components thus obtained are expressed in terms of kernels associated with ev(cid:173) ery training vector. This paper shows that by approximating the covariance matrix in feature space by a reduced number of exam(cid:173) ple vectors, using a maximum-likelihood approach, we may obtain a highly sparse form of kernel PCA without loss of effectiveness.

'Kernel' principal component analysis (PCA) is an elegant nonlinear generalisation of the popular linear data analysis method, where a kernel function implicitly defines a nonlinear transformation into a feature space wherein standard PCA is performed. Unfortunately, the technique is not'sparse', since the components thus obtained are expressed in terms of kernels associated with every training vector. This paper shows that by approximating the covariance matrix in feature space by a reduced number of example vectors, using a maximum-likelihood approach, we may obtain a highly sparse form of kernel PCA without loss of effectiveness. 1 Introduction Principal component analysis (PCA) is a well-established technique for dimensionality reduction, and examples of its many applications include data compression, image processing, visualisation, exploratory data analysis, pattern recognition and time series prediction.

'Kernel' principal component analysis (PCA) is an elegant nonlinear generalisation of the popular linear data analysis method, where a kernel function implicitly defines a nonlinear transformation into a feature space wherein standard PCA is performed. Unfortunately, the technique is not'sparse', since the components thus obtained are expressed in terms of kernels associated with every training vector. This paper shows that by approximating the covariance matrix in feature space by a reduced number of example vectors, using a maximum-likelihood approach, we may obtain a highly sparse form of kernel PCA without loss of effectiveness. 1 Introduction Principal component analysis (PCA) is a well-established technique for dimensionality reduction, and examples of its many applications include data compression, image processing, visualisation, exploratory data analysis, pattern recognition and time series prediction.

'Kernel' principal component analysis (PCA) is an elegant nonlinear generalisationof the popular linear data analysis method, where a kernel function implicitly defines a nonlinear transformation intoa feature space wherein standard PCA is performed. Unfortunately, thetechnique is not'sparse', since the components thus obtained are expressed in terms of kernels associated with every trainingvector. This paper shows that by approximating the covariance matrix in feature space by a reduced number of example vectors,using a maximum-likelihood approach, we may obtain a highly sparse form of kernel PCA without loss of effectiveness. 1 Introduction Principal component analysis (PCA) is a well-established technique for dimensionality reduction,and examples of its many applications include data compression, image processing, visualisation, exploratory data analysis, pattern recognition and time series prediction.